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Home , Albert Marden, Alex Eskin, Bhama Srinivasan, Bryna Kra, Cathy Kessel, Dan Abramovich

Mathematical Sciences Research Institute Annual Report for 2007-2008

1. Overview of Activities...... 3 1.1 New Developments ...... 3 1.2 Major Programs & Associated Workshops...... 8 1.3 Scientific Activities Directed at Underrepresented Groups in ...... 18 1.4 Other Scientific Activities...... 19 1.5 Program Highlights ...... 21 1.6 Programs Consultant List ...... 27 1.7 MSRI Experiences ...... 28 ♦ Organizers ...... ♦ Postdoctoral Fellows ......

2. Program and Workshop Participation ...... 38 2.1 Program Participant List ...... 38 2.2 Program Participant Summary...... 45 2.3 Program Participant Demographic Data ...... 46 2.4 Workshop Participant List ...... 49 2.5 Workshop Participant Summary...... 50 2.6 Workshop Participant Demographic Data ...... 52 2.7 Program Publication List ...... 55

3. Postdoctoral Fellows ...... 82 3.1 Postdoctoral Fellow Placement List...... 99 3.2 Postdoctoral Fellow Placement Summary ...... 102 3.3 Postdoctoral Fellow Placement Demographic Data ...... 103 3.4 Postdoctoral General Member List ...... 106 3.5 Postdoctoral General Member Summary...... 108

4. Graduate Students ...... 109 4.1 Summer Graduate Workshops ...... 109 4.2 Program Associates ...... 117 4.3 Summer Graduate Workshops List...... 119 4.4 Summer Graduate Workshops Summary...... 123 4.5 Summer Graduate Workshops Demographic Data...... 124 4.6 Program Associates List ...... 127 4.7 Program Associates Summary ...... 128 4.8 Program Associates Demographic Data ...... 129 4.9 Graduate Student List ...... 132 4.10Graduate Student Summary ...... 133

5. Undergraduate Program...... 135 5.1 Undergraduate Program Participant Summary ...... 140

1 6. Financial Support List...... 141 7. Institute Directors Meeting Report (MIDS)...... 142 8. Interim Reports and Updates...... 156 9. Committee Membership ...... 152 10. Appendix - Final Reports...... 155

Program Reports...... ♦ Teichmüller Theory and Kleinian Groups ...... ♦ Geometric Theory...... ♦ Combinatorial ...... ♦ Representation Theory of Finite Groups and Related Topics......

Workshop Reports ...... ♦ Connections for Women: Teichmüller Theory and Kleinian Groups...... ♦ Topics in Teichmüller Theory and Kleinian Groups ...... ♦ Introduction to Teichmüller Theory and Kleinian Groups ...... ♦ Connections for Women: Geometric ...... ♦ Introduction to ...... ♦ Topics in Geometric Group Theory...... ♦ Connections for Women: Introduction to the Spring, 2008 programs...... ♦ Introductory Workshop on Combinatorial Representation Theory ...... ♦ Lie Theory...... ♦ Topics in Combinatorial Representation Theory...... ♦ Introductory Workshop on the Representation Theory of Finite Groups and ♦ Related Topics ...... ♦ Homological Methods in Representation Theory...... ♦ Cyber Enable Discovery and Innovation: Computation and Complex ...... ♦ Systems ...... ♦ Contact Structures, Dynamics and the Seiberg-Witten Equations in ...... ♦ Dimension 3...... ♦ MSRI Undergraduate Program: Experimental Mathematics...... ♦ Critical Issues in Education Workshop: Teaching and Learning Algebra...... ♦ Mathematical Systems Biology of Cancer II...... ♦ MSRI 25th Anniversary Workshop ...... ♦ Exterior Differential Systems and the Method of Equivalence ...... ♦ CMI/MSRI Workshop: Modular Forms and Arithmetic ......

Summer Graduate Workshop Reports...... ♦ IAS/PCMI: Statistical Mechanics...... ♦ Data Assimilation for the Carbon Cycle...... ♦ Continuous Optimization and its Applications...... ♦ Deformation Theory and Moduli in Algebraic ...... ♦ A Window into Zeta and Modular Physics......

2 1. Overview of Activities

This annual report covers MSRI projects and activities that concluded during the third year of the NSF core grant, 2007-2008.

1.1 New Developments and Demographic Data

Groups and discrete structures were fundamental research themes for the year 2007-2008. It was a year rich in cross-disciplinary collaborations and notable for the consistently high volume of seminars and study groups. This was also the year that we deepened our commitment to the mathematics of Climate Change, one of the most interdisciplinary and compelling set of problems that we now face. The goal is to grab graduate student’s attention and give them the tools and training they need to react creatively to this world crisis. We supported a three-week summer school in 2008, but the dates of the event sit on the other side of our boundary delineating academic years. This will be detailed on more fully in next year’s report, along with plans for an interesting sequence of workshops to address how mathematics is involved in widely differing aspects of climate change.

The next few paragraphs give some highlights of workshops and events that took place during the 2007-8 year.

25th Anniversary Celebration: While we constantly look to the future in our scientific programs, this year was also a time to reflect on MSRI’s history. Last year was MSRI’s 25th birthday, and, to celebrate, we held a week-long scientific workshop, the focus of which was currently active areas of mathematics in which MSRI played a significant role. We invited old friends, to give their accounts of past activities and the founding of MSRI, and young faces, to remind us of our future directions. There were 16 research-expository talks of extremely high quality, a panel on mathematics education chaired by Deborah Ball, and a panel on the past, present, and future of MSRI. A few of the distinguished speakers: Persi Diaconis, , Bryna Kra, Richard Melrose, Inez Fung, , and Michael Hopkins. All spoke of their time here at MSRI, and the developments in their respective fields over the years. The range of topics was quite broad and the talks were of uniformly high quality. Following our usual practice, these talks were recorded and are available for viewing on our VMath site for streaming videos.

Cyber-enabled Discovery and Innovation (CDI): MSRI hosted a one-day workshop (at the behest of the NSF) to advertise a major new initiative, Cyber-enabled Discovery and Innovation. This initiative was intended to foster American competitiveness through research contributing to “a new generation of computationally-based discovery concepts and tools to deal with complex, data-rich, and interacting systems”. It was the first time in MSRI’s history that the math institutes were asked to gather the math community’s forces to think in terms of these large scale projects for possible funding. Masoud Nikravesh and Robert Bryant organized this workshop around the theme of Computation and Complex Systems. There were talks on Climate Models,

3 Astrophysics, Scientific Data Visualization and Analysis, just to mention a few. There were 109 registered participants.

MSRI Biology Colloquia: We held two colloquia in this series (funded by a generous grant from the ) during the 2007-8 year. The audience was a combination of in residence at MSRI and those in the greater Bay Area interested in mathematical biology.

Dr. Garrett Odell from the gave three fascinating talks on agent-based modeling. He first spoke of his most recent work, on the sea urchin embryo and how it gets its furrow in the right place, and used this as an illustration in the following talks about the benefits of agent-based modeling. His models of how diverse embryos establish spatial gene expression patterns revealed these networks to be astonishingly robust in that they continue to make the correct pattern in the face of thousand-fold variations in the strengths and functional forms of interactions among participating genes. Odell stressed his fundamental point that such robustness is crucial to make networks functionally heritable in polymorphic populations.

Dr. Alan Perelson (from Los Alamos National Labs) also came and spoke about his work on the analysis of several disease models: How Mathematics Provides Information about HIV/AIDS, Modeling Immune Responses: Coping with Diversity Using Shape-Space Formulations, and Modeling Immune Responses: Mathematical Modeling of Viral Infections of Humans: Influenza and Hepatitis.

MSRI-UP program: This was the first year that MSRI–UP, the undergraduate research experience targeted for underrepresented minorities, was funded by the NSF. This supplement helped to increase the number of students who could be supported from the original 12 to 18 (although the funding came a little late to attract the full 18 students this past year). We hosted 15 students last summer and will seek 18 in the future. Since the NSF award is for four years, it lends stability to the program, for which the organizers are grateful. See Section 5 for a full report, with more details and pictures.

Networking Experiment: Recruitment of underrepresented minorities is a perennial problem for all the Math Institutes, and MSRI, in particular, struggles with it constantly. During 07-08, we started a networking tree in an effort to complie a list of names and contact information for underrepresented minorities, women, or well-connected people who were researchers in fields centered on our upcoming programs. We started with each program’s organizers. We asked them to identify appropriate people for our list. We then contacted the organizers’ nominations and asked them the same question. We continued iterating this process until we ran out of nominations. The resulting final list was given to the organizers to use as a database for information emails and phone calls. For a first try, we gathered approximately 20 names for each program. MSRI will continue the experiment next year and try to improve the algorithm.

Modern Math Workshop: The original idea of the Modern Math workshop has evolved. In the past, MSRI organized a set of speakers to go to HBCUs to advertise upcoming programs and workshops that would be held within our doors. In the more recent past, we have shifted our host sites from HBCUs to a pre-conference workshop associated with the annual SACNAS

4 meeting. For the first time, all the US-based math institutes (MSRI, AIM, IMA, IPAM, MBI, PCMI, and SAMSI) joined to develop a two-day program at the 2008 SACNAS meeting in Salt Lake City. The idea was to identify a managing institute each year that would handle the infrastructure: writing the grant, acting as liaison with SACNAS and the hosting convention center, and keeping all the institutes informed of the plans. MSRI performed this function for the 2008 SACNAS, and IMA will take over for the next one. (Others will step in the following years.)

China Girls Math Olympiad: At the urging of Zuming Feng, one of the coaches of the International Math Olympiad (IMO), and a little push from MSRI, a consortium of professional organizations, corporate sponsors, and private donors came together to support two teams of high school girls at the China Girls Math Olympiad for the past two summers. Planning and holding these events is a year-long process, with tutoring for the girls, national test taking to qualify, summer camp, and then visa, flight reservations, and blogs to arrange. The girls have had spectacular results, so much so that MSRI is looking for sustained funding for future CGMOs. The medal winners from the 2007 team were Sherry Gong (gold), Wendy Hou (silver), Patricia Li (bronze), Marianna Mao (bronze), and Wendy Mu (bronze). In 2008, Carolyn Kim, Jenny Jin, Inyoung Cho, Colleen Lee, and Joy Zheng all earned bronze medals, Wendy Mu earned silver, and Jenny Iglesias and Lynnelle Ye earned gold. The word is now out among our top-tier girls that this is an experience worth working towards.

Oakland/East Bay Math Circle: The academic year 07-08 was also the first for our Oakland/East Bay Math Circle. These programs are always hard to start. One can’t get funding until almost all the details are in place, but one can’t advertise the opening until one gets the funding. However, after a few bumps, the Circle was up and running, funded, and supported by the Oakland Public School District and Laney College, the downtown site of the Circle. Attendance improved over the course of the year, and, now that people see how this works, we hope for smoother sailing next year.

The Math Festivals at Google and Pixar: These are one-day events held at a corporate sponsor’s campus, where hundreds of middle school and high school children and their parents come to engage in mathematical activities. Lots of tables are set up, each staffed by a local , and each having a set of assigned mathematical problems, puzzles, and/or hands-on activities.

Public Understanding of Mathematics: MSRI continued to host several public events to deepen our relationship with our neighbors and cultivate an appreciation of the people and ideas that form the mathematics community.

In particular, we hosted the showing of two films of mathematicians: George Csicery’s “Julia Robinson and Hilbert’s Tenth Problem” and Agnes Handwerk and Harrie Willems’ “Wolfgang Doeblin – a mathematician rediscovered”, along with several musical events, including a conversation among Christopher Taylor, David Benson, and Bob Osserman. Mr. Taylor was on the Berkeley campus to perform Messaien’s Vingt regards sur l’enfant Jesus, and during the conversation with Benson and Osserman, he explored the mathematical ideas and themes that underly the structure of this monument of the 20th century piano literature.

5 Demographic Data.

During the academic year 2007-08, MSRI hosted 30 one-semester NSF Postdoctoral Fellows, 266 program members (members that came for period of at least one month), and 2047 workshop participants.

The Postdoctoral program was particularly successful and is described in detail in Section 3. Of the Fellows, 40% were female, 38% were US Citizens or Permanent Residents, and 73% listed a US university as their home institution. Of those institutions, 41% are located in the Northeast, 32% in the West, 18% in the Midwest, and the remaining 9% are in the South. Detailed demographic tables can be found in Section 3.

MSRI had a total of 266 (long-term) members. An ‘average’ member spends 80 days at MSRI, and the average number of members present everyday is roughly 75 (83% of our capacity), with peak attendance in November and March. Of the (long-term) members, 45 (17%) were female, 2 were black, 5 belonged to the Hispanic/Latino community, and 1 member was a Native American. It should be noted that half of the members declined to state their ethnicity and Hispanic/Latino ancestry, while all revealed their gender. Of the (long-term) members, 132 (51%) reported being US Citizen or Permanent Resident and 158 (60%) listed a US university as their home institution. Of those institutions, 37% are located in the Midwest, 23% in the West, a similar number in the Northeast, and 16% in the South. Of the (long-term) members, 12% were graduate students, 30% had received their Ph.D degree after 2000, 24% received theirs between 1990 and 2000, and the remaining 34% had received their Ph.D. prior to 1990. Detailed demographic data can be found in Section 2.

In its 2007-08 workshops, MSRI hosted 2047 separate visits (some visitors attended multiple events). We have gender data from1724 participants. Of these, 520 (30%) were female. There were 113 (10%) Self-Reported under-represented minorities. Of the participants, 38% were US Citizen or Permanent Residents, and 75% of all workshop participants have their home institution in the US. The ‘year of Ph.D.’ distribution is somewhat different from the one for long-term visitors. In particular, 30% of the workshop participants were graduate students and 25% received their Ph.D. after 2000. The US regional distribution was also somewhat different: 22 % were from the Midwest, 40% from the West, 22% from the Northeast, and the remaining 15% were from the South. Data on workshop participant demographics can be found in Sections 2 and 4.

6 Statistical Summary Total Member Days 21152 # of Distinct Member 266 Average # of Days per Member 80 Average # of Members per Day 75

2007-2008 Member Visit Length

40

35

30

25

20

Members 15

10

5

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Week

Month Members 2007-2008 Member in MSRI by Month Aug-07 81 Sep-07 93 140

Oct-07 107 120 Nov-07 120 Dec-07 89 100

Jan-08 72 80 Feb-08 92 Mar-08 116 60

Apr-08 99 40 May-08 81 20 0 Aug-07 Sep-07 Oct-07 Nov-07 Dec-07 Jan-08 Feb-08 Mar-08 Apr-08 May-08

7 1.2 Major Programs and their Associated Workshops

There were four major Programs for the MSRI fiscal year 2007-08, and ten workshops associated to them:

Program 1: Geometric Group Theory

August 20, 2007 to December 14, 2007 Organized By: Mladen Bestvina, Jon McCammond, Michah Sageev, Karen Vogtmann

Geometric group theory is a relatively young field but with older and deeper roots in which groups are studied from combinatorial and topological perspectives. Combinatorial group theory traces back to the work of Dehn, and focuses on the combinatorial nature of cell complexes associated to groups. Topological methods in group theory focuses on the cohomology of groups and their finiteness properties, and hence views groups as essentially topological objects. In the mid 1980's, spurred by ideas of Jim Cannon and Misha Gromov, group theorists began to pay attention to the geometric structures which cell complexes can carry. This attention shed a great deal of light on the earlier combinatorial and topological investigations into group theory, and stimulated other innovative ideas which have been developing at a rapid pace. As it has grown over the past 20 years, geometric group theory has developed many different facets, including geometry, , analysis, logic.

These facets are often studied in the context of specific groups or classes of groups: Artin groups, Coxeter groups, braid groups, mapping class groups, the Torrelli group, Out(Fn ), Aut(Fn ), lattices in Lie groups, square-complex groups, Thompson's group, automata groups etc.

The new, more geometric, perspectives enabled rapid progress on many of these fronts. A tremendous solidification of previously disparate results has also occurred. The semester program at MSRI capitalized on this recent surge of activity. The program brought people from the various branches of geometric group theory together to work on some of the many longer- standing open questions in the field that are now being studied from fresh and promising perspectives, and to further strengthen the connections the field has to the other branches of mathematics.

Workshops Associated with the Geometric Group Theory program:

Connections for Women: Geometric Group Theory August 23, 2007 to August 24, 2007 Organized By: , Indira Chatterji, and Karen Vogtmann

This 2-day workshop consisted of four mini-courses on classical topics in geometric group theory, each consisting of two hours of lectures plus associated discussion sessions. Participants were asked to contribute a one or two-page PDF poster advertising their own research. These were compiled into a booklet and distributed to all registered participants.

8 Introduction to Geometric Group Theory August 27, 2007 to August 31, 2007 Organized By: Mladen Bestvina, Jon McCammond, Michah Sageev, Karen Vogtmann

This workshop consisted of several mini-courses, 3 hours each, as well as a program of one-hour lectures. The planned mini-course speakers and their topics were as follows:

Jim Cannon: Non-positive and negative curvature in group theory Ruth Charney: Coxeter and Artin groups Benson Farb: Mapping class groups Bruce Kleiner: Quasi-isometric rigidity Ian Leary: Finiteness properties and Bestvina-Brady Morse Theory

Topics in Geometric Group Theory November 05, 2007 to November 09, 2007 Organized By: Noel Brady, Mike Davis, Mark Feighn

This conference was devoted to cutting-edge developments in geometric group theory. Talks covered aspects of the following topics: CAT(0)-spaces and CAT(0)-cubical complexes, hyperbolic and relatively hyperbolic groups, automorphism groups of free groups and surface groups, self-similar groups, Coxeter groups and Artin groups, lattices in Lie groups and Kac- Moody groups, asymptotic dimension, measurable group theory, and limit groups.

This workshop was partially supported by a grant from Collaborative Research Center 701, University of Bielefeld, funded by the DFG (German Research Foundation).The organizers and MSRI acknowledge the assistance of Professor Herbert Abels in this project.

9

Program 2: Teichmüller Theory and Kleinian Groups August 20, 2007 to December 14, 2007 Organized By: Jeffrey Brock, Richard Canary, Howard Masur, , Alan Reid

The fields of Kleinian groups and Teichmüller theory have each seen dramatic changes in recent years. Many new techniques have been developed, major conjectures have been solved, and new directions and connections have been forged. Yet to a large extent progress has been made in parallel without the level of direct communication across these two fields that is clearly warranted. The MSRI program in Teichmüller theory and Kleinian groups addresses the need to strengthen connections between these two fields, and reassess new directions for each at a critical time in its history.

The recent solutions of the tameness conjecture, density conjecture and the ending lamination conjecture put the study of hyperbolic 3-manifolds and Kleinian groups at a transitional point. Information about the mapping class group and the curve complex that has arisen out of this is already bearing in important ways on questions in Teichmüller theory and the dynamics and geometry of the mapping class group, and the geometric component of the ending lamination conjecture suggests the possibility of effective models and bounds for closed hyperbolic manifolds. Such developments are playing an important role in strengthening our understanding of parameter spaces of Kleinian groups, including their local and global topology.

Likewise, an important development in theory has been the discovery of fertile connections between rational billiards, translation surfaces and flows on Teichmüller space and . A major focus of the program was to explore this subject and its connections to , and the of the complex of curves on a surface. More generally, there have been recent breakthroughs in understanding the extent of the analogies between the mapping class group and Kleinian groups, and the connections to Veech surfaces and the geometry of the mapping class group which makes this area one of particular intererest for researchers in flows on moduli space and in hyperbolic geometry alike.

This program served as a proving ground for the extent of the connections and analogies between these two areas, and generated new threads of inquiry.

Workshops Associated with the Teichmüller Theory and Kleinian Groups program:

Connections for Women: Teichmüller Theory and Kleinian Groups August 16, 2007 to August 17, 2007 Organized By: ,

Each day of this two-day workshop featured two mainly expository lectures in the morning aimed at the level of advanced graduate students and new postdocs in geometry and topology. These talks on the themes of the Teichmuller theory and Kleinian groups program introduced viewpoints and tools with broad applicability. In the afternoons, women working in fields in or around geometric topology (including hyperbolic geometry, low-dimensional topology, metric

10 geometry, and geometric group theory) gave shorter talks in their areas of expertise. Learning who's doing what was a main goal of the workshop. Introduction to Teichmüller Theory and Kleinian Groups August 20, 2007 to August 24, 2007 Organized By: Jeff Brock, Richard Canary, Howard Masur, Alan Reid, and Maryam Mirzakhani

This five-day workshop was comprised of six three-hour mini-courses, run by experts in the field, intended to give a summary of recent results in their various areas of expertise and frame new directions for future research. The mini-courses brought participants up-to-date on recent work in the topology and deformation theory of hyperbolic 3-manifolds, dynamics of flows on moduli spaces of translation surfaces, and the geometry of Teichmüller space and the mapping class group, and their various combinatorial models. There was time in between these mini- courses for substantial discussion and interaction. The mini-courses were also supplemented by one-hour talks.

Topics in Teichmüller Theory and Kleinian Groups November 12, 2007 to November 16, 2007 Organized By: Jeff Brock, Ken Bromberg, Richard Canary, Howard Masur, Alan Reid, Maryam Mirzakhani, and John Smillie

This five-day conference, served as the main research conference for the MSRI program in Teichmüller theory and Kleinian groups, and took stock of the work and results of participants of the program, as well as incorporated outside perspectives. The conference, included roughly 25 main speakers from a broad variety of focus areas, and made the case for a continued modernization of the connections between these two fields which have undergone such dramatic changes in recent years. Ample time was provided for discussion in between talks, and communicating across these two fields was emphasized.

11 Program 3: Combinatorial Representation Theory January 14, 2008 to May 23, 2008 Organized By: P. Diaconis, A. Kleshchev, B. Leclerc, P. Littelmann, A. Ram, A. Schilling, R. Stanley

In representation theory, abstract algebraic structures are represented using matrices or geometry. These representations provide a bridge between the abstract symbolic mathematics and its explicit applications in nearly every branch of mathematics as well as in related fields such as physics, chemistry, engineering, and statistics.

In Combinatorial Representation Theory, combinatorial objects are used to model these representations. These are refined enough to help describe, count (how many there are), enumerate (how to generate them all), and understand the representation theory. Furthermore, the interplay between the algebra and the combinatorics goes both ways: the combinatorics helps answer algebraic questions and the algebra helps answer combinatorial questions.

Particularly in the last couple of decades, the field of Combinatorial Representation Theory has become a thriving discipline. Some recent catalysts stimulating the growth of this field have been the "crystals" discovered by Kashiwara and Lusztig and the development of the combinatorics of affine Lie groups and their connection to . In the 21st century Combinatorial Representation Theory lies at the intersection of several fields: combinatorics, representation theory, analysis, , Lie theory, and mathematical physics. These fields often operate under separate umbrellas, and the primary goal of this program was to bring together the experts of the various flavors of Combinatorial Representation Theory together in one interdisciplinary setting.

The very recent connections between path models for crystals and the geometry of the loop Grassmanian and between complex reflection groups and p-compact groups are indicators that the future holds even more gems for this field. The program made an effort to focus on main problems of the field such as:

• positive combinatorial formulas: q-weight multiplicities, Kazhdan-Lusztig polynomials, generalized Littlewood-Richardson coefficients; • combinatorial indexings and constructions of irreducible representations: Springer correspondences, Langlands classifications, path models, tableaux; • the Virasoro conjecture: counting branched covers of Riemann surfaces, Hurwitz numbers, cycle types, symmetric functions, determinant formulas; • representation theoretic interpretations of Macdonald polynomials: Hilbert scheme, diagonal invariants, affine and graded Hecke algebra modules: • distributions and convergence of random processes: random matrices, subsequences of permutations, statistical mechanics models; • decomposition numbers for representations: affine Lie algebras, modular representations, highest-weight categories, homology representations, finite goups of Lie type; • product structure in cohomology and K-theory: Schubert varieties, quiver varieties, toric varieties, loop Grassmanians. • cluster algebras: generalized associahedra associated with root systems, coordinate rings of flag varieties and their double Bruhat cells.

12 Workshops Associated with the Combinatorial Representation Theory program:

Connections for Women: Introduction to the Spring, 2008 programs January 16, 2008 to January 18, 2008 Organized By: and Monica Vazirani

This intensive three-day workshop for women introduced advanced graduate students and recent PhDs to current areas of research in Representation Theory.

It consisted of introductory mini-courses and talks, as well as a poster session where all participants were encouraged to present their work. An important purpose of the workshop was to establish a professional network for participants by introducing them to each other and each others' work. To this end there was also a dinner and social activities, as well as a panel discussion on issues related to women in mathematics.

The workshop was part of the semester programs Combinatorial Representation Theory and Representation Theory of Finite Groups and Related Topics and participants were encouraged to attend associated Introductory Workshops on Combinatorial Representation Theory from January 21 to January 25, and Representation Theory of Finite Groups from January 28 to February 1. The Connections for Women workshop was a good introduction to these two workshops.

Introductory Workshop on Combinatorial Representation Theory January 22, 2008 to January 25, 2008 Organized By: Persi Diaconis, Arun Ram, Anne Schilling (Chair)

The soul of Combinatorial Representation Theory (CRT) lies in the interplay between combinatorics and various branches of mathematics. Combinatorial methods are applied to solve problems in representation theory, Lie theory, geometry, and mathematical physics and, in symbiosis, deep combinatorial problems also arise from these areas.

The goal of the Introductory Workshop was to survey current and recent developments in the field. The talks focused on tableaux, reflection groups, finite groups, geometry and mathematical physics in the realm of Combinatorial Representation Theory.

Lie Theory March 10, 2008 to March 14, 2008 Organized By: Alexander Kleshchev, Arun Ram, Richard Stanley (chair), Bhama Srinivasan

The emphasis was on the interplay of combinatorics, Lie theory and theory. Connections between these areas go back at least to Schur and Weyl: representations of the , polynomial representations of , Weyl’s character formula. In the last 20 years the field has seen great development as the combinatorics of Young tableaux has been generalized to any Lie type via the theory of crystals. Littlemann’s path model approach to crystals makes a strong connection between this theory and the geometry of the flag variety

13 and recent activity is exploring the connection between representation theory, the geometry of the loop Grassmannian and affine Hecke algebras. In the modular representation theory of finite groups of Lie type, connections with complex geometry have arisen in the defining characteristic case and with affine Kac-Moody algebras in the non-defining characteristic case.

Topics covered by the workshop included finite groups of Lie type, algebraic groups, quantum groups, affine Lie algebras, Hecke algebras, cluster algebras, W-algebras, and modular Lie algebras.

Topics in Combinatorial Representation Theory March 17, 2008 to March 21, 2008 Organized By: Sergey Fomin, Bernard Leclerc, Vic Reiner (Chair), Monica Vazirani

Representation theory has often been a key to unlocking problems of enumeration and structure for our favorite combinatorial objects. In the reverse direction, answering many of the central questions of representation theory required development of sophisticated combinatorial techniques and constructions. This interplay, which has only intensified in recent years, was the focus of the workshop.

Topics discussed include (the combinatorial aspects of): quiver representations; cluster algebras; Macdonald and LLT polynomials; representation-theoretic techniques in quantum/statistical mechanics; generalized Littlewood-Richardson rules, Schur-positivity, and connections with Schubert calculus; crystal bases and graphs; affine Grassmannians, Mirkovic- Vilonen cycles; dual canonical and semi-canonical bases; Horn and Deligne-Simpson problems; tropical calculus in representation theory.

A relatively small number of talks left ample time for informal discussions and presentations.

14 Program 4: Representation Theory of Finite Groups and Related Topics January 14, 2008 to May 23, 2008 Organized By: J. L. Alperin, M. Broue, J. F. Carlson, A. Kleshchev, J. Rickard, B. Srinivasan

Founded by Frobenius and Schur more than a century ago, the representation theory of finite groups is today a thriving field with many recent successes. Current research centers on the many open questions, particularly regarding representations over the or rings of positive characteristic. Brauer developed block theory to understand better such representations, and it proved important in solving some problems in the classification of finite simple groups. In the last few years the area has been driven by a panoply of exciting new conjectures concerning correspondence of characters and derived equivalences of blocks. A key feature is the interplay between the research on general finite groups and important special classes of groups. Some major advances have been made in the representation theories of symmetric groups and groups of Lie type.

Around the same time as Brauer, Eilenberg and MacLane gave an algebraic definition of group cohomology, analogous to similar constructions in topology, and it has been an important tool for those studying group representations. There are many fruitful interactions among mathematicians from diverse backgrounds who use group cohomology, including those who work in representation theory and . More recently we have seen very active interactions between homotopy theory, commutative algebra, group actions and modular representation theory. Topics such as p-local groups, group actions on finite complexes and homotopy representations blend algebra and topology in novel and productive ways.

The goals of the semester focused the research on some of the conjectures and also fostered emerging interdisciplinary connections between several related areas in algebra and topology.

The introductory workshop concentrated on some of the many fundamental open problems in group representations. Topical workshops emphasized the connections with the theory of Lie algebras and algebraic groups and with algebraic topology.

Workshops Associated with the Representation Theory of Finite Groups and Related Topics program:

Connections for Women: Introduction to the Spring, 2008 programs (Co-sponsored by the program in Combinatorial Representation Theory. See workshop description above).

Introductory Workshop on the Representation Theory of Finite Groups February 04, 2008 to February 08, 2008 Organized By: Jonathan Alperin(chair), Robert Boltje, Markus Linckelmann

The workshop focused on surveying main active areas of representation theory of finite groups, especially highlighting major unsolved problems. It was meant to be accessible for graduate students and non-specialists with some background in representation theory. The bulk of the week's program consisted of four short series of lectures:

15 Block theory and counting conjectures The course introduced the basic ideas of modular representations, including block theory, the main theorems of Brauer and the Green correspondence. Special theories for cyclic and nilpotent blocks were covered. Subsequently, several counting conjectures were discussed. These included the Alperin-McKay conjecture, Alperin's weight conjecture, the Knorr-Robinson synthesis via alternating sums, Dade's conjecture and recent subtle refinements.

Representation theory of groups of Lie type While emphasizing the general linear group, this course covered topics including representations in characteristic zero, p and "el" and related structures such as Hecke algebras.

Representation theory and topology The purpose of this course was to describe some of the important tie-ins between representation theory and algebraic topology through topics from cohomology of groups applied to representation theory, homological algebra (e.g. derived categories), fusion systems and p-local finite groups.

Broue's abelian defect group conjecture This course focused on equivalences between derived categories of blocks and on Broue's isotypies between blocks. In the case of finite groups of Lie type, related geometric structures enter the picture. These include Deligne-Lusztig varieties and complex reflection groups. The case of the symmetric groups was also discussed.

The four courses were supplemented by a number of single lectures on a variety of topics.

Lie Theory (Co-sponsored by the program in Combinatorial Representation Theory. See workshop description above).

Homological Methods in Representation Theory March 31, 2008 to April 04, 2008 Organized By: David Benson, Daniel Nakano (chair), Raphael Rouquier

Over the last century, algebraic invariants like cohomology have been a fundamental tool in studying properties of topological spaces. In the last 40 years, this trend has been reversed, cohomology and other homological methods have been used to study algebraic objects by introducing geometry (i.e., algebraic varieties, derived categories) that captures information about the algebras and their representations. The theme of this workshop involved exploring the deep connections between representations and their underlying geometry. The main topics included: Cohomology Theory: Varieties for Modules (for finite group schemes, quantum groups and other types of algebras), Endopermutation and Endotrivial Modules, p-local Finite Groups;

Derived Categories: Broue's Conjecture, Structure of Triangulated Categories, Representation Dimension;

16 Representations and Cohomology of Specific Groups and Algebras: Symmetric Groups, Finite Chevalley Groups, Reductive Algebraic Groups and associated Frobenius kernels.

The lectures at this meeting were aimed at presenting new developments in the area in a manner accessible for young researchers in the field.

17 1.3 Scientific Activities Directed at Underrepresented Groups in Mathematics

(These activities are in addition to the 3 “Connections for Women” workshops.)

Modern Mathematics: An Introduction to MSRI's 2008-09 Programs October 10, 2007 to October 11, 2007 Organized By: Ricardo Cortez, Kathleen O'Hara, Ivelisse Rubio

This workshop was held at the Kansas City Marriott Downtown located at 200 West 12th Street, Kansas City, Missouri, directly preceding the Annual Meeting of SACNAS. The focus was on the Analysis of Singular Spaces, and Additive Combinatorics, and Algebraic Geometry.

MSRI-UP 2008 research topic: Experimental Mathematics June 14, 2008 to July 27, 2008 Organized By: Ivelisse Rubio (University of Puerto Rico, Humacao), Duane Cooper (Morehouse College), Ricardo Cortez (Tulane University), Herbert Medina (Loyola Marymount University), and Suzanne Weekes (Worcester Polytechnic Insitute).

The MSRI-UP is a comprehensive program for undergraduates that aims at increasing the number of students from underrepresented groups in mathematics graduate programs. MSRI-UP included summer research opportunities, mentoring, workshops on the graduate school application process, and follow-up support.

18 1.4 Other Scientific Activities

Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in dimension 3 June 09, 2008 to June 13, 2008 Organized By: Helmut Hofer, Michael Hutchings, Peter Kronheimer, Tom Mrowka and Cliff Taubes

This workshop concentrated on recently discovered relationships between Seiberg-Witten theory and contact geometry on 3 dimensional manifolds. One consequence of these relationships is a proof of the Weinstein conjecture in dimension 3. Another is an isomorphism between the Seiberg-Witten Floer (co)homology and embedded contact homology, the latter a form of Floer homology that was defined by Michael Hutchings. The over arching plan was to introduce the salient features of both the contact geometry side of the story and the Seiberg-Witten side, and then discuss how they are related.

Computation and Complex Systems October 12, 2007 Organized By: Robert Bryant (MSRI) and Masoud Nikravesh (UC Berkeley)

Beginning in 2008, The National Science Foundation (NSF) has begun to fund a major new initiative on Cyber-enabled Discovery and Innovation (CDI). This initiative is intended to foster American competitiveness through research contributing to "a new generation of computationally based discovery concepts and tools to deal with complex, data-rich, and interacting systems." The NSF notes several examples of themes in this research category: data mining of large sets, interacting complex systems, high-performance computational experimentation, virtual environments, and educating researchers and students in computational discovery.

It is clear that mathematics must play a key role in research in the areas outlined for this initiative. The Division of Mathematical Sciences at the National Science Foundation is strongly supportive of the involvement of mathematical and cross-disciplinary groups in research projects connected with CDI. As a first step in promoting mathematics involvement, the NSF Mathematical Sciences Institutes have begun a coordinated effort to inform the mathematical community about this initiative and to facilitate the development of research proposals.

Initial workshops

Between October 12, 2007 and November 1, 2007, there were four one-day NSF workshops on CDI at NSF Institutes, of which this was the first.

The plan for this workshop was to include key lectures about large scale interdisciplinary problems, round table discussions about mathematical challenges in these and related areas, and Q & A sessions about the structure of the CDI initiative and the NSF's expectations for proposals. Subsequent workshops were held as follows: on October 29 at the Institute for Pure and Applied Mathematics (IPAM), which focused on the knowledge extraction aspect of CDI; on October 30 at the Mathematical Biosciences Institute (MBI), which focused on opportunities for mathematicians who are interested in doing interdisciplinary work related to biology; and on

19 November 1 at the Statistical and Applied Mathematical Sciences Institute (SAMSI), which focused on issues and challenges in the areas of Knowledge Extraction, Interacting Elements, Computational Experimentation and Virtual Environments.

Exterior Differential Systems and the Method of Equivalence May 05, 2008 to May 09, 2008 Organized By: Jeanne Clelland, William F. Shadwick (Chair) and George Wilkens

The workshop in Exterior Differential Systems and the Method of Equivalence surveyed state of the art applications of these techniques and celebrated the contributions of Robby Gardner to our current understanding of Cartan’s powerful machinery.

CMI/MSRI Workshop: Modular Forms and Arithmetic June 28, 2008 to July 02, 2008 Organized By: Frank Calegari, Samit Dasgupta, David Ellwood, , and Richard Taylor

This conference, jointly funded by MSRI and the Clay Mathematics Institute, brought together researchers on many aspects of the arithmetic applications of modular (and automorphic) forms. This is currently a very broad and very active subject. Our intention was to encourage interaction between those working in different sub-disciplines. To this end we limited lectures to 4 hours a day, allowing plenty of time for informal interactions. On Tuesday, July 1, 2008 at 7pm, a dinner to honor on his 60th birthday was held at the University of , Berkeley Faculty Club.

Mathematical Systems Biology of Cancer II October 24, 2007 to October 26, 2007 Organized By: Joe Gray, Elizabeth Purdom, Terry Speed and Paul Spellman.

This workshop was designed to encourage and support the mathematical community's involvement in the effort to study cancer using system approaches. Conference presenters included mathematicians and computer scientists presently involved in systems approaches to cancer and more general fields of biology. These presenters covered general approaches to systems biology including analysis of genome scale data as well as statistical, continuous, and hybrid methods for pathway modeling. The workshop also provided tutorials covering the use of tools and methods in systems biology as well as on the fundamental biological processes involved in cancer. In addition, the workshop provided travel support for students and postdocs from the mathematical sciences to foster interest in this field.

20 1.5 Program Highlights

1. Geometric Group Theory (Fall 2007).

The four organizers, Mladen Bestvina (University of Utah), Jon McCammond (UC Santa Barbara), Michah Sageev (Technion) and Karen Vogtmann (), were present during the entire program, which resulted in a very well organized and lively program. In addition to the 8 NSF postdocs (3 females, 5 males) there were a large number of general members still in the post-doctoral stage of their careers and more than twenty graduate students (split between the two programs) in residence during the entire semester, virtually all accompanying their dissertation advisors. The organizers were keenly aware of the large number of early career mathematicians who were in residence, and they planned their activities accordingly. In addition to the three workshops usually associated with the program, the organizers created a number of ‘local institutions’ that helped structure the time of the members in residence for the entire semester. Many of these activities were designed with the MSRI Postdoctoral fellows and other younger mathematicians in mind. They included a series of seven minicourses, a weekly research seminar, a weekly post-doc seminar and a weekly grad student seminar, a communal lunch, a Thursday lunch question-and-answer session, and a couple of special lectures. The communal lunch, referred to as ‘commie lunch’, was such a success that the organizers recommended it for future programs. The format consisted of an informal potluck featuring, among other things, Cheese Board bread transported up the hill by bicycle each morning by one of the program organizers (Bestvina or Sageev). An additional notable event was the one-day Centennial Birthday Conference celebrating the fact that both Peter Kropholler and Lee Mosher reached the age of 50 during 2007. The highlight of the conference was Benson Farb’s description of his joint work with Lee Mosher as “an encounter with a genius.” A special cake was made in the shape of a surface of genus two and decorated with a long exact sequence. Jon McCammond’s abode served as the location for the conference party, which ran long into the night. The party eventually broke up due to an argument over the correct use of the term ‘right coset’.

Throughout the semester, the interaction between the two Fall programs, Geometric Group Theory and Teichmüller Theory and Kleinian Groups proved to be quite fructuous. For example, the overlap between the areas enabled the organizers to run a joint post-doc seminar, a joint graduate student seminar and a joint question-and-answer session. Several collaborations between members of the two programs were started at MSRI. For example, Bestvina, Bromberg, and Fujiwara produced a joint work on the asymptotic dimension of Teichmüller space. Lastly, the organizers described a few scientific highlights: During the Introductory Workshop, Bruce Kleiner, after hearing an inspiring lecture by Alain Valette, found another proof of Gromov’s polynomial growth theorem that uses harmonic functions on groups and isometric actions on Hilbert space in place of the Montgomery-Zippin characterization of Lie groups. Kleiner’s proof was presented by David Fisher at the Topics in Geometric Group Theory workshop. Other results that were established during the semester include the following: (1) Ian Leary figured out that CAT(0) cube complexes are complete precisely when every ascending sequence of nested cubes terminates and (2) Bestvina, Bromberg, and Fujiwara computed the asymptotic dimension of Teichmüller space.

21

2. Teichmüller Theory and Kleinian Groups (Fall 2007)

The organizers, Jeffrey Brock (Brown), Richard Canary (Michigan), Howard Masur (UIC), Maryam Mirzakhani (Princeton), and Alan Reid (Texas) describe their program as “a tremendous success; one that exceeded their expectations in virtually every respect. Indeed, one Research Professor in the program, Ursula Hamenstädt, described the program as “the best professional experience of her career.” This was gratifying for the organizers to hear, but, in fact, such sentiments express well the particular blend of camaraderie, enthusiasm, and topical relevance and timing of the program and its sibling program in geometric group theory. Participants felt they were all in the right place at the right time. The principal regret of the organizers and of many participants was the inability of the program to run for the full year, as many developments were just gaining traction as the program drew to a close.

One of the research highlights of the program was the presence of Mahan Mitra, who was funded to visit from India for six weeks. He had recently established that the of any freely indecomposable is locally connected and the experts in the field were eager to understand his proof. His proof has the further impact of giving a Cannon-Thurston map from the Gromov boundary of the abstract group to the limit set of the Kleinian group. He gave a long series of talks on the argument, which were convincing and exciting for the experts that remained through to the end (which included Jeff Brock, Ken Bromberg, Dick Canary, Yair Minsky, and Lee Mosher). An upshot of this series of talks was a new collaboration Mitra engaged in with Saul Schleimer and Chris Leininger, which eventually evolved into an argument that the boundary of the curve complex of a once-punctured surface is locally path connected and path connected. Their preprint is posted on the arXiv at arXiv:0808.3521.

A further international success was the month-long visit of Cyril Lecuire. During this time, Lecuire worked with Javier Aramayona on their study of geodesics in the pants complex (a combinatorial model for the Weil-Petersson metric on Teichmüller space). Cyril also discussed his work on a general characterization of when Kleinian surface groups converge and diverge, which became a joint project with Jeff Brock, Ken Bromberg, and Dick Canary.

Another important success was the work of Maryam Mirzakhani with in counting geodesics in the thin part of the moduli space of quadratic differentials. During the program, they worked on the general case for different strata and began a collaboration with Kasra Rafi on this. which is ongoing and played a significant role in furthering the goals of the project.

A new result of Jeff Brock with Howard Masur and Yair Minsky emerged out of their considerations of ending laminations for Weil-Petersson geodesics. Their consideration of the case of geodesics that recur to the thick part of Teichmüller space gave a first entry point into a systematic study of the geodesic flow. They proved an ending lamination theorem for recurrent geodesics and then used these ending laminations to prove that the geodesic flow is topologically transitive and that the set of closed orbits is dense. Using this result, they also enhanced their bounded geometry theorem (bounded combinatorics and bounded geometry are equivalent) to apply to geodesic lines rays and segments. The first paper, Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence and flows, is under revision at Geom. Funct. Anal 6. Anna Lenzhen’s talk in the postdoctoral seminar about the limiting behavior of Teichmüller

22 geodesics in the Thurston compactification gave rise to a new collaboration between Brock and Lenzhen to show explicitly the failure of convergence of Weil-Petersson geodesics in the Thurston compactification of Teichmüller space (despite the existence of the ending lamination). Their project is still underway, but is currently involved with seeking convergence criteria for the Thurston boundary. A collaboration between Anna Lenzhen and Howard Mazur that was conducted at MSRI during this program was the study of the asymptotic geometry of Teichmüller geodesic rays. They showed that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent but are not absolutely continuous with respect to each other, then the rays diverge in Teichmüller space. Those are only a few examples among the10 scientific highlights listed by the organizers that resulted in collaborative success among the participants. The organizers also felt that a key element of the success of the program was the topical overlap with the concurrently running program in Geometric Group Theory. Many of their mini-courses and seminar talks were directly relevant to the Teichmüller Theory program, and there was considerable intellectual cross-fertilization that took place between participants. As a key example, a project that intertwined these fields was the collaboration between Ken Bromberg, Mladen Bestvina, and Koji Fujiwara to compute the asymptotic dimension of Teichmüller space, as mentioned in the previous section. This example is one of many, but it serves to emphasize the important benefits of running thematically similar programs concurrently. This aspect of the semester at MSRI was uniformly praised.

3. Representations of Finite Groups and Related Topics

Three of the organizers, Michel Broue, John Carlson, and Alexander Kleshchev, were in residence for the entire program. The three other organizers, Jonathan Alperin, Jeremy Rickard, and Bhama Srinivasan were present for periods of 2 to 3 months. Their presence greatly contributed to the sense of cohesiveness that the researchers expressed in their comments to us. There were four workshops associated to the program. Two of these, Connections for Women, and Lie Theory, were shared with the program on Combinatorial Representation theory. The heart of the research part of the program was revealed in the several (five, all told) seminars that were held weekly during the program. The seminar on Representations of Groups of Lie Type, organized by Zongzhu Lin and his postdoc mentee, Daniel Juteau, produced a few exciting results. A notable one was Juteau’s counter-example to a conjecture by Mirkovic and Vilonen. In addition, his main idea, a geometric approach to modular representations involving modular character sheaves that was expounded in his dissertation, attracted a lot of attention.

The seminar on Representations of Symmetric Groups and Closely Related Topics was organized by David Hemmer and his postdoctoral mentee, Sinead Lyle. It also included lectures by some members of the program on Combinatorial Representation Theory, namely Anatoly Vershik, Hyohe Miyachi, Francesco Brenti, and Olly Ru. It was a perfect opportunity for interactions between programs that yielded several important results in both areas of research. One of the surprising results was Dave Hemmer’s stability theorem for symmetric group Specht

23 module cohomology. Kleshchev and Brundan found new presentations of blocks of symmetric groups and cyclotomic Hecke algebras. These presentations establish an isomorphism between the blocks and the cyclotomic Khovanov-Lauda algebras introduced three months ago.

The seminar on Biset Functors was organized by Serge Bouc, with all of the lectures presented by Bouc, Boltje, Ragnarsson, and Webb. One of the postdocs in the program, Kari Ragnarsson, made some progress defining Mackey functors and Burnside rings for fusion systems. Bouc succeeded in proving one his own conjectures. He showed that for a group G, the cohomological Mackey functor for G over the base field k have projective resolutions with polynomial growth if and only if the Sylow p- of G are cyclic, in the case p > 2, or have sectional rank at most 2, if p = 2. The seminar on Homological Methods in Representation Theory was organized by David Benson and Nadia Mazza. The seminar featured lectures by Vera Serganova, from the University of California, Berkeley, as well as lectures by members Ragnarsson, Webb, Nakano, Grodal, Lin, Webb, Symonds, Rickard, and Carlson. There were a couple of notable advances to come from this area of the program. Dave Benson and Julia Pevtsova discovered methods for constructing vector bundles over projective space in infinite characteristics, using the modules of constant Jordan type. The properties of these modules were developed by Carlson, Friedlander, Pevtsova, and Suslin. Peter Symonds proved several theorems related to Castelnuovo-Mumford regularity. In particular, he settled some conjectures of Kemper and others on the regularity of rings of polynomial invariants and proved Benson’s conjecture on the regularity of cohomology rings. This last result has significant implications for the computation of cohomology.

There was an informal working group on character theory led Martin Isaacs, Gabriel Navarro, Pham Huu Tiep, and others. This is an area that has focused on important conjectures by Alperin, Dade, Isaacs, Navarro, and others. One of the most striking results to come from the semester at MSRI is a proof of Brauer’s height zero conjecture in the case of blocks of maximal defect in characteristic 2. This conjecture states that all complex irreducible characters in a p-block B of a finite group G have height zero if and only if the defect group of B is abelian. It was first proposed by more than 50 years ago and has been confirmed for many specific groups. However, up until now, there had been few results of any generality on the subject. While it may in general be difficult to predict the overall impact of a program on the future of a research area, the signs were very positive. The program featured a large diverse group of young researchers. The research in the area has expanded into some new and unexpected directions of study. At the same time, some significant progress was made on a few of the old questions that have been driving research in the area.

4. Combinatorial Representation Theory

Five of the organizers, Bernard Leclerc, Persi Diaconis , Alexander Kleshchev, Arun Ram, and Anne Schilling, were present for most of the duration of the program. Richard Stanley and Peter Littleman visited for periods of one to two months. Combinatorial Representation Theory is the interaction of combinatorics and representation theory. It lies at the intersection of several fields: combinatorics, representation theory, harmonic analysis, algebraic geometry, and mathematical physics. Many experts in these various fields came together under the interdisciplinary heading

24 of Combinatorial Representation Theory. The facilities at MSRI were ideal for bringing this group together for a focused semester, and the interaction with the concurrent program Representation Theory of Finite Groups and Related Topics was so intense that it was never clear which members were officially members of which program. The natural overlap between these fields was beneficial to both. The program saw great interplay between combinatorics, geometry, finite groups, Lie theory, and probability in their applications to representation theory. There was a focused excitement in the air throughout the program and an environment in which there was intense work on problems such as • Interaction of geometry, representation theory, and combinatorics, • Macdonald polynomials • Applications of combinatorial representation theory • Computational advances and development of Sage-Combinat (a computer package) • Cluster algebras, quivers, and quantum affine algebras.

One of the exciting moments came when Arun Ram and Martha Yip discovered a new combinatorial formula for Macdonald polynomials. This new formula is valid for all root systems. One of the most exciting aspects of the Ram-Yip formula is the fact that it is in terms of the path model, which also has an algebro-geometric interpretation in terms of galleries in a building. The form of the new formula is the same as that of the formula of Haglund-Haiman- Loehr for type GL(n), but there is a fascinating and not very well understood collapsing of terms that relates the two formulas. Recent preprints of Cristian Lenart study this collapsing of terms. The connection between the path model combinatorics and the algebro-geometric interpretation was the centerpiece of discussions at MSRI between Peter Littelmann and Cristian Lenart. The compression of terms seems to have an algebro-geometric background related to the interpretation of galleries (or alcove walks) in the framework of affine buildings and the affine Grassmannian. Several significant projects provided beautiful applications of combinatorial representation theory. One of the striking results was the discovery by Lauren Williams, J.C. Novelli, and J.Y. Thibon of a connection between the asymmetric exclusion process and combinatorial Hopf algebras. Several important features of the stationary distribution of the process can be read directly from the combinatorial Hopf algebra perspective, and there is further data available on the Hopf algebra side that, so far, is not yet understood in terms of the asymmetric exclusion process. Lauren Williams was a Viterbi postdoctoral fellow at MSRI for the whole semester and J.C. Novelli visited MSRI for a short period. Sami Assaf, Persi Diaconis and Kannan Soundararajan are completing a beautiful project on the study of random walks on cosets. A particular case of interest, where the group is the symmetric group and the is a Young subgroup, corresponds to the analysis of shuffles of a bicolored deck of cards. They show that log n shuffles are sufficient to mix up a deck with n cards that are half red and half black. The proof of these results uses representation theory (character formulas for the symmetric group evaluated at transpositions), combinatorics (to get formulas for which decks are mostly likely to appear), and probabilistic and analytic methods (to get asymptotics for distance to uniformity). The Focused Research group on ”Affine Schubert calculus” had a significant presence at MSRI with many members (Jason Bandlow, Francois Descouens, Anne Schilling, Mark Shimozono, Nicolas Thiéry, Mike Zabrocki) being in residence for varying amounts of time during the semester. On of the goals of this research group is to share computational software development

25 efforts between the participants, and, at the end, to make it freely available. Under the leadership of Florent Hivert and Nicolas Thiéry, the open source algebraic combinatorics package MuPAD- Combinat (http:://mupad-combinat.sf.net/) has been developed since 2001. The rapid growth of Sage (www.sagemath.org) makes it a much more viable alternative for a combinatorics package. Sage was started in 2005 by William Stein (now at the University of Washington), and it already consists of over two million lines of code. It incorporates several of the best free, open-source mathematics software packages available (GAP, Singular, Macaulay, GMP, and MPFR, just to name a few), as well as a huge original library, including several new algorithms not yet found elsewhere. (Incidentally, MSRI is holding a 4-day workshop in April 2009 on the SAGE software.)

26 1.6 Programs Consultant List

Consultant Consultant Disciplinary Program Name Specialty Consultant Employer Probability, Algorithms and Statistical Physics Probability & Applications University of California, Berkeley Computational and Statistical University of California, Aspects of Analysis Algebraic Geometry Los Angeles Combinatorial, Enumerative and Toric Geometry William Fulton Algebraic Geometry , Ann Arbor Climate Change Summer School Inez Fung Climate Change University of California, Berkeley Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in Differential Geometry & dimension 3 Helmut Hofer Geometric Analysis New York University, New York Climate Change Dynamical Systems & University of North Summer School Chris Jones Climate Change Carolina,Chapel Hill Geometric and Analytical Aspects of Nonlinear Dispersive Equations Carlos Kenig PDE Analysis , Chicago Number Theory, Advances in Algebra and Automorphic Forms, Geometry & Algebraic Geometry , Cambridge Representation Theory,Algebraic Combinatorial Representation Geometry, Mathematical Theory Andrei Okounkov Physics, & Probability , Princeton PDE, Analysis, Applied Probability, Geometry and George Mathematics, Integrable Systems Papanicolaou & Finance , Stanford Analytic and Computational Aspects of Differential Geometry & Elliptic and Parabolic Equations Geometric Analysis Stanford University, Stanford Teichmuller Theory and Kleinian Groups/ Geometric Group Theory Karen Vogtmann Topology Cornell University, Ithaca Symposium on the Mathematical Challenges The Jackson Laboratory, Bar of Systems Genetics Richard Woychik Genetics Harbor Geometric dynamical systems, mathematical biology, population Climate Change Summer dynamicsand & Climate School Mary Lou Zeeman Change Bowdoin College, Brunswick Representation Theory,Algebraic Combinatorial Representation Geometry, & Special Theory Andrei Zelevinsky Functions Northeastern University, Boston

27 1.7 MSRI Experiences

Each year we write to organizers and postdoctoral fellows from programs held two, four, and ten years ago (2005-06, 2003-04, 1997-98) to ask for an update on the effect of the program/workshop on the disciplines and their careers. Below are the responses we received; program and workshop organizers are followed by postdoctoral fellows.

Organizers

2005-2006

Inez Fung (Data Assimilation for the Carbon Cycle)

Two workshops on Carbon Data Assimilation were co-sponsored by MSRI and NCAR. The workshops brought two communities together. The carbon cycle science community has a new scientific challenge with the upcoming launch (January 2009) of the new satellite - the Orbiting Carbon Observatory - which will provide an unprecedentedly large volume of asynoptic observations of CO2 in the atmosphere. The mathematics community has new techniques to interrogate, compress the data and to assimilate them into atmospheric circulation models. Several of the students from the first workshop (held at MSRI) returned to the second workshop (at NCAR). The instructors at the two workshops have succeeded in starting several new grants and new projects to carry out assimilation of the OCO data.

The MSRI support was utterly critical to the start of the new field of carbon data assimilation. It was an occasion for the instructors from the carbon science and the mathematics communities as much as for the students to learn from one another. Such a workshop would not have been successful within an individual NSF, NASA, NOAA or DOE program. The facilities at MSRI are superb - and the ability to have computer projects was a great component.

Bjorn Poonen (Cohomological Approaches to Rational Points)

Postdoctoral Fellows: Timothy Browning is a "Reader in pure mathematics" at Bristol. Mirela Ciperiani is a postdoc at Columbia. Alina Cojocaru has a tenure-track position at U. Illinois at Chicago. Matilde Lalin has a tenure-track position at U. Alberta. Aaron Levin: not sure, but he had a visiting position at the De Giorgi Center in Pisa at least until this summer. Maria Sabitova is a postdoc at U. Illinois Urbana-Champaign. Ronald van Luijk is now a research associate at U. Warwick. Olivier Wittenberg is now permanently (I think) at Strasbourg and is supported by the CNRS. As far as I can see, all are continuing successful academic careers.

As for new results etc. coming out of the semester, there are many, but let me focus on one area that I know well.

28

In March 2006 our program held a workshop on cohomological approaches to rational points, focusing on such things as the Brauer-Manin obstruction and the descent obstruction. In the first half of 2008, research in this area culminated with a clear picture of the relationship between these two and a proof that these obstructions are insufficient to answer the basic question of which multivariable polynomial equations have rational solutions.

These results can be seen as the next step in the line of research starting with the results of Lind and Reichardt in the 1940s showing that the Hasse principle can fail, and the result of Skorobogatov in 1999 showing that the Brauer-Manin obstruction is insufficient to explain this failure.

They are obtained by combining a preprint of my own with preprints of two other mathematicians, one of whom (Alexei Skorobogatov) was a Research Professor, and another of which (Cyril Demarche) is a graduate student of one of our General Members (David Harari).

I think it is fair to say that it was the MSRI semester and workshop that led to this breakthrough, even if there is no direct connection.

Barbara Keyfitz (Women in Mathematics: The Legacy of Ladyzhenskaya and Oleinik)

The event in question was the "two olgas" conference -- I was a co- organizer. Because this was not a technical conference focused on my research area, I did not expect to make new research collaborations, and I did not. In my opinion, the main value of such events -- and they are valuable -- is to expose young women to role models and to showcase the achievements of more senior women. In this case, the meeting was successful. I heard talks from several people I had not heard before -- Natasa Pavlovic and stand out in particular -- who I will make a point of including in future events.

I'm sorry that I don't have any particular "success stories" -- women who told me that the event was important to them -- but I'm sure they exist! Everyone who was there thought it was a great occasion.

Chuck Newman (Probability, Geometry and Integrable Systems)

Although I was a member of the organizing committee for the 2005 workshop on Probability, Geometry and Integrable systems, my own areas of probability theory are somewhat off the main topics of the meeting. Hence I will restrict my comments to some issues related to the joint paper with Federico Camia (entitled "SLE(6) and CLE(6) from critical percolation") that we contributed to the proceedings, but I do think that these comments help indicate the important role played by MSRI.

Federico, who had been my PhD student, was, at the time of the MSRI meeting a postdoc in the Netherlands (he is now an assistant professor

29 there) and was unable to attend, but we used the opportunity of the meeting to prepare a talk and paper specifically intended to be accessible to a much wider mathematical audience than our highly technical paper published in Probab. Theory Rel. Fields in 2007. Both papers involve giving (or sketching) a detailed proof of one of the basic convergence results in the subject of Schramm-Loewner Evolutions (SLE), first conjectured by Schramm, and then partially proved by Smirnov. The subject of SLEs, which is about 10 years old, represents a major breakthrough on the border between probability theory and statistical mechanics. Indeed one of the important relatively early workshops on the subject took place in April/May 2001 at MSRI, where Smirnov announced his results about conformal invariance for the scaling limit of critical percolation. I also attended (and spoke at) that workshop which influenced my decision to work in the area. Incidentally, this area continues to be highly active both in the mathematics and theoretical physics communities, as was clear in a very recent (August 2008) workshop in Montreal co- organized by (2006 Fields medalist), John Cardy (distinguished physicist from Oxford) and myself. The importance of workshops such as that one and the ones at MSRI cannot be overemphasized.

Reinhard Laubenbacher (MSRI Summer Graduate Workshop: Mathematical aspects of computational biology)

This feedback is about the graduate student summer workshop 2006. I was one of the two organizers and one of the principal lecturers.

The program was very well attended, with over 50 participants. The organizational support that MSRI provided was excellent, and the program went very smoothly. I have since had interactions with several of the participants. Just in the last week I received an e-mail from one student who sent me her M.S. thesis. After the workshop she changed her research interests and completed a thesis closely related to one of the workshop topics.

Several other participants e-mailed me over the last year to discuss career choices and research problems. One participant significantly changed her Ph.D. thesis work as a result of the workshop, including computational approaches discussed during the workshop.

My graduate assistant, Brandy Stigler, who helped conduct the workshop, has had extensive communications with workshop participants over the last couple of years, and has helped with specific research questions as well as general advice about career choices.

In summary, I think that the workshop was a resounding success, and MSRI was instrumental in making is that. I was very impressed with the organization and support.

30 2003-2004

Bob Megginson (MSRI/Howard Workshop on Geometry: An Introduction to 2003-04 Programs at the Mathematical Sciences Research Institute)

As Deputy Director I was co-organizer of the "MSRI/Howard Workshop on Geometry: An Introduction to 2003-04 Programs at the Mathematical Sciences Research Institute". One of the participants in that workshop was an African American mathematician (and also accomplished chemist), Troy L. Story, from Morehouse College. It was that workshop that convinced Professor Story that he should apply to participate in the then-upcoming program in modern differential geometry. One outcome of his participation in the differential geometry program was a book in the area (Introduction to Differential Geometry with Applications to Navier-Stokes Dynamics, iUniverse, 2005). This book has been generally well received, and I quote a reviewer's comments on Amazon.com:

"This book has the clearest and most direct introduction to differential geometry that I've seen, thus, I highly recommend it.

"One of the book's highlights is chapter 2, which presents a discussion of exterior calculus assuming only a knowledge of conventional calculus. Unlike most books on this topic (exterior calculus), this book includes a definition of the exterior derivative rather than a few examples.

"Another highlight is the chapter on dynamics, where it is shown that many areas of dynamics can be described by differential one-forms, including Navier-Stokes dynamics for incompressible fluids."

It is notable that one of the MSRI participants and presenters in that workshop, whose encouragement and enthusiasm for Professor Story's work helped seal the deal in Professor Story's mind, was Robert Bryant, now MSRI Director. This is just one example of the impact that MSRI has had on addressing the underrepresentation of minorities in mathematics at many different levels, through the very highest.

Richard A. Olshen (Genetics of Complex Disease)

Our workshop on the “Genetics of Complex Disease” was different from some others in that there were many speakers, 36 in all. By “complex disease” we mean disease such as cardiovascular disease and cancer that are largely heritable, but not by simple Mendelian mechanisms. While some participants were from the , some were from abroad. They include Agnes Hsiung, head of Biostatistics and Bioinformatics at the National Health Research Institutes in Taipei and Benjamin Yakir of the Hebrew University. Terry Speed splits his time between the University of California, Berkeley and the Walter and Eliza Hall Institute of Medical Research in Melbourne, Australia. Hsiung spoke on cDNA microarray experimental design. She is one of the reigning authorities regarding the genetics of hypertension, a topic regarding which I (Richard Olshen) spoke. Since the workshop Yakir has coauthored the book The Statistics of Gene Mapping with co-organizer David Siegmund of Stanford University.

31 Siegmund is John D. and Sigrid Banks Professor at Stanford and recipient of many other honors, for example election to the U.S. National Academy of Sciences. He spoke on mapping quantitative trait loci in the presence of gene-covariate interactions. Professor Speed’s talk was on finding genes associated with multiple sclerosis.

Joe Gray, Member of the Board of Directors of the U.S. National Institutes of Health and Co- Director of the Lawrence Berkeley National Laboratory, spoke on, “Genetic in Cancer.” Much work on cancer makes use of SAM and PAM software and “nearest shrunken centroids,” coauthored by Robert Tibshirani of Stanford University, as well as Tibshirani’s “lasso” approach to inference about many parameters. Tibshirani spoke at the workshop on, “Sample Classification from Protein Mass Spectroscopy by Probability Contrasts.” This was in keeping with our view that genes and the environment, and interactions among them, bear upon complex disease, genes through single polymorphic sites (SNPs), expression, and levels of proteins in the systemic circulation. Protein secondary structure was addressed by Chip Lawrence, now of . Regarding cancer, Adam Olshen of Memorial Sloan- Kettering Cancer Center spoke on his coauthored approach to CGH (comparative genomic hybridization), in particular the CBS (circular binary segmentation) method. As well, Fred Wright, then of State University and now of the University of North Carolina, spoke on mapping tumor suppressor genes. We note that Wright was perhaps the first to quantify the number of genes in the human genome as well under 30,000, not the more than 100,000, as had been thought.

Dr. Eddy Rubin of LBL, Director of the U.S. Department of Energy’s Joint Genome Institute, spoke on his celebrated comparisons of genomes in the study of complex disease. Karl Broman of Johns Hopkins University spoke on that topic as well, “Gene Mapping in Model Organisms.” The famous but still somewhat controversial HapMap was discussed from mathematical points of view by Fengzhu Sun of the University of Southern California and celebrated computer scientist Richard Karp of the University of California, Berkeley. Reviewers please note that Karp has won a National Medal of Science and the Turing Award.

Warren Ewens of the University of Pennsylvania and author of the famous Ewens sampling formula gave a review of the transmission-disequilibrium test and tests that have come after it.

Matthew Stephens, then of the University of Washington and now of the University of Chicago, has coauthored much fundamental software used in studying the genetics of complex disease. They include BIMBAM, fastPHASE, SCAT, and HOTSPOTTER.

It has been noted more than several times that a serious problem in studying the genetics of complex disease concerns failure of results to replicate. This can owe to population “admixture,” a topic addressed at our workshop by Hua Tang, then of the Fred Hutchinson Cancer Research Center in Seattle, now of the Department of Genetics at Stanford.

Ingilief Hallgrimsdottir addressed the workshop on algebraic aspects of linkage analysis. She was then a Ph.D. student at the University of California, Berkeley and is now doing postdoctoral work at Oxford University.

32 Reviewers will note from major journals and newspapers that many leading figures in the ongoing, increasingly important study of the genetics of complex disease were among the 100+ participants in our lively workshop. There are too many publications among subsets of participants and others of their collaborators to list here.

1997-1998

Carl Mueller (Stochastic Partial Differential Equations)

I was involved in the workshop on Stochastic PDE in 97-98. MSRI is a wonderful place, and I certainly benefited from talking to many colleagues during that period. Unfortunately, I have nothing specific to report. My work evolves over time, an it's often hard for me to pinpoint a specific conference that that was crucial. Likewise, I work with a number of collaborators who I've known for a long time, and it's hard to pick a certain time which was crucial to our work.

Eric Friedlander (Workshop on Homotopy theory for algebraic varieties with applications to K-theory and quadratic forms)

As I recall, the meeting for which you wish a report was the first MSRI "Hot Topics" workshop, dedicated to recent advances in algebraic K-theory and especially the just announced proof of the Milnor Conjecture by . Vladimir presented details of his very recent work - - which ultimately resulted in his -- and many other mathematicians actively participated. I believe that this meeting brought together Vladimir Voevodsky and Markus Rost for the first time, ultimately leading to the recently completed proof of the Bloch-Kato Conjecture (which extends to odd primes the Milnor Conjecture).

The meeting was attended by many of the up-and-coming algebraic K-theorists of the time. I believe that this meeting has been viewed retrospectively as a watershed event.

Felipe Voloch (Model Theory, Algebra and Arithmetic)

I was an organizer of the program on Model Theory of Fields in the Spring of 98, but Model Theory is not my specialty. My role was to foster interaction with other areas, primarily Number Theory. In this, the program was very successful, thanks to the help of many people. I know the interaction of Model Theory and Number Theory is flourishing, for instance, in the work of Thomas Scanlon and Rahim Moosa (who were postdoctoral/student participants at the time) as well as influenced the work of Bjorn Poonen, even though he was in residence just for a short period at the end. Poonen and I recently wrote a paper which has received some attention and which uses the relatioship of Model Theory and Number Theory in a fundamental way. I believe the program was very successful overall but I haven't kept in touch with pure model- theoretic side of things.

33 Carlos Kenig (Harmonic Analysis)

The only update that I have is that one of our postdocs, Terry Tao, has been awarded a Fields Medal since the last response. A couple of the results mentioned in his citation can be traced to his time at MSRI.

Anand Pillay (Model Theory of Fields)

The 97-98 model theory of fields program was an important one for our subject.

Concerning postdocs, it marked the beginning of Tom Scanlon's (now at Berkeley) successful career, and I believe that many of the contacts he made and discussions he had there influenced much of his subsequent work. Also David Pierce's postdoc served him well. (Now at METU, Ankara).

The program was built in many ways around the interactions between model theory and diophantine geometry, but in a sense it represented the end rather than beginning. Contacts with people like Voloch and Buium lessened after the program (Scanlon being an exception).

On the other hand, motivic integration was a new topic in 97-98. There were talks about it in MSRI and it has since blossomed within model theory. The old model theory/diophantine geometry work has also now resurfaced, but with other collaborators on the number theory/geometry side. People like van den Dries could comment more on the o-minimal side of the program.

Personally I was worn out by the MSRI semester. We tried to have too many activities and seminars, and the atmosphere was far from relaxed.

34 Postdoctoral Fellows

2005-2006

Aaron Levin (Rational and Integral Points on Higher-Dimensional Varieties)

I am writing regarding my time spent as a postdoc at MSRI in Spring '06. Papers that were written (or influenced) as a result of time I spent at MSRI include:

Ideal class groups, Hilbert's irreducibility theorem, and integral points of bounded degree on curves. J. Théor. Nombres Bordeaux 19 (2007), no. 2, 485--499

Variations on a theme of Runge: effective determination of integral points on certain varieties (submitted)

Ideal class groups and torsion in Picard groups of varieties (submitted)

As for more general items of long-term impact, which while not as specific, I think are just as important, I would include getting to meet for the first time many important researchers in my subject area and the many lectures I attended on topics which haven't yet made their way into my research, but will undoubtedly come up at some point in the course of my work. My experience at MSRI was very positive, and I can't think of any specific criticisms or suggestions at the moment.

Nora Ganter (New Topological Structures in Physics)

The stay was very enjoyable and productive. Especially at the opening conference in Morelia, I heard a talk by Lupercio which triggered my interest in string topology. Most of the rest of my semester at MSRI, I spent writing down my ideas about how his joint work with Uribe and Xicotencatl connects to my former work on the Dijkgraaf-Moore-Verlinde-Verlinde formula. The result was the following paper:

Title: Stringy power operations in Tate K-theory Authors: Nora Ganter Categories: math.AT Algebraic Topology (physics.math-ph Mathematical Physics) Comments: 41 pages, 5 figures, improved exposition MSC: 55N34; 55P35; 58J26 http://front.math.ucdavis.edu/0701.5565

(submitted to Topology and Geometry, under review). Having the string topology community around when writing this was of course great. I had the chance to talk to Ralph Cohen a lot and to spend a week at Stanford visiting him, and I met Craig

35 Westerland for the first time at MSRI. Craig and I are now colleagues; we just arrived at Melbourne University and the common experience of the MSRI program makes it very natural for us to talk math to each other.

2003-2004

Emma Carberry (Differential Geometry)

I was a postdoctoral fellow at MSRI in the differential geometry programme of 2003 -- 2004. The year not only gave me a respite from teaching to concentrate on my research, it also gave me the chance to interact with a community of people doing similar research. I learnt a great deal from informal conversations and talks given at this time and also built many valuable professional contacts. I recently posted a paper arXiv:0805.3732 with my co-author Erxiao Wang whom I met during this time. I am also currently engaged in a collaboration with Daniel Fox who might also met during this time. The opportunity to interact with Robert Bryant was particularly helpful and was followed up with a semester spent at the University of Duke in order to visit him. We mathematicians need nothing so much as one another, and by bringing together people working on similar problems the Mathematical Sciences Research Institute does a tremendous service to our community.

1997-1998

HSIAN-HUA TSENG (New topological structures in Physics)

I was a postdoctoral fellow at MSRI affilated with the program “New topological structures in Physics” held during Spring 2006 semester. I became quite familiar with MSRI when I was a graduate student in Berkeley. During those days I often went up-hill for seminars and workshops. When I first learnt that I was awarded a postdoctoral fellowship at MSRI, I was very excited. My experience as a postoctoral fellow at MSRI turned out to be more fruitful than I had anticipated. The organizers of the program invited many experts in the field, and I benefited a lot from discussing with them. For example, discussions with Yongbin Ruan, Tyler Jarvis, and Takashi Kimura had influenced my research later on. Communications with other postdoctoral fellows turned out to be even more important to my research. At MSRI I started collaborating with Tom Coates, Hiroshi Iritani (both of whom were MSRI postdoctoral fellows at the time), and Alessio Corti. This ongoing collaboration had resulted several papers already. In my opinion, our works on crepant roslution conjecture in Gromov-Witten theory [1], computations of twisted orbifold Gromov-Witten invariants in genus 0 [2], and computations of quantum cohomology of Fano toric stacks [3] are worth mentioning. At MSRI I also started collaborating with Todor Milanov (a postdoctoral fellow at Stanford at the time) on projects related to integrable systems and Gromov-Witten theory. Our paper [4], which grew out of the collaboration at MSRI, identifies integrable hierarchies that govern the Gromov-Witten theory of P1-stacks.

36 In my experience, being a postdoctoral fellow at MSRI has benefited my research in two aspects. First, I was given the chance to meet with many world experts in the subjects of my interests. This not only helped me shape my view towards research tpoics, but also broadened my research horizon. Second, being in contact with other MSRI postdocs was really a unique experience that greatly advanced my research. MSRI made many efforts to put postdocs in contact with each others. The weekly MSRI postdoc seminar served as one such effort, for instance. In my case this resulted in very productive collaborations which I mentioned above. I believe that I wouldn’t have worked on these directions had I not been given the opportunity to visit MSRI.

In summary, my days as a postdoctoral fellow at MSRI had played a key role in my career as a research mathematician. I am sure that MSRI postdoctoral fellowship will continue to help advance researchs of mathmaticians in their early career.

37 2. Programs and Workshop Participation

2.1 Program Participant List (See attached file for full detail) (O:\0708AnnualReport\NSF Report 07-08\a. Program Participant List\ a. Program Participant List 07-08)

Family Name First Name Home Institute Name Position Program Andersen Henning Aarhus Universitet Professor CRT Andre Carlos University of Lisbon Associate Professor CRT postdoctoral research Assaf Sami Massachusetts Institute of Technology fellow CRT Barcelo Hélène Arizona State University Professor CRT Researcher (full-time Baumann Pierre Universit\'e de Strasbourg I (Louis Pasteur) research position) CRT Benkart Georgia University of Wisconsin, Madison Professor CRT Bergeron Nantel York University Professor CRT Bessenrodt Christine Leibniz Universitaet Hannover Professor CRT Centre National de la Recherche Researcher (full-time Bonnafe Cedric Scientifique research position) CRT Brenti Francesco Università di Roma, II Professor CRT Brundan Jonathan University of Oregon Professor CRT École Polytechnique Fédérale de Chlouveraki Maria Lausanne (EPFL) Post Doc CRT Daugherty Zajj University of Wisconsin Graduate Student CRT Davis Matt University of Wisconsin, Madison Graduate Student CRT Commissariat \`a l'\'Energie Atomique (CEA)-- Centre d'\'Etudes Nucl\'eaires de Saclay di Francesco Philippe (CENS) Research member CRT Diaconis Persi Stanford University Professor CRT Laboratoire de Mathematiques de Dudas Olivier Besançon CNRS (UMR 6623) UFR - ST Graduate Student CRT Emsiz Erdal Universidad deTalca Graduate Student CRT Fomin Sergey University of Michigan Professor CRT Fourier Ghislain Universität zu Köln Post Doc CRT Frenkel Edward UCB - University of California, Berkeley Professor CRT Garsia Adriano University of California Professor CRT UNAM - Universidad Nacional Autonoma Geiss Christof de Mexico Professor CRT Goodman Frederick University of Iowa Professor CRT Professor of Gordon Iain University of Edinburgh Mathematics CRT Graber John University of Iowa Graduate Student CRT Guralnick Robert University of Southern California Professor CRT Halverson Tom Macalester College Professor CRT Hansen Mike Harvey Mudd College Graduate Student CRT Universit\'e Versailles/Saint Quentin-en- Hernandez David Yvelines Researcher CRT jacon Nicolas Universite de Franche comte Maitre de conferences CRT

38 Kamnitzer Joel American Institute of Mathematics Postdoctoral Fellow CRT Kato Syu Kyoto University Post Doc CRT Kedem Rinat University of Illinois at Urbana-Champaign Associate Professor CRT Alexander Kleshchev (Sasha) University of Oregon Associate Professor CRT Kujawa Jonathan University of Oklahoma Assistant Professor CRT Lam Thomas Harvard University Assistant Professor CRT Lascoux Alain Université de Paris-Est Professor CRT Lau Michael University of Windsor Assistant Professor CRT Leclerc Bernard Université de Caen Professor CRT Lecouvey Cedric Laboratoire Joseph Liouville Calais Maitre de conferences CRT Lehrer Gustav University of Sydney Professor CRT Littelmann Peter Universit\"at zu K\"oln Professor CRT Institute for Theoretical and Experimental Loktev Sergei Physics Professor CRT MIT - Massachusetts Institute of Lusztig George Technology Professor CRT Lyle Sinead University of East Anglia Lecturer CRT Mathas Andrew University of Sydney Associate Professor CRT Mbirika Abukuse (Aba) University of Iowa Graduate Student CRT Orellana Rosa Dartmouth College Associate Professor CRT Purbhoo Kevin University of Waterloo Assist. Prof. CRT Pylyavskyy Pavlo University of Michigan Post Doc CRT Ram Arun University of Melbourne Professor CRT Rhoades Brendon Graduate Student CRT Saxl Jan University of Cambridge Professor CRT Schilling Anne University of California, Davis Associate Professor CRT Schroeer Jan University of Bonn Professor CRT MIT - Massachusetts Institute of Professor of Applied Stanley Richard Technology Mathematics CRT Stembridge John University of Michigan Professor CRT Thiem Nat University of Colorado Assistant Professor CRT Thiéry Nicolas UC Davis Maître de conférence CRT Vazirani Monica University of California, Davis Assistant Professor CRT Vershik Anatoly Russian Academy of Sciences Professor CRT Virk Rahbar University of Wisconsin Graduate Student CRT Wang Weiqiang Professor CRT Warnaar S The University of Melbourne Senior Research Fellow CRT Williams Lauren Harvard University Post Doc CRT Yip Martha University of Wisconsin Graduate Student CRT Zabrocki Mike York University Professor CRT Zalesski Alexandre University of East Anglia Professor CRT Beck Matthias San Francisco State University Assistant Professor CP 07-08 Friedlander Susan Professor CP 07-08 Hain Richard Professor CP 07-08 Hengesbach Conrad Duke University Graduate Student CP 07-08 Juan Lourdes Texas Tech University Assistant Professor CP 07-08 H.C.Wang Assistant Lim Seonhee Cornell University Professor CP 07-08

39 NSF Posdoctoral Lotay Jason Fellow CP 07-08 Lott John University of Michigan Professor CP 07-08 Matsumoto Makoto Hiroshima University Professor CP 07-08 Maria Moors Cabot McMullen Curtis Harvard University Professor CP 07-08 Pearlstein Gregory Michigan State University Assistant Professor CP 07-08 Smith Abraham Duke University Graduate Student CP 07-08 Springer Tonny Rijksuniversiteit te Utrecht Professor CP 07-08 Terasoma Tomohide University of Tokyo Professor CP 07-08 Xu Feng Duke University Graduate Student GGT Abels Herbert Universität Bielefeld Professor GGT Algom Kfir Yael University of Utah Graduate Student GGT Allcock Daniel University of Texas, Austin Associate Professor GGT Arzhantseva Goulnara Université de Genève Professerue Adjointe GGT Behrstock Jason ritt assistant professor GGT Berkove Ethan Lafayette College Associate Professor GGT Bestvina Mladen University of Utah Professor GGT Brendle Tara Louisiana State University Assistant Professor GGT Universitat politecnica de Catalunya Burillo José EPSC Assistant Professor GGT Bux Kai-UWe University of Virginia Assistant Professor GGT Pierre- Institut des Hautes Études Scientifiques Caprace Emmanuel (IHES) Researcher GGT Cashen Christopher University of Utah Post Doc GGT Chatterji I Assistant Professor GGT Cleary Sean City College, CUNY Associate Professor GGT Coulbois Thierry Universite Aix-Marseille III () Assistant Professor GGT Davis Michael Ohio State University Professor GGT Delucchi Emanuele Binghamton University (SUNY) Post Doc GGT Delzant Thomas Université de Strasbourg (Louis Pasteur) Professor GGT Dymarz Tullia Post Doc GGT Farb Benson University of Chicago Professor GGT Feighn Mark Rutgers University, Newark Professor GGT Hedrick and NSF Fernos Talia University of California, Los Angeles Postdoctoral Fellow GGT Fujiwara Koji Tohoku University Associate Professor GGT Geoghegan Ross SUNY, Binghamton Professor GGT Grigortchuk Rostislav Texas A \&\ M University Professor GGT Groves Daniel California Institute of Technology Instructor GGT Guirardel Vincent Universit\'e de Toulouse III (Paul Sabatier) Maitre de Conferences GGT Hambleton Ian McMaster University Professor GGT Hsu Timothy San Jose State University Associate Professor GGT Irmak Elmas Bowling Green State University Assistant Professor GGT Januszkiewicz Tadeusz Ohio State University Professor GGT Kapovich Ilya University of Illinois at Urbana-Champaign Associate Professor GGT Kar Aditi Ohio State University Graduate Student GGT Kim Sang-hyun University of Texas at Austin Post Doc GGT Kropholler Peter University of Glasgow Professor GGT

40 Leary Ian Ohio State University Professor GGT Levitt Gilbert Université de Caen Professor GGT Louder Larsen University of Utah Post Doc GGT Lustig Martin Universite P. Cezanne - Aix Marseille III Full professor GGT Malone William University of Utah Graduate Student GGT McCammond Jon University of California, Santa Barbara Professor GGT Meier John Lafayette College Professor GGT Min Honglin Rutgers University-Newark Graduate Student GGT THE INSTITUTE OF MATHEMATICAL Mitra Mahan SCIENCES Professor GGT Mosher Lee Rutgers University, Newark Professor GGT Okun Boris University of Wisconsin-Milwaukee Associate Professor GGT Osajda Damian University of Wroclaw Post Doc GGT Papazoglou Panagiotis University of Athens Associate Professor GGT Szego assistant Pettet Alexandra Stanford University professor GGT Pfaff Catherine Rutgers University Graduate Student GGT Putman Thomas Massachusetts Institute of Technology Post Doc GGT REMY Bertrand Institut Camille Jordan - Universite Professor GGT Riley Tim University of Bristol Academic Faculty GGT Sabalka Lucas UC Davis Assistant Professor GGT Sageev Michah Technion--- Institute of Technology Professor GGT Sapir Mark Centennial Professor GGT Scott Richard Santa Clara University Associate Professor GGT Sela Zlil Hebrew University Professor GGT Swenson Eric Brigham Young University Associate Professor GGT Thomas Anne Cornell University Post Doc GGT Ventura Enric Universitat Politecnica de Catalunya Associate Professor GGT Vogtmann Karen Cornell University Professor GGT Whyte Kevin University of Illinois at Chicago Associate Professor GGT Wiest Bert University of Rennes 1 tenured junior professor GGT Wilton Henry University of Texas R. H. Bing Instructor GGT Wise Daniel McGill University Professor GGT Xie Xiangdong Georgia Southern University Professor GGT Alperin Jon University of Chicago Professor RTFG Benson Dave University of Aberdeen Professor RTFG Blomgren Martin Royal Institute of Technology (KTH) Phd Student RTFG Boltje Robert University of California Professor RTFG Centre National de la Recherche Researcher (full-time Bonnafe Cedric Scientifique research position) RTFG Directeur de Bouc Serge Universit\'e de Picardie (Jules Verne) Recherches RTFG Broué Michel Universit\'e de Paris VII (Denis Diderot) Professor RTFG Cabanes Marc Universit\'e de Paris VII (Denis Diderot) Professor RTFG Carlson Jon University of Georgia Professor Emeritus RTFG Coskun Olcay Bilkent University Graduate Student RTFG Craven David University of Oxford Junior Research Fellow RTFG Danz Susanne University of Jena research assistant RTFG

41 Digne François Universit\'e de Picardie (Jules Verne) Professor RTFG Doty Stephen Loyola University Chicago Professor RTFG Fong Paul University of Illinois at Chicago Professor RTFG Noyes Professor of Friedlander Eric Northwestern University Mathematics RTFG Chair in Pure Geck Meinolf University of Aberdeen, King's College Mathematics RTFG Geline Michael University of Chicago Grad Student RTFG Glesser Adam University of Aberdeen Post Doc RTFG Grodal Jesper University of Copenhagen Professor RTFG Guralnick Robert University of Southern California Professor RTFG Hemmer David State University at Buffalo, SUNY Assistant Professor RTFG Himstedt Frank Technische Universität Munchen Professor RTFG Hiss Gerhard RWTH Aachen Professor RTFG Isaacs I. Martin University of Wisconsin, Madison Professor RTFG jacon Nicolas University of Chicago Maitre de conferences RTFG Jones Vaughan UCB - University of California, Berkeley Professor RTFG Juteau Daniel Universit\'e de Paris VII (Denis Diderot) Post Doc RTFG Keller Bernhard University Paris 7 Professor RTFG Kessar Radha University of Aberdeen Senior Lecturer RTFG Krause Henning University of Paderborn Professor RTFG Kuelshammer Burkhard University of Jena Professor RTFG Kujawa Jonathan University of Oklahoma Assistant Professor RTFG Lehrer Gustav University of Sydney Professor RTFG Lin Zongzhu Kansas State University Professor RTFG Linckelmann Markus University of Aberdeen Professor RTFG Lyle Sinead University of East Anglia Lecturer RTFG Magaard Kay Wayne State University Professor RTFG Malle Gunter TU Kaiserslautern Professor (W3) RTFG Marcus Andrei Babes-Bolyai'' University of Cluj-Napoca Professor RTFG Maroti Attila University of Southern California Post Doc RTFG Mathas Andrew University of Sydney Associate Professor RTFG Mazza Nadia University of Aberdeen Research fellow RTFG Michel Jean University Paris VII Director of research RTFG Nakano Daniel University of Georgia Professor RTFG Navarro Gabriel University of Valencia Professor RTFG postdoctoral research Noeske Felix RWTH Aachen University fellow RTFG Professor of O'Brien Eamonn University of Auckland Mathematics RTFG Olsson Jørn University of Copenhagen Professor RTFG Acting Assitant Pevtsova Julia University of Washington Professor RTFG Researcher Director, Puig Lluis Universit\'e de Paris VII (Denis Diderot) First Class RTFG Ragnarsson Kari University of Illinois Assistant Professor RTFG Rainbolt Julianne St. Louis University Associate Professor RTFG Professor of Pure Rickard Jeremy University of Bristol Mathematics RTFG

42 Robinson Geoffrey University of Aberdeen Professor RTFG Saxl Jan University of Cambridge Professor RTFG Scott Leonard University of Virginia Professor RTFG Späth Britta RWTH Aachen University RTFG Srinivasan Bhama University of Illinois at Chicago Professor RTFG Stancu Radu University of Copenhagen Post Doc RTFG Swenson Daniel University of Minnesota Graduate Student RTFG Symonds Peter Professor RTFG Thiem Nat University of Colorado Assistant Professor RTFG Tiep Pham University of Florida Professor RTFG Webb Peter University of Minnesota Professor RTFG Zalesski Alexandre University of East Anglia Professor RTFG Zhang Jiping Peking Univiversity Professor RTFG Assistant Professor Adeboye Ilesanmi University of Southern California (NTT) TTKG Agol Ian University of California Associate Professor TTKG Visiting Assistant Aramayona Javier University of Illinois at Urbana-Champaign Professor TTKG Associate Professor of Brock Jeffrey Brown University Mathematics TTKG Bromberg Kenneth University of Utah Assistant Professor TTKG Calderin Ivo Florida State University PhD Graduate Student TTKG Calta Kariane Vassar College Post Doc TTKG Canary Richard University of Michigan Professor TTKG Cavendish William Brown University Graduate Student TTKG Cheung Yitwah Northwestern University Assistant Professor TTKG DeBlois Jason University of Illinois Post Doc TTKG Duchin Moon UC Davis VIGRE Fellow TTKG Tamarkin Assistant Dumas David Brown University Professor TTKG Earle Clifford Cornell University Professor TTKG Eskin Alex University of Chicago Professor TTKG Professor of Fenley Sergio Florida State University Mathematics TTKG RTG Postdoctoral Futer David Michigan State University Instructor TTKG Gabai David Princeton University Professor TTKG Gokturk Ali Brown University Graduate Student TTKG Goldman William University of Maryland, College Park Professor TTKG Gordon Cameron University of Texas at Austin Professor TTKG Greenberg Michael Brown University Graduate Student TTKG Hamenstaedt Ursula Universität Bonn Professor TTKG Hensel Sebastian University of Bonn Graduate Student TTKG Huang Zheng University of Michigan Post Doc TTKG Centre International de Recontres Hubert Pascal Math\'ematiques (CIRM), Luminy Professor TTKG Kent Richard Brown University Assistant Professor TTKG Kerckhoff Steven Stanford University Professor TTKG Kim Inkang Seoul National University Associate professor TTKG

43 Charge de recherche Lecuire Cyril Universit\'e de Toulouse III (Paul Sabatier) CNRS TTKG Leininger Christopher University of Illinois at Urbana-Champaign Assistant Professor TTKG postdoc assistant Lenzhen Anna University of Michigan professor TTKG Magid Aaron University of Michigan Graduate Student TTKG Mangahas Johanna University of Michigan Graduate Student TTKG Marden Albert University of Minnesota Professor TTKG Masur Howard University of Illinois Professor TTKG Minsky Yair Yale University Professor TTKG Mirzakhani Maryam Harvard University Assistant professor TTKG THE INSTITUTE OF MATHEMATICAL Mitra Mahan SCIENCES Professor TTKG Namazi Hossein Princeton University Instructor TTKG Nipper Emanuel Universität Bonn Graduate Student TTKG Parlier Hugo IGAT Institute Post-doctoral assistant TTKG Rafi Kasra University of Chicago Instructor TTKG Reid Alan University of Texas at Austin Professor TTKG Schleimer Saul University of Warwick Assistant Professor TTKG Schumacher Georg Philipps-Universität Marburg Professor TTKG Series Caroline University of Warwick Professor TTKG Smillie John Cornell University Professor TTKG Soehl Jakob University of Bonn Graduate Student TTKG Soma Teruhiko Tokyo Metropolitan University Professor TTKG Souto Juan University of Chicago Assistant professor TTKG Tao Jing University of Illinois at Chicago Graduate Student TTKG Thompson Josh University of Utah Graduate Student TTKG Tsai Chia-yen University of Illinois at Urbana-Champaign Graduate Student TTKG Ulcigrai Corinna Princeton University Assistant Professor TTKG Weiss Barak Ben Gurion University of the Negev Associate Professor TTKG Wolf Michael Rice University Professor TTKG Wolpert Scott University of Maryland, College Park Professor TTKG Zorich Anton University of Rennes 1 Full Professor TTKG

44 2.2 Program Participant Summary

# of # of %* % %* Program Citizens Decline/ # of # of Decline/ Name of Activity Participants & Per Res No Reply Female Minorities No Reply

69 31 46% 1 12 17% 2 6% 34 Combinatorial Representation Theory

15 8 62% 2 3 20% 1 11% 6 Complementary Program (07- 08)

67 33 50% 1 13 19% 1 3% 35 Geometric Group Theory

Representation Theory of 67 26 40% 2 8 12% - 0% 39 Finite Groups and Related Topics

Teichmuller Theory and 59 37 66% 3 10 17% 3 8% 20 Kleinian Groups

277 135 50% 9 46 17% 7 5% 134 Total

Total No. of Distinct Program Participants 266 132 51% 9 45 17% 7 5% 129

*Percentage for Citizens & Minorities are computed out of participants that provided info on their citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

45 2.2 Program Participant Demographic Data

Demongraphic Statistics Male v s . Female % Over % Respon- Over # dents Total Distinct Program 16.9% 0.4% Participants 266 Male 220 83.02% 82.7% Female 45 16.98% 16.9% Decline to State

Gender 1 0.38% 0.4% 82.7%

% Over % Respon- Over Ethnicities # dents Total Male Female Decline to State Gender Native American 1 0.73% 0.4% Ethnicities Statistic Asian 17 12.41% 6.2% Black 2 1.46% 0.7% 0.7% Hispanic 5 3.65% 1.8% 1.8% 0.4% 6.2% Pacific 0 0.00% 0.0% 0.0% 121 88.32% 44.0% White 46.9% Decline to State Ethnicities 129 48.50% 46.9% 44.0% Minorities 7 5.1% 2.6%

US Citizen 117 45.5% 44.0% Decline to State Citizenship 9 3.4% 3.4% Per Resident 15 5.6% 5.6% Native American Asian US Citizen & Per Black Hispanic Pac if ic White Resident 132 51.4% 49.6% Decline to State Ethnicities Home Inst. in US 158

Year of Degree # % Year of Highest Degree 2009 31 11.7% 2005-2008 39 14.7% 2000-2004 40 15.0% 17.7% 0.0% 11.7% 1995-1999 35 13.2% 1990-1994 30 11.3% 6.8% 14.7% 1985-1989 26 9.8% 1980-1984 18 6.8% 9.8% Yr <1980 47 17.7% 15.0% 11.3% Decline to state 0 0.0% 13.2% Total 266 100.00% **Statistic Calculation based on participant that replied to each category. Ethnicities' percentage might be over stated since some participants 2009 2005-2008 2000-2004 selected to be included in more than one 1995-1999 1990-1994 1985-1989 ethnicity groups. 1980-1984 Yr <1980 Decline to state

46 Home Institution Classified by States Home Inst By Region State # % # % South AL - 0.0% 26 16.5% AR - 0.0% DE - 0.0% Home Institution Classified by States FL 3 1.9% Midw est Northeast South West GA 3 1.9% KY - 0.0% LA 1 0.6% MD 2 1.3% West 23.42% MS - 0.0% Midw est 36.71% NC 4 2.5% OK 1 0.6% SC - 0.0% South 16.46% TN 1 0.6% Northeast TX 8 5.1% 23.42% VA 3 1.9% WV - 0.0% West AK - 0.0% 37 23.4% AZ 1 0.6% CA 24 15.2% CO 1 0.6% HI - 0.0% ID - 0.0% MT - 0.0% NV - 0.0% NM - 0.0% OR 2 1.3% UT 8 5.1% WA 1 0.6% WY - 0.0% Midwest IL 23 14.6% 58 36.7% IN - 0.0% IA 3 1.9% KS 1 0.6% MI 12 7.6% MN 5 3.2% MO 1 0.6% ND - 0.0% NE - 0.0% OH 6 3.8% SD - 0.0% WI 7 4.4% Northeast CT 2 1.3% 37 23.4% ME - 0.0% MA 8 5.1% NH 1 0.6% NJ 7 4.4% NY 11 7.0% PA 2 1.3% RI 6 3.8% VT - 0.0% Home Inst. in US 158 158

47 Home Institution Countries Classified by Region

Home Home Institution Countries Classified by Region Inst By Region Country # % # % Australia AU 4 1.5%5 1.9% NZ 1 0.4% Australia 5 Eastern Asia CN 1 0.4%7 2.6% JP 5 1.9% Eas ter n A s ia 7 KR 1 0.4% Northern Asia RU 2 0.8%2 0.8% Northern Asia 2 Southern Asia IN 1 0.4%1 0.4% Middle America MX 1 0.4%1 0.4% Southern Asia 1 North America CA 6 2.3%164 61.7% US 158 59.4% Middle America 1 South America CL 1 0.4%1 0.4% Middle North A merica 164 East IL 3 1.1%3 1.1% Central Europe CH 3 1.1%22 8.3% South America 1

DE 18 6.8% Middle Eas t 3 PL 1 0.4% Northern Europe DK 4 1.5%5 1.9% Central Europe 22 SE 1 0.4% Southern Europe RO 1 0.4%8 3.0% TR 1 0.4% Northern Europe 5 GR 1 0.4% IT 1 0.4% ES 3 1.1% Southern Europe 8 PT 1 0.4% Western Europe FR 27 10.2%47 17.7% Western Europe 47 GB 19 7.1% NL 1 0.4% 0 50 100 150 Total 266 266

48 2.3 Workshop Participant List

49 2.4 Workshop Participant Summary

No. of # of Decline/ Decline/ Workshop Citizens & %* No # of %* # of # of %* No Name of Activity Participants Per Res Reply Female Decline Minorities Reply

A Window into Zeta and 43 6 55% 32 13 35% 6 - 0% 38 Modular Physics

CMI/MSRI Workshop: Modular Forms 68 24 44% 13 11 18% 8 6 15% 27 and Arithmetic

Computation and Complex 62 17 71% 38 10 21% 14 2 7% 32 Systems

Connections for Women: 57 23 53% 14 37 80% 11 5 16% 25 Geometric Group Theory

Connections for Women: Introduction to 47 21 49% 4 35 88% 7 3 13% 23 the Spring, 2008 programs Connections for Women: Teichmuller Theory 48 24 69% 13 26 67% 9 4 13% 16 and Kleinian Groups

Continuous Optimization 39 - 0% 38 12 33% 3 - 0% 37 and Applications Critical Issues in Education Workshop: 131 41 93% 87 64 63% 29 12 12% 34 Teaching and Learning Algebra Deformation Theory and Moduli in Algebraic 63 7 39% 45 14 23% 3 - 0% 55 Geometry

Exterior Differential Systems and 48 13 52% 23 7 17% 7 1 5% 27 the Method of Equivalence

Homological Methods in 81 32 48% 14 13 19% 12 1 3% 46 Representation Theory

Hot Topics: Contact structures, dynamics and 53 24 50% 5 1 2% 6 2 6% 19 the Seiberg-Witten equations in dimension 3

IAS/PCMI summer conference: 14 3 60% 9 4 31% 1 1 25% 10 Statistical Mechanics

Introduction to Geometric 117 53 58% 26 31 30% 15 5 8% 55 Group Theory

Introduction to Teichmuller Theory and 96 51 65% 17 30 35% 11 5 9% 40 Kleinian Groups

50

No. of # of Decline/ Decline/ Workshop Citizens & %* No # of %* # of # of %* No Name of Activity Participants Per Res Reply Female Decline Minorities Reply Introductory Workshop on Combinatorial 134 62 56% 24 32 28% 18 5 7% 64 Representation Theory Introductory Workshop on the Representation 95 39 48% 14 19 23% 14 3 7% 50 Theory of Finite Groups

Lie Theory 150 58 47% 27 31 24% 22 5 7% 76

Math Fest 12 7 100% 5 2 22% 3 - 0% 5

Mathematical Systems 60 8 62% 47 10 36% 32 1 4% 33 Biology of Cancer II

Modern Mathematics: An Introduction to 45 19 66% 16 15 35% 2 24 82% 17 MSRI's 2008-09 Programs

MSRI Summer Microprogram on Nonlinear 47 18 43% 5 7 16% 2 5 15% 13 Partial Differential Equations

MSRI's 25th Anniversary 153 75 76% 54 27 21% 23 9 112% 76 Celebration

MSRI-UP 2008 research topic: 20 1 100% 19 5 45% 9 1 50% 18 Experimental Mathematics Topics in Combinatorial Representation 129 45 47% 33 21 21% 30 5 8% 65 Theory

Topics in Geometric Group 145 62 56% 35 25 20% 18 5 6% 68 Theory Topics in Teichmuller Theory and 90 40 62% 25 18 22% 8 4 8% 38 Kleinian Groups

Total Number of Workshops Participants 2,047 773 57% 682 520 30% 323 113 11% 1,007

*Percentage for Female, Citizens & Minorities are computed out of participants that provided info on their gender, citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

51 2.6 Workshop Participant Demographic Data Demographics Statistics Male v s . Female % Over % Respon- Over # dents Total 15.8% Total Program Participants 2047 Male 1204 69.84% 58.8% Female 520 30.16% 25.4% 25.4% 58.8% Decline to State Gender 323 15.78% 15.8%

% Over % Respon- Over Male Female Decline to State Gender Ethnicities # dents Total Native American 11 1.06% 0.5% Ethnicities Statistic Asian 172 16.54% 8.1% 1.3%

Black 27 2.60% 1.3% 0.5% 8.1% 4.0% Hispanic 84 8.08% 4.0% 0.1% 2 0.19% 0.1% Pacific 47.5% White 815 78.37% 38.5% Decline to State

Ethnicities 1007 49.19% 47.5% 38.5%

Minorities 113 10.87% 5.5%

US Citizen 710 52.0% 34.7% Decline to State Citizenship 682 33.32% 33.3% Native American Asian Per Resident 63 3.1% Black Hispanic Pac if ic White US Citizen & Per Decline to State Ethnicities Resident 773 37.8% Year of Highest Degree Home Inst. in US 1534

% Over % 12.8% 30.3% Respon- Over 12.1% Year of Degree # dents Total 2009 621 34.81% 30.3% 3.3% 2005-2008 255 14.29% 12.5% 2000-2004 243 13.62% 11.9% 4.4% 12.5% 7.0% 11.9% 1995-1999 143 8.02% 7.0% 5.7% 1990-1994 117 6.56% 5.7% 1985-1989 90 5.04% 4.4% 1980-1984 67 3.76% 3.3% 2009 2005-2008 2000-2004 Yr <1980 248 13.90% 12.1% 1995-1999 1990-1994 1985-1989 Decline to state 263 12.8% 1980-1984 Yr <1980 Decline to state Total 2047 100.0% **Statistic Calculation based on participant that replied to each category. Ethnicities' percentage might be over stated since some participants selected to be included in more than one ethnicity groups.

52 Home Institution Countries Classified by Home Institution Countries Classified by Region Region Home Inst By Region Country # % # %

Australia AU 16 0.8% 17 0.8% Australia 17 NZ 1 0.0% Eastern Asia CN 5 0.2% 75 3.7% JP 41 2.0% East ern A sia 75 KR 27 1.3% TW 2 0.1% Northern Asia 1 Northern Asia RU 1 0.0% 1 0.0% Southeastern Sout heast ern A sia 7 Asia IN 5 0.2% 7 0.3% SG 2 0.1% Middle America MX 6 0.3% 8 0.4% M iddle America 8 PR 2 0.1% North America CA 60 3.0% 1594 78.9% Nort h A merica 159 4 US 1534 75.9% South America AR 1 0.0% 16 0.8% BR 7 0.3% Sout h A merica 16 UY 7 0.3% VE 1 0.0% Middle East 15 Middle East IL 12 0.6% 15 0.7% SA 3 0.1% Central Europe 79 Central Europe AT 3 0.1% 79 3.9% CH 18 0.9% DE 53 2.6% Eastern Europe 1 PL 5 0.2% Eastern Europe BY 1 0.0% 1 0.0% Northern Europe 8 Northern Europe DK 6 0.3% 8 0.4% Sout heast ern 7 NO 1 0.0% Europe SE 1 0.0% Southeastern Sout hern Europe 19 Europe RO 3 0.1% 7 0.3% TR 4 0.2% Sout hwest ern 19 Southern Europe Europe GR 5 0.2% 19 0.9% IT 14 0.7% Western Europe 15 4 Southwestern Europe ES 16 0.8% 19 0.9% 0 500 1000 1500 PT 3 0.1% Western Europe BE 1 0.0% 154 7.6% Length of Workshops

FR 62 3.1% ≤ 3 days 4 to 7 days 1 - 2 w eeks GB 84 4.2% 2 - 3 w eeks > 3 w eeks IE 7 0.3% Total 2020 2020 > 3 w eeks 2 - 3 w eeks 3% 1% Length of Workshop # % ≤ 3 days 1 - 2 w eeks 23% ≤ 3 days 462 23% 7% 4 to 7 days 1359 66% 1 - 2 weeks 145 7% 4 to 7 days 2 - 3 weeks 61 3% 66% > 3 weeks 20 1% Total 2047

53 Home Institution Classified by States Home Inst By Region State # % # % South AL 5 0.3% 231 15.2% AR - 0.0% DE - 0.0% Home Institution Classified by Countries FL 28 1.8% Midw est Northeast South West GA 31 2.0% KY 2 0.1% LA 29 1.9% Midw est 23.43% MD 16 1.0% West MS 3 0.2% 39.30% NC 23 1.5% OK 14 0.9% SC 1 0.1% Northeast TN 4 0.3% South 22.11% TX 49 3.2% 15.16% VA 26 1.7% WV - 0.0% West AK - 0.0% 599 39.3% AZ 20 1.3% CA 463 30.4% CO 20 1.3% HI 9 0.6% ID 2 0.1% MT - 0.0% NV 1 0.1% NM 1 0.1% OR 24 1.6% UT 37 2.4% WA 22 1.4% WY - 0.0% Midwest IL 112 7.3% 357 23.4% IN 16 1.0% IA 21 1.4% KS 13 0.9% MI 73 4.8% MN 20 1.3% MO 19 1.2% ND - 0.0% NE 10 0.7% OH 31 2.0% SD - 0.0% WI 42 2.8% Northeast CT 19 1.2% 337 22.1% ME 2 0.1% MA 113 7.4% NH 6 0.4% NJ 44 2.9% NY 103 6.8% PA 29 1.9% RI 20 1.3% VT 1 0.1%

Total 1,524 1524

54 2.7 Program Publication List

Last Name First Name Publication Title Co-Authors Abels Herbert Proper invariant pseudometrics A. Manoussos and G.Noskov Abels Herbert Undistorted solvable linear groups Roger Alperin Lower bounds for the volume of Adeboye Ilesanmi hyperbolic n-orbifolds Adeboye Ilesanmi On volumes of hyperbolic 4-orbifolds Andersen Henning Sum formulas and Ext-groups Endomorphism Algebras of Tensor Modules for Quantum Groups at Roots Andersen Henning of Unity G. I. Lehrer, R. B. Zhang Supercharacters of the p-Sylow subgroups of the finite symplectic and Andre Carlos orthogonal groups Ana Margarida Neto A supercharacter theory for the p- Sylow subgroups of the finite Andre Carlos symplectic and orthogonal groups Ana Margarida Neto On the factorization of supercharacters Andre Carlos of finite algebra groups Olga Pinho On the linear characters of finite Andre Carlos algebra groups Totally geodesic subgraphs of the Hugo Parlier, Kenneth J. Aramayona Javier pants complex Shackleton A characterisation of plane Aramayona Javier quasiconformal maps using triangles Peter Haissinsky A quick approach to pants complex Aramayona Javier automorphisms Jeffrey F. Brock, Cyril Lecuire, Hugo Parlier and Kenneth J. Aramayona Javier Finite geodesicity of the pants complex Shackleton Simplicial Embeddings of pants Aramayona Javier complexes Arzhantseva Goulnara Random groups Thomas Delzant M. Bridson, T. Januszkiewicz, I. Infinite groups with fixed point Leary, A. Minasyan, J. \'Swi\c Arzhantseva Goulnara properties atkowski Dovetail shuffles of bicolored decks Persi Diaconis, Kannan Assaf Sami (working title) Soundararajan A kicking basis for Garsia-Haiman Assaf Sami modules in two columns (working title) Adriano Garsia The Discrete fundamental group of te Barcelo Helene Associahedron of type A. Chris Severs, JAcob White Subspace arrangements and discrete Barcelo Helene homotopy groups. Chris Severs, Jacob White Barcelo Helene Basis graphs of shifted complexes Abdul Jarrah, Susanna Fishel Barcelo Helene Coloring Complexes Einar Steingrimson MV cycles, good basis, and the ring of Baumann Pierre functions on the unipotent subgroup Joel Kamnitzer

55 MV polytopes and the semicanonical Baumann Pierre basis Joel Kamnitzer mapping class groups are quasi- bruce kleiner, yair minsky, lee Behrstock Jason isometrically rigid mosher Growth of intersection numbers for free Behrstock Jason group automorphisms Mladen Bestvina, Matt Clay Behrstock Jason Subgroups of mapping class groups Cornelia Drutu and Mark Sapir Whittaker Modules for Generalized Benkart Georgia Weyl Algebras Matthew Ondrus Benkart Georgia Multiparameter Weyl Algebras Harish-Chandra Modules for Viktor Bekkert and Vyacheslav Benkart Georgia Generalized Weyl Algebras Futorny Yetter-Drinfeld Modules and Cocycle Mariana Pereira and Sarah Benkart Georgia Twists Witherspoon Benkart Georgia (Preliminary - Centralizers for SO(3) ) Thomas Halverson Varieties and cohomology of infinitely Benson Dave generated modules Jon Carlson Gluing representations via idempotent modules and constructing endotrivial Benson Dave modules Paul Balmer and Jon Carlson Modules of constant Jordan type with Benson Dave one non-projective block Modules of constant Jordan type and Benson Dave the Horrocks-Mumford bundle Localising subcategories of the stable Srikanth Iyengar and Henning Benson Dave module category of a finite group Krause Modules of constant Jordan type and Benson Dave vector bundles on projective space Julia Pevtsova A generalization of the notion of a Benson Dave complex Robert Boltje Benson Dave Schur-Weyl duality over finite fields Steve Doty The pre-lie non-symmetric operad is a Bergeron Nantel free M. Livernet Collumn operator on Diagonal Bergeron Nantel harmonics and k-Schur functions M. Zabrocki and A. Garsia A classification of Combinatorial Hopf Bergeron Nantel Algebras Thomas Lam and Huilan Li The Cohomology Group of the Berkove Ethan Whitehead Group of a Free Product John Meier Berkove Ethan The L2 Invariants of Clean Complexes Tim Hsu Quasi-isometry classification of right Bestvina Mladen angled Artin groups Bruce Kleiner, Michah Sageev Nonembedding of higher rank lattices Bestvina Mladen into Out(F_n) Mark Feighn Asymptotic dimension of Teichmuller Bestvina Mladen space Ken Bromberg, Koji Fujiwara Boltje Robert Fibred biset functors Olcay Coskun Boltje Robert r-fold complexes Dave Benson Complexity and cohomology of Bouc Serge cohomological Mackey functors

56 Bouc Serge Biset functors for finite groups Heegaard splittings and the Johnson- , Nathan Broaddus, Brendle Tara Morita representation Andy Putman Finiteness properties of the symmetric Brendle Tara Torelli group Dan Margalit Generators of the third term in the Brendle Tara Johnson filtration Dan Margalit, Alexandra Pettet Configuration spaces of rings and Brendle Tara wickets Allen Hatcher Parabolic Kazhdan-Lusztig polynomials for quasi-minuscule Brenti Francesco quotients F. Incitti, M. Marietti Parabolic Kazhdan-Lusztig R- polynomials for quasi-minuscule Brenti Francesco quotients Kazhdan-Lusztig polynomials for Brenti Francesco Deodhar permutations B. Jones Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and Brock Jeffrey relative hyperbolicity Howard Masur Asymptotics of Weil-Petersson geodesics I: ending laminations, Brock Jeffrey recurrence, and flows Howard Masur and Yair Minsky Asymptotics of Weil-Petersson geodesics II: bounded geometry and Brock Jeffrey combinatorics Howard Masur and Yair Minsky The classification of finitely generated Brock Jeffrey Kleinian groups Richard Canary and Yair Minsky Brock Jeffrey Inflexibility of hyperbolic 3-manifolds Ken Bromberg Simple points on boundaries of deformation spaces of hyperbolic 3- Ken Bromberg, Richard Canary, Brock Jeffrey manifolds and Yair Minsky Maximal twists and pseudo-Anosov Brock Jeffrey shuttles in the Weil-Petersson metric Bounded geometry and combinatorics Yair Minsky, Hossein Namazi, Brock Jeffrey for hyperbolic Heegaard splittings and Juan Souto Pants decompositions, Heegaard splittings and volumes of hyperbolic 3- Brock Jeffrey manifolds Juan Souto Ken Bromberg, Richard Canary, Brock Jeffrey Divergence of Kleinian surface groups and Cyril Lecuire Geometric inflexibility of hyperbolic 3- Bromberg Kenneth manifolds Jeff Brock Simple points on boundaries of Jeff Brock, Dick Canary, Yair Bromberg Kenneth deformation space Minsky Aymptotic dimension of Teichmuller Bromberg Kenneth space Mladen Bestvina, Koji Fujiwara Paul FONG, Bhama Broué Michel Blocks of finite reductive groups SRINIVASAN Broué Michel A Course on Reflection Groups Broué Michel Semisimple elements and Spetses

57 The degenerate analogue of Ariki's Brundan Jonathan categorification theorem A. Kleshchev Young's semi-normal form for higher Brundan Jonathan levels A. Kleshchev Connectivity of Hemispheres in Euclidean Buildings with Applications Bux Kai-UWe to Finiteness Properties (tentative) Kevin Wortman Amir Mohammadi and Kevin Bux Kai-UWe Sl_n{Z[t]) is not of type FP_{n-1} Wortman Mladen Bestvina and Dan Bux Kai-UWe The Dimension of the Torelli Group Margalit Some Remarks on the Braided Bux Kai-UWe Thompson Group BV Dmitriy Sonkin Automorphisms of tree-based RAAGS and partially symmetric automorphisms Ruth Charney and Karen Bux Kai-UWe of free groups Vogtmann Okuyama contractions and infinite Cabanes Marc Coxeter groups. Calta Kariane Coding and the horocycle flow Calta Kariane Algebraic Periodicity John Smillie A Borel-Serre bordification of the Calta Kariane moduli space of Abelian differentials John Smillie, Tom Schmidt Algebraically periodic translation Calta Kariane surfaces arXiv:math/0703567 John Smillie The classification of finitely generated Canary Richard Kleinian groups Jeff Brock, Yair Minsky Untouchable points on boundaries of Jeff Brock, Ken Bromberg, Yair Canary Richard deformation spaces of Kleinian groups Minsky A hierarchical criterion for convergence Jeff Brock, Ken Bromberg, Cyril Canary Richard of Kleinian groups Lecuire On the structure of locally compact Caprace Pierre-Emmanuel CAT(0) groups Nicolas Monod Compactifications combinatoire et de Caprace Pierre-Emmanuel Chabauty des immeubles Jean Lecureux Rank one isometries of buildings and Caprace Pierre-Emmanuel applications Koji Fujiwara Carlson Jon Blocks and support varieties J. Rickard A polynomial-time reduction algorithm for groups of semilinear or subfield Max Neunhoeffer and Colva Carlson Jon class Roney-Dougal Carlson Jon Finite generation of Tate cohomology S. Chebolu and J. Minac Freyd's generating hypothesis for the stable module category of a finite Carlson Jon group S. Chebolu and J. Minac Varieties and cohomology for infinitely Carlson Jon generated modules D. J. Benson The group of endotrivial modules for Carlson Jon the symmetric and alternating groups D. J. Hemmer and Nadia Mazza Canonical constructions for modules Carlson Jon over groups of order $p^2$ E. Friedlander and A. Suslin

58 Generic kernels and other module Carlson Jon constructions E. Friedlander and J. Pevtsova Quasi-isometries Between Tubular Cashen Christopher Groups Amalgamation of Virtually Abelian Groups Along Virtually Cyclic Cashen Christopher Subgroups Cashen Christopher Graphs of Virtually Abelian Groups Patterns of Geodesics in the Cashen Christopher Hyperbolic Plane Cashen Christopher Asymptotic Invariants of Groups Hausdorff dimension of the set of Cheung Yitwah singular vectors Topological Dichotomy and Strict Cheung Yitwah Pascal Hubert, Howard Masur Cheung Yitwah Slow divergence and Unique ergodicity Alex Eskin Divergent trajectories in SL(4,R) and a Cheung Yitwah Conjecture of W.Schmidt Barak Weiss Rouquier blocks of the cyclotomic Chlouveraki Maria Ariki-Koike algebras Rouquier blocks of the cyclotomic Chlouveraki Maria Hecke algebras of G(de,e,r) Random subgroups of Thompson's Cleary Sean group F Elder, Rechnitzer and Taback Commensurators and subgroups of Cleary Sean finite index in Thompson's group F Burillo, Roever Analysis of haplotype inference Cleary Sean algorithms St.John Random subgroups of Thompson's Cleary Sean group F Elder, Rechnitzer and Taback Coskun Olcay Fibred biset functors Robert Boltje R-trees and compact systems of Coulbois Thierry isometries Arnaud Hilion, Martin Lustig Coulbois Thierry Index of laminations for free groups Arnaud Hilion Coulbois Thierry Actions of free groups on dendrite Asli Yaman R-trees for iwip automorphisms Coulbois Thierry of free groups Vertices, sources and Green correspondents of the simple modules Danz Susanne for the large Mathieu groups Vertices, sources and Green correspondents of the simple modules for the covering groups of the Mathieu Danz Susanne group M22 Burkhard Kuelshammer On the vertices of the basic spin module for the symmetric group in Danz Susanne characteristic 2 Burkhard Kuelshammer Daugherty Zajj Affine and graded BMW algebras Arun Ram, Rahbar Virk Two-boundary graded diagram Daugherty Zajj algebras

59 Compactly supported cohomology of Jan Dymara, Tadeusz Davis Michael Buildings Januszkiewicz, Boris Okun Representations of Rank Two Affine Davis Matt Hecke Algebras at Roots of Unity On the doubled tetrus DeBlois Jason arXiv:0804.3984 DeBlois Jason Limit groups and property "tau" , Henry Wilton DeBlois Jason Virtually special link complements Eric Chesebro, Henry Wilton Combinatorics of covers of Delucchi Emanuele complexified arrangements Combinatorics of polar orderings and Delucchi Emanuele Follow-up arrangements Simona Settepanella (U Pisa) Delucchi Emanuele "&c" M.D. Froidcoeur On the volume of hyperbolic three manifolds and the complexity of their Delzant Thomas fundamental groups Leonid Potyagailo Proof of the combinatorial Kirillov- Di Francesco Philippe Reshetikhin conjecture Rinat Kedem Q-systems as cluster algebras II: Cartan matrix of finite type and the Di Francesco Philippe polynomial property Rinat Kedem Bases of generalized Temperley-Lieb Digne François algebras Gus Lehrer Good roots of central elements in braid Digne François groups Jean Michel Doty Stephen Schur-Weyl duality for finite fields Dave Benson Doty Stephen Annihilators of permutation modules Kathryn Nyman Polynomial representations of Doty Stephen Chevalley groups Quantized mixed tensor space and Richard Dipper and Friedericke Doty Stephen Schur-Weyl duality Stoll On the defining relations for Anthony Giaquinto and John B. Doty Stephen generalized q-Schur algebras Sullivan Duchin Moon The flat-length spectrum (working title) Chris Leininger, Kasra Rafi Duchin Moon Stars at infinity in Teichmuller space Joseph Maher Filling at infinity in groups (working Duchin Moon title) Anne Thomas The Schwarzian derivative and measured laminations on Riemann Dumas David surfaces Projective structures, grafting, and Dumas David measured laminations Michael Wolf Grafting lines fellow travel Teichmuller Dumas David geodesics Young-Eun Choi, Kasra Rafi Extension of grafting to boundaries of Dumas David Teichmuller space Inkang Kim Large scale geometry of certain Dymarz Tullia solvable groups Holomorphic coordinates on Teichmueller and compactified Moduli Earle Clifford space

60 On the moduli of closed Riemann Earle Clifford surfaces with symmetries, II Trigonometric Cherednik algebra at critical level and quantum many-body Emsiz Erdal problems E. M. Opdam, J. V. Stokman Point Counting and Volume J. Athreya, S. Bufetov and M. Eskin Alex asymptotics in Teichmuller space Mirzakhani Ciounting closed geodesics in Eskin Alex Teichmuller space M. Mirzakhani Eskin Alex Counting closed geodesics in strata K. Rafi, M. Mirzakhani Feighn Mark Lattices in Out(F_n) Mladen Bestvina Feighn Mark Definable subsets of free groups Mladen Bestvina Negative curvature and the free factor Feighn Mark complex Mladen Bestvina Feighn Mark The conjugacy problem in Out(F_n) Michael Handel Surface groups and mapping tori of Feighn Mark endomorphisms Michah Sageev Harmonic functions on R-covered foliations and group actions on the Renato Feres and Kamlesh Fenley Sergio circle Parwani Axis cocompact subgroups of the Fenley Sergio mapping class group Lee Mosher Cluster algebras and triangulated Fomin Sergey surfaces. Part II: Lambda lengths Dylan Thurston (tentative) An Equivariant Bijection Michel Broue and Bhama Fong Paul Conjecture in Finite Reductive Groups Srinivasan Generalized Weyl modules for loop Fourier Ghislain algebras C.Chari, T.Pal Fourier Ghislain KR crystals for type C_n A.Schilling, M.Okado On the endomorphisms of Weyl modules over affine Kac-Moody Boris Feigin and Leonid Frenkel Edward algebras at the critical level Rybnikov Andrey Losev and Nikita Frenkel Edward Instantons beyond topological theory II Nekrasov Langlands duality of representations of Frenkel Edward quantum groups David Hernandez Constructions for infinitesimal group Friedlander Eric schemes Julia Pevtsova Weil restriction and cohomological Friedlander Eric varieties Special properties of certain modules Friedlander Eric for elementary abelian $p$-groups J. Carlson and A. Suslin Friedlander Eric Higher order rank varieties J. Carlson and J. Pevtsova An inviscid dyadic model of turbulence: Alexey Cheskidov and Natasa Friedlander Susan the global Pavlovic Kolmogorov's law for a dyadic model of Friedlander Susan turbulence Alexey Cheskidov Nonlinear instability for the surface Friedlander Susan quasi geostrophic equations Natasa Pavlovic and Vlad Vicol

61 A characterization of higher rank symmetric space via bounded Fujiwara Koji cohomology Mladen Bestvina Free subgroups generated by pseudo- Anosov elements in mapping class Fujiwara Koji groups Cusp areas of Farey manifolds and applications to Efstratia Kalfagianni, Jessica S. Futer David arXiv:0808.2716 Purcell cluster algebra strucutres and semicanonical bases for unipotent Geiss Christof groups B. Leclerc, J. Schroeer Geoghegan Ross Modules over CAT(0) spaces Robert Bieri K-theoretic invariants of certain Geoghegan Ross Marco Varisco Geoghegan Ross (untitled) Herbert Abels Control of transfer and weak closure in Antonio Diaz, Nadia Mazza, Glesser Adam fusion systems Sejong Parkk Glesser Adam Trivial Fusion Systems The commuting category of a fusion Glesser Adam system Markus Linckelmann Proper affine actions and geodesic Goldman William flows on hyperbolic surfaces Labourie, Margulis Trace coordinates on Fricke spaces of Goldman William some elementary hyperbolic surfaces Affien cubic surfaces and the character Goldman William variety of the 4-holed sphere Domingo Toledo Goldman William Projective Geometry on Manifolds Torelli action on moduli spaces of Goldman William SU(2)-representations of surfaces Eugene Xia, Joe Previte posiibly Dick Canary, Misha Goldman William No Title Kapovich Affine deformations of the three-holed Goldman William sphere Virginie Charette, Todd Drumm Cyclotomic Birman--Wenzl--Murakami Algebras II: Admissibility Relations and Goodman Frederick Freeness Holly Hauschild Cellularity of Cyclotomic Birman-- Goodman Frederick Wenzl--Murakami Algebras Cellularity and the Jones basic Goodman Frederick construction John Graber Gordon Cameron Reducible and Finite Dehn Fillings S. Boyer and X. Zhang Gordon Cameron Dehn Surgery and 3-Manifolds Grodal Jesper Classification of G-actions on spheres Jeff Smith Grodal Jesper Fundamental groups of p-local groups Bob Oliver Integral homology of p-local finite Grodal Jesper groups Grodal Jesper Classification of G-actions on spheres Jeff Smith Grodal Jesper Fundamental groups of p-local groups Bob Oliver Grodal Jesper Integral homology of p-local finite

62 groups The quadratic isoperimetric inequality for mapping tori of free group automorphisms, I: Positive Groves Daniel automorphisms Martin Bridson Free-group automorphisms, train Groves Daniel tracks and the Beaded Decomposition Martin Bridson The quadratic isoperimetric inequality for mapping tori of free group Groves Daniel automorphisms, II: The general case Martin Bridson Conjugacy classes of solutions to equations and inequations over Groves Daniel hyperbolic groups Henry Wilton The structure of limits groups over Groves Daniel hyperbolic groups Henry Wilton Effective shortening and constructing Groves Daniel Makanin-Razborov diagrams Henry Wilton Residual finiteness, QCERF, and Groves Daniel fillings of hyperbolic groups Ian Agol, Jason Manning Homomorphisms to mapping class Groves Daniel groups Groves Daniel Subgroups of the mapping class group Guirardel Vincent Equations for residually free groups Levitt Guirardel Vincent Automorphisms of orbifold groups Fujiwara Cohomology of alternating and Guralnick Robert symmetric groups P. Tiep Primitive Monodromy Groups of Genus Guralnick Robert at most Two Dan Frohardt and Kay Magaard A classification of variation of mixed Makoto Matsumoto, Gregory Hain Richard Hodge structure (tentative) Pearlstein, tomohide Terasoma Infinitesimal relations between Galois Hain Richard automorphisms Makoto Matsumoto Crystals and Casimirs of Classical Lie Halverson Thomas Algebras Arun Ram Halverson Thomas A q-Partition Algebra Arun Ram and Nat Theiem Combinatorics of the q-Partition Halverson Thomas Algebra Nat Thiem Combinatorics of the q-Partition Halverson Tom Algebra Nat Thiem Halverson Tom Partition algebras and their q analogs Arun Ram and Nat Theiem Halverson Tom Motzkin Algebras Model Characters in the Symmetric Halverson Tom Group Michael Decker (not at MSRI) Combinatorics of the q-Partition Halverson Tom Algebra Nat Thiem Halverson Tom Partition algebras and their q analogs Arun Ram and Nat Theiem Halverson Tom Motzkin Algebras Georgia Benkart Model Characters in the Symmetric Halverson Tom Group Michael Decker (not at MSRI)

63 Topological classification of 4- manifolds with Baumslag-Solitar Hambleton Ian fundamental groups Peter Teichner, Matthias Kreck Dress induction and the Burnside Hambleton Ian quotient Green ring Larry Taylor, Bruce Williams P. A. Smith theory and coarse Hambleton Ian geometry Lucian Savin Invariant measures for the Hamenstaedt Ursula Teichmueller flow Bowen's construction for the Hamenstaedt Ursula Teichmueller flow Second bounded cohomology for Hamenstaedt Ursula Out(Fn) Hemmer David On the cohomology of Young modules Nakano and Cohen A classification of the group of endotrivial modules for the symmetric Hemmer David and alternating groups Carlson and Mazza Cohomology and generic cohomology for Specht modules of the symmetric Hemmer David group. Hernandez David Geometry of the analytic N. Reshetikhin, C. de Concini Cluster algebras and quantum affine Hernandez David algebras B. Leclerc On simple representations of quantum Hernandez David affine algebras V. Chari Smallness problem for quantum affine Hernandez David algebras and quiver varieties Irreducible Characters of Sylow-p- Subgroups of Exceptional Groups of Himstedt Frank Lie Type Tung Le, Kay Magaard Iwahori-Hecke Algebras for Cuspisdal Modules in Extended Groups With Split Hiss Gerhard BN-Pairs Gunter Malle Decomposition Numbers for Certain Projective Modules for Finite Chevalley Hiss Gerhard Groups Alexander Zalesski Hsu Timothy The l^2 invariants of clean complexes Ethan Berkove Artin HNN-extensions virtually embed Hsu Timothy in Artin groups Ian Leary Matchwebs I: Fundamentals and Katherine Shelley, San Jose Hsu Timothy uniqueness State Univ. Hsu Timothy Cubulating amalgams Daniel Wise, McGill Univ. Mark Logan, U. Minn.-Morris; Methods for nesting rank 3 normalized Shahriar Shahriari, Pomona Hsu Timothy matching rank-unimodal posets College Cubulating graphs of free groups with Hsu Timothy cyclic edge groups Daniel Wise, McGill Univ. Average curvatures of Weil-Petersson Huang Zheng geodesics in Teichmuller space Hopf Differentials and Weil-Petersson Huang Zheng geodesics

64 Injective Simplicial Maps of the Arc Irmak Elmas Complex on Nonorientable Surfaces Homomorphisms of Mapping Class Irmak Elmas Groups Chris Leininger Irmak Elmas Injections Between Outer Spaces Gabriela Schmithusen Isaacs I. Martin Character sums and double cosets Gabriel Navarro Isaacs I. Martin Rational elements in finite groups Gabriel Navarro Mullineux involution for Ariki-Koike jacon Nicolas algebras Cedric Lecouvey Representation theory of affine Hecke Susumu Ariki and Cedric jacon Nicolas algebra of type A Lecouvey Infinite groups with fixed point Arzhantseva, Bridson, Leary, Januszkiewicz Tadeusz propoerties Minasyan, Swiatkowski Groups possesing complex Januszkiewicz Tadeusz hierarchical decompositions Kropholler, Leary Jones Vaughan Algebraic decomposition of bimodules. Picard-Vessiot extensions with Juan Lourdes specified Galois groups Ted Chinburg, Andy Magid Generic Picard-Vessiot extensions for Juan Lourdes non-connected groups Arne Ledet Decomposition numbers for perverse Juteau Daniel sheaves. Modular representations of reductive groups and geometry of affine Juteau Daniel Grassmannians. Modular Springer correspondence and Juteau Daniel decomposition matrices. Decomposition numbers for perverse Juteau Daniel sheaves. Modular representations of reductive groups and geometry of affine Juteau Daniel Grassmannians. Modular Springer correspondence and Juteau Daniel decomposition matrices. Intersection form, laminations and Kapovich Ilya currents on free groups Martin Lustig Domains of discontinuity for subgroups Kapovich Ilya of Out(F_N) Martin Lustig tempered modules in exotic Deligne- Langlands correspondence (tentative Dan Ciubotaru (projected Kato Syu title) coauthor) Positivity for the cluster algebras Kedem Rinat associated with Q-systems Philippe Di Francesco Q-systems as cluster algebras II: Polynomiality and non-simply laced Kedem Rinat case Philippe Di Francesco Kedem Rinat Q-systems as cluster algebras Cluster algebras and quiver Keller Bernhard representations Keller Bernhard Calabi-Yau completions and their duals

65 Zamolodchikov's periodicity conjecture Keller Bernhard via triangulated categories Christopher Leininger, Saul Kent Richard Trees and mapping class groups Schleimer Kent Richard Subgroups of mapping class groups Christopher Leininger Congruence subgroups of the mapping Marco Boggi, Dan Abramovich, Kent Richard class group Ben Wieland Bers slices are Zariski dense Kent Richard arXiv:0807.4509 David Dumas Kent Richard Bounded Image Theorems Kent Richard Subgroups of free groups Lars Louder Kent Richard Slicing, Skinning, and Grafting David Dumas Deformation of 4-D hyperbolic orbifold Kerckhoff Steven (temporary title) Pete Storm On duality inducing automorphisms and sources of simple modules in Kessar Radha classical groups David Craven, Charles Eaton, Kessar Radha Blocks with a Klein four defcet group Markus Linckelmann On the endomorphism algebras of modular Gelfand-Graev Kessar Radha representations Cedric Bonnafe The graded center of the stable Kessar Radha category of a Brauer tree algebra Markus Linckelmann Kessar Radha On blocks with one simple module Markus Linckelmann Co-contractions of Graphs and Right- Kim Sang-hyun Anlged Artin Groups Surface subgroups of Graph Product of Kim Sang-hyun Groups Representations of the general linear Alexander groups which are irreducible over Kleshchev (Sasha) subgroups Pham Tiep Alexander Kleshchev (Sasha) Higher level Young's seminormal form Jonathan Brundan Krause Henning Stratifying triangulated categories Benson and Iyengar Classification of localizing subcategories of the stable module Krause Henning category of a finite group Benson and Iyenagar Cohomological Finiteness Conditions Brita Nucinkis and Conchita Kropholler Peter for Elementary Amenable Groups Martinez Groups with complicated hierarchical Ian Leary and Tadeusz Kropholler Peter decompositions Januszkiewicz Group Actions on Algebraic Cell Kropholler Peter Complexes C T C Wall Cohomological Dimension of Soluble Kropholler Peter Groups Martin Bridson A note on blocks with dihedral defect Kuelshammer Burkhard groups Sylvia Andersch

66 Vertices, sources and Green correspondents of the simple modules for the covering groups of the Mathieu Kuelshammer Burkhard group M_22 Susanne Danz On vertices of the basic spin module for the symmetric group in Kuelshammer Burkhard characteristic 2 Susanne Danz Nilpotent blocks and products of Kuelshammer Burkhard normal subgroups Gabriel Navarro Cohomology and Support Varieties for Kujawa Jonathan Lie II Boe and Nakano Cohomology and Support Varieties for Kujawa Jonathan Lie Superalgebras of type W(n Cohomology and Support Varieties for Kujawa Jonathan Lie Superalgebras II Boe and Nakano Cohomology and Support Varieties for Kujawa Jonathan Lie Superalgebras of type W(n) Bagci and Nakano Representations of the degenerate Kujawa Jonathan affine Sergeev algebra Hill and Sussan Combinatorial Hopf algebras and Lam Thomas towers of algebras Nantel Bergeron, Huilan Li Luc Lapointe, Jennifer Morse, Lam Thomas k-Shape poset and k-Schur functions Mark Shimozono Total positivity for isotropic Pavlo Pylyavskyy, Lauren Lam Thomas Grassmannians Williams Total positivity for isotropic Pavlo Pylyavskyy, Lauren Lam Thomas Grassmannians Williams Combinatorial Hopf algebras and Lam Thomas towers of algebras Nantel Bergeron, Huilan Li Lam Thomas Total positivity for loop groups Pavlo Pylyavskyy Schubert and Macdonald polynomials: Lascoux Alain a parallel not yet fixed: generalization of Lascoux Alain macdonald polynomials Ole Warnaar Schubert and Macdonald polynomials: Lascoux Alain a parallel not yet fixed: generalization of Lascoux Alain macdonald polynomials Ole Warnaar Lau Michael Forms of conformal superalgebras Victor Kac, Arturo Pianzola Leary Ian A metric Kan-Thurston theorem None. G Arzhantseva, M Bridson, T Infinite groups with fixed point Januszkiewicz, A Minasyan, J Leary Ian properties Swiatkowski Groups with extensive hierarchical Leary Ian decompositions T Januszkiewicz, P H Kropholler Artin HNN-extensions virtually embed Leary Ian in Artin groups T Hsu T Januszkiewicz, R Valle, R Leary Ian Simple platonic polygonal complexes Vogeler Leary Ian A metric Kan-Thurston theorem None.

67 G Arzhantseva, M Bridson, T Infinite groups with fixed point Januszkiewicz, A Minasyan, J Leary Ian properties Swiatkowski Groups with extensive hierarchical Leary Ian decompositions T Januszkiewicz, P H Kropholler Artin HNN-extensions virtually embed Leary Ian in Artin groups T Hsu T Januszkiewicz, R Valle, R Leary Ian Simple platonic polygonal complexes Vogeler Leary Ian A metric Kan-Thurston theorem None. G Arzhantseva, M Bridson, T Infinite groups with fixed point Januszkiewicz, A Minasyan, J Leary Ian properties Swiatkowski Groups with extensive hierarchical Leary Ian decompositions T Januszkiewicz, P H Kropholler Artin HNN-extensions virtually embed Leary Ian in Artin groups T Hsu T Januszkiewicz, R Valle, R Leary Ian Simple platonic polygonal complexes Vogeler Leary Ian A metric Kan-Thurston theorem None. G Arzhantseva, M Bridson, T Infinite groups with fixed point Januszkiewicz, A Minasyan, J Leary Ian properties Swiatkowski Groups with extensive hierarchical Leary Ian decompositions T Januszkiewicz, P H Kropholler Artin HNN-extensions virtually embed Leary Ian in Artin groups T Hsu T Januszkiewicz, R Valle, R Leary Ian Simple platonic polygonal complexes Vogeler Cluster algebra structures and semicanonical bases for unipotent Leclerc Bernard groups Christof Geiss, Jan Schröer Preprojective algebras and cluster Leclerc Bernard algebras Christof Geiss, Jan Schröer Y-systems, T-systems and q- Leclerc Bernard characters of quantum affine algebras David Hernandez On the Mullineux involution for Ariki- Lecouvey Cedric Koike algebras Nicolas Jacon The modular branching rule for affine Lecouvey Cedric Hecke algebras Susumu Ariki and Nicoas Jacon Strong convergence : Thurston's Lecuire Cyril cracked eggshell J. Anderson Endomorphism algebras of tensor modules for quantum groups at roots Lehrer Gustav of unity. H. Andersen and R Zhang An explicit presentation for a quantum Lehrer Gustav endomorphism algebra. R Zhang Uniform convergence in the mapping Richard P Kent IV and Saul Leininger Christopher class group Schleimer D.B. McReynolds, A.W. Reid, Leininger Christopher Length and eigenvalue equivalence and W.D. Neumann

68 Richard P Kent IV and Saul Leininger Christopher Trees and mapping class groups Schleimer Connectivity of the space of ending Leininger Christopher laminations Saul Schleimer Leininger Christopher The flat-length spetrum Moon Duchin and Kasra Rafi Local connectivity and the curve Mahan (Mitra) Mj and Saul Leininger Christopher complex Schleimer Leininger Christopher Subgroups of the mapping class group Richard P Kent IV Universal Cannon--Thurston maps and the boundary of the curve complex Leininger Christopher arXiv:0808.3521 Schleimer, Saul Teichmuller geodesics that do not have Lenzhen Anna a limit in PMF Divergence of Teichmuller geodesics Lenzhen Anna arXiv:0803.1867 Howard Masur covering theory for complexes of Lim Seonhee groups Anne Thomas elements of finite order and uniform lattices in the autormorphism group of Lim Seonhee trees Panos Papasoglu finite generation properties of Lin Zongzhu cohomology rings of infinite groups Lizhen Ji Frobenius twisted cojugacy classes on Lin Zongzhu $G$-stable pieces Xuhua he Compatible Frobenius splitting and Lin Zongzhu lifting of projective modules D. Nakano Linckelmann Markus Antipodes for source algebras D. A. Craven, C. Eaton, R. Linckelmann Markus Blocks with Klein four defect Kessar Blocks with one simple module and Linckelmann Markus defect 2 R. Kessar Commuting categories for blocks and Linckelmann Markus fusion systems A. Glesser On graded centers of stable categories Linckelmann Markus of finite p-groups R. Stancu On the combinatorics of Hall- Littelmann Peter Littlewoord polynomials Stephan Gaussent A Lie-theoretic construction of P. Etingof, A. Oblomkov, L. Loktev Sergey spherical symplectic reflection algebras Rybnikov Weight Multiplicity Polynomials of Loktev Sergey Multi-variable Weyl Modules Deformations of Compact Lotay Jason Coassociative 4-folds with Boundary A. Kovalev Asymptotically Conical Associative 3- Lotay Jason folds Lagrangians in S^6: Deformations, Lotay Jason Embeddings and Examples Dimensional reduction and the long- Lott John time behavior of Ricci flow Lott John Locally collapsed 3-manifolds Bruce Kleiner Louder Larsen Krull dimension for limit groups I

69 Louder Larsen Krull dimension for limit groups II sorry, I don t hqve tinme qny more, the Lustig Martin next talk starts ... A generalized q-analogue of the Lyle Sinead Carter-Payne theorem Andrew Mathas Lyle Sinead Classifying reducible Specht modules Matthew Fayers Magaard Kay Primitive Groups of Genus 2 Frohardt,D. and Guralnick, R. Constructive Recognition of Tensor Magaard Kay Products O'Brien Irreducible Characters of Sylow p- subgroups of Groups of Exceptional Magaard Kay Lie Himstedt, F. and Le, T. Examples of deformation spaces of hyperbolic 3-manifolds that are not Magid Aaron locally connected. Iwahori-Hecke algebras for cuspidal Malle Gunter modules in extended groups Gerhard Hiss Constructing representations of Malle Gunter cyclotomic Hecke algebras Jean MIchel Malle Gunter Zeros of Brauer characters Gabriel Navarro Uniform uniform exponential growth of Mangahas Johanna subgroups of the mapping class group Clifford theory and Galois theory for Marcus Andrei indecomposable modules Holomorphic Coordinates on Teichmueller and Compactified Moduli Marden Albert Space (tentative title) Cilfford Earle Marden Albert The View from Above On the clique and chromatic numbers of the generating graph of a finite Maroti Attila group Andreas Lucchini On the extraspecial case of the non- Maroti Attila coprime k(GV) problem Robert M. Guralnick Asymptotics of Weil-Petersson geodesics I: ending laminations, Masur Howard recurrence and flows. Jeffrey Brock, Yair Minsky Counting reducibles in the mapping Masur Howard class group Masur Howard Divergent Teichmuller geodesics Anna Lenzhen Masur Howard Teichmuller geometry of moduli space Benson Farb Masur Howard The Geometry of the disc complex Saul Schleimer The irreducible characters of Mathas Andrew alternating Hecke algebras Leah Ratliff Generalized Carter-Payne Mathas Andrew homomorphsism for Hecke algebras Sinead Lyle and John Murray completion of arithmetic mapping class Matsumoto Makoto group Richard Hain the group of endotrivial modules for the Mazza Nadia symmetric and alternating groups Jon Carlson, Dave Hemmer Control of transfer and weak closure in Antonio Diaz, Adam Glesser, Mazza Nadia fusion systems Sejong Park

70 Mazza Nadia The dade functor Serge Bouc Mazza Nadia On elemetary abelian p-subgroups McCammond Jon Pulling apart orthogonal groups McCammond Jon Factoring Affine Isometries Noel Brady Handlebody bundles and free-by-cyclic McCammond Jon groups Noel Brady and Dan Groves McCammond Jon Commutator Geometry Jason Manning Geometric presentations for pure Artin McCammond Jon groups Emanuele Delucchi McCammond Jon An unusual affine Artin group Sam Kim Connectivity at infinity of robot McCammond Jon configuration space John Meier Visualization tools for 4-dimensional McCammond Jon regular polytopes John Meier Uniformly Diophantine numbers in a McMullen Curtis fixed real quadratic field McMullen Curtis Barycentric subdivision (working title) P. Diaconis Spetses for split primitive reflection Michel Jean groups Michel Broue, Gunter Malle Representations of cyclotomic algebras for primitive complex Michel Jean reflection groups Gunter Malle Representations of exceptional Michel Jean cyclotomic Hecke algebras Gunter Malle Garside categories and Deligne- Michel Jean Lusztig varieties Francois Digne Counting closed geodesics on moduli Mirzakhani Maryam spaces of surfaces Alex Eskin Lattice Point Asymptotics and Volume Mirzakhani Maryam Growth on Teichmuller space Alex Counting closed geodesics on moduli Mirzakhani Maryam spaces of surfaces Alex Eskin Lattice Point Asymptotics and Volume Mirzakhani Maryam Growth on Teichmuller space J. athreya, A. Bufetov, A. Eskin Mapping Class Groups and Mitra Mahan Interpolating Complexes: Rank Boundary of Curve Complexes of Mitra Mahan Surfaces with one puncture C. Leininger, S. Schleimer Quasi-isometric rigidity for mapping Jason Behrstock, Bruce Kleiner, Mosher Lee class groups (working title) Yair Minsky Subgroups of Outer Automorphisms of Mosher Lee Free Groups (working title) Michael Handel Train Tracks and the Masur-Minsky Mosher Lee Sum (working title) Subgroups of Mapping Class Groups, Mosher Lee Revisited (working title) Michael Handel (no others... I should have chosen only Mosher Lee 4 papers instead of 5) On the cohomology of Young modules Nakano Daniel for the symmetric group Fred Cohen, David Hemmer

71 Cohomology and Support Varieties for Nakano Daniel the Lie W(n) Irfan Bagci, Jonathan Kujawa Realizing rings of regular functions via Nakano Daniel quantum group cohomology Zongzhu Lin Non-realizability and edning Namazi Hossein laminations Juan Souto Revisiting Thurston's Uniform Injectivity Namazi Hossein Theorem Juan Souto Bounded Combinatorics and Heegaard Namazi Hossein splittings Bounded geometry in Hyperbolic 3- Jeff Brock, Yair Minsky, Juan Namazi Hossein manifolds Souto Hyperbolization and Heegaard Jeff Brock, Yair MInsky, Juan Namazi Hossein distance Souto Brauer's Height Zero Conjecture for Navarro Gabriel the 2-Blocks of Maximal Defect P. H. Tiep Navarro Gabriel Characters Sums and Double Cosets M. Isaacs Degrees of Rational Characters of Navarro Gabriel Finite Groups, P. Tiep Navarro Gabriel Rational Elements of Finite Groups M. Isaacs A natural Correspondence for Navarro Gabriel Characters in Blocks, M.W. Liebeck, A. Shalev and P. O'Brien Eamonn On the Ore Conjecture Tiep On some questions about a family of Alberto Cavicchioli and Fulvia O'Brien Eamonn cyclically presented groups Spaggiari Decomposing tensor products of O'Brien Eamonn groups of Lie type Kay Magaard Decompositions of Hecke-von Okun Boris Neumann algebras R. Scott Compactly supported cohomology of M. Davis, J. Dymara, T. Okun Boris buildings Januszkiewicz Generalized weighted L^2 homology and automatic growth series of Coxeter Okun Boris groups R. Scott Sign conjugacy classes in symmetric Olsson Jørn groups Olsson Jørn Cores of partitions and block coverings Emmanuel Briand and Orellana Rosa Kronecker Polynomials Mercedes Rosas Reduced Kronecker polynomials and Emmanuel Briand and Orellana Rosa Polyhedra Mercedes Rosas Osajda Damian Boundaries of systolic groups Piotr Przytycki Osajda Damian On simplicial non-positive curvature Peripheral subgroups of relatively Papazoglou Panagiotis hyperbolic groups Parlier Hugo "Filling surfaces with simple curves" Jim Anderson, Alexandra Pettet "Shortest point projections in the Pants Parlier Hugo Complex" Javier Aramayona, Cyril Lecuire

72 Richard Hain, Makoto Matsumoto, Tomohide Pearlstein Gregory Not yet determined. Terasoma Constructions for infinitesimal group Pevtsova Julia schemes E. Friedlander Cohomology of finite dimensional M. Mastnak, P. Schauenburg, S. Pevtsova Julia pointed Hopf algebras Witherspoon The prime spectrum of the tensor Pevtsova Julia triangulated category D^{perf}[X/C] P. Smith Pevtsova Julia Generalized support varieties E. Friedlander Generic Kernels and other module Pevtsova Julia constructions J. Carlson, E. Friedlander Modules of constant Jordan type and Pevtsova Julia vector bundles D. Benson Compression of root systems and the Purbhoo Kevin E-sequence A Littlewood-Richardson rule for Purbhoo Kevin Grassmannian Permutations Frank Sottile Groebner cycles in toric varieties via Purbhoo Kevin tropicalization A note on the connectivity of certain Putman Thomas complexes associated to surfaces The Casson invariant and the word Putman Thomas metric on the Torelli group N. Broaddus, B. Farb Symplectic Heegaard splittings and Putman Thomas linked abelian groups Joan Birman, Dennis Johnson On the second cohomology group of certain finite-index subgroups of the Putman Thomas mapping class group On the abelianization of the Torelli Putman Thomas group and its subgroup Brendon Rhoades, Kyle Pylyavskyy Pavlo Promotion via webs and cyclic sieving Petersen Parametrizing totally positive isotropic Pylyavskyy Pavlo Grassmanian Lauren Williams, Thomas Lam Curve complexes with connected Rafi Kasra boundary are rigid Saul Schleimer Moon Duchin and Chris Rafi Kasra The flat length spectrum Leininger Grafting lines and Teichmuller Youn-Eun Choi and David Rafi Kasra geodesics Dumas Obtainable sizes of on finite Ragnarsson Kari sets Bridget Tenner Completion of the Burnside ring and Ragnarsson Kari Mackey functors Ragnarsson Kari Completion of the representation ring Detecting saturation in the double Ragnarsson Kari Burnside ring Radu Stancu Topology of the Boolean complex of a Ragnarsson Kari Coxeter system Bridget Tenner

73 Notes on the Norm Map Between the Hecke Algebras of the Gelfand-Graev Representations of GL(2,q^2) and Rainbolt Julianne U(2,q) A Comparison of Shintani Descent and Rainbolt Julianne the Curtis-Shoji Norm Notes on the Norm Map Between the Hecke Algebras of the Gelfand-Graev Representations of GL(2,q^2) and Rainbolt Julianne U(2,q) A Comparison of Shintani Descent and Rainbolt Julianne the Curtis-Shoji Norm Weyl Groups and Basis Elements of Hecke Algebras of Gelfand-Graev Rainbolt Julianne Representations Regular Elements and Minimal Rainbolt Julianne Polynomials A combinatorial formula for Macdonald Ram Arun Polynomials Martha Yip MathML for mathematics research Ram Arun articles The center of the affine and graded Ram Arun BMW algebras Rahbar Virk, Martha Yip Constructing positively folded alcove Ram Arun walks James Parkinson Ram Arun Fock space and Jantzen filtrations Peter Tingley Covolume des groupes S- arithmetiques et faux plans projectifs (d'apres Mumford, Prasad, Klingler, REMY Bertrand Yeung...) Cyclic Sieving, Promotion, and Rhoades Brendon Representation Theory Rhoades Brendon Promotion and Cyclic Sieving via Webs P. Pylyavskyy, K. Petersen Rickard Jeremy Blocks and support varieties Jon Carlson W. Dison, M. Elder and R. Riley Tim The of Stallings' group Young Riley Tim Hydra Groups M. Bridson The Geometry of the Conjugacy Riley Tim Problem M. Bridson Rigidity and the isomorphism problem Sabalka Lucas for tree braid groups Online robot navigation in multiple Sabalka Lucas dimensions The permutation characters of symplectic groups in even Saxl Jan characteristic on orthogonal forms Multiplicity free permutation Saxl Jan representations of classical groups J van Bon and NFJ Inglis Hecke group algebras as degenerate Florent Hivert, Anne Schilling, Schilling Anne affine Hecke algebras Nicolas Thiery

74 Characterization of promotion Jason Bandlow, Anne Schilling. Schilling Anne operators on tensor products Nicolas Thiery Kirillov-Reshetikhin crystals of type C_n^{(1)}, D_{n+1}^{(2)} and Ghislain Fourier, Masato Okado, Schilling Anne A_{2n}^{(2)} Anne Schilling Schleimer Saul Rigidity of the curve complex Kasra Rafi Schleimer Saul Covers and the curve complex Kasra Rafi Connectivity of the boundary of the Schleimer Saul curve complex Chris Leininger Local path-connectivity of the the Schleimer Saul space of ending laminations Chris Leininger, Mahan Mj Schleimer Saul The geometry of the disk complex Howard Masur Cluster algebra structures and semicanonical bases for unipotent Schroeer Jan groups Christof Geiss, Bernard Leclerc Moduli of weighted punctured Riemann Schumacher Georg surfaces II Harmonic maps and conical Schumacher Georg singularities Zheng Huang Automatic growth series for right- Scott Richard angled Coxeter groups Rebecca Glover (student) Reciprocity of growth series for right- Scott Richard angled Coxeter groups Generalized Hecke von Neumann algebras and automatic growth series Scott Richard for Coxeter groups Boris Okun Scott Leonard From Specht resolutins to standard On a theorem of Archipv- Terrell Hodge, Paramasamy Scott Leonard Bezrukavnikov-Ginzburg Kapparuchuk Some nontrivial Kazhdan-Lusztig coeffeicents of affine Weyl group of Scott Leonard type \tildeA_n Nanua Xi Diophantine geometry over groups IX: Sela Zlil Envelopes and imaginaries (tentative) On associativity and Sela Zlil geometry Eliyahu Rips Pleating rays for the Maskit embedding Series Caroline of the twice punctured torus. Rauzy reduction and simple curves on Series Caroline the twice punctured torus. Corinna Ulgicrai Plotting Plotting pleating rays in the Maskit embedding of the twice punctured Series Caroline torus. David Dumas The bending measure conjecture for Series Caroline the Maskit embedding. Algebraically periodic translaton Smillie John surfaces Kariane Calta Borel-Serre compactification of spaces Smillie John of translation surfaces Tom Schmidt and Kariane Calta Smillie John Algebraic periodicity and discriminants Kariane Calta

75 Finiteness results for flat surfaces: Smillie John large cusps and short geodesics Barak Weiss Symbolic dynamics of the translation Smillie John flow on the octagon Corinna Ulcigrai Smillie John Homotopy shadowing Yutaka Ishii Smillie John Characterizations of lattice surfaces Barak Weiss Homotopy pseudo-orbits and iterated Suzanne Hruska and Rodrigo Smillie John monodromy groups Perez GL(2,R) structures on 5-dimensional Smith Abraham manifolds Bryant (adviser) GL(2,R) structures on manifolds and Smith Abraham their integrability Robert Bryant Integrability of 2nd order PDEs and Smith Abraham GL(2,R) geometry Geometry and topology of geometic Soma Teruhiko limits Ken'u Geometry and topology of geometic Soma Teruhiko limits Ken'ich Ohshika Geometric approach to Ending Lamination Conjecture Soma Teruhiko arXiv:0801.4236 Around the local structure of Späth Britta exceptional groups of Lie type The McKay Conjecture for Classical Späth Britta Groups and some Primes The McKay Conjecture for exceptional Späth Britta groups and odd primes Remarks on parabolic character Springer Tonny sheaves (provisional) Decompositions related to Springer Tonny symmetric varieties Global to local bijections in blocks of Srinivasan Bhama finite reductive groups Paul Fong, Michel Broue Isolated blocks in twisted general linear Srinivasan Bhama groups Graded center of the stable module Stancu Radu category over the Klein four group Markus Linckelmann Stancu Radu Transfer theorems on fusion systems Adam Glesser, Nadia Mazza On the Glauberman theorem on fusion Stancu Radu systems Silvia Onofrei Stancu Radu Fusion systems on pro-p groups Peter Symonds On the characteristic idempotent of a Stancu Radu fusion system Kari Ragnarsson Stanley Richard Promotion and evacuation Stembridge John Admissible W-graphs Swenson Eric semiatblity in CAT(0) cube complexes Sageev, Micah Swenson Eric Convex subgroups of CAT(0) groups Panos Papasoglou Swenson Eric On a theorem of Farrells Ian Leary Rank rigidity in CAT(0) cube Swenson Eric complexes Dan Guralnik

76 Swenson Daniel Ph.D. Thesis: The Steinberg Complex Castelnuovo Mumford regularity of Symonds Peter rings of polynomial invariants Castelnuovo Mumford regularity of Symonds Peter cohomology rings of finite groups Symonds Peter Fusion systems for profinite groups An element of order 4 in the Symonds Peter Nottingham group at the prime 2 T. Chinburg Fixed points of actions of finite group Symonds Peter schemes Geometric construction of Relative completeion of fundamental group and Terasoma Tomohide its addmisibilty R. Hain, G.Pearlstein Thiem Nat q-Partition algebra combinatorics T. Halverson Thiem Nat Partition algebras and their q-analogs T. Halverson and A. Ram A supercharacter theory for finite Thiem Nat reductive groups C.R. Vinroot Applications of the supercharacter Thiem Nat theory of algebra groups P. Diaconis, M. Isaacs The super-representation theory of the finite group of upper-triangular Thiem Nat matrices Hecke group algebras as degenerate Thiéry Nicolas affine Hecke algebras Florent Hivert, Anne Schilling Characterization of promotion Thiéry Nicolas operators on tensor products Jason Bandlow, Anne Schilling Existence and covolumes of lattices for Thomas Anne Davis complexes Density of commensurators for right- Thomas Anne angled buildings Angela Kubena Barnhill Cocompact discrete subgroups of Lisa Carbone, Leigh Cobbs and Thomas Anne SL_2 over nonarchimedean local fields Inna Korchagina Covolumes of lattices in Kac-Moody Thomas Anne groups Talia Fernos Covering theory for complexes of Thomas Anne groups Seonhee Lim Counting overlattices for polyhedral Thomas Anne complexes Seonhee Lim Thomas Anne Higher divergence functions for groups Moon Duchin Real Schotty Complex Projective Thompson Josh Structures Brauer's height zero conjecture for the Tiep Pham 2-blocks of maximal defect G. Navarro Symmetric powers and a problem of Tiep Pham Kollar and Larsen R. M. Guralnick Degrees of rational characters of finite Tiep Pham groups G. Navarro M. Liebeck, E. O'Brien, A. Tiep Pham The Ore conjecture Shalev Representations of finite special linear Tiep Pham groups in non-defining characteristic A. S. Kleshchev

77 Representations of finite general linear Tiep Pham groups which are irreducible over a Brauer's height zero conjecture for the Tiep Pham 2-blocks of maximal defect G. Navarro Symmetric powers and a problem of Tiep Pham Kollar and Larsen R. M. Guralnick Degrees of rational characters of finite Tiep Pham groups G. Navarro M. Liebeck, E. O'Brien, A. Tiep Pham The Ore conjecture Shalev Representations of finite special linear Tiep Pham groups in non-defining characteristic A. S. Kleshchev Representations of the general linear groups which are irreducible over Tiep Pham subgroups A. S. Kleshchev Tiep Pham Linear groups of bounded deviation R. M. Guralnick Tiep Pham Bounds on $H^2$ R. M. Guralnick Hall-Higman type theorems for semisimple elements of finite classical Tiep Pham groups A. E. Zalesski Irreducible restriction problem for some Tiep Pham Ree groups F. Himstedt, H. N. Nguyen Tiep Pham Block coverings C. Bessenrodt Special character values of group Tiep Pham elements G. Navarro, J. B. Olsson The bound of least dilatation of pseudo-Anosov homeomorphisms of Tsai Chia-yen punctured surfaces Christopher Leininger Symbolic dynamics of the translation Ulcigrai Corinna flow on the octagon John Smillie Ulcigrai Corinna Coding and the horocycle flow Kariane Calta Trigonometric sums and diophantine Ulcigrai Corinna approximation Symbolic dynamics of the translation Ulcigrai Corinna flow on the octagon John Smillie Ulcigrai Corinna Coding and the horocycle flow Kariane Calta Trigonometric sums and diophantine Ulcigrai Corinna approximation Yakov Sinai Symbolic dynamics of the translation Ulcigrai Corinna flow on the octagon John Smillie Ulcigrai Corinna Coding and the horocycle flow Kariane Calta Trigonometric sums and diophantine Ulcigrai Corinna approximation Yakov Sinai Renewal-type Limit Theorem for the Gauss Map and Continued Fractions Ulcigrai Corinna arXiv:0710.1283 Yakov G. Sinai A bijection on core partitions and a parabolic quotient of the affine Vazirani Monica symmetric group Chris Berg, Brant Jones

78 $(\ell,0)$-Carter partitions, a generating function, and their crystal Vazirani Monica theoretic interpretation Chris Berg decidability and the conjugacy Ventura Enric problem for some extensions of groups O. Bogopolski, A. Manrtino Ventura Enric Conjugacy in free-by-cyclic groups T. Riley The behaviour of the Laplace transform of the invariant measure on Vershik Anatoly the hypersphere The behaviour of the Laplace transform of the invariant measure on Vershik Anatoly the hypersphere The behaviour of the Laplace transform of the invariant measure on Vershik Anatoly the hypersphere Invariant measures o the set of Vershik Anatoly universal graphs F.Petrov Globalisation of partial isometries of Vershik Anatoly metric spaces Integral representations of current Vershik Anatoly groups. M.Graev All invarinat measures with resect to Vershik Anatoly continual Cartan subsgorup Vogtmann Karen Automorphisms of tree-based RAAGs Kai-Uwe Bux and Ruth Charney Actions of automorphism groups of Vogtmann Karen free groups Martin Bridson Heather Anderson and Brad Vogtmann Karen A presentation for Aut(F_n) Forrest Representations of Lie superalgebras Wang Weiqiang in prime characteristic I Lei Zhao (tentative) Kostant homology formulas Shun-Jen Cheng, Jae-Hoon Wang Weiqiang for Lie superalgebras Kwon (Tentative title): Theta functions, elliptic hypergeometric series, and Kawanaka's macdonald polynomial Warnaar S conjecture R. Langer and M. Schlosser (Tentative title): Interpolation Macdonald polynomials and basic Warnaar S hypergeometric series A. Lascoux Warnaar S No working title yet A. Lascoux The module structure of the coinvariant Webb Peter algebra of a finite group representation A. Broer, V. Reiner and L. Smith Extending the coinvariant theorems of Chevalley, Shepard-Todd, Mitchell and Webb Peter Springer A. Broer, V. Reiner and L. Smith Stratifications of Mackey functors II: Webb Peter globally defined Mackey functors Weiss Barak Characterizations of lattice surfaces John Smillie Periodic product of trees in CAT(0) Whyte Kevin square complexes Sageev Whyte Kevin Some fake S-arithmetic lattices Mosher

79 Whyte Kevin Coarse Fibrations Whyte Kevin On quasi-isometry groups The conjugacy problem in right-angled Wiest Bert Artin groups and their subgroups John Crisp, Eddy Godelle Patrick Dehornoy, Ivan Wiest Bert Why are braids orderable Dynnikov, Dale Rolfsen The totally non-negative part of G/P is K. Rietsch Williams Lauren a CW complex Combinatorial Hopf algebras, Hall- Littlewood polynomials, permutation Williams Lauren tableaux. J.C. Novelli and J.Y Thibon Discrete Morse theory and totally non- Williams Lauren negative flag varieties Parameterizations of Cominuscule flag Williams Lauren varieties P. Pylyavksy and T. Lam Bergman complexes of Coxeter arrangements and the tropical Williams Lauren Grassmannian F. Ardila Williams Lauren Type B alternating sign matrices A. Lascoux Conjugacy classes of solutions to equations and inequations over Wilton Henry hyperbolic groups Daniel Groves The structure of limit groups over Wilton Henry hyperbolic groups Daniel Groves Algorithmic construction of Makanin- Wilton Henry Razborov diagrams Daniel Groves The profinite topology and JSJ Wilton Henry decompositions of 3-manifolds Pavel Zalesskii Wilton Henry Limit groups and property tau Ian Agol and Jason DeBlois Certain hyperbolic manifolds are Eric Chesebro and Jason Wilton Henry virtually special DeBlois SU(3) structures and special Xu Feng Lagrangian Xu Feng Instantons on nearly Kahler manifolds Harmonic morphisms with totally Xu Feng geodesic fibers A combinatorial formula for Macdonald Yip Martha polynomials Arun Ram Addendum to `The product of the Weil character and the Steinberg character Zalesski Alexandre in finite classical groups' G. Hiss Hall-Higman type theorems for semisimple elements of exceptional Zalesski Alexandre groups of Lie type Pham Huu Tiep Decomposition numbers for certain projective modules for finite Chevalley Zalesski Alexandre groups Gerhard Hiss Projective modues for finite Chevalley Zalesski Alexandre groups Gerhard Hiss

80 The Weil character for finite symplectic Zalesski Alexandre groups Elements of maximal order in finite Zalesski Alexandre groups of Lie type I.D. Suprunenko Regular semisimple elements and cross characteristic representations of Zalesski Alexandre finite groups of Lie type Zhang Jiping Block separations and coverings Christine Bessenrodt Zhang Jiping groups and Zeta functions Explicit Jenkins-Strebel representatives of all strata of Abelian Zorich Anton and quadratic differentials

81 3. Postdoctoral Fellows

The postdoctoral program at MSRI is central to MSRI’s mission of continued excellence in mathematics research. MSRI organizes and hosts semester-long, and two-semester-long programs that, during the time of the program, become the leading edge in that field of study. MSRI’s postdocs engage with fellow mathematicians from all over the world to develop their interests and contribute to the Science community. During the 2007-2008 academic year, MSRI selected 53 postdoctoral scholars with research interests in the programs that MSRI offers. 28 of those were NSF Postdoctoral Fellows, and 2 were funded by the NSA. We also had our first Viterbi Postdoctoral Fellow. (MSRI received an endowment for postdoc support from the Viterbi Family Foundation that permit us to appoint two semester-long postdoctoral fellows each year beginning in 2009-10. We received enough endowment funds in 2008 to provide significant support for one semester-long postdoc. Lauren Williams was the first Viterbi Endowed Postdoctoral Scholar.)

There were many more excellent postdoc applicants than we could fund with our NSF Postdoctoral Fellowship budget line. The program organizers took additional funds from their allocated (NSF) budget to support an additional 23 participants that were within 5 years of having completed their Ph.D. Those were ‘Postdoc Research Members’ (as opposed to NSF Postdoctoral Fellows) and received a per diem of $2,400 per month. While they were not monetarily compensated at the same level as the NSF Postdoctoral Fellows, they received all other privileges. That is, all Postdocs were assigned a mentor upon their arrival, they participated in a weekly Postdoc seminar, and they were a vibrant part of the research community. They also had the same logistic privileges (office, library access, bus pass, etc…)

Of the 30 Postdocs (of all kinds), 41% were female, the highest ratio (and total number) in the last 5 years. Of the 23 Postdoc Research Members, 6 were female (26%). The numbers of US Citizen and Permanent Residents were 11 (38%) and 8 (36%), respectively. These numbers also represented an increase in percentage over the past years. It is fair to say that all organizers were extremely satisfied with the Postdoctoral program and believed that it was an enormous success. Looking at the Institution Placement list (below), one sees that, of the 20 NSF Postdocs who stayed in the US, 15 went on to Group I Universities and 3 went to Group II Universities, while the other two went to Vassar College and De Paul University. As for the Postdoc members, 8 of the 11 went on to Group I Institutions, 2 to Group II Universities, and the last one went to Suffolk University. Of the postdocs who were not US Citizens (or Permanent Residents), most went on to equally prestigious institutions, such as Oxford University, l’École Polytechnique, and Bristol University. Here are additional details on the NSF Postdoctoral Fellows for each program.

82 Geometric Group Theory

Name Placement Institution Ph.D. Mentor Christopher Cashen University of Utah 2007 Kevin Whyte Emanuele Delucchi SUNY-Binghamton 2006 Jon McCammond Tullia Dymarz Yale University 2007 Kevin Whyte Sang-hyun Kim University of Texas at 2007 Micah Sageev Austin Larsen Louder University of Michigan 2007 Mark Feighn Damian Osajda University of 2004 Jon McCammond Wroclawski Thomas Putnam Massachusetts Institute 2007 Mladen Bestvina of Technology Anne Thomas Cornell University 2007 Karen Vogtmann

Christopher received his Ph.D. from the University of Illinois, Chicago in 2007, under the supervision of Kevin Whyte. His dissertation was titled “Quasi-isometries among tubular groups.” Cashen studied the quasi-isometry classification of what are known as "tubular groups". Culler remarks on how novel his approach is and that it "may well provide the best sort of result that is within reach of current techniques."

Christopher Cashen Emanuele received his Ph.D. from the Swiss Federal Institute of Technology in 2006. His thesis was titled “Combinatorics and topology of arrangement covers and of nested set complexes.” He is a geometric combinatorialist with a strong interest in Coxeter groups, Artin groups and hyperplane arrangements. His recent work has tended more towards geometric group theory - particular Garside structures and other connections with the work of Davis Bessis.

Enamuele Delucchi

83 Tullia received her Ph.D. in 2007 from the University of Chicago, under the supervision of Benson Farb. . Her work was on the geometry of solvable Lie groups, which played a key role in establishing recent, long sought-after quasi-isometric rigidity results for lattices in solvable groups.

Tullia Dymarz

Sang-Hyun received his Ph.D. from Yale University in 2007, under the supervision of . His dissertation was titled “Surface Subgroups of Graph Products of Groups and Right- angled Artin Groups.” Kim has developed new techniques for embedding right angled Artin groups in others. In this way, he has succeeded in settling a long-outstanding open question about the existence of surface subgroups in right-angled Artin groups. Sang-hyun Kim

Larsen received his Ph.D. from the University of Utah in 2007, under the supervision of Mladen Bestvina. His dissertation was titled “Krull Dimension for Limit Groups.” Lars’ thesis breaks ground on the problem of the existence of the Krull dimension for limit groups, and solves it in an important special case. Zlil Sela writes "In tackling the existence of the Krull dimension, Lars has demonstrated a combination of technical abilities, dare Larsen Louder and overall perspective that are rare for a graduate student, and not that common even among professional mathematicians."

84

Damian received his Ph.D. from the University of Wroclaw in 2004, under the supervision of Tadeusz Januszkiewicz. While his thesis work was in equivariant infinite dimensional topology, his latest work is in the new and burgeoning area of simplicial non-positive curvature. In particular, he settled a conjecture of Januskiewicz and Swiatkowski on the Gromov boundary of 7-systolic spaces. Damian Osajda

Thomas received his Ph.D. from the University of Chicago in 2007, under the supervision of Benson Farb. Putman studies mapping class groups, with an emphasis on the Torelli group. Farb writes "Andy Putman is probably the best Geometry/Topology student among the more than 40 I have seen since I arrived at Chicago in 1994."

Thomas Putnam

Anne received her Ph.D. from the University of Chicago in 2007, under the supervision of Benson Farb. She is motivated by the theory of lattices in semi-simple Lie groups, as well as the theory of Bass and Lubotzky of lattices in automorphism groups of trees, Thomas has broken new ground in understanding lattices in automorphism groups of buildings and related spaces. Bass writes "Thomas is clearly a very promising and already quite productive young researcher, with a far reaching research agenda for which she has Anne Thomas pioneered many of the important techniques."

85 Teichmuller Theory and Kleinian Groups

Name Placement Institution Ph.D. Mentor Ilesanmi Adeboye University of Southern 2006 Ian Agol California Javier Aramayona National University of 2005 Jeff Brock Ireland Kariane Calta Vassar College 2004 Howard Masur Moon Duchin University of California, 2005 Dick Canary Davis Zheng Huang University of Michigan 2003 Howard Masur Anna Lenzhen University of Michigan 2006 Jeff Brock Hossein Namazi University of Texas, 2005 Dick Canary Austin

Ilesanmi received his Ph.D. from the University of Michigan at Ann Arbor in 2006, under the supervision of Richard Canary. His thesis was titled “Volumes of hyperbolic orbifolds.” Ilesanmi is a promising young geometer whose work so far has focused on obtaining explicit lower bounds on volumes of higher dimensional hyperbolic orbifolds. The topological consequences of this work include bounds on sizes of outer automorphism groups of fundamental groups of hyperbolic manifolds. It is well-known, abstractly, that such volume bounds must exist, but explicit bounds Ilesanmi Adeboye are only known in dimension 3 and in the special case of manifolds in all dimensions. For orbibolds, the presence of more complicated torsion subgroups makes the situation significantly more difficult in higher dimensions. Ilesanmi has made significant progress in all dimensions and is close to obtaining an explicit bound in 4 dimensions. His techniques likely generalize to other rank 1 symmetric spaces.

86 Javier received his Ph.D. from the University of Southampton in 2004, under the supervision of Brian Bowditch. His thesis was titled “The coarse geometry of Teichmuller space.” Javier Aramayona is an excellent and promising young mathematician with many strong results already under his belt and very interesting projects underway. He gave a new proof of the Gromov hyperbolicity of the Weil- Petersson metric in dimension two that was very technically Javier Aramayona interesting, in that it made use of a generalized “flat plane” type result in a nonlocally compact setting. Since then his joint work with Anderson and Parlier gave rise to the notion of a “thick metric space” later popularized by Behrstock, Drutu and Mosher. These spaces have no collection of subsets with respect to which they are strongly relatively hyperbolic. He is an expert in pushing through subtle geometric arguments in the setting of combinatorial models for Teichmüller space, where notably he and his collaborators have shown that Farey-graphs in the pants complex are totally geodesically embedded.

Kariane received her Ph.D. from the University of Chicago in 2004. Amongst all the postdocs whose area can be described roughly as working in flows on moduli spaces of translation surfaces and geometry of Teichmuller space, Calta was recognized as the strongest. Her Ph.D. dissertation opened up a new area of research in the field and has had a deep impact. She found in genus 2 certain SL(2,R) closed invariant subspaces inside the whole moduli space. These were described by certain equations defined over number fields. C. McMullen described the Kariane Calta same space as the locus of real multiplication. She has begun work with J. Smillie to extend these ideas to higher genus, in what they call algebraic periodicity. This work holds great promise to extend our knowledge of translation surfaces in higher genus. She has recently done work with Wortman to classify the horocycle invariant measures on the locus of real multiplication.

87 Moon received her Ph,D. from the University of Chicago. Her thesis was titled "Geodesics track random walks in Teichmüller space." Moon’s work concerns large scale behavior of geodesics in the Teichmüller metric. Her thesis concerned a thin-triangles result for the Teichmüller metric, appropriately interpreted (it is a result of Masur that the Teichmüller metric is not negatively curved, and due to Masur-Wolf that it is not Gromov-hyperbolic). Her recent result with Rafi on the quadratic

Moon Duchin divergence of geodesics is particularly interesting, and gets to central features of the mapping class group and its interaction with the Teichmüller metric.

Zheng received his Ph.D. from Rice University in 2003, under the supervision of Mike Wolf. His thesis was titled “Harmonic maps and the geometry of Teichmueller space.” Zheng is an expert on differential geometry of the Weil-Petersson metric on Teichmüller space. He verified a conjecture of J. Brock and B. Farb that while the Weil-Petersson metric is Gromov hyperbolic in dimension 2, it nevertheless does not have curvature bounded away from zero (despite having negative curvature). He brought a unique perspective to the program, and collaborated with others in the program, notably David Dumas and Mike Wolf. Zheng Huang

Anna received her Ph.D. from the University of Illinois, Chicago in

2006, under the supervision of Howard Masur. Her thesis was titled

“Teichmüller geodesics which do not have a limit in PMF.” In her thesis

she found Teichmüller geodesics that do not converge in theThurston

compactification of Teichmüller space. This was a very good thesis that

reopened and answered questions concerning the interaction of the

Teichmüller metric and the Thurston compactification of Teichmüller

space. In her current work she is trying to find similar phenomena in the

context of the Weil-Petersson metric on Teichmuller space. Her interests Anna Lenzhen are central to the program at MSRI.

88 Hossein received his Ph.D. from Stony Brook University, under the supervision of Yair Minsky. Hossein is one of two or three leading postdocs in the field of hyperbolic geometry in the last 3 years. His thesis was a tour-de-force, giving a new inroad into understanding the geometry of closed hyperbolic 3-manifolds via their Heegaard splittings. He has continued in this thread in joint work with Juan Souto analyzing the geometry of the case when the glueing map is pseudo-Anosov completely. In new work with J.Brock, Minsky and Souto, Hossein has given a necessary and sufficient condition for a

Hossein Namazi closed manifold to have bounded geometry, a generalization to the closed case of the bounded geometry theorem of Minsky. Hossein sits

at the center of an important crossroads in 3-manifold geometry and

topology, relating the combinatorial study of curves on surfaces to

developing a complete theory of the geometry of hyperbolic 3-maifolds

in general.

89 Combinatorial Representation Theory

Name Placement Ph.D. Mentor Institution Sami Assaf Massachusetts 2007 Persi Diaconis Institute of Technology Maria Ecole Polytechnique 2007 Cedric Bonnafe Chlouveraki Ghislain Fourier Universitat Koln 2007 Edward Frenkel/Anne Schilling Syu Kato Kyoto University 2003 Springer/Littelmann/Anderson

Jonathan Kujawa University of 2003 Jan Saxl/Gustav Leher Oklahoma Sinead Lyle University of East 2003 David Hemmer Anglia Kevin Purbhoo University of 2004 Hélène Barcelo/Francesco Waterloo Brenti

Nat Thiem University of 2004 Bhama Srinivasan/Anatoly Colorado Vershik Lauren Williams Harvard University 2005 Persi Diaconis/Georgia Benkart

Sami received her Ph.D. from the University of California, Berkeley in 2007, under the supervision of Mark David Haiman. Her dissertation was titled “Dual equivalence graphs, ribbon tableaux and Macdonald polynomials.” Sami has conjectured a combinatorial interpretation for the Schur expansion of an arbitrary LLT polynomial, which automatically gives an interpretation for the Schur expansion of Macdonald polynomials. Sami's thesis is one of the Sami Assaf most exciting theses in symmetric function theory in recent memory.

90 Maria received her Ph.D. from the Université Paris VII in 2007, under the supervision of Michel Broué. Her dissertation was titled “On the cyclotomic Hecke algebras of complex reection groups." Maria has a postdoctoral position at the Ecole Polytechnique in Lausanne. While at MSRI she began and finished the calculation of the Rouquier blocks for the complex reflection groups of all of the infinite series, thus completing their determination for all complex reflection groups. She has written two papers on the subject: \Rouquier blocks of the cyclotomic Ariki-Koike algebras" and Maria Chlouveraki \Rouquier blocks of the cyclotomic Hecke algebras of G". Both of the papers have been submitted and posted on the archiv. During the time at MSRI she served as coordinator for the seminars for both programs.

Ghislain received his Ph.D. from the University of Cologne in 2007, under the supervision of Peter Littelmann. His dissertation was titled “On the combinatorics of finite dimensional representations of loop Algebras.” Ghislain is an extremely talented young mathematician. He was the driving force behind the project which has understood how Demazure crystals sit inside the Kirillov-Reshetikhin (KR) crystal. In his thesis he has discovered a very close connection between the "Kyoto path model" and the "Littelmann path model" and, in planned

work with Hernandez in Autumn of 2007, they will use these new ideas Ghislain Fourier to search for a bijection between the "Kashiwara / Nakajima monomial model" and the "Littelmann path model" of these crystals. For somebody who is at such an early stage of his career, he has been remarkably productive. He has already established collaborations with Schilling, Chari, Littelmann and Hernandez and seems on a perfect track to a very productive future.

Syu received his Ph.D. in 2003 under the supervision of Matumoto. His dissertation was titled “Equivariant bundles on group completions.” He is currently at University of Tokyo as a COE Researcher. Syu’s recent works on exotic Deligne-Langlands correspondence and exotic Springer correspondence are of special importance. His first main result is an analog of Bott's theorem (which describes the cohomology groups of all line bundles on flag Syu Kato manifolds) for wonderful compactifications of adjoint semisimple groups. Another remarkable achievement of Kato is his description of all equivariant vector bundles on wonderful group completions and, more generally, on equivariant compactifications of reductive groups. 91 Jonathan received his Ph.D. in 2003 from the University of Oregon, under the supervision of Johnathan Brundan. His dissertation was titled “The representation theory of the supergroup GL(m|n)”. He was formerly an NSF postdoctoral fellow at University of Georgia. There are very few people in his age group who would be able to appraoch problems through both the geometric/functorial viewpoint as well as

Jonathan Kujawa through combinatorics. Kujawa has written 7 papers; These papers are all very interesting and significant. Kujawa is one of the top junior researchers in representation theory, both nationally and internationally.

Sinead received her Ph.D. from Imperial College in 2003, under the supervision of Gordan James. She is working on the representation theory of symmetric groups. She has already produced some outstanding work including a solution to a conjecture of Andrew Mathas.

Sinead Lyle

Kevin received his Ph.D. from the University of California, Berkeley in 2004 under the supervision of Allen Knutson. His dissertation was titled “Vanishing and nonvanishing criteria for branching Schubert calculus”. He was formerly a postdoc at University of British Columbia. Kevin’s most important theorem is his Nullstellensatz for amoebas, which is a foundational result in tropical geometry. His work Kevin Purbhoo on nonvanishing products in the cohomology rings of Grassmannians is also important, and he is an expert in the area of Horn inequalities, the Hermitian sum problem and its generalizations. The word that best describes Purhboo and his work is “ingenious”. Purbhoo is the deepest and most creative person on the job market this year who is working at the interface of combinatorics and algebraic geometry.

92

Nat received his Ph.D. from the University of Wisconsin, Madison in 2004, under the supervision of Arun Ram. His dissertation was titled “Unipotent Hecke algebras.” He is currently a Szegö Assistant professor at Stanford. Nat works in combinatorial aspects of representation theory of finite groups of Lie type and is one of the very best young people in this area. He has a broad expert’s knowledge of combinatorics, perhaps specializing algebraic combinatorics, symmetric function theory, and the like. He also has a Nat Thiem broad, expert’s knowledge of finite groups of Lie type, their structure, and, particularly, their representation theory. He is very smart, very deep, and has something extra: a true originality that makes you stop and ask “Where did that come from?”. After moving to Stanford for a postdoctoral position under Persi Diaconis, Nat developed his work in several directions. In one by Nat, joint with Ryan Vinroot, some mysterious facts related to Ennola duality get very natural explanations, a dimension formula analogous to the famous Green formula for GL(n, q) is obtained, a new construction of model is discovered (generalizing Klyachno’s work on GL(n, q)). Nat has exciting plans for the future, of which the most intriguing one is to understand the relation between unipotent Hecke algebras, Kawanaka’s generalized Gelfand-Graev representations, and supercharacters of Andre.

Lauren received her Ph.D. from the Massachusetts Institute of Technology in 2005, under the supervision of Richard Stanley. Her dissertation was titled “Combinatorial aspects of total positivity”. She is currently at Harvard. Lauren has made seven significant contributions to algebraic combinatorics: Four of them are related to the idea of the "positive part" of an , motivated by Lusztig's definition of the totally postive part of a real flag variety. This Lauren Williams work involves the positive Bergman complex of an oriented matroid, tropical algebraic geometry and cluster algebras (with D. Speyer), and the shellability of certain posets. Her other three contributions concern the structure of generalized permutohedra, permutation enumeration,

93 Representation Theory of Finite Groups and Related Topics

Names Placement Institution Ph.D. Mentor Maria Ecole Polytechnique 2007 Cedric Bonnafe Chlouveraki Daniel Juteau University of Caen 2007 Zongzhu Lin Jonathan University of Oklahoma 2003 Jan Saxl/Gustav Leher Kujawa Sinead Lyle University of East Anglia 2003 David Hemmer Nadia Mazza University of Aberdeen 2003 Peter Symonds Attila Morati University of Southern 2004 Dave Benson California Julia Pevtsova University of Washington 2002 Jon Carlson Kari Ragnarsson Depaul University 2004 Peter Webb Nat Thiem Oxford University 2004 Bhama Srinivasan/Anatoly Vershik

Maria Chlouveraki (See above)

Juteau received his Ph.D. from the Université Paris VII in 2007, under the supervision of Cédric Bonnafé and Raphael Rouquier. His dissertation was titled “Modular Springer correspondence and decomposition Matrices.” They are working on modular character sheaves. This is the problem of using the cohomology rings of varieties to study the representation theory of groups of Lie type. Juteau has made some very good progress in this difficult area. Daniel Juteau

94

Jonathan Kujawa (See above)

Sinead Lyle (See above)

Nadia received her Ph.D. from the Université de Lausanne in 2003, under the supervision of Jacques Thévenaz. Her thesis was titled “Modules d’endo-permutation.” Her work on endopermutation modules contributed to the complete classification of the modules by Bouc. More recent work has been on the classification of endotrivial modules for finite groups.

Nadia Mazza

95

Attila received his Ph.D. (University) in 2004, under the supervision of Geoffrey Robinson. He has proved some striking results on counting conjugacy classes. He is also interested in combinatorial and probabilistic problems in group theory.

Attila Morati

Julia received her Ph.D. from Northwestern University in 2002. Julia begins an assistant professorship at the University of Washington in the fall of 2008. She presented one of the Evan's Lectures during the semester at MSRI as well as giving plenary lectures at the Connections for Women workshops and the workshop on Homological Methods in Representation Theory. During the semester at MSRI, she completed a paper on "Spectrum of the tensor triangulated category of perfect Julia Pevtsova complexes over a stack, with P. Smith and continued work on a project on finite dimensional pointed Hopf algebras with with M. Mastnak, P. Schauenburg, and S. Witherspoon. She completed another paper on "Constructions for Infinitesimal group schemes", with E. Friedlander. The results were proved for a paper with D. Benson on "Vector bundles and modules of constant Jordan type". In addition, she worked with Jon Carlson and Friedlander on a project new to develop invariants for modular representations. A paper on this subject with the title "Higher rank varieties and generic kernels" should be written soon. There may be at least one more paper to come from this project.

96

Kari received his Ph.D. from the Massachusetts Institute of Technology in 2004, under the supervision of Haynes Miller. His thesis was titled “Frobenius transfers and p-local finite groups.” Kari had a visiting position at the University of Illinois at Chicago in the fall of 2007. He will have a visiting position at DePaul University, also in Chicago, in the fall of 2008. His main focus during the semester at MSRI was an ongoing project to define Burnside rings and Mackey functors for fusion systems. The object was to generalize specific properties exhibited by classical Mackey functors when examined at a prime, and Kari Ragnarsson much of his time was spent on a preliminary paper illustrating these p- local properties. He benefited in this regard from conversations with his postdoc mentor Peter Webb and with Serge Bouc. The preliminary paper is in the advanced stages of writing, while the fusion version is still work in progress. Two papers were completed and submitted during his time at MSRI. Ragnarsson spoke in the seminar on Homological Methods in Representation Theory and the seminar on Biset Functors as well as the Postdoc Seminar. Beyond his work on Mackey functors and fusion systems, he has submitted two paper "Obtainable sizes of finite topologies" and "Homotopy type of the boolean complex of a Coxeter system" both written with Bridget Eileen Tenner. Most of the work for these papers was done during the semester at MSRI. Another paper on "Fusion data in the Burnside ring" concerns results obtained at MSRI in work with Radu Stancu.

Nat Thiem (See above)

97 Complementary Program

Name Placement Institution Ph.D. Mentor Seonhee Lim Cornell University 2006 Ursula Hamenstadt

Seonhee received her Ph.D. from Yale University and Université Paris-Sud in 2006, under the supervision of G. Margulis and Frederic Paulin. Her thesis was titled “Enumeration and entropy rigidity of lattices in automorphism groups of trees and buildings.” Seonhee’s research interests lie in the area of geometric group theory and theory, dynamics of group actions on manifolds and more generally on polyhedral complexes. She is especially interested in group Seonhee Lim actions on trees and buildings, lattices of the automorphism groups of trees and buildings, lattices of algebraic groups over local fields, their rigidity and their dynamics.

98

3.1 Postdoctoral Placement List (See attached file for detail Postdoctoral List) (O:\0708AnnualReport\NSF Report 07-08\d. Postdoctoral Placement List\d. Postdoctoral Fellow Placement List 07-08 Detail) Last Placement Placement Placement Placeme Placement Name First Name Institution Department Position MSRI Mentor nt State Country Program University of Southern Department of Assistant Adeboye Ilesanmi California Mathematics Professor Ian Agol CA US TTKG National Aramayo University of Department of na Javier Ireland Mathematics Lecturer Jeff Brock IR TTKG Massachuss etts Institute of Department of CLE Moore Persi Assaf Sami Technology Mathematics Instructor Diaconis MA US CRT Vassar Department of Assistant Howard Calta Kariane College Mathematics Professor Masur NY US TTKG VIGRE University of Department of postdoctor Cashen Christopher Utah Mathematics al fellow Kevin Whyte UT US GGT Le Département d'Enseignemen t et de École Recherche de Chlouver Polytechniqu Mathématiques Postdoctor Cedric CRT/RTF aki Maria e al Fellow Bonnafe CH G Visiting SUNY- Department of Assistant Jon Delucchi Emanuele Binghamton Mathematics Professor McCammond NY US GGT University of California, Department of Postdoctor Duchin Moon Davis Mathematics al Fellow Dick Canary CA US TTKG Gibbs Yale Department of Assistant Dymarz Tullia University Mathematics Professor Kevin Whyte CT US GGT Edward Universitat Department of Assistant Frenkel/Anne Fourier Ghislain Koln Mathematics Professor Schilling DE CRT University of Department of Assistant Howard Huang Zheng Michigan Mathematics Professor Masur MI US TTKG University of Department of Charge de Juteau Daniel Caen Mathematics Recherche Zongzhu Lin FR RTFG Research Institute for Springer/Litte Kyoto Mathematical Assistant lmann/Ander Kato Syu University Sciences Professor son JP CRT University of Texas, Department of Bing Micah Kim Sang-hyun Austin Mathematics Instructor Sageev TX US GGT

99 Jan University of Department of Assistant Saxl/Gustav CRT/RTF Kujawa Jonathan Oklahoma Mathematics Professor Lehrer OK U.S. G Postdoctor ate University of Department of Assistant Lenzhen Anna Michigan Mathematics Professor Jeff Brock MI US TTKG H.C. Wang Cornell Department of Assistant Ursula Complim Lim Seonhee University Mathematics Professor Hamenstadt NY US entary NSF University of Department of Postdoctor Louder Larsen Michigan Mathematics al Fellow Mark Feighn MI US GGT University of Department of David CRT/RTF Lyle Sinead East Anglia Mathematics Lecturer Hemmer UK G Adjunct University of Research Southern Department of Assistant Maroti Attila California Mathematics Professor Dave Benson CA US RTFG University of Department of Postdoctor Peter Mazza Nadia Aberdeen Mathematics al Fellow Symonds UK RTFG University of Texas, Department of Assistant Namazi Hossein Austin Mathematics Professor Dick Canary TX US TTKG

University of Department of Jon Osajda Damian Wroclawski Mathematics Instructor McCammond PL GGT University of Department of Assistant Pevtsova Julia Washington Mathematics Professor Jon Carlson WA US RTFG Hélène Barcelo/ University of Department of Assistant Francesco Purbhoo Kevin Waterloo Mathematics Professor Brenti CA CRT Massachuss etts Institute of Department of CLE Moore Mladen Putman Thomas Technology Mathematics Instructor Bestvina MA US GGT Ragnarss DePaul Department of Visiting on Kari University Mathematics Professor Peter Webb IL US RTFG Bhama Srinivasan/A University of Department of Assistant natoly CRT/RTF Thiem Nat Colorado Mathematics Professor Vershik CO US G H.C. Wang Cornell Department of Assistant Karen Thomas Anne University Mathematics Professor Vogtmann NY US GGT

100 Benjamin Peirce Assistant Professor and NSF Persi Harvard Department of Postdoctor Diaconis/Geo Williams Lauren University Mathematics al Fellow. rgia Benkart MA US CRT

101 3.2 Postdoctoral Fellow Participant Summary

# of # of Citizens Postdoc & Per Decline/ # of # of Decline/ Name of Activity Fellow Res % No Reply Female % Minorities % No Reply Combinatorial Representation 9 4 44% - 4 44% - 0% 5 Theory

Complementary Program (07-08) 1 - 0% - 1 100% - 0% -

Geometric Group Theory 8 3 38% - 2 25% - 0% 5

Representation Theory of Finite Groups and 9 4 44% - 4 44% - 0% 7 Related Topics Teichmuller Theory and Kleinian 7 2 33% 1 3 50% 1 33% 4 Groups

Total 34 13 39% 1 14 42% 1 8% 21 Total Number of Distinct Postdoc Fellow 30 11 38% 1 12 41% 1 8% 18

*Percentage for Citizens & Minorities are computed out of participants that provided info on their citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

102 3.3 Postdoctoral Fellow Demographic Data Demographic Statistic

# % Total Program Participants 30 Male 18 60.00% Female 12 40.00% Decline to State Gender 0 0.00%

% Over % Respon- Over Ethnicities # dents Total Native American 0 0.00% 0.0% Asian 2 16.67% 6.7% Black 1 8.33% 3.3% Hispanic 0 0.00% 0.0% Pacific 0 0.00% 0.0% White 10 83.33% 33.3% Decline to State Ethnicities 18 60.0%

Minorities 1 8.33% 3.3%

US Citizen 9 31.0% 30.0% Decline to State Citizenship 1 3.33% 3.3% Per Resident 2 6.7% US Citizen & Per Resident 11 36.7%

Home Inst. in US 22

Year of Degree # % 2009 0 0.00% 2005-2008 18 60.00% 2000-2004 12 40.00% 1995-1999 0 0.00% 1990-1994 0 0.00% 1985-1989 0 0.00% 1980-1984 0 0.00% Yr <1980 0 0.00% Decline to state 0 0.00% Total 30 100.00% **Statistic Calculation based on participant that replied to each category. Ethnicities' percentage might be over stated since some participants selected to be included in more than one ethnicity groups.

103

Home Institution Classified by Countries Home Inst By Region State # % # % South AL - 0.0% 2 9.1% AR - 0.0% DE - 0.0% FL - 0.0% GA - 0.0% KY - 0.0% LA - 0.0% MD - 0.0% MS - 0.0% NC - 0.0% OK 1 4.5% SC - 0.0% TN - 0.0% TX 1 4.5% VA - 0.0% WV - 0.0% West AK - 0.0% 7 31.8% AZ - 0.0% CA 3 13.6% CO 1 4.5% HI - 0.0% ID - 0.0% MT - 0.0% NV - 0.0% NM - 0.0% OR - 0.0% UT 2 9.1% WA 1 4.5% WY - 0.0% Midwest IL 2 9.1% 4 18.2% IN - 0.0% IA - 0.0% KS - 0.0% MI 2 9.1% MN - 0.0% MO - 0.0% ND - 0.0% NE - 0.0% OH - 0.0% SD - 0.0% WI - 0.0% Northeast CT 1 4.5% 9 40.9% ME - 0.0% MA 3 13.6% NH - 0.0% NJ 1 4.5% NY 4 18.2% PA - 0.0% RI - 0.0% VT - 0.0% Total 22 22

104

Home Institution Countries Classified by Region Home Inst By Region Country # % # % Eastern Asia JP 1 3.3% 1 3.3% North America CA 1 3.3% 23 76.7% US 22 73.3% Central Europe CH 1 3.3% 3 10.0% DE 1 3.3% PL 1 3.3% Western Europe FR 1 3.3% 3 10.0% GB 2 6.7% Total 30 30

105

3.4 Postdoctoral General Member List (See attached file for detail General Member List) (O:\0708AnnualReport\NSF Report 07-08\d. Postdoctoral Placement List\d. Postdoctoral General Member List 07-08 Detail) Last First Placement Placement Placement Placement Placement Name Name Institution Department Position Mentor State Country Program Behrstoc Columbia Department of Ritt Assistant No k Jason University Mathematics Professor Mentor NY US GGT Louisiana State Department of Assistant No Brendle Tara University Mathematics Professor Mentor LA US GGT Research Fellow of Oxford Department of Christ Church Gabriel Craven David University Mathematics College Navarro UK RTFG Oxford Department of Assistant Radha Danz Susanne University Mathematics Professor Kessar UK RTFG Department of Mathematics, Statistics, and Research Kenneth University of Computer Assistant Bromber DeBlois Jason Illinois, Chicago Science Professor g IL US TTKG Department of Mathematics, Statistics, and University of Computer Assistant Steve Dumas David Illinois, Chicago Science Professor Kerckhoff IL US TTKG Persi Diaconis/ Universidad de Department of Postdoctoral Rinat Emsiz Erdal Talca Mathematics Fellow Kedem CL CRT Hedrick University of Assistant California, Los Department of Adjunct Mark Fernos Talia Angeles Mathematics Professor Sapir CA US GGT Temple Department of Assistant Sergio Futer David University Mathematics Professor Fenley PA US TTKG Markus Suffolk Department of Assistant Linkelma Glesser Adam University Mathematics Professor nn MA US RTFG Edward Laboratoire de Frenkel/A Hernand University of Mathématiques Chercheur au nne ez David Versailles de Versailles CNRS Schilling FR CRT University of Department of Maître de Monica Jacon Nicolas Besancon Mathematics conférences Vazirani FR CRT Tamarkin Ursula Brown Department of Assistant Hamenst Kent Richard University Mathematics Professorship ädt RI US TTKG Kenneth Université Paul Laboratoire Bromber Lecuire Cyril Sabatier Emile Picard g FR TTKG

106 Junior Oxford Department of Research Robert Lotay Jason University Mathematics Fellow Bryant UK Comp University in Aachen, Department of Assistant Jon Noeske Felix Germany Mathematics Professor Carlson DE RTFG Karen Alexandr University of Department of Postdoctoral Vogtman Pettet a Michigan Mathematics Fellow n MI US GGT T.H.Hildebra Barcelo/ Pylyavsk University of Department of ndt Assistant Brenti/St yy Pavlo Michigan Mathematics Professor embridge MI US CRT University of Department of Senior No Riley Tim Bristol Mathematics Lecturer Mentor UK GGT RWTH Aachen Department of Postdoctoral Michel Späth Britta University Mathematics Fellow Broue DE RTFG University of Department of Postdoctoral Michel Stancu Radu Copenhagen Mathematics Fellow Broue DK RTFG University of Department of Assistant John Ulcigrai Corinna Bristol Mathematics Professor Smillie UK TTKG University of Department of R.H. Bing Mark Wilton Henry Texas, Austin Mathematics Instructor Feign TX US GGT

107 3.5 Postdoctoral General Member Participant Summary

# of Decline/ Decline/ General # of No # of # of No Name of Activity Members Citizens % Reply Female % Minorities % Reply

4 - 0% - - 0% - 0% 4 Combinatorial Representation Theory

6 3 50% - 3 60% 1 33% 3 Geometric Group Theory

7 1 14% - 2 29% - 0% 5 Representation Theory of Finite Groups and Related Topics

6 4 67% - 1 17% - 0% 1 Teichmuller Theory and Kleinian Groups

1 - 0% 1 - 0% - 0% - Complementary

24 8 35% 1 6 26% 1 9% 13 Total

Total Distinct Postdoc General Members 23 8 36% 1 6 27% 1 9% 12

*Percentage for Citizens & Minorities are computed out of participants that provided info on their citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

108 4. Graduate Students

Most graduate students who visit MSRI have been invited to take part in one of our Summer Graduate Workshops. A much smaller number of graduate students are invited each year as “Program Associates” in our semester and year-long scientific programs. Program Associates benefit greatly from the chance to interact with a wide variety of mathematicians, gaining intense exposure to current ideas and trends in their area of specialization. While MSRI does not have the financial resources to fund the Program Associates it provides as much support as we can.

4.1 Summer Graduate Workshops

Every summer MSRI organizes several summer graduate workshops (usually two weeks each), most of which are held at MSRI. Attending one of these workshops can be a very motivating and exciting experience for a student: participants have often said that it was the first experience where they felt like real mathematicians, interacting with other students and mathematicians in their field. Each sponsoring institution is invited to send two students (in total) to participate in these programs, and can send a third if the group includes a woman or a member of an under- represented minority. MSRI covers the travel (up to $700 USD for foreign sponsoring institutions) and local expenses of all the students. The procedure is as follows: MSRI’s deputy director informs the Sponsor's Representative and the Director of Graduate Studies of the available Summer Graduate Programs for the following year. The Director of Graduate Studies submits nominations of students for particular programs. If the chosen program is already full, the Sponsoring Institution may make additional nominations to other programs until its quota of two or three accepted participants is reached. Students from none sponsoring institutions are also welcome to apply. They need to be nominated by their department’s chair of graduate studies.

The following are descriptions of the 5 Summer Graduate workshops which took place during the 2007-2008 academic year. Altogether 159 graduate students from all over the US participated in those workshops, of which 29% were female. See the table below for detailed demographic data.

109

SGW 1: IAS/PCMI Summer Conference: Statistical Mechanics Date: July 01, 2007 to July 21, 2007 Location: IAS/Park City Mathematics Institute, Salt Lake City, UT Organizers: Scott Sheffield, Thomas Spencer

The Graduate Summer School is designed to bridge the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, the students had completed their first year, and in some cases, had already begun working on a thesis. While a majority of the participants were graduate students, some postdoctoral scholars and researchers also attended.

The main activity of the Graduate Summer School was a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures were not duplicate standard courses available elsewhere. Each course consisted of lectures with problem sessions. Course assistants were available for each lecture series. The participants of the Graduate Summer School met three times each day for lectures, with one or two problem sessions scheduled each day as well.

Course Descriptions: The Renormalisation Group and Equilibrium Statistical Mechanics; David Brydges, University of British Columbia In the theory of critical phenomena in statistical mechanics, the idea of a scaling limit is exemplified by observing a very long self-avoiding walk from far away so that individual steps become invisible and one sees (the occupation density of) a path in the continuum. The scaling limit is the probability law for this random continuum path.

The Renormalisation Group (RG) is a nascent program to construct and classify scaling limits in d dimensions based on the Nobel prize work of Ken Wilson on the renormalisation group (RG). RG is a map acting on a space of statistical mechanical models. Models are probability measures on random fields and RG acts on a model by integrating out the short distance fluctuations giving rise to a new model whose typical random field has the same long distance fluctuations but suppressed short distance fluctuations. Finding scaling limits corresponds to determining the fixed points of RG. Statistical mechanical models were introduced, in particular dipoles on Zd, formulate the RG precisely, and show that the fixed point for RG in the case of the dipole gas is the massless Gaussian field.

Background reading for lectures by David Brydges: (download .pdf)

Lectures on Random matrices; Alice Guionnet, Ecole Normale Supérieure de Lyon The theory of random matrices has developed rapidly during the last 15 years in connection with fields as diverse as statistics, theoretical physics, number theory and combinatorics. This course was an introduction to part of this theory. After a general overview of random matrices, we focused on the so-called Wigner matrices which are self-adjoint matrices with independent entries (modulo the symmetry constraint). We studied the empirical measure of the eigenvalues of such matrices, or more generally traces of words in such matrices. We showed that it

110 converges (Wigner's theorem), study its fluctuations and prove that concentration of measure holds. We then restricted ourselves to Gaussian entries and discuss large deviations and matrix models in connection with the enumeration of certain graphs called maps. The prerequisites were basic probability theory.

Background reading: see Lecture Notes on St. Flour at www.umpa.ens-lyon.fr/~aguionne/

Dimers and random surfaces; Richard Kenyon, University of British Columbia The goal of the lectures was to introduce the dimer model and discuss the role it plays in recent results about limit shapes for crystal surfaces. The dimer model can be viewed as a model of random surfaces, and we intended to show how in the scaling limit (when the lattice spacing tends to zero) the random surfaces can have non-random limit shapes which arise from energy minimization considerations. We also discussed connections with SLE and the Gaussian free field.

Recommended course reading: Chapters 1-4, An introduction to the dimer model at http://arxiv.org/PS_cache/math/pdf/0310/0310326v1.pdf

An Introduction to the Schramm-Loewner Evolution; Gregory Lawler, University of Chicago

An introduction to the mathematics of SLE (Schramm-Loewner evolution):

Topics included: --- Basics of univalent functions --- Loewner differential equation --- Definition of Schramm-Loewner evolution --- Phases and dimension of the path --- Conformal transformations of SLE --- Restriction, locality, and the fundamental martingales --- Relation with Brownian loops

I assumed stochastic calculus through Ito's formula and complex variables through the Riemann mapping theorem. Much of the material came from my book Conformally Invariant Processes in the Plane (2005). Zeros of Gaussian Analytic Functions, determinantal processes and gravitational allocation; Yuval Peres, Microsoft Research and University of California, Berkeley

Lecture 1: Point Processes and Repulsion. Point processes (random scatters of points in space) have applications in many areas, including statistics and cosmology. Recently, there has been increasing interest in processes that exhibit "repulsion". We saw why zeros of random polynomials have this property, and described the effect of repulsion on matching and allocation problems.

Lecture 2: Zeros of Gaussian Analytic Functions. Zeros of Gaussian analytic functions have a remarkable rigidity property, discovered by M. Sodin: The first order intensity determines the whole process. For each of the classical

111 , planar, spherical and hyperbolic, there is a one-parameter family of Gaussian analytic functions with isometry-invariant zeros. Lecture 3: Determinantal Processes. Discrete and continuous point processes where the joint intensities are determinants arise in Combinatorics (Random spanning trees) and Physics (Fermions, eigenvalues of Random matrices). For these processes the number of points in a region can be represented as a sum of independent, zero-one valued variables, one for each eigenvalue of the relevant operator.

Lecture 4: Zeros of the I.I.D. Gaussian Power Series. The power series with i.i.d. complex Gaussian coefficients has zeros that form an isometry- invariant determinantal process in the disk model of the hyperbolic plane. (Joint work with B. Virag). This allows an exact calculation of the law of the number of zeros in a subdisk. We also analyzed the dynamic version where the coefficients perform Brownian motion.

Lecture 5: The Translation-Invariant Planar Gaussian Zeros. Sodin-Tsirelson analyzed the zeros of the Gaussian power series with Euclidean symmetry. Their results reveal a surprising analogy with a four-dimensional Poisson process. In particular, the probability of a large disk of radius R to be free of zeros decays like exp(-cR^4). A remarkable "gravitational allocation" that allots a unit of area to each zero in a translation invariant way was discovered by Sodin and Tsirelson. Nazarov, Sodin and Volberg ahowed that the diameters of the domains of attraction have exponential tails.

Lecture 6: Gravitational Allocation for Poisson Points. While the method of gravitational allocation is not applicable to the planar Poisson process, it does apply to the Poisson process in dimensions 3 and higher; (This is Joint work with S. Chatterjee, R. Peled, D. Romik). The argument starts with the classical calculation by Chandrasekar of the total gravity force acting on a point, which has a stable law. Here also the domains of attraction have an exponential tail, and the proof uses ideas of dependent percolation. See http://pcmi.ias.edu/current/Peresimages.htm for images related to Peres’ course. Gaussian Analytic Functions Book - PDF Download (3.2MB) Conformal invariant models; Wendelin Werner, Université Paris-Sud We first discussed some two-dimensional discrete models from statistical physics (critical percolation, uniform spanning trees, etc.) and studied their large-scale properties and in particular their conformal invariance. We saw how and why the SLE (Schramm-Loewner Evolutions) that was studied in Greg Lawler's course can then be used to derive results concerning the discrete models. Finally, we described continuous two-dimensional systems in which SLE loops are naturally embedded.

Background/preparation reading: G.R. Grimmett, Percolation, Springer Basic material on conformal invariance (Riemann's mapping theorem & Morrera's theorem) -- for example in Ahlfors' Complex Analysis. The course was otherwise self-contained. Material related: My lecture notes from Saint-Flour http://arxiv.org/abs/math/0303354 and Les Houches http://front.math.ucdavis.edu/math.PR/0511268

112

SGW 2: Data Assimilation for the Carbon Cycle Date: July 08, 2007 to July 13, 2007 Location: Boulder, CO. Organizers: James Clark (Duke University), Inez Fung (University of California, Berkeley), Eugenia Kalnay (University of Maryland), Jeffrey Anderson, , Douglas Nychka, and David Schimel, (National Center for Atmospheric Research)

This workshop exposed students in the geosciences, ecology, and mathematics to multidisciplinary science through a focus on estimating the sources and sinks of carbon for the Earth system. One goal was to train the next generation of researchers to work within a multidisciplinary science team that combines geoscientists, ecologists, applied mathematicians, and statisticians. Participants obtained an overview of this problem but also some specific skills in tackling inverse problems and working with geophysical and biogeochemical models.

Prospective participants applied online at The Institute for Mathematics Applied to Geosciences (IMAGe). MSRI also provided funding for several students from MSRI Academic Sponsoring Institutions who were selected through the application process as participants.

113 SGW 3: Continuous Optimization and Applications Date: July 09, 2007 to July 20, 2007 Location: MSRI Organizer: Henry Wolkowicz (University of Waterloo)

This workshop introduced to graduate students the main ideas of Continuous Optimization and its Applications. In particular, we emphasized the major developments in the last ten years. This included the use of interior point methods in the solution of large scale linear and nonlinear programs. The workshop included a hands-on approach. Numerical tests were done using the NEOS Server for Optimization and the large group of NEOS Solvers.. Solution interpretation and sensitivity analysis were emphasized.

The workshop was divided into three series of lectures and hands-on labs:

• The first series included an introduction to the modern theory of convex programming, its extensions and applications. This included separation and support theorems, and Lagrange multiplier results. This series emphasized that: the great watershed in optimization is not between linearity and nonlinearity, but convexity and nonconvexity (Rockafellar, 1993.)

• The main series of lectures involved numerical algorithms for general nonlinear optimization. This included both modern interior point approaches as well as classical Lagrange multiplier methods such as sequential quadratic programming, SQP. We included applications to engineering and financial problems and emphasize the large scale case.

• The final series concentrated on specialized topics and applications. In particular, this included optimization over convex sets described as the intersections of the set of symmetric, positive semidefinite matrices with affine spaces, i.e. Semidefinite Programming. This area has attracted a lot of interest due to the number of important applications, to e.g. Discrete Optimization and more general Engineering Problems. We studied and used several current solvers that are in the public domain.

114 SGW 4: Deformation Theory and Moduli in Algebraic Geometry Date: July 23, 2007 to August 03, 2007 Location: MSRI Organizers: Max Lieblich (Princeton University), Martin Olsson (University of California, Berkeley), Brian Osserman (University of California, Berkeley), Ravi Vakil (Stanford University)

This workshop introduced to graduate students the main ideas of deformation theory and moduli spaces in algebraic geometry. We illuminated the general theory through extensive discussions of concrete examples and applications. The intended audience was the graduate student with a strong interest in algebraic geometry, having at least some familiarity with the language of schemes, and ideally comfortable with the content of Hartshorne's book Algebraic Geometry.

The workshop was anchored by three lecture series on different aspects of the subject, as well as student projects aimed at furthering an understanding of the lectures and pursuing more advanced topics. The lecture series were:

• Moduli spaces: functors, algebraic spaces, stacks, algebraic stacks (Lieblich). • Deformations (a): tangent and obstruction spaces (Olsson). • Deformations (b): representability and Schlessinger's criterion (Osserman).

Accompanying the lectures was a collection of tightly integrated exercises. A large portion of the workshop was devoted to these exercises, with the students expected to work on them in the afternoon and evening. In addition, students were expected to give short presentation on background material, which will be arranged in advance.

115 SGW 5: A Window into Zeta and Modular Physics Date: June 16, 2008 to June 27, 2008 Location: MSRI Organizers: Floyd Williams (University of Massachusetts) and Klaus Kirsten (Baylor University)

Speakers: Klaus Kirsten (co-organizer, Baylor University), Geoffrey Mason (UCSC), (UCSD), Michael Tuite (National University of Ireland), and Floyd Williams (organizer,University of Massachusetts). In recent years, a noteworthy and very fruitful interlacing of number theory and physics has emerged. As indicated in the September 2007 issue of the AMS Notices, for example, a new journal "Communications in Number Theory and Physics" has just been launched to follow significant interactions and dynamics between these two fields. Several books are now available, in addition to an array of conference and workshop activity, that accent this fortunate merger of "pure" mathematics and physical theory-with applications that range from field theory (conformal and topological), extended objects (strings and branes) cosmology and black hole physics, to Bose-Einstein condensation and the theory of relativistic gases.

The workshop was designed to provide students a bridge, or a window, into this vast, interesting, rapidly-developing, interactive arena. Some special attention was given to zeta and modular aspects of the interactions. The students were provided with some lecture notes and hand-outs, and with very ample opportunities to engage in discussion/question sessions with the lecturers, apart from the two hours per day lectures. In some cases, web material was available prior to the workshop-all in a concerted effort to afford maximal learning situations. Prof. Terras, for example, already had some introductory notes on line to help students get a head start towards her lectures on : http://www.math.ucsd.edu/~aterras/newchaos.pdf; also compare http://www.math.ucsd.edu/~aterras/newbook.pdf

The students were required to have a good working knowledge of complex variables, including familiarity with infinite products and the gamma function. It was suggested they should read some introductory material on the Riemannn zeta function (say, from a text on complex variables), and it would be helpful for them to do some minimal reading on the Dedekind eta functions. The material in Tom Apostol's Springer Graduate Text on "Modular Functions and Dirichlet Series in Number Theory" (chapter 3), for example, was more than needed for a basic understanding of the Dedekind eta function. Alternately, one could obtain the same information on zeta and eta (without proofs) by simply going to Google.

116 4.2 Program Associates

While most graduate students who visit MSRI have been invited to take part in one of our Summer Graduate Workshops, a smaller number of students are invited each year as “Program Associates” in our semester and year-long scientific programs. Program Associates benefit greatly from the chance to interact with a wide variety of mathematicians, gaining intense exposure to current ideas and trends in their area of specialization. While MSRI does not have the financial resources to fund the program Associates they they are closely supervised and essentially benefit from all members’ privileges. They are provided with access card to the building, allowing them to use the premises at any time and days of the week. They are given a bus pass, as well as library and sports facilities access pass. Thirty two graduate students spent a semester at MSRI during the academic year 2007-08, of those 26% were female. See the table below for a detailed description of the demographic data.

During the Fall 2007, in an effort to help young mathematicians (post-docs and graduate students) become familiar with areas of research in the field of geometric group theory, several series of introductory minicourses (6 lectures each) were given. Students from the 2 programs (Geometric Group Theory, and Teichmuller Theory Kleinian Groups) were attending those courses. Each minicourse ran for half of the semester. Students and post-docs were assigned as note-takers and the notes were made publicly available. The minicourses were,

• Mark Feighn - limit groups. • Zlil Sela - algebraic geometry over groups • Kevin Whyte - quasi-isometric rigidity • Peter Kropholler - Cohomology of groups • Gilbert Levitt – Out(Fn) • Mark Sapir – Asymptotic cones • Lee Mosher - Mapping class groups

In addition, there was a weekly graduate student seminar. It was run and organized by two graduate students, one from each program: Aditi Kar (GGT) and Will Cavendish (TTKG). The organizers routinely polled their attendees and then approached various faculty members to give introductory talks on topics of interest. One of the graduate students, William Cavendish, solved a problem concerning the quasi-isometry type of path metrics on the Mumford-Deligne compactification lifted to the Teichmuller space: they are all quasi-isometric to the pants complex.

The Spring 2008 programs organizers also closely supervised their graduate students. There were 14 graduate students in residence at MSRI for the bulk of the Spring programs. Arun Ram led a meeting of the graduate students each nonworkshop Friday 9:30-11:00. During these meetings they discussed mathematics, community, culture, teaching, job searches and many other topics. Primarily the discussions seemed to focus on explaining and discussing mathematics terms that the students had heard “in the air" but did not understand. The feeling that both the professionals and the students were doing the same kinds of work nurtured maturity and stimulated the students to discover, and be surprised by, their own mathematical potential. For many of the students the semester was particularly valuable as a “job

117 development" workshop. “We wanted a sense of what a job in mathematics is ...This was an intense dose." This “career workshop" was an unplanned, supplementary, outcome which came from the community and the natural vertical integration of all of the participants at MSRI. The students also felt that the separation between faculty and graduate students is more blurred at MSRI than at their home department.

Lastly, students reported positive progress and new results in their own research directly resulting from their interaction with senior members who were not their advisor. In general, they felt that the student/faculty ratio should not be changed and that the resources for research, particularly the non-circulating library, were very beneficial to their work.

118 4.3 Summer Graduate Workshops List (See Attached file for Full Detail List) (O:\0708AnnualReport\NSF Report 07-08\h. Graduate Student Program Summary\h. Summer Graduate Workshops List Family Name First Name Home Institute Name Position Activitytitle Ashley Caleb Graduate Student AWZMP Bakhova Maiia Louisiana State University Graduate Student AWZMP Banerjee Abhishek Johns Hopkins University Graduate Student AWZMP University of Massachusetts, Beheshti Shabnam Amherst Graduate Student AWZMP Bi Shuchau University of California Graduate Student AWZMP Boettner Stefan Tulane University Graduate Student AWZMP Cohen Sean North Carolina State University Graduate Student AWZMP Conway Alex Princeton University Graduate Student AWZMP Crompton Catherine Emory University Graduate Student AWZMP D'Ambroise Jennie Graduate Student AWZMP Farrington Eleanor Graduate Student AWZMP Franze Craig Central Michigan University Graduate Student AWZMP Gharahbeigi Sara Washington University Graduate Student AWZMP Hurley Donny Graduate Student AWZMP Jensen Erik University of North Carolina Graduate Student AWZMP Kharel Savan Indiana University Graduate Student AWZMP Kim Myoungil Boston University Graduate Student AWZMP Kirsten Klaus Baylor University Graduate Student AWZMP Kleinman Aaron University of California Graduate Student AWZMP Kohl Karen Tulane University Graduate Student AWZMP Krauel Kayden University of California Graduate Student AWZMP Malikiosis Romanos University of California Graduate Student AWZMP Malmskog Elizabeth Colorado State University Graduate Student AWZMP Marion Samantha University of Alberta Graduate Student AWZMP Marks Chris University of California Graduate Student AWZMP Mason Geoff University of California Professor AWZMP Nelson Paul California Institute of Technology Graduate Student AWZMP Nitz Ted Graduate Student AWZMP Pejic Michael Graduate Student AWZMP Powell Kevin Graduate Student AWZMP Quddus Safdar Washington University Graduate Student AWZMP Roy Michael University of Colorado Graduate Student AWZMP Shankar Arul Princeton University Graduate Student AWZMP George Steele Alexander Boston University Graduate Student AWZMP Sun Jie University of Alberta Graduate Student AWZMP Terras Audrey University of California Professor AWZMP National University of Ireland, Tuite Michael Galway Senior Lecturer AWZMP Vinogradov Ilya Princeton University Graduate Student AWZMP California Institute of Walji Nahid Technology Graduate Student AWZMP Wechter Matthew University of Illinois Graduate Student AWZMP Whitcher Ursula University of Washington Graduate Student AWZMP Williams Floyd University of Massachusetts, Graduate Student AWZMP

119 Amherst Wittenborn Erika University of Colorado Graduate Student AWZMP Alipanahi Ramandi Babak University of Waterloo Graduate Student COA Allmaras Moritz Texas A & M University Graduate Student COA Aydin Burcu University of North Carolina Graduate Student COA Bihun Oksana University of Missouri-Columbia Graduate Student COA Calef matthew Vanderbilt University Graduate Student COA Chis Voicu McMaster University Graduate Student COA Cru David State University College, SUNY Graduate Student COA Dudek Andrzej Emory University Graduate Student COA Emmerling Thomas Boston University Graduate Student COA Ghobadi Kimia McMaster University Graduate Student COA Gomez Rita Portland State University Graduate Student COA Gu Shiyuan Louisiana State University Graduate Student COA Guevara Alvaro Louisiana State University Graduate Student COA Hristova Dessislava Boston University Graduate Student COA Indratno Sapto Kansas State University Graduate Student COA Iraniparast Maryam University of Waterloo Graduate Student COA Kazmi Syed University of Iowa Graduate Student COA Kim Edward University of California Graduate Student COA Klinke Olaf Tulane University Graduate Student COA Krislock Nathan University of Waterloo Graduate Student COA Lin Qiuying University of Washington Graduate Student COA Lin Shaowei University of California Graduate Student COA University of Southern Lin Wei California Graduate Student COA Mastin Matt University of Georgia Graduate Student COA Palta Hasan Emory University Graduate Student COA Pintilie Stephan University of Waterloo Graduate Student COA Poerschke Annika Emory University Graduate Student COA Powell John Portland State University Graduate Student COA University of Southern Seliger Philip California Graduate Student COA Shen Wei Arizona State University Graduate Student COA Spjut Richard University of California Graduate Student COA Sun Yannan Washington State University Graduate Student COA Wei Fengrong University of Iowa Graduate Student COA Wolkowicz Henry University of Waterloo Graduate Student COA Yang Limin Washington State University Graduate Student COA Yu Jie University of Georgia Graduate Student COA Zhu Jiaping McMaster University Graduate Student COA Zhu Minyue University of Hong Kong Graduate Student COA Zollinger Elizabeth Boston University Graduate Student COA Arap Maxim University of Georgia Graduate Student DTMAG Au Suanne University of Nebraska Graduate Student DTMAG Bakker Benjamin Princeton University Graduate Student DTMAG Berlekamp David University of California Graduate Student DTMAG Bhatt Bhargav Princeton University Graduate Student DTMAG

120 Bowne- Anderson Hugo University of New South Wales Graduate Student DTMAG Budreau Dan University of California Graduate Student DTMAG Chan Kenneth University of New South Wales Graduate Student DTMAG Chen Dawei Harvard University Graduate Student DTMAG Cheng Chuangxun Northwestern University Graduate Student DTMAG California Institute of Cheong Wan-Keng Technology Graduate Student DTMAG Coskun Emre Michigan State University Graduate Student DTMAG Dan-Cohen Ishai University of California Graduate Student DTMAG Deusler Bradley Rice University Graduate Student DTMAG Dundon Ariana University of Washington Graduate Student DTMAG EL Fassy Fihry Youssef Cornell University Graduate Student DTMAG Fang Bohan Northwestern University Graduate Student DTMAG Fedorchuk Maksym Harvard University Graduate Student DTMAG Geraschenko Anton University of California Graduate Student DTMAG Gillam William Columbia University Graduate Student DTMAG Gong Shengjun University of Hong Kong Graduate Student DTMAG Gupta Shuvra University of Pennsylvania Graduate Student DTMAG Hall Jack Stanford University Graduate Student DTMAG Ho Wei Princeton University Graduate Student DTMAG Hodge Andrew University of California Graduate Student DTMAG Kalayci Serhan University of Alberta Graduate Student DTMAG Kim Wansu University of Michigan Graduate Student DTMAG Kissounko Veniamine University of Toronto Graduate Student DTMAG Kitchen Sarah University of Utah Graduate Student DTMAG Konstantinovskiy Lev University of Georgia Graduate Student DTMAG Kooistra Remkes University of Alberta Graduate Student DTMAG Lee Brandyn University of North Carolina Graduate Student DTMAG Lee Hwayoung University of California Graduate Student DTMAG Li Shuijing Rice University Graduate Student DTMAG Li Si Harvard University Graduate Student DTMAG Lin Jan-Li Indiana University Graduate Student DTMAG Liu Yu-Han Ohio State University Graduate Student DTMAG Mazur Justin Indiana State University Graduate Student DTMAG Moore Michael Brigham Young University Graduate Student DTMAG Muller Greg Cornell University Graduate Student DTMAG Obus Andrew University of Pennsylvania Graduate Student DTMAG Patakfalvi Zsolt University of Washington Graduate Student DTMAG Pradhan Neeraj University of California Graduate Student DTMAG Sanborn Barbara Arizona State University Graduate Student DTMAG Schnell Christian Ohio State University Graduate Student DTMAG University of Illinois at Urbana- Sheshmani Artan Champaign Graduate Student DTMAG Sibilla Nicolo Northwestern University Graduate Student DTMAG Siegel Charles Rutgers University Graduate Student DTMAG Smyth David Harvard University Graduate Student DTMAG University of Minnesota Twin Taipale Kaisa Cities Graduate Student DTMAG Thompson Rob Portland State University Graduate Student DTMAG

121 Turkelli Seyfi University of Wisconsin Graduate Student DTMAG Tzeng Yu-Jong Stanford University Graduate Student DTMAG University of Massachusetts, Visiting Assistant Urzua Giancarlo Amherst Professor DTMAG Van der Wyck Fred Harvard University Graduate Student DTMAG Wang Jie Ohio State University Graduate Student DTMAG Wise Jonathan Brown University Graduate Student DTMAG Wood Melanie Princeton University Graduate Student DTMAG Xu Da University of Iowa Graduate Student DTMAG Xu Fei Rice University Graduate Student DTMAG Yampolskiy Yevgen University of Missouri Graduate Student DTMAG Yu Na University of British Columbia Graduate Student DTMAG Zhang Ziyu Stanford University Graduate Student DTMAG Armentano Diego Centro de Matematica Graduate Student IAS/PCMI Carrasco Matias Centro de Matematica Graduate Student IAS/PCMI Handy Jon University of California Graduate Student IAS/PCMI Kirkpatrick Kay University of California Graduate Student IAS/PCMI Kumar Rohini University of Wisconsin Graduate Student IAS/PCMI Lian Zeng Brigham Young University Graduate Student IAS/PCMI Matic Ivan University of California Graduate Student IAS/PCMI Mester Peter Indiana University Graduate Student IAS/PCMI Mkrtchyan Sevak Graduate Student IAS/PCMI Prescott Timothy University of California Graduate Student IAS/PCMI Somersille Stephanie University of California Graduate Student IAS/PCMI Tingley Peter University of California Graduate Student IAS/PCMI Wang Zhiren Princeton University Graduate Student IAS/PCMI Zemlyanova Anna Louisiana State University Graduate Student IAS/PCMI

122 4.4 Summer Graduate Workshops Summary

# of No. of Citizens Decline/ Decline/ SGW & Per %* No # of % # of # of %* No Name of Activity Participants Res Reply Female Decline Minorities Reply

A Window into Zeta and 43 6 55% 32 13 35% 6 - 0% 38 Modular Physics

Continuous Optimization 39 - 0% 38 12 33% 3 - 0% 37 and Applications

Deformation Theory and 63 7 39% 45 14 23% 3 - 0% 55 Moduli in Algebraic Geometry

IAS/PCMI summer 14 3 60% 9 4 31% 1 1 25% 10 conference: Statistical Mechanics Total Distinct SGW Participants 159 16 46% 124 43 29% 13 1 5% 140

*Percentage for Citizens & Minorities are computed out of participants that provided info on their citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

123 4.5 Summer Graduate Workshops Demographic Data

Demographic Statistics % Over % Respo Over # n-dents Total Total SGW Participants 159 Male 103 70.55% 64.8% Female 43 29.45% 27.0% Decline to State Gender 13 8.2%

% Over % Respo Over Ethnicities # n-dents Total Native American 0 0.00% 0.0% Asian 3 15.79% 1.9% Black 1 5.26% 0.6% Hispanic 0 0.00% 0.0% Pacific 0 0.00% 0.0% White 15 78.95% 9.4% Decline to State Ethnicities 140 88.1%

Minorities 1 5.26% 0.6%

US Citizen 15 42.9% 9.4% Decline to State Citizenship 124 77.99% Per Resident 1 US Citizen & Per Resident 16 45.7% 10%

Home Inst. in US 132

Year of Degree # % 2009 154 96.86% 2005-2008 2 1.26% 2000-2004 0 0.00% 1995-1999 0 0.00% 1990-1994 0 0.00% 1985-1989 0 0.00% 1980-1984 1 0.63% Yr <1980 2 1.26% Decline to state 0 0.00% 100.00 Total 159 %

124

Home Institution Classified by Countries Home Inst By Region State # % # % South AL - 0.0% 26 19.7% AR - DE - 0.0% Home Institution Classified by Countries FL - 0.0% Midw es t Northeast South West GA 8 6.1% KY - 0.0% LA 7 5.3% Midw est MD 1 0.8% West 20.45% MS - 0.0%37.12% NC 4 3.0% OK - 0.0% SC - 0.0% Northeast 22.73% TN 1 0.8% South TX 5 3.8% 19.70% VA - 0.0% WV - 0.0% West AK - 0.0% 49 37.1% AZ 2 1.5% CA 32 24.2% CO 3 2.3% HI - 0.0% ID - 0.0% MT - 0.0% NV - 0.0% NM - 0.0% OR 3 2.3% UT 3 2.3% WA 6 4.5% WY - 0.0% Midwest IL 5 3.8% 27 20.5% IN 4 3.0% IA 3 2.3% KS 1 0.8% MI 3 2.3% MN 1 0.8% MO 4 3.0% ND - 0.0% NE 1 0.8% OH 3 2.3% SD - 0.0% WI 2 1.5% Northeast CT - 0.0% 30 22.7% ME - 0.0% MA 14 10.6% NH - 0.0% NJ 9 6.8% NY 4 3.0% PA 2 1.5% RI 1 0.8% VT - 0.0% Total 132 132

125

Home Institution Countries Classified by Region Home Institution Countries Classified by Region

Home Inst By Region Country # % # % Australia AU 2 1.3%2 1.3% Australia 2 Eastern Asia CN 2 1.3%2 1.3% North America CA 14 9.2%146 95.4% US 132 86.3% South America UY 2 1.3%2 1.3% Western Europe IE 1 0.7%1 0.7% Eastern Asia 2 Total 153 153

Length of Workshop # ≤ 3 days 0 4 to 7 days 0 1 - 2 weeks 145 North America 146 2 - 3 weeks 14 > 3 weeks 0 Total 159

Home Institution Countries Classified by Region South America 2

Western Europe Australia 0.65% 1.31% South Eastern Asia Am erica 1.31% 1.31% Western Europe 1

North Am erica 0 50 100 150 95.42%

126 4.6 Program Associates List

Family Inst Inst Name First Name Home Institute Name Home Inst City State Country Algom Kfir Yael University of Utah Salt Lake City UT US Blomgren Martin Royal Institute of Technology (KTH) Stockholm SE Calderin Ivo Florida State University Tallahassee FL US Cavendish William Brown University Providence RI US Coskun Olcay Bilkent University Ankara TR Daugherty Zajj University of Wisconsin Whitewater WI US Davis Matt University of Wisconsin Madison WI US Laboratoire de Mathematiques de Dudas Olivier Besançon CNRS (UMR 6623) UFR - ST 25000 BESANÇON FR Geline Michael University of Chicago Chicago IL US Gokturk Ali Brown University Providence RI US Graber John University of Iowa Iowa City IA US Greenberg Michael Brown University Providence RI US Hansen Mike Harvey Mudd College Claremont CA US Hengesbach Conrad Duke University Durham NC US Rheinische Friedrich-Wilhelms-Universit\"at Hensel Sebastian Bonn Bronheim NRW DE Kar Aditi Ohio State University Lima OH US Magid Aaron University of Michigan Ann Arbor MI US Malone William University of Utah Salt Lake City UT US Mangahas Johanna University of Michigan Ann Arbor MI US Abukuse Mbirika (Aba) University of Iowa Iowa City IA US Min Honglin Rutgers University Newark NJ US Nipper Emanuel Universität Bonn Bonn DE Pfaff Catherine Rutgers University Piscataway NJ US Smith Abraham Duke University Durham NC US Soehl Jakob Universität Bonn Bonn DE Swenson Daniel University of Minnesota Twin Cities Minneapolis MN US Tao Jing University of Illinois Chicago IL US Thompson Josh University of Utah Salt Lake City UT US University of Illinois at Urbana- Tsai Chia-yen Champaign Champaign IL US Tsai Chung-Jun Harvard University Cambridge MA US Virk Rahbar University of Wisconsin Whitewater WI US Yip Martha University of Wisconsin Whitewater WI US

127 4.7 Program Associates Summary

# of Decline/ Decline/ Decline/ Program #of %* No # of % No # of %* No Workshop name Associates Citizens Repley Female Repley Minorities Repley

Connections for Women: 10 6 60% - 6 67% 1 1 17% 4 Geometric Group Theory

Connections for Women: Introduction to 4 3 75% - 2 50% - 1 33% 1 the Spring, 2008 programs Connections for Women: Teichmuller Theory 7 6 86% - 3 43% - 1 17% 1 and Kleinian Groups

Exterior Differential Systems and the 2 1 100% 1 - 0% - - 0% 2 Method of Equivalence

Homological Methods in 4 3 75% - - 0% - 1 33% 1 Representation Theory

Hot Topics: Contact structures, dynamics and 1 - 0% - - 0% - - 0% - the Seiberg-Witten equations in dimension 3

Introduction to Geometric Group 9 6 75% 1 5 56% - 1 17% 3 Theory

Introduction to Teichmuller 12 9 75% - 5 42% - 1 11% 3 Theory and Kleinian Groups

Introductory Workshop on Combinatorial 9 6 67% - 2 22% - 1 17% 3 Representation Theory

Introductory Workshop on the Representation Theory of Finite 6 4 67% - - 0% - 1 25% 2 Groups

Lie Theory 9 6 67% - 2 22% - 1 17% 3

MSRI's 25th Anniversary Celebration 2 2 100% - 1 50% - 1 50% -

Topics in Combinatorial 6 5 83% - 2 33% - 1 20% 1 Representation Theory

Topics in Geometric Group Theory 11 5 56% 2 5 50% 1 1 20% 6 Topics in Teichmuller Theory and 12 7 58% 4 33% 1 14% 5 Kleinian Groups Total 104 69 69% 4 37 36% 2 13 19% 35

Total Number of Distinct Program Associates 32 17 59% 3 8 26% 1 2 12% 15

*Percentage for Citizens & Minorities are computed out of participants that provided info on their citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

128 4.8 Program Associate Demographic Data Demographic Statistics % Over % Respon- Over # dents Total Distinct Program Participants 32 Male 23 74.19% 71.9% Female 8 25.81% 25.0% Decline to State Gender 1 3.1%

% Over %

Respon- Over Ethnicities # dents Total Native American 0 0.00% 0.0% Asian 4 23.53% 11.8% Black 1 5.88% 2.9% Hispanic 1 5.88% 2.9% Pacific 0 0.00% 0.0% White 13 76.47% 38.2% Decline to State Ethnicities 15 46.88% 44.1%

Minorities 2 11.8% 6.3%

US Citizen 17 58.6% 53.1% Decline to State Citizenship 3 9.4% 9.4% Per Resident 0 0.0% 0.0% US Citizen & Per 53.1% Resident 17 58.6% Home Inst. in US 26

Year of Degree # % 2009 28 100.0% 2005-2008 0 0.0% 2000-2004 0 0.0% 1995-1999 0 0.0% 1990-1994 0 0.0% 1985-1989 0 0.0% 1980-1984 0 0.0% Yr <1980 0 0.0% Decline to state 0 0.0% Total 28 100.00% **Statistic Calculation based on participant that replied to each category. Ethnicities' percentage might be over stated since some participants selected to be included in more than one ethnicity groups.

129 Home Institution Classified by States

Home Inst By Region State # % # % South AL - 0.0% 3 11.5% AR - 0.0% DE - 0.0% FL 1 3.8% GA - 0.0% KY - 0.0% LA - 0.0% MD - 0.0% MS - 0.0% NC 2 7.7% OK - 0.0% SC - 0.0% TN - 0.0% TX - 0.0% VA - 0.0% WV - 0.0% West AK - 0.0% 4 15.4% AZ - 0.0% CA 1 3.8% CO - 0.0% HI - 0.0% ID - 0.0% MT - 0.0% NV - 0.0% NM - 0.0% OR - 0.0% UT 3 11.5% WA - 0.0% WY - 0.0% Midwest IL 3 11.5% 13 50.0% IN - 0.0% IA 2 7.7% KS - 0.0% MI 2 7.7% MN 1 3.8% MO - 0.0% ND - 0.0% NE - 0.0% OH 1 3.8% SD - 0.0% WI 4 15.4% Northeast CT - 0.0% 6 23.1% ME - 0.0% MA 1 3.8% NH - 0.0% NJ 2 7.7% NY - 0.0% PA - 0.0% RI 3 11.5% VT - 0.0% Home Inst. in US 26 26

130

Home Institution Countries Classified by Region

Home Inst By Region Country # % # % North America US 26 81.3% 26 81.3% Central Europe DE 3 9.4% 3 9.4% Northern Europe SE 1 3.1% 1 3.1% Southeastern Europe TR 1 3.1% 1 3.1% Western Europe FR 1 3.1% 1 3.1% Total 32 32

131 4.9 Graduate Student List

Aside from the graduate students who participate in the Summer Graduate Workshops, and the Program Associates who spend longer period of time at the institute, several other graduate students attended the scientific workshops. MSRI carefully monitors the list of workshops applicants and sees that a high ratio of graduate students are funded. Attached is a list of all the graduate students (635 with repetitions) that participated in various workshops during the 2007- 08 academic year.

132

4.10 Graduate Student Summary

No. # of Decline/ Decline/ of Grad. Citizens & No # of # of # of No Name of Activity Students Per Res % Reply Female % Decline Minorities % Reply A Window into Zeta and Modular Physics 40 4 50% 32 12 35% 6 0 0% 36 CMI/MSRI Workshop: Modular Forms and Arithmetic 24 4 18% 2 6 32% 5 1 14% 17

Computation and Complex 100 Systems 6 3 % 3 0 0% 1 0 0% 2 Connections for Women: Geometric Group Theory 24 11 61% 6 15 88% 7 2 15% 11 Connections for Women: Introduction to the Spring, 2008 programs 16 6 40% 1 14 93% 1 1 17% 10 Connections for Women: Teichmuller Theory and Kleinian Groups 19 10 67% 4 9 56% 3 3 27% 8 Continuous Optimization and Applications 39 0 0% 38 12 33% 3 0 0% 37 Critical Issues in Education Workshop: 100 Teaching and Learning Algebra 6 5 % 1 2 40% 1 0 0% 1 Deformation Theory and Moduli in Algebraic Geometry 62 7 41% 45 14 24% 3 0 0% 54 Exterior Differential Systems and the Method of Equivalence 13 5 56% 4 1 8% 1 0 0% 8 Homological Methods in Representation Theory 15 6 46% 2 0 0% 1 1 14% 8 Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in dimension 3 23 7 37% 4 0 0% 3 2 15% 10 IAS/PCMI summer conference: Statistical Mechanics 14 3 60% 9 4 31% 1 1 25% 10 Introduction to Geometric Group Theory 48 20 54% 11 13 33% 8 2 8% 22 Introduction to Teichmuller Theory and Kleinian Groups 37 18 62% 8 12 39% 6 3 14% 16 Introductory Workshop on Combinatorial Representation Theory 49 18 49% 12 12 29% 8 1 4% 26 Introductory Workshop on the Representation Theory of Finite Groups 29 13 50% 3 4 15% 3 2 12% 12

Lie Theory 30 11 48% 7 6 27% 8 2 12% 13 Mathematical Systems Biology of Cancer II 9 2 67% 6 3 60% 4 0 0% 3 Modern Mathematics: An Introduction to MSRI's 2008-09 Programs 17 8 50% 1 7 41% 0 12 92% 4 MSRI Summer Microprogram on Nonlinear Partial Differential Equations 13 6 50% 1 1 8% 0 1 11% 4

133

No. # of Decline/ Decline/ of Grad. Citizens & No # of # of # of No Name of Activity Students Per Res % Reply Female % Decline Minorities % Reply MSRI's 25th Anniversary Celebration 13 4 36% 2 1 10% 3 1 25% 9 MSRI-UP 2008 research topic: Experimental Mathematics 2 0 0% 2 0 0% 2 0 0% 2 Topics in Combinatorial Representation Theory 32 7 39% 14 6 30% 12 3 20% 17 Topics in Geometric Group Theory 28 14 67% 7 13 50% 2 1 6% 11 Topics in Teichmuller Theory and Kleinian Groups 26 13 62% 5 6 24% 1 3 20% 11

Total 634 205 51% 230 173 32% 93 42 15% 362

Total Number of Distinct Graduate Students 463 126 47% 196 108 27% 65 23 13% 289

*Percentage for Female, Citizens & Minorities are computed out of participants that provided info on their gender, citizenship status & Ethnicity. (# of Minorites / (# of Participants – # of Decine or No Reply))

134 5. Undergraduate Program

Research Topic: Experimental Mathematics Date: June 14, 2008 to July 27, 2008 Organizers: Ivelisse Rubio (University of Puerto Rico, Humacao), Duane Cooper (Morehouse College), Ricardo Cortez (Tulane University), Herbert Medina (Loyola Marymount University), and Suzanne Weekes (Worcester Polytechnic Insitute).

The MSRI-UP is a comprehensive program for undergraduates that aims at increasing the number of students from underrepresented groups in mathematics graduate programs. This years program included summer research opportunities, mentoring, workshops on the graduate school application process, and follow-up support.

MSRI-UP trained undergraduates in mathematical research through a six-week summer program at MSRI in Berkeley, CA. • Provided participating students opportunities to present their research at national conferences in the year following the summer program. • Introduced participating students to a network of mentors through national societies known for their mentoring activities and professional support for students. • Guided students in the process of applying to graduate programs and fellowships.

Prerequisites: Participating students had already taken the Calculus sequence and a course in Linear Algebra. A course in numerical analysis was considered helpful. For maximum benefit, it was recommended that the students take a course in physics, chemistry or biology.

Overview of the summer program:

The MSRI-UP summer program was designed for undergraduate students who had completed two years of university-level mathematics courses and wanted to conduct research in the mathematical sciences.

During the summer, each of the 18 student participants:

• participated in the mathematics research program under the direction of Dr. Moll • completed a research project done in collaboration with other MSRI-UP students • gave a presentation and write a technical report on his/her research project • attended a series of colloquium talks given by leading researches in their field

135 • attended workshops aimed at developing skills and techniques needed for research careers in the mathematical sciences; and • learned techniques that will maximize a student's likelihood of admissions to graduate programs as well as the likelihood of winning fellowships

After the summer, each student:

• had an opportunity to attend a national mathematics or science conference where students will present their research • became part of a network of mentors that will provide continuous advice in the long term as the student makes progress in his/her studies

Graduating seniors Of the twelve student participants in 2007, six entered their senior year following the summer research program and graduated in May of 2008. Five of the six graduates have enrolled in graduate programs as specified in the table. One graduate has decided to take time to do work for his Church before considering graduate school. All of the students have given poster or oral presentations on their work at MSRI since the 2007 summer program. Those students who have not yet graduated also remain in contact with Prof. Cortez and all have participated in some other research program or internship since MSRI-UP. The 2008 summer program just ended on July 25, 2008. There were 17 student participants and all of them are scheduled to attend the SACNAS conference this coming October to present their work.

Graduated Student Undergraduate Institution Graduate Institution

Carmen Smith Spelman College University of Iowa

Louis de la Torre University of CA, Davis Northwestern University Sean Ewing-Owens Morehouse College University of CA, Riverside Sofia Garcia De Paul University University of Iowa Talea Mayo Grambling State University University of TX, Austin

136

Short Biographies of the 2008 MSRI-UP

Erin Beyerstedt, Graduate Student, Tulane University, 2008 MSRI-UP Research Assistant

Erin Beyerstedt did her undergraduate work at Carthage College in Wisconsin, majoring in mathematics and biology. After graduating, she continued her studies at Western Washington University and earned her master's degree in mathematics. While there, she worked as a teaching assistant for several courses and taught two classes independently. She is currently in her third year of graduate school at Tulane University. Her future plans include completing her doctorate degree and pursuing a teaching position at the university level.

Luis A. Medina, Postdoctoral Fellow, Rutgers University, 2008 MSRI-UP Research Associate

Luis A. Medina was born in Humacao, Puerto Rico and raised in the town of Yabucoa. In 2003, he received his B.A. in Computational Mathematics from the University of Puerto Rico, Humacao. Luis decided to continue studying Mathematics and in 2003 he joined the Math Ph.D. program at Tulane University. He received his Ph.D. in Mathematics in May 2008. Currently, Luis has a three year postdoctoral position at Rutgers University. His research interests are Experimental Math, Number Theory, and Special Functions.

Victor H. Moll, Professor, Tulane University, 2008 MSRI-UP Research Advisor

Victor Hugo Moll Becker was born in Santiago, Chile. His interest in mathematics was observed by his elementary school teachers that encourage him to pursue a career in engineering. After realizing that he was not able to put wires together, he became a mathematic major at Universidad Santa Maria in Valparaiso. He arrived in the U.S. in 1980 to study mathematics at the Courant Institute, New York University. His advisor was Henry P. McKean. At the Courant Institute he met his (future) wife Lisa J. Fauci and, after a brief postdoctoral work at Temple University, they joined the Department of Mathematics at Tulane University in 1986. They have two sons: Alexander and Stefan.

Victor Moll's mathematical interests are in the area of Experimental Mathematics. He is mainly interested in issues connected to Special Functions in its many facets. During the summers of 2000 and 2002 he was in charge of an undergraduate research group that was part of SIMU. He enjoys collaborating and exchanging mathematical ideas with colleagues, graduate and undergraduate students.

Candice Price, Graduate Student, University of Iowa, 2008 MSRI-UP Research Assistant

Candice Price was born in Long Beach, CA but moved to Sacramento, CA at the age of 7. She finished her undergraduate education at California State University Chico where she was an active participate in LS-AMP. In 2004, Candice receive the LS-AMP Bridge to the Doctorate Fellowship and continued her education at San Francisco State University, where she received an M.A in mathematics under the guidance of Prof. Mariel Vazquez. She continued on to a PhD program in pure mathematics at the University of Iowa. Candice is a student of Prof. Isabel Darcy studying Topology with an emphasis in Knot Theory. She is hoping to finish her degree in

137 2010 and continue on to a postdoctoral position before, hopefully, receiving a faculty position as a mathematics professor.

Ivelisse Rubio, Professor, University of Puerto Rico – Rio Piedras, 2008 MSRI-UP Summer Director

Ivelisse M. Rubio was born and raised in Puerto Rico. She received her B.S. and M.S. in Mathematics from the University of Puerto Rico-Río Piedras and her Ph.D. in Applied Mathematics from Cornell University. In 1998 she co-founded the NSF-REU Summer Institute in Mathematics for Undergraduates (SIMU) at the UPR-Humacao. Ive is currently a Professor in the Computer Science Department at the UPR-Rio Piedras. Her research interests are finite fields and applications to error-correcting codes.

138 2008 MSRI-UP Students

Berrizbeitia, Ana Univ of Texas, Austin Cayco, Natasha (Alex) California Institute of Technology Enrigue, Cindy Univ of California, Berkeley Feliciano-Semidei, Ricela Univ of Puerto Rico, Mayaguez Garcia, Richard Univ of Puerto Rico, Rio Piedras Kallus, Nathan Univ of California, Berkeley Koffi, Gerard Univ of Massachusetts, Boston Moll, Alexander Columbia University Nguyen, Aileen Cal State Pomona Noble, Laine Tulane University Ojeda, Ivan Univ of Puerto Rico, Rio Piedras Ortiz, Marcos State Univ of New York, Buffalo Rosenberg, Jason Tulane University Sigilie, Jessica Washington University Torres-Castro, Loraine Univ of Puerto Rico, Rio Piedras Wilson, Bobby Morehouse College Wingfield, Kevin Morehouse College

139 5.1 Undergraduate Program Participant Summary

First Given Family Name Name Berrizbeitia Ana Cayco Gajjic Natasha Enrrigue Cindy Feliciano- semidei Ricela Garcia Richard Kallus Nathan Koffi Gerard Nguyen Aileen Noble Laine Ojeda Ivan Ortiz Marcos Rosenberg Jason Stigile Jessica Torres Castro Loraine Wilson Bobby Wingfield kevin

Moll Alexander

140 6. Financial Support List

141 7 Institute Directors Meeting Report (MIDS)

Meeting of the NSF Math Institute Directors May 2-3, 2008 Minutes

In attendance: Doug Arnold IMA [email protected] Jim Berger SAMSI [email protected] IAS [email protected] Robert Bryant MSRI [email protected] Russel Caflisch IPAM [email protected] Brian Conrey AIM [email protected] Avner Friedman MBI [email protected] Marty Golubitsky MBI [email protected] Mark Green IPAM [email protected] David Levermore Facilitator [email protected] Tony Nance MBI [email protected] Fadil Santosa IMA [email protected]

Tony Chan NSF [email protected] Dean Evasius NSF [email protected] Joanna Kania-Bartoszynska NSF [email protected] Hans Kaper NSF [email protected] Deborah Lockhart NSF [email protected] Peter March NSF [email protected] Chris Stark NSF [email protected]

May 2, 2008 (Institute Directors only)

1. Discussions i. We briefly reviewed the Institutes' role in publicizing CDI to the mathematical sciences community. Once the competition is complete we plan to evaluate the impact of these activities. We will discuss with NSF the usefulness of continuing such activities. ii. We reviewed the page we received from DMS on a proposed report "Mathematical Sciences 2025". There was general support for the idea, and interest in learning more and discussing it with Peter and other NSF representatives. We feel that the Institutes can help this effort through community forums to gather input and promote discussion. iii. We discussed the rising cost of air transportation and our reimbursement policies for it.

142 iv. Herb Clemens reported on the US National Committee for Mathematics and the IMU Developing Countries Strategy Group activities, and led a discussion on institute support for mathematicians in the developing world. Herb offered to help the Institute Directors identify mathematicians and mathematical activities in developing countries.

2. Action Items from May 2nd

i. Technical Committee

• Technical committee should draft a page with links to subscription pages for all the Institutes newsletters, etc. [Mark Green] • Each year starting with the annual meeting, the chair of the technical committee will be the representative from the institution hosting the next meeting. The chair will report to the MIDs at the next meeting. [Jim Berger] ii. AWM mentoring network

• The Institutes will continue their financial support of the AWM Mentor Network at roughly $550/year each. iii. Diversity events/committee

• At each annual meeting, the representative to the Math Institutes Diversity Committee from the institution hosting the meeting will report on the activities of the Diversity Committee over the preceding year. [Jim Berger] • Changes to the Diversity Committee web page: a) The link on the sidebar should be identified with text. b) There should be a list of the members (people not institutes) of the Committee. c) The list of member institutes should be identified as such. d) There should be links to a diversity page at each member institute, either linked from the list of member institutes or elsewhere on the page. A fortiori every member institute must have such a page. e) The link to the 2005 Blackwell-Tapia press release should be replaced by a Blackwell-Tapia page, listing all previous winners, preferably with photos and citations, and links to all past Blackwell-Tapia conferences. f) See about adding IAS to the Diversity Committee web page. [Jim Berger] • Members could help each other with NSF reporting appendices, as needed

iv. Math Institute Website

• Update links to MBI at mathinstitutes.org. [Doug Arnold] • Add highlight (formerly “nugget”) submission schedule to highlight instructions. Send mathinstitutes.org login instruction to all institutes. [Doug Arnold] • Post MID minutes somewhere in a protected part of mathinstitutes.org [Doug Arnold]

143 vii. Data Collection for NSF Reporting

• NSF/DMS will send each institute a letter amending their grants to formalize the reporting procedure agreed to at the May 2006 MID meeting.. Brief report due May 1 (to be submitted via FastLane as annual report), full report due in the fall after fiscal year closing." [NSF]

2. Brainstorming

• Investigate possibilities for a joint searchable video archive for the institutes. [Brian Conrey]

May 3 (NSF included)

** = Action Item from May 3rd

1. Report to NSF

Institute Directors reported on their May 2 meeting and presented the minutes to NSF.

2. Presentation by NSF

NSF Budget Breakdown and Overview Tony Chan presented a budget breakdown starting at the Directorate level then moving into MPS. The American Competitiveness Initiative (ACI) is driving the budget. MPS is an ACI directorate, and DMS is an ACI division, but Tony had to make a case for the latter status.

Mathematical Sciences 2025 Based on a two-page document prepared by Peter March, there was a discussion on where mathematics should be positioned by 2025. Tony Chan mentioned that NRC may be asked to produce a report on this topic.

“What More Can We Get From The MID Group?” The MID group sees and thinks about emerging developments in the Mathematical Sciences as well as connections with other fields. Tony Chan mentioned that he would like to take advantage of the directors’ knowledge by having them take on a trend-spotting role, and to also include personal opinions.

**MPS/DMS will initiate the request for this voluntary and confidential report.

2007 Committee of Visitors (COV) report revisited Peter March mentioned that on both 2004 and 2007 the COV wanted better DMS clarity on the two issues of:

• Coverage – Are there areas of math not served? • Overlap – Are the institutes stepping on each others’ toes? Is there coordination?

144 Addressing this question must also include an answer to the question “How does DMS know?”

**Peter suggested a workshop of stakeholders addressing the above and other questions DMS may have, including “What data do we have (collectively)?

Location and Date of MID 2009 April 17-18, 2009 was approved, hosted by SAMSI.

145 8. Interim Reports and Updates

Mathematical Sciences Research Insitute May 2008

§0. Introduction As agreed with program representatives at the Mathematics Institute Directors’ Meeting in Spring 2007, we are now presenting an interim report on our scientific activities and financial status, pending the detailed report that will be generated in Fall 2008. This supplements the data of the previously filed Annual Reports. The first part of the report consists of data on the scientific programs. A brief summary of the financial data is at the end. More details, including participant lists and programs, can be found on MSRI’s web pages. See http://www.msri.org.

§1. Scientific Activities

This is a brief account of the programs that were held (or are projected to be held) at least in part during the period July 1, 2007 through June 30, 2008.

§1.1. Major Programs

§1.1.1. Geometric Group Theory. August 20, 2007 to December 14, 2007 Organized By: Mladen Bestvina, Jon McCammond, Michah Sageev, Karen Vogtmann In the 1980s, attention to the geometric structures that cell complexes can carry shed light on earlier combinatorial and topological investigations into group theory, stimulating other provocative and innovative ideas over the past 20 years. As a consequence, geometric group theory has developed many different facets, including geometry, topology, analysis, logic.

§1.1.2. Teichmuller Theory and Kleinian Groups. August 20, 2007 to December 14, 2007 Organized By: Jeffrey Brock, Richard Canary, Howard Masur, Maryam Mirzakhani, Alan Reid These fields have each seen recent dramatic changes: new techniques developed, major conjectures solved, and new directions and connections forged. Yet progress has been made in parallel without the level of communication across these two fields that is warranted. This program will address the need to strengthen connections between these two fields, and reassess new directions for each.

§1.1.3. Combinatorial Representation Theory. January 14, 2008 to May 23, 2008 Organized By: P. Diaconis, A. Kleshchev, B. Leclerc, P. Littelmann, A. Ram, A. Schilling, R. Stanley Recent catalysts stimulating growth of this field in the last few decades have been the discovery of ‘crystals’ and the development of the combinatorics of affine Lie groups. Today the subject intersects several fields: combinatorics, representation theory, analysis, algebraic geometry, Lie theory, and mathematical physics. The goal of this program is to bring experts in these areas together in one interdisciplinary setting.

§1.1.4. Representation Theory of Finite Groups and Related Topics. January 14, 2008 to May 23, 2008 Organized By: J. L. Alperin, M. Brou´e, J. F. Carlson, A. Kleshchev, J. Rickard, B. Srinivasan Current research centers on many open questions, i.e., representations over the integers or rings of positive characteristic, correspondence of characters and derived equivalences of blocks. Recently we have seen active interactions in group cohomology involving many areas of topology and algebra. The focus of this program will be on these areas with the goal of fostering emerging interdisciplinary connections among them.

146 §1.2. Workshops

The workshops listed below are of two essentially different kinds: The ‘programmatic workshops’, i.e., the workshops that are planned in conjunction with one of the concurrently running programs (as listed above). These are usually organized by a group that has substantial overlap with the organizing committee(s) of its parent program(s). The themes of such workshops usually draw on those of the parent program(s), so no extra description will be given below. The other workshops are ‘non-programmatic’ and hence require a bit more description, which is supplied as needed.

§1.2.1. MSRI Summer Microprogram on Nonlinear Partial Differential Equations July 23, 2007 to August 10, 2007 Organized By: L. C. Evans (UC Berkeley, Chair), C. Gutierrez (Temple), C. Sogge (Johns Hopkins), D. Tataru (UC Berkeley) This three week program emphasized the overlapping research areas of nonlinear dispersive equations (NDE) and nonlinear elliptic equations (NEE), and was intended as an extension of the MSRI programs in these fields from Fall 2005, with a focus on subsequent research developments. Weeks 1 and 3 comprised informal but focused workshops on NEE and NDE, respectively. Topics of emphasis included regularity estimates for elliptic PDE, with applications to mass transport problems and to image processing. The NDE workshop focussed upon eigenfunction estimates, resonances, and spectral resolutions for elliptic operators on manifolds. Week 2 was devoted to several expository minicourses of general interest, especially for younger mathematicians. These were organized into four lecture series:

• Craig Evans, ‘Survey of weak convergence methods for nonlinear PDE’ • Cristian Gutierrez, ‘The Monge-Ampere equation’ • Chris Sogge, ‘Dispersion and existence theorems for nonlinear wave equations’ • Daniel Tataru, ‘Nonlinear wave equations’

§1.2.2. Connections for Women: Teichmuller Theory and Kleinian Groups August 16, 2007 to August 17, 2007 Organized By: Moon Duchin, Caroline Series Parent Program: Teichmuller Theory and Kleinian Groups

§1.2.3. Introduction to Teichmuller Theory and Kleinian Groups August 20, 2007 to August 24, 2007 Organized By: Jeff Brock, Richard Canary, Howard Masur, Alan Reid, and Maryam Mirzakhani Parent Program: Teichmuller Theory and Kleinian Groups

§1.2.4. Connections for Women: Geometric Group Theory August 23, 2007 to August 24, 2007 Organized By: Ruth Charney, Indira Chatterji, and Karen Vogtmann Parent Program: Geometric Group Theory

§1.2.5. Introduction to Geometric Group Theory August 27, 2007 to August 31, 2007 Organized By: Mladen Bestvina, Jon McCammond, Michah Sageev, Karen Vogtmann Parent Program: Geometric Group Theory

§1.2.6. Mathematical Systems Biology of Cancer II October 24, 2007 to October 26, 2007 Organized By: Joe Gray, Elizabeth Purdom, Terry Speed and Paul Spellman This workshop was designed to encourage and support the mathematical community’s involvement in the effort to study cancer using system approaches. Conference presenters included mathematicians and computer scientists presently involved in systems approaches to cancer and more general fields of

147 biology. These presenters covered general approaches to systems biology including analysis of genome scale data as well as statistical, continuous, and hybrid methods for pathway modeling. The workshop provided tutorials covering the use of tools and methods in systems biology as well as on the fundamental biological processes involved in cancer. In addition, the workshop provided travel support for students and postdocs from the mathematical sciences to foster interest in this field.

§1.2.7. Topics in Geometric Group Theory November 05, 2007 to November 09, 2007 Organized By: Noel Brady, Mike Davis, and Mark Feighn Parent Program(s): Geometric Group Theory

§1.2.8. Topics in Teichmuller Theory and Kleinian Groups November 12, 2007 to November 16, 2007 Organized By: Jeff Brock, Ken Bromberg, Richard Canary, Howard Masur, Alan Reid, Maryam Mirzakhani, and John Smillie Parent Program: Teichmuller Theory and Kleinian Groups

§1.2.9. Connections for Women: Introduction to the Spring 2008 programs January 16, 2008 to January 18, 2008 Organized By: Bhama Srinivasan and Monica Vazirani Parent Programs: Combinatorial Representation Theory, Representation Theory of Finite Groups and Related Topics.

§1.2.10. Introductory Workshop on Combinatorial Representation Theory January 22, 2008 to January 25, 2008 Organized By: Persi Diaconis, Arun Ram, Anne Schilling (Chair) The goal of the Introductory Workshop was to survey current and recent developments in the field. The talks focussed on tableaux, reflection groups, finite groups, geometry and mathematical physics in the realm of Combinatorial Representation Theory. Parent Program: Combinatorial Representation Theory

§1.2.11. MSRI’s 25th Anniversary Celebration January 26, 2008 to January 30, 2008 Organized By: Alejandro Adem, Robert Bryant, and Isadore Singer This was a high-level, general conference on a range of mathematical topics that had been important in the first 25 years of MSRI’s history. As befitting the broad mission of the Institute, these talks included not only mathematical exposition by some of the leaders who have been and are about to be involved with MSRI programs, but also an opening program of mathematics and music and some panels to reflect on the most important directions for future development.

§1.2.12. Introductory Workshop on the Representation Theory of Finite Groups February 04, 2008 to February 08, 2008 Organized By: Jonathan Alperin(chair), Robert Boltje, Markus Linckelmann Parent Program: Representation Theory of Finite Groups and Related Topics

§1.2.13. Lie Theory March 10, 2008 to March 14, 2008 Organized By: Alexander Kleshchev, Arun Ram, Richard Stanley (chair), Bhama Srinivasan Parent Programs: Combinatorial Representation Theory, Representation Theory of Finite Groups and Related Topics

§1.2.14. Topics in Combinatorial Representation Theory March 17, 2008 to March 21, 2008 Organized By: Sergey Fomin, Bernard Leclerc, Vic Reiner (Chair), Monica Vazirani Parent Program: Combinatorial Representation Theory

148 §1.2.15. Homological Methods in Representation Theory March 31, 2008 to April 04, 2008 Organized By: David Benson, Daniel Nakano (chair), Raphael Rouquier Parent Program: Representation Theory of Finite Groups and Related Topics

§1.2.16. Exterior Differential Systems and the Method of Equivalence May 05, 2008 to May 09, 2008 Organized By: Jeanne Clelland, William F. Shadwick (Chair) and George Wilkens Exterior Differential Systems and the Method of Equivalence surveys state-of-the-art applications of these techniques and celebrates the contributions of Robby Gardner to our current understanding of Cartans powerful machinery.

§1.2.17. Critical Issues in Education Workshop: Teaching and Learning Algebra May 14, 2008 to May 16, 2008 Organized By: Al Cuoco, chair, (Center for Mathematics Education), Deborah Ball, ex officio (University of Michigan), (University of Michigan), Herb Clemens (Ohio State University), James Fey (University of Maryland), Megan Franke (UCLA), Roger Howe (Yale University), Alan Schoenfeld (UC Berkeley), and Ed Silver (University of Michigan) For over two decades, the teaching and learning of algebra has been a focus of mathematics education at the precollege level. This workshop examined issues in algebra education at two critical points in the continuum from elementary school to undergraduate studies: at the transitions from arithmetic to algebra and from high school to university. In addition, the workshop involved participants in discussions about various ways to structure an algebra curriculum across the entire K-12 curriculum.

§1.2.18. Hot Topics: Contact structures, dynamics, and the Seiberg-Witten equations in dimension 3 June 09, 2008 to June 13, 2008 Organized By: Helmut Hofer, Michael Hutchings, Peter Kronheimer, Tom Mrowka and Cliff Taubes This workshop will concentrate on recently discovered relationships between Seiberg-Witten theory and contact geometry on 3 dimensional manifolds. One consequence of these relationships is a proof of the Weinstein conjecture in dimension 3. Another is an isomorphism between the Seiberg-Witten Floer (co)homology and embedded contact homology, the latter a form of Floer homology that was defined by Michael Hutchings. The overarching plan is to introduce the salient features of both the contact geometry side of the story and the Seiberg-Witten side and then discuss how they are related.

§1.2.19. CMI/MSRI Workshop: Modular Forms and Arithmetic June 28, 2008 to July 02, 2008 Organized By: Frank Calegari, Samit Dasgupta, David Ellwood, Bjorn Poonen, and Richard Taylor This conference, jointly funded by MSRI and the Clay Mathematics Institute, will bring together researchers on many aspects of the arithmetic applications of modular (and automorphic) forms. This is currently a very broad and very active subject. Our intention is to encourage interaction between those working in different sub-disciplines. To this end it is hoped to limit lectures to 4 hours a day, allowing plenty of time for informal interactions. On Tuesday, July 1, 2008 at 7pm there will be a dinner to honor Ken Ribet on his 60th birthday.

149 §1.3. Summer Graduate Workshops

§1.3.1. IAS/PCMI summer conference: Statistical Mechanics July 01, 2007 to July 21, 2007 Location: IAS/Park City Mathematics Institute in Park City, UT Organized By: Scott Sheffield, Thomas Spencer The Graduate Summer School bridged the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, the participating students have completed their first year, and in some cases, were already working on a thesis. The main activity of the Graduate Summer School was a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures did not duplicate standard courses available elsewhere. Each course consisted of lectures with problem sessions. Course assistants were available for each lecture series. The participants of the Graduate Summer School met three times each day for lectures, with one or two problem sessions scheduled each day as well.

§1.3.2. Summer Graduate Workshop on Data Assimilation for the Carbon Cycle July 08, 2007 to July 13, 2007 Location: National Center for Atmospheric Research (NCAR) in Boulder, CO. This workshop exposed students in the geosciences, ecology, and mathematics to multidisciplinary science through a focus on estimating the sources and sinks of carbon for the Earth system. One goal was to train the next generation of researchers to work within a multidisciplinary science team that combines geoscientists, ecologists, applied mathematicians, and statisticians. Participants obtained an overview of this problem but also some specific skills in tackling inverse problems and working with geophysical and biogeochemical models. Organizers and lecturers included: • James Clark (Duke University) • Inez Fung (University of California - Berkeley) • Eugenia Kalnay (University of Maryland) • Jeffrey Anderson, David Baker, Douglas Nychka, and David Schimel (NCAR)

§1.3.3. Continuous Optimization and Applications July 09, 2007 to July 20, 2007 Organized By: Henry Wolkowicz. (University of Waterloo) This workshop introduced graduate students to the main ideas of Continuous Optimization and its Applications. In particular, it emphasized the major developments in the last ten years. This included the use of interior point methods in the solution of large scale linear and nonlinear programs. The workshop included a hands-on approach. Numerical tests were done using the NEOS Server for Optimization and the large group of NEOS Solvers. Solution interpretation and sensitivity analysis were emphasized. The workshop was divided into three series of lectures and hands-on labs. The first series included an introduction to the modern theory of convex programming, its extensions and applications. This included separation and support theorems and Lagrange multiplier results. This series emphasized that the great watershed in optimization is not between linearity and nonlinearity, but between convexity and nonconvexity (Rockafellar, 1993.) The main series of lectures involved numerical algorithms for general nonlinear optimization. This included both modern interior point approaches as well as classical Lagrange multiplier methods, such as sequential quadratic programming, SQP. We included applications to engineering and financial problems and emphasized the large scale case. The final series concentrated on specialized topics and applications. In particular, this included optimization over convex sets described as the intersections of the set of symmetric, positive semidefinite matrices with affine spaces, i.e. Semidefinite Programming. This area has attracted a lot of interest due to the number of important applications, to e.g. Discrete Optimization and more general Engineering Problems. We studied and used several current solvers that are in the public domain.

150 §1.3.4. Deformation Theory and Moduli in Algebraic Geometry July 23, 2007 to August 03, 2007 Organized By: Max Lieblich (Princeton), Martin Olsson (Berkeley), Brian Osserman (Berkeley), Ravi Vakil (Stanford) This workshop is intended to introduce to graduate students the main ideas of deformation theory and moduli spaces in algebraic geometry. We hope to illuminate the general theory through extensive discussions of concrete examples and applications.

§1.3.5. A Window into Zeta and Modular Physics June 16, 2008 to June 27, 2008 Organized By: Floyd Williams (University of Massachusetts) and Klaus Kirsten (Baylor University) In recent years, a noteworthy and very fruitful interlacing of number theory and physics has emerged. As indicated in the September 2007 issue of the AMS Notices, for example, a new journal Communications in Number Theory and Physics has just been launched to follow significant interactions and dynamics between these two fields. Several books are now available, in addition to an array of conference and workshop activity, that accent this fortunate merger of ‘pure’ mathematics and physical theory—with applications that range from field theory (conformal and topological), extended objects (strings and branes), cosmology and black hole physics to Bose-Einstein condensation and the theory of relativistic gases.

§1.4. Recruitment of Underrepresented Minorities

§1.4.1. Modern Mathematics: An Introduction to MSRI’s 2008-09 Programs October 10, 2007 to October 11, 2007 Organized By: Ricardo Cortez, Kathleen O’Hara, Ivelisse Rubio This workshop was held at the Kansas City Marriott Downtown located at 200 West 12th Street, Kansas City, Missouri, directly preceding the Annual Meeting of SACNAS. The focus was on introducing the programs to be held at MSRI in the Fall of 2008 and the Spring of 2009: Analysis of Singular Spaces, Ergodic Theory and Additive Combinatorics, and Algebraic Geometry.

§1.4.2. MSRI-UP 2008 research topic: Experimental Mathematics June 14, 2008 to July 27, 2008 Organized By: Ivelisse Rubio (University of Puerto Rico, Humacao), Duane Cooper (Morehouse College), Ricardo Cortez (Tulane University), Herbert Medina (Loyola Marymount University), and Suzanne Weeks (worcester Polytechnic Insitute) The MSRI-UP is a comprehensive program for undergraduates that aims at increasing the number of students from underrepresented groups in mathematics graduate programs. MSRI-UP includes summer research opportunities, mentoring, workshops on the graduate school application process, and follow-up support.

§1.5. Other scientific activities §1.5.1. Computation and Complex Systems October 12, 2007 Organized By: Robert Bryant (MSRI) and Masoud Nikravesh (UC Berkeley) This workshop included key lectures about large scale interdisciplinary problems, round table discussions about mathematical challenges in these and related areas, and Q & A sessions about the structure of the Cyper-enabled Discovery and Innovation (CDI) initiative and the NSF’s expectations for proposals.

151 9. Committee Membership

MATHEMATICAL SCIENCES RESEARCH INSTITUTE BOARD OF TRUSTEES (2008-2009) NAME TERM INSTITUTIONAL AFFILIATION • Edward Baker (Secretary, 2008-2011) 2005-2009 The Cambridge Strategy Ltd Deborah Loewenberg Ball 2007-2011 University of Michigan Elwyn Berlekamp 2008-2012 University of California, Berkeley Andrea Bertozzi 2006-2010 University of California, Los Angeles Ruth Charney 2007-2011 Brandeis University Jennifer Chayes 2008-2012 Microsoft Corporation • (Chair, 2007-2010) 2006-2010 Princeton University Jerry Fiddler 2008-2012 Investor and Board Director Dan Freed 2007-2011 University of Texas, Austin Jeff Goodby 2006-2010 Goodby, Silverstein & Partners Phillip A. Griffiths 2008-2012 Institute for Advanced Study William R. Hearst III 2007-2011 Kleiner, Perkins, Caufield & Byers Maria M. Klawe 2008-2012 Harvey Mudd College Donald Knuth 2006-2010 Stanford University • Julius R. Krevans (Vice Chair, 2008-2011) 2008-2012 Univ. of Calif., San Francisco (retired) Tom Leighton 2005-2009 Akamai Technologies Dusa McDuff 2005-2009 SUNY at Stony Brook Prabhakar Raghavan 2007-2011 Yahoo! Inc. Lucy Sanders 2007-2011 NCWIT, Univ. of Colorado at Boulder 2005-2009 Princeton University Myron Scholes 2007-2011 Platinum Grove Asset Management James H. Simons 2007-2011 Renaissance Technologies Corporation Nathaniel Simons 2007-2011 Renaissance Technologies Corporation Hugo Sonnenschein 2006-2010 University of Chicago • Roger A. Strauch (Treasurer, 2006-2009) 2006-2010 The Roda Group Sandor Straus 2008-2012 Merfin, LLC Andrew Viterbi 2005-2009 The Viterbi Group • Robert Bryant, Director, and Hélène Barcelo, Deputy Director, Ex-officio Trustees; Kathleen O’Hara, Associate Director; (Directorate also on SAC, HRAC, EAC, and Trustees Committees) • Helmut Hofer (2006-09*) and Carlos Kenig (2007-10*), Co-Chairs of the Scientific Advisory Committee, Ex-officio Trustees • Morris Kalka, Chair of the Committee of Academic Sponsors (2008-2011), Ex-officio Trustee Ricardo Cortez, Chair of the Human Resources Advisory Committee (2007-2009), Ex-officio Trustee

152 MSRI TRUSTEES’ COMMITTEES, 2008-2009 AUDIT COMMITTEE – Calvin Moore, Chair; , Roger Strauch (Treasurer) COMMITTEE ON TRUSTEES – Charles Fefferman, Chair; Robert Bryant, , Will Hearst, Morris Kalka (ex officio), Sheldon Katz (ex officio), Lucy Sanders, Hugo Sonnenschein COMMITTEE ON PUBLIC UNDERSTANDING – Dusa McDuff, Chair; Andrea Bertozzi, Ruth Charney, Dan Freed, Jeff Goodby, Julius Krevans COMMITTEE ON WOMEN IN MATHEMATICS – Ruth Charney, Chair; Andrea Bertozzi, Jennifer Chayes, Ricardo Cortez, Tom Leighton, Dusa McDuff, Peter Sarnak CORPORATE PARTNERS COMMITTEE – Edward Baker, Chair; Elwyn Berlekamp, Jennifer Chayes, Jerry Fiddler, Maria Klawe, Julius Krevans, Lucy Sanders, Myron Scholes, Nat Simons, Hugo Sonnenschein, Andrew Viterbi DEVELOPMENT COMMITTEE – Julius Krevans, Chair; Edward Baker, Elwyn Berlekamp, Jennifer Chayes, Charles Fefferman, William R. Hearst III (ex officio), Maria Klawe, Tom Leighton, Douglas Lind (ex officio), Andrew Viterbi EDUCATION COMMITTEE – Deborah Loewenberg Ball, Chair; Maria Klawe, Julius Krevans, Tom Leighton

MSRI TRUSTEES’ COMMITTEES, 2008-09 (continued) FINANCE COMMITTEE – Roger Strauch, Chair; Charles Fefferman, Jerry Fiddler, Enrico Hernandez (CFO, ex officio), Julius Krevans, Sandor Straus INVESTMENT COMMITTEE – William R. Hearst III, Chair; Myron Scholes, Sandor Straus RECOMPETITION COMMITTEE – Charles Fefferman, Chair; Hélène Barcelo, Robert Bryant, Robert Calderbank, Phillip Griffiths, Maria Klawe, Kathy O’Hara, Roger Strauch STEERING COMMITTEE – Charles Fefferman, Chair; Edward Baker, Hélène Barcelo, Robert Bryant, Helmut Hofer, Carlos Kenig, Morris Kalka, Julius Krevans, Kathy O’Hara, Roger Strauch

MSRI Advisory Committees, 2008-2009 SCIENTIFIC ADVISORY COMMITTEE NAME TERM* INSTITUTIONAL AFFILIATION David Aldous 2006-2010 University of California, Berkeley William Fulton 2008-2012 University of Michigan • Helmut Hofer (Co-Chair, 2006-2009) 2005-2009 Courant Institute • Carlos Kenig (Co-Chair, 2007-2010) 2006-2010 University of Chicago Barry Mazur 2005-2009 Harvard University Andrei Okounkov 2007-2011 Princeton University George Papanicolaou 2008-2012 Stanford University Rick Schoen 2007-2011 Stanford University Karen Vogtmann 2006-2010 Cornell University Andrei Zelevinsky 2006-2010 Northeastern University *Terms start on July 1st and terms end on June 30th.

153 EDUCATIONAL ADVISORY COMMITTEE

NAME TERM** INSTITUTIONAL AFFILIATION Michèle Artigue 2003-2012 Université Paris Deborah Loewenberg Ball, Chair 2003-2011 University of Michigan Hélène Barcelo 2008-2010 MSRI Hyman Bass 2003-2012 University of Michigan Sybilla Beckmann 2008-2012 University of Georgia Robert Bryant 2007-2012 MSRI Herb Clemens 2006-2010 Ohio State University Amy Cohen 2008-2012 Rutgers University Ricardo Cortez 2008-2012 Tulane University Ted Courant 2007-2012 Bentley School David Eisenbud 2008-2012 University of California, Berkeley Roger Howe 2004-2012 Yale University Maria Klawe 2008-2012 Harvey Mudd College Julius Krevans 2008-2012 Univ. of Calif., San Francisco (retired) Tom Leighton 2005-2009 Akamai Technologies Jim Lewis 2003-2012 University of Nebraska-Lincoln Robert Megginson 2008-2012 University of Michigan Robert Moses 2003-2012 The Algebra Project Inc. Kathy O’Hara 2006-2009 MSRI Alan Schoenfeld 2003-2012 University of California, Berkeley Hung-Hsi Wu 2008-2012 University of California, Berkeley **Terms start on March 1st and terms end on March 1st.

MSRI Advisory Committees, 2008-09 (continued) HUMAN RESOURCES ADVISORY COMMITTEE NAME TERM*** INSTITUTIONAL AFFILIATION Sylvia Bozeman 2008-2011 Spelman College Ricardo Cortez (Chair, 2007-2009) 2004-20091 Tulane University James H. Curry 2006-2009 University of Colorado Victor de la Peña 2006-2009 Columbia University Trachette Jackson 2006-2009 University of Michigan Victor Moll 2007-2010 Tulane University Ivelisse M. Rubio 2007-2010 University of Puerto Rico David Scott 2007-2010 University of Puget Sound Richard Tapia 2006-2009 Rice University ***Terms start on April 1st and terms end on March 31st.

154

10. Appendix – Final Report

155 PROGRAM REPORT

TEICHMULLER¨ THEORY AND KLEINIANGROUPS

ORGANIZING COMMITTEE: Jeffrey Brock (Brown), Richard Canary (Michigan), Howard Masur (UIC), Maryam Mirzakhani (Princeton), and Alan Reid (Texas)

1 Introduction

The program in Teichmuller¨ theory and Kleinian groups, by all accounts, was a tremendous success; one that exceeded the expectations of the organizers in virtually every respect. Indeed, one Research Professor in the program, Ursula Hamenstadt,¨ described the program as “the best professional experience of her career.” Though gratifying for the organizers to hear, such sentiments express in large measure the particular blend of cameraderie, enthusiasm and topical rel- evance and timing of the program, and its sibling program in geometric group theory. We were all in the right place at the right time. The role of early career mathematicians in the program and the benefits to these mathematicians was a particularly rousing example of the important role an institution such as MSRI can play in the professional development of early ca- reer mathematicians. Likewise, many new connections were created and fostered across the fields of Teichmuller¨ theory and Kleinian groups, many new collab- orations between unlikely parties emerged, and an overall air of excitement and enthusiasm was palpable throughout the program. The principal regret of the or- ganizers and of many participants was the inability of the program to run for the full year, as many developments were just gaining traction as the program drew to a close. A particular benefit and to our program was the concurrently running pro- gram in Geometric Group Theory. The fields of geometric group theory, Kleinian groups, and Teichmuller¨ theory are sufficiently related and interrelated that it was only a matter of time before new connections were built between participants in these apparently distinct programs. We list some of the particular successes below. Our report would be remiss, in addition, were it to fail to include the sig- nificant intellectual impact and energizing influence of the substantial cadre´ of

1 graduate students in attendance through the support of their advisors. Their num- ber was sufficient to provide a critical mass for the organization of a graduate student seminar, in which postdoctoral fellows were often the primary speakers, and to which senior faculty general members were respectfully discouraged from attending. These seminars often ran well over the standard allotted one-hour time frame, some times running over two hours. All reports were that these seminar were enormously beneficial to the graduate students particularly in regard to their ability to understand the seminar talks, mini-courses and workshop lectures. Their enthusiasm and ability to network and connect across field and location remains an important outgrowth of our program (together with Geometric Group Theory) of which the organizers are uniformly proud. The topics that maintained primary focus in the workshops and seminars had largely to do with the participants and their collaborations. A central feature of our program was the many cross-pollinating collaborations that arose between different fields and different career-stages.

2 Research Developments

A hallmark of the program was its many collaborations across levels. But rather than breaking up into working groups for the semester, these collaborations shifted notably as different projects took precedence.

1. One of the research highlights of the program was the presence of Mahan Mitra who was funded to visit from India for six weeks. He had recently established that the limit set of any freely indecomposable Kleinian group is locally connected and the experts in the field were eager to understand his proof. His proof has the further impact of giving a Cannon-Thurston map from the Gromov boundary of the abstract group to the limit set of the Kleinian group. He gave a long series of talks on the argument, which were convincing and exciting for the experts that remained through to the end (which included Jeff Brock, Ken Bromberg, Dick Canary, Yair Minsky and Lee Mosher). An upshot of this series of talks was as new collaboration Mitra engaged in with Saul Schleimer and Chris Leininger which eventually evolved into an argument that the boundary of the curve complex of a once-punctured surface is locally path connected and path connected. The key technical tool here is a Universal Cannon-Thurston map from a subset of the circle

2 at infinity for the associated closed surface group to the boundary of the curve complex of the once-punctured surface. Their preprint is posted on the arXiv at arXiv:0808.3521.

2. A further international success was the program’s support of Cyril Lecuire to visit MSRI for a month. During this time, Lecuire worked with Javier Aramayona on their study of geodesics in the pants complex (a combina- torial model for the Weil-Petersson metric on Teichmuller¨ space). During this period, Cyril discussed his work on a general characterization of when Kleinian surface groups converge and diverge which became a joint project with Jeff Brock, Ken Bromberg and Dick Canary. This is currently being completed.

3. The presence of Teruhiko Soma in our program was an important benefit to workers in Kleinian groups, particularly in regard to his simplifications of the proof of the tameness conjecture (due to Agol and Calegari-Gabai) and his work on the structure of geometric limits. He presented a proof of the topological and geometric classification of geometric limits of quasi- Fuchsian groups (in joint work with Ohshika). An outgrowth of his time at MSRI was a more geometric approach to the combinatorial problems present in Brock-Canary-Minsky proof of the ending lamination conjec- ture. He recently circulated a preprint that demonstrates an interesting new perspective on the problem. It is posted on the arXiv at arXiv:0801.4236.

4. An important success was work of Maryam Mirzakhani with Alex Eskin in counting geodesics in the thin part of moduli space of quadratic differ- entials. During the program, they worked on the general case for different strata, and began collaboration with Kasra Rafi on this which is ongoing and played a significant role in furthering the goals of the project.

5. A new result Jeff Brock with Howard Masur and Yair Minsky emerged out of their considerations of ending laminations for Weil-Petersson geodesics. Their consideration of the case of geodesics that recur to the thick part of Teichmuller¨ space gave a first entry point into a systematic study of the geodesic flow. They proved an ending lamination theorem for recurrent recurrent geodesics, and then used these ending laminations to prove that the geodesic flow is topologically transitive and that the set of closed or- bits is dense. Using this result, they also enhanced their bounded geometry

3 theorem (bounded combinatorics and bounded geometry are equivalent) to apply to geodesic lines rays and segments. The first paper Asymptotics of Weil-Petersson geodesics I: ending lamina- tions, recurrence and flows is under revision at Geom. Funct. Anal..

6. Anna Lenzhen’s talk in the postdoctoral seminar about the limiting behav- ior of Teichmuller¨ geodesics in the Thurston compactification gave rise to a new collaboration between Brock and Lenzhen to show explicitly the failure of convergence of Weil-Petersson geodesics in the Thurston compactifica- tion of Teichmuller¨ space (despite the existence of the ending lamination). Their project is still underway, but is currently involved with seeking con- vergence criteria for the Thurston boundary.

7. Moon Duchin wrote a paper together with Chris Leininger and Kasra Rafi where they proved that the lengths of simple closed curves associated to a translation surface determine the surface, but in contrast to the situation with hyperbolic geometry, a finite number of lengths do not suffice to determine the surface.

8. Kariane Calta worked with John Smillie on their paper Algebraically pe- riodic translation surfaces (see also below). They study the relationship between this algebraically defined concept and the dynamics of the flows on the surfaces. Their paper has been accepted by the Journal of Modern Dynamics.

9. Anna Lenzhen and Howard Masur collaborated on a project that studies properties of Teichmuller geodesics. Their paper Divergence of Teichmuller¨ rays has been submitted.

10. Jeff Brock and Ken Bromberg continued their work on their paper Ge- ometric inflexibility of hyperbolic 3-manfiolds, and its resulting new proof of the double limit theorem for pseudo-Anosov iteration. This paper is a precursor to their work with Juan Souto giving a new proof of the end- ing lamination conjecture using approximations by maximal cusps. Much progress was made on these projects.

These research results represent a sampling of activities at MSRI that resulted in collaborative successes among participants.

4 3 Postdoctoral Mentoring and Results

We were especially pleased by the very strong group of young mathematicians who participated in our program as postdoctoral fellows. This group consisted of: Ilesanmi Adeboye, Javier Aramayona, Kariane Calta, Jason Deblois, Moon Duchin, David Dumas, David Futer, Zheng Huang, Richard Kent, Anna Lenzhen, Hossein Namazi and Corinna Ulcigrai. They generated a tremendous amount of energy and contributed new ideas to the program. Each postdoctoral fellow was assigned one of the more senior members as a mentor. We list each postdoc, their mentor, their professional placement beyond the program at MSRI, and specific research accomplishments in the form of papers and preprints below. The mentors met regularly with their assigned postdoctoral fellows to discuss mathematics and offer career advice. The weekly postdoctoral research seminar gave these fellows the opportunity to give a focused thirty-minute research talk on their work, with the aim of famil- iarizing the senior members and their fellow postdocs with their research projects. These were very well attended (perhaps in part due to the pizza made available by MSRI after the talks concluded!). Placements, mentors, and publications. The professional placements of these postdoctoral fellows is listed below. We also list their postdoctoral mentors and publications arising out of work at MSRI. 1. Ilesanmi Adeboye – mentor: Ian Agol, U.C. Berkeley.

Professional Placement. Adeboye begins a position at U.C. Santa Barbrara this fall (2008). Ian Agol worked closely with Ilesanmi Adeboye to assist with Adeboye’s derivation of an explicit lower bound for the volume of any hyperbolic 4- manifold. Adeboye completed and submitted his first paper, Lower bounds for the vol- ume of hyperbolic n-orbifolds, during the MSRI program. It has since been published in the Pacific Journal of Mathematics. According to Adeboye, “whenever it is that I get my next paper completed, some credit will be due to the program. I was able to discuss work that will appear in that paper with Ian Agol, Mischa Kapovich and others.”

2. Javier Aramayona – mentor: Jeff Brock, Brown University.

5 Professional Placement. Aramayona took a permanent position at the Na- tional University of Ireland in Galway. Aramayona began collaboration with Chris Leininger which later resulted in a paper with Juan Souto which exhibited embeddings of one mapping class group in another with curious properties. Aramayona also met often with Jeff Brock to discuss his project with partic- ipants Lecuire and Parlier on the behavior of geodesics in the pants complex. An upshot was the development by Aramayona of a short proof of the The- orem of Dan Margalit that the automorphism group of the pants complex is the mapping class group. This has taken the form of the paper Simplicial embeddings of pants com- plexes which is a preprint. Of the program, Javier says, “Participating in the ”Teichmuller Theory and Kleinian Groups” at MSRI during the Fall 2007 was, without a doubt, the best professional experience I have had to this date. Being among some of the top-notch experts in the field was immensely beneficial for me, as it was being around some people at a similar stage of their careers as I was. I believe the stay at MSRI came at the right moment for me, and I am still discovering how much I learned during that time. I also met a lot of people, or at least got to spend some time with people I knew a bit, with whom I am collaborating or are potential collaborators. The seminar programme there was very good, since there were many spe- cialised seminars but also some more general ones, and some given by post- docs. One thing I really appreciated was that we were not overwhelmed by seminars and we could get our work done.”

3. Kariane Calta – mentor: Howard Masur, UIC.

Professional Placement. Calta took a tenure-track position at Vassar Col- lege. Her joint paper Algebraically periodic translation surfaces with John Smil- lie written at MSRI has appeared in the the Journal of Modern Dynamics.

4. Jason Deblois – mentor: Kenneth Bromberg, Utah.

6 Professional Placement. Deblois returned to his postdoctoral position at the University of Illinois, Chicago (UIC). While at MSRI, he worked on the papers On the doubled tetrus, arXiv:0804.3984, Rank gradient of cyclic covers, in preparation, and Hyperbolic manifolds tiled by right-angled ideal polyhedra are virtually special, joint with Eric Chesebro and Henry Wilton, who was a participant in the Geometric Group Theory program.

5. Moon Duchin – mentor: Dick Canary, Michigan.

Professional Placement. Duchin took a postdoctoral position at the Uni- versity of Michigan. While at MSRI, Duchin worked on her papers The flat-length spectrum (working title) with Chris Leininger and Kasra Rafi, which is close to sub- mission. She also started a collaboration with Anne Thomas from the Geometric Group Theory program on Filling at infinity in groups. This is work in progress. Of her experience at MSRI, she writes, “I’d say the MSRI semester was enormously helpful to me both in terms of research progress and in terms of getting better connected. I did a postdoc-focused job search during/right after the program, and I definitely think the success of that search was con- nected to the exposure I got at MSRI. After my talk in the topics workshop, for instance, a French colleague more or less offered me a visiting position on the spot, and I expect to spend this Spring visiting Marseille.”

6. David Dumas – mentor: Steve Kerckhoff, Stanford.

Professional Placement. Dumas begins a tenure-track position at UIC this fall. While at MSRI, Richard Kent and he submitted their paper Slicing, Skin- ning, and Grafting to the American Journal of Mathematics, where it was later accepted for publication. It has not yet appeared. They also started work on a follow-up project, in which they show that Bers slices are Zariski dense in the character variety. A preprint on this topic is now available on the arxiv (identifier 0807.4509).

7 Young-Eun Choi, Kasra Rafi, and he completed a project on grafting lines and the Teichmuller metric during the program. They are in the final stages of writing a paper on this. During the Fall semester he wrote a substantial part of his survey on com- plex projective structures, which he completed in early 2008, and which will appear as a chapter in the Handbook of Teichmuller Theory, Volume II, published by the European Mathematical Society (edited by A. Pa- padopoulos). Of his experience at MSRI, he writes: “The weekly postdoc seminar was really useful, as it gave me a way to find out what the other young people at the MSRI program were thinking about. Other than this, I think the single most beneficial aspect of the program was having so many experts in one place, available for informal discussions during the unstructured time built into the schedule.”

7. David Futer – mentor: Sergio Fenley, U. Florida.

Professional Placement. Futer begins a tenure-track position at Temple University this fall. While at MSRI, Futer worked on the papers: Cusp areas of Farey manifolds and applications to knot theory. (Submitted) with Efstratia Kalfagianni and Jessica S. Purcell, and Cusp volume of fibered 3-manifolds. (In preparation) with Saul Schleimer.

8. Zheng Huang – mentor: Howard Masur, UIC.

Professional Placement. Huang took a position at CUNY Staten Island. While at MSRI, Zeno worked on the paper Average Curvatures of Weil- Petersson In Teichmuller¨ Space, which is in submission.

9. Richard Kent – mentor: Ursula Hamenstadt,¨ Bonn.

Professional Placement. Kent returned to his position as a Tamarkin As- sistant Professor at Brown University. While at MSRI, Kent worked on his joint papers with David Dumas, listed above.

8 10. Anna Lenzhen – mentor: Jeff Brock, Brown.

Professional Placement. Anna returned to her postdoctoral position at the University of Michigan, and left for a one-year position in Lille (France) this fall. While at MSRI, Anna worked on and completed her paper Divergence of Teichmuller¨ rays with Howard Masur, which is currently under submission. She also began work on the joint paper Divergence of Teichmuller¨ rays with Kasra Rafi, which is in preparation. 11. Hossein Namazi – mentor: Dick Canary, Michigan.

Professional Placement. Namazi begins a tenure-track position at U.T. Austin this fall. While at MSRI, Namazi worked toward completion of the papers Non- realizability and ending laminations, with Juan Souto Revisiting Thurston’s Uniform Injectivity Theorem, also with Souto and Quasiconvexity and shrinkwrap- ping all of which are close to submission. He also continued his work to- ward the joint paper with Brock, Minsky and Souto Bounded geometry and combinatorics for hyperbolic Heegaard splittings. Of his experience at MSRI, Namazi writes, “...it was certainly amazing to have all of these amazing people around and I think the organizers did very well and more importantly were very cool. I could not attend the workshop in the beginning but I thought the conferences in November were well orga- nized and gathered excellent combination of people. Also I believe it was a great idea to have the Group Theory program and the Kleinian Groups- Teichmuller¨ Theory one at the same time and I personally benefitted a lot from this.” 12. Corinna Ulcigrai – mentor: John Smillie, Cornell.

Professional Placement. Ulcigrai begins a tenure-track position at the Uni- versity of Bristol. While at MSRI, Ulcigrai worked on the publication A Renewal-Type Limit Theorem for Continued Fractions and the Gauss Map (2008) with Yakov G. Sinai, in Ergodic Theory and Dynamical Systems. The postdoctoral fellows also interacted closely with the graduate students. They often gave long expository talks in the weekly graduate student seminar.

9 4 Graduate Student Interactions

The graduate students in attendance benefitted greatly from the chance to interact with a wide variety of mathematicians in the area. Our graduate students and postdoctoral fellows also had the added benefit of getting intense exposure to ideas from geometric group theory, which is likely to have a large payoff later on in their careers. Chris Leininger worked closely with Juan Souto and Dick Canary’s student Johanna Mangahas and his insights and comments played an instrumental role in her paper Uniform uniform exponential growth for subgroups of the mapping class group. Her preprint is posted on the arXiv at arXiv:0805.0133. Ken Bromberg and Steve Kerckhoff worked closely with Dick Canary’s stu- dent Aaron Magid. Their explanation of the application of cone manifold defor- mation theory to Kleinian groups was a key inspiration in Magid’s proof (accom- plished later in the year) that spaces of Kleinian surface groups are not locally connected. His result was inspired by and generalizes Bromberg’s earlier result that the space of punctured torus groups is not locally connected. Magid’s paper Examples of relative deformation spaces that are not locally connected is under revision at Math. Annalen. William Cavendish, a graduate student of Jeff Brock, was very active in the program, preparing the notes from the introductory workshop, and serving as a primary organizer for the graduate student seminar. During the workshop he solved a problem concerning the quasi-isometry type of path metrics on the Mumford- Deligne compactification lifted to the Teichmuller¨ space: they are all quasi-isometric to the pants complex.

5 Diversity

We note that a third (four of twelve) of our postdoctoral fellows were women (Calta, Duchin, Lenzhen and Ulcigrai), and one was African-American (Ade- boye). Of the research level participants, three senior participants were women (Hamenstadt,¨ Mirzakhani, and Series) and two affiliated visitors were also women (Elmas Irmak and Asli Yaman).

10 6 Synergistic Aspects

The organizers feel that a key element of the success of the program was the top- ical overlap with the concurrently running program in Geometric Group Theory. Many of their mini-courses and seminar talks were directly relevant to our pro- gram, and there was considerable intellectual germination that took place between participants. As a key example, a project that intertwined these fields was the collaboration between Ken Bromberg, Mladen Bestvina, and Koji Fujiwara to compute the asymptotic dimension of Teichmuller¨ space. It is hard to imagine a better exam- ple of synergy between the programs: (1) the question of asymptotic dimension is one that arises typically in the field of geometric group theory, (2) it is be- ing addressed toward Teichmuller¨ space and (3) the methods employed to solve the problem now involve the Masur-Minsky hierarchies in the curve complex, a key tool in the solution to the Ending Lamination Conjecture of Brock-Canary- Minsky, a central question in Kleinian groups. Their collaboration has continued on since the program, and has now produced, almost accidentally, a new proof of the Nielsen realization theorem (in additional collaboration with Juan Souto) by showing that length functions are convex along a particular choice of Fenchel Nielsen coordinates. This example is one of many, but it serves to emphasize in important ways the benefits of running thematically similar programs concurrently. This aspect of the semester at MSRI was uniformly praised.

11 REPORT ON THE GEOMETRIC GROUP THEORY PROGRAM MSRI FALL 2007

MLADEN BESTVINA, JON MCCAMMOND, MICHAH SAGEEV, AND KAREN VOGTMANN

1. Introduction The semester-long research program in geometric group theory was held at MSRI during the fall of 2007 (August 23 - December 15). The semester was proposed and organized by Mladen Bestvina (Univer- sity of Utah), Jon McCammond (UC Santa Barbara), Michah Sageev (Technion) and Karen Vogtmann (Cornell University). All of the orga- nizers were in residence for the entire semester and took active organi- zational roles in the program. The program focused on several topics in geometric group theory, including spaces of negative and non-positive curvature, asymptotics (boundaries, asymptotic cones), cohomological invariants and finiteness properties of groups, analtyic aspects (e.g. amenability and property (T)), and connections with logic. In the sections below we report on the MSRI post-doctoral fellows, the work- shops and conferences held, the other activites we organized during the semester, our interactions with the other program and a few nuggets and breakthroughs by our members during their time at MSRI.

2. Postdoctoral fellows There were eight official MSRI postdoctoral fellows in the geometric group theory program. Here they are in alphabetical order, together with a brief description of their research area, their mentor at MSRI and their current academic affliation. Christopher Cashen. Chris Cashen graduated from the University of Illinois, Chicago and is currently a VIGRE postdoctoral fellow at the University of Utah. Emanuele Delucchi. Emanuele Delucchi graduated from E.T.H. in Z¨urich and is currently a visiting assistant professor at SUNY Bingham- ton. His research focuses on Artin groups and the fundamental groups

Date: June 23, 2008. 1 2 BESTVINA, MCCAMMOND, SAGEEV, AND VOGTMANN of other complex hyperplane arrangements. His mentor at MSRI was Jon McCammond. Tulia Dymarz. Dymarz studies the large-scale geometry of finitely generated groups, in particular solvable groups. Her mentor at MSRI was Kevin Whyte. She is Gibbs Assistant Professor at Yale University. Sam Sang-hyun Kim. Sam Kim graduated from Yale University and is currently Bing Instructor at the Universtiy of Texas at Austin. His research focuses on braid groups and ]-angled Artin groups. His mentor at MSRI was Michah Sageev. Lars Louder. (NSF postdoc at Rutgers, Newark) Louder studies limit groups and the first order theory of free groups. His mentor was Mark Feighn. He is now NSF Postdoctoral Fellow at the University of Michigan. Damian Osajda. Damian Osajda’s research focuses of the simpli- cial nonpositive curvature conditions introduced by Januszkiewicz and Swiatkowski. His mentor at MSRI was Jon McCammond. Andrew Putman. His mentor was Mladen Bestvina. He is CLE Moore Instructor at M.I.T. His research focuses on mapping class groups and Torelli groups. Anne Thomas. Anne Thomas studies lattices in automorphism groups of locally finite polyhedral complexes. Her mentor at MSRI was Karen Vogtmann. She is H. C. Wang Assistant Professor at Cornell Univer- sity. In addition to the official MSRI post-docs, there were a large number of general members still in the post-doctoral stage of their careers, and more than twenty graduate students (split between the two programs) in residence the entire semester, virtually all accompanying their dis- sertation advisors. As we planned the activities for the semester we were acutely aware of the large number of early career mathematicians who were in residence and we planned accordingly.

3. Workshops and Conferences The geometric group theory program included three major work- shops and a one-day conference. The first was a “Connections for Women Workshop” run by Ruth Charney, Indira Chatterji and Karen Vogtmann (August 23-24). This workshop was immediately followed by an “Introductory Workshop” (August 27-31) organized by the program organizers. These workshops were among the most widely attended GEOMETRIC GROUP THEORY 3 workshops in the history of MSRI with approximately 150 people at- tending each of them (not all of whom were officially registered). To- wards the end of the semester (November 5-9), a “Topics in Geometric Group Theory” workshop was held focusing on more recent advances in the field. This workshop was organized by Noel Brady, Mike Davis and Mark Feign. Finally, a special centennial birthday conference was held on November 28 for Peter Kropholler and Lee Mosher, who both turned 50 during 2007. For details on the Connections for Women con- ference, see the separate report submitted by Karen Vogtmann. The other three are discussed below. Introductory Workshop on Geometric Group Theory. This workshop was organized by the main organizers of the program (Bestv- ina, McCammond, Sageev and Vogtmann). In an effort to encourage young people in the field, a theme we continued throughout the semes- ter, the workshop was constructed around several series of introductory lectures. • Nonpositive and negative curvature - Jim Cannon • Coxeter and Artin groups - Ruth Charney • Mapping class groups - Benson Farb • Quasi-isometric rigidity - Bruce Kleiner • Cohomological group theory - Ian Leary Each main speaker gave a series three hour-long lectures, aimed at exposing the young researcher to the a collection of major results and tools used in geometric group theory. In addition to these lecture series, several introductory hour lectures were given. As noted earlier, the workshop was extremely well attended. MSRI member Daniel Groves hosted a party for all participants on Thursday evening. Topics in Geometric Group Theory. This workshop focused on recent advances in the field. The speakers and their topics were: Monday November 5, 2007 Alex Eskin: Coarse differentiation and the geometry of poly- cyclic groups Yehuda Shalom: Almost normal subgroups of arithmetic groups and the structure of totally disconnected groups Alexander Dranishnikov: On asymptotic dimension of Cox- eter groups Indira Chatterji: Median spaces and applications Tuesday November 6, 2007 Karen Vogtmann: Automorphism groups of right-angled Artin groups 4 BESTVINA, MCCAMMOND, SAGEEV, AND VOGTMANN

Panagiotis Papazoglou: Higher isoperimetric inequalities for com- plexes and groups Anne Thomas: Lattices acting on polyhedral complexes Denis Osin: Normal automorphisms of relatively hyperbolic groups Koji Fujiwara: Rank-1 isometries on CAT(0) spaces and quasi- homomorphisms Wednesday November 7, 2007 Bertrand Remy: A family of simple groups acting on buildings Dan Margalit: Dimension of Torelli groups Thomas Putman: On the homology of finite index subgroups of the mapping class group Jason Behrstock: Quasi-isometric rigidity of the mapping class groups Jan Dymara: L2 cohomology of buildings Thursday November 8, 2007 Vincent Guirardel: Geometric Makanin algorithm for solving equations in virtually free groups Michael Handel: Global fixed points for centralizers and Morita’s Theorem (joint work with John Franks) Larsen Louder: Krull dimension for limit groups Volodymyr Nekrashevych: Space of marked groups and non- uniform exponential growth Friday November 9, 2007 Martin Bridson: Finitely presented, residually-free groups Tadeusz Januszkiewicz: Groups with fixed point properties Jason Manning: Residual finiteness and separability of quasi- convex subgroups David Fisher: Kleiner’s proof of the polynomial growth theorem The workshop included a conference “banquet” held one evening in the foyer of MSRI, complete with Cheese Board Pizza and the music of the band “Feighning Enthusiasm” fronted by member-in-residence Mark Feighn. Kropholler-Mosher Fest A one-day Centennial Birthday Coference was held to celebrate the fact that both Peter Kropholler and Lee Mosher reaching the age of 50 during 2007. The highlight of the con- ference was Benson Farb’s description of his joint work with Lee Mosher as “an encounter with a genious.” A special cake was made in the shape of a surface of genus two, decorated with a long exact sequence. Jon McCammond’s abode served as the location for the conference party, GEOMETRIC GROUP THEORY 5 which ran long into the night. The party ended when participants came to blows over the correct use of the term “right coset.”

4. Organizational Structure In addition to the three workshops and the one-day conference, the organizers created a number of “local institutions” that helped struc- ture the time of the members in residence for the entire semester. Many of these were designed with the MSRI Postdoctoral fellows and other younger mathematicians in mind. They included a series of seven mini- courses, a weekly research seminar, a weekly post-doc seminar and a weekly grad student seminar, a communal lunch, a Thursday lunch question and answer session, and a couple of special lectures. Minicourses. In an effort to help young mathematicians (post-docs and graduate students) become familiar with areas of research in the field, as well as to help older mathematicians learn new tricks and additional subfields, several series of introductory minicourses (6 lec- tures each) were given. Each minicourse ran for half of the semester. Students and post-docs were assigned as note-takers at each of these minicourses and these notes were made publicly available. First half: • Mark Feighn - limit groups. • Zlil Sela - algebraic geometry over groups • Kevin Whyte - quasi-isometric rigidity Second half: • Peter Kropholler - Cohomology of groups • Gilbert Levitt – Out(Fn) • Mark Sapir – Asymptotic cones Both halves: • Lee Mosher - Mapping class groups Lee Mosher’s course was specifically targeted at participants in both the geometric group theory program and the Kleinian groups and Te- ichm¨ullertheory program, but many of the other courses were also regularly attended by people from both programs. Weekly Research Seminar. A weekly research seminar in Geometric Group Theory ran the entire semester. This gave an opportunity for the more senior members to present their current research. Post-doc seminar. Every Friday at noon a seminar was held featur- ing short lectures by post-docs from each of the programs. This was followed by a pizza lunch. The seminar itself was run and organized 6 BESTVINA, MCCAMMOND, SAGEEV, AND VOGTMANN by two post-docs, one from each program: Anne Thomas (GGT) and David Futer (KGTT). Graduate student seminar. For the graduate students in atten- dence, we created a seminar with a similar structure. It was run and organized by two graduate students, one from each program: Aditi Kar (GGT) and Will Cavendish (KGTT). The organizers routinely polled their attendees and then approached various faculty members to give introductory talks on topics of interest. Communal Lunch. Shortly after the semester began, an informal pot luck lunch in the member’s lounge was instituted featuring, among other things, Cheese Board bread transported up the hill each morning by one of the program organizers (Bestvina or Sageev) on their bicy- cles. This potluck, referred to as “commie lunch” served to forestall any potential mid-day retreat to downtown Berkeley. Commie lunch involved a simple weekly sign-up and after a while pretty much ran itself. We highly recommend it to future programs. Thursday Lunch Q and A. In addition to the communal lunch, every Thursday at noon a lunchtime question-and-answer session was instituted. This was known as “stupid questions time.” During this hour anyone could ask very basic questions and the senior researchers in the audience would be forced to go to the board and give unprepared answers. We found that the relaxed, informal setting worked very well, and it made it easier for the younger mathematicians to interact with the more senior ones. Jorgen Andersen’s special lectures. Finally, there was a timely breakthrough on a central problem in the geometry of mapping class groups by Jorgen Anderson. He proved that mapping class groups do not have property (T). Anderson gave a series of two special lectures on his proof late in the semester at MSRI.

5. Interactions with the other program As should be clear from our descriptions above, one of the factors that we believe led to the success of the geometric group theory pro- gram this past fall was the overlap with the parallel Kleinian Groups and Teichmuller Theory program at MSRI. The common denominator of both programs is the interplay between group theory and geometry. Thus, while the objects of study in the two groups are different, there is much overlap in techniques and there is a common language, which includes such notions as geodesics, curvature, boundaries, etc. There are several collaborations between members of the two programs that GEOMETRIC GROUP THEORY 7 were started at MSRI (for example, the Bestvina-Bromberg-Fujiwara work on the asymptotic dimension of Teichm¨uller space), but in addi- tion, the overlap between the areas enabled us to run a joint post-doc seminar, a joint graduate student seminar and a joint question and answer session.

6. Nuggets and breakthroughs Finally, we conclude with a few “nuggets” about our time at MSRI. During the Introductory Workshop, Bruce Kleiner, after hearing an inspiring lecture by Alain Valette, found another proof of Gromov’s polynomial growth theorem that uses harmonic functions on groups and isometric actions on Hilbert space in place of the Montgomery- Zippin characterization of Lie groups. Kleiner’s proof was presented by David Fisher at the Topics in Geometric Group Theory Workshop. Other results that were established during the semester include the following: (1) Ian Leary figured out that CAT(0) cube complexes are complete precisely when every ascending sequence of nested cubes ter- minates, and (2) Bestvina, Bromberg and Fujiwara computed the as- ymptotic dimension of Teichmuller space.

Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090 E-mail address: [email protected]

Department of Mathematics, University of California, Santa Bar- bara, CA 93106 E-mail address: [email protected]

Department of Mathematics, The Technion, Haifa, Israel E-mail address: [email protected]

Department of Mathematics, Cornell University, Ithaca, NY 14853- 4201 E-mail address: [email protected] Final report on the One-Semester Program in Combinatorial Representation Theory MSRI, Spring 2008

Organizing Committee

Persi Diaconis (Stanford University) Alexander Kleshchev (University of Oregon) Bernard Leclerc (Universit´ede Caen) Peter Littelmann (Universit¨atzu K¨oln) Arun Ram, Chair (University of Melbourne/University of Wisconsin–Madison) Anne Schilling (University of California–Davis) Richard Stanley (Massachusetts Institute of Technology)

Overview The semester-long research program on Combinatorial Representation Theory was held at MSRI from January 15 to May 23, 2008. It was based on a proposal accepted by the SAC of MSRI in 2004. Combinatorial Representation Theory is the interaction of combinatorics and representation theory. It lies at the intersection of several fields: combinatorics, representa- tion theory, harmonic analysis, algebraic geometry, and mathematical physics. Many experts in these various fields came together under the interdisciplinary heading of Combinatorial Representation Theory. The facilities at MSRI were ideal for bringing this group together for a focused semester and the interaction with the concurrent program Representation Theory of finite groups and related topics (denoted RFG below) was so intense that it was never clear which members were officially members of which programs. The natural overlap between these fields provided positive input for both programs. The program saw great interplay between combinatorics, geometry, finite groups, Lie theory, and probability in their applications to representation theory. There was a focused excitement in the air, throughout the program, an environment where there was intense work on problems such as • Interaction of geometry, representation theory and combinatorics, • Macdonald polynomials • Applications of combinatorial representation theory • Computational advances and development of Sage-Combinat • Cluster algebras, quivers and quantum affine algebras

Personnel The program had 7 members on the organising committee, all of whom were in residence at MSRI for a significant portion of the semester: Kleshchev, Leclerc, Ram, Schilling (full semester), Diaconis (3 months), Littelmann and Stanley (1 month each). Present as senior 2 msri spring 2008: final report researchers were Georgia Benkart (3 months), Sergey Fomin (3 months), Adriano Garsia (1 month), Alain Lascoux (1 month) and Anatoly Vershik (4 months) were present as senior researchers. There were 9 postdoctoral fellows, 4 of which were joint with the RFG program, all in residence at MSRI for the whole semester. The postdocs David Hernandez, Erdal Emsiz, Joel Kamnitzer, Nicolas Jacon, Pavlo Pylyavskyy were also at MSRI for extended stays as general members. There were 37 additional general members and 10 graduate students participating in the program with long term (1 month or more) stays. Of special note was the cooperation between the Combinatorial Representation Theory program and the Representations of Finite Groups program. Among the participants who were jointly funded by the two programs were organiser: Alexander Kleshchev; general mem- bers: Nicolas Jacon, C´edricBonnaf´e,Jon Brundan, Robert Guralnick, Gus Lehrer, , Andrew Mathas, Jan Saxl, Alexandre Zalesski; and postdoctoral fellows: Maria Chouvleraki, Jon Kujawa, Sinead Lyle, Nat Thiem. In practice the interaction between the programs was so fluent that the participants were never sure who was in which program. The best indicator was whether a given individual’s office was on the 2nd floor or the 3rd floor. • Simons Professors: Persi Diaconis, Georgia Benkart, Anatoly Vershik. Generous support from the Simons foundation funded the stays of Persi Diaconis, Georgia Benkart and Anatoly Vershik. Persi Diaconis, an organizer of the combinatorial representation theory program, is Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University and a promoter and leader of the field of Combinatorial Representation Theory both in its theory and in its applications. One of the highlights of the semester was the Applied Representation Theory Seminar he organised, which had extensive participation from both programs and speakers from MSRI, UC Berkeley, Stanford, etc. Georgia Benkart retired as Van Vleck Professor of mathematics at the University of Wisconsin in 2006. She won the University of Wisconsin Distinguished Teaching Award in 1987 and the Mid-Career Faculty Research Award in 1996. She was the Mathematical Asso- ciation of America P´olya Lecturer for 2000-2002. She is one of the leaders in Combinatorial Representation Theory with more than 20 Ph.D students, 50 coauthors, and 100 journal articles. She is now president elect of the Association for Women in Mathematics and her presence and many stimulating lectures added great energy to our program. Anatoly Vershik is President of the St.Petersburg Mathematical Society and Head of the Laboratory of Representation Theory and Computational Mathematics at the St.Petersburg Department of Steklov Institute of Mathematics. He has been a leader in the field for almost 40 years with over 200 papers emphasizing the interaction between asymptotics, combinatorics and representation theory. His presence provided a very important link to the “Russian school” of thought in the field and gave life to many of the probabilistic applications of the field that were being discussed during the semester. • Senior research scientists: Sergey Fomin, Adriano Garsia and Alain Lascoux. It was an honor for us to have these three senior researchers in residence. Adriano Garsia and Alain Lascoux were in residene for one month each and Sergey Fomin for three months. All of them can be considered to have, over and over, given birth, life, encouragement, and energy to so many parts of the field over the past 4 decades. They did it, once again, with their presence at MSRI in Spring 2008. combinatorial representation theory 3

Sergey Fomin is one of the longtime worldwide leaders in algebraic combinatorics and the creator, with Andrei Zelevinsky, of a whole subfield of algebraic combinatorics: cluster algebras. This topic was one of the primary topics of the MSRI semester and it was partic- ularly beneficial to have Fomin in residence for an extended period. During his stay Sergey gave several inspiring talks and had discussions with an immense number of participants. Adriano Garsia has pioneered many different parts of combinatorial representation the- ory. This semester, at MSRI, we saw his influence and teachings throughout every aspect of the very active research on Macdonald polynomials that was going on. He remains an inspiring force to so many in the field, particularly the younger researchers. As usual, Alain Lascouxs reminded us of the value of the integration of combinatorics, polynomials, representations, geometry and mathematical physics. He began new exciting projects postdocs and younger researchers and stimulated others by his insightful observa- tions. Of course, he has been doing this for our community since the appearance of his PhD thesis in 1974. • UC Berkeley’s points program: Edward Frenkel. The MSRI “points” program with UC Berkeley made it possible to have Edward Frenkel in residence throughout the program. He is the winner of the 2002 Hermann Weyl prize and a leader in research in the Geometric Langlands Program and its relationship to other areas of mathematics and physics. He co-manages, with K. Vilonen, a DARPA ”Focus Area in Theoretical Mathematics” on the topic of the Langlands program. His books ”Langlands Correspondence for Loop Groups” and ”Vertex Algebras and Algebraic Curves” (with David Ben-Zvi) are quickly becoming classics in the field. The MSRI program in Combinatorial Representation Theory benefited greatly from Edward Frenkel’s expertise in the representation theory of affine Lie algebras and quantum groups. In particular, he began a new collaboration with MSRI member David Hernandez to study the categories of “integrable” representations of affine Lie algebras and double affine Lie algebras. • Viterbi Endowed Postdoctoral Scholar: Lauren Williams. Lauren Williams’ ap- pointment as Viterbi Endowed Postdoctoral Scholar was possible through generous support from the Viterbi Family Foundation. Dr. Andrew J. Viterbi is a pioneer in the field of Wire- less Communications and a member of the National Academy of Sciences. He created the Viterbi Algorithm for interference suppression and efficient decoding of a digital transmis- sion sequence which is used by all four international standards for digital cellular telephony. QualComm, cofounded by Dr. Viterbi and Irwin Jacobs in 1985, is the recognized pioneer of the Code Division Multiple Access (CDMA) digital wireless technology which allows many users to share the same radio frequencies and thereby increase system capacity many times over analog system capacity. In spite of being only 2 years away from her Ph.D., Lauren Williams already has written more than 11 papers. Prominent senior researchers in the field use, for their own research, the matrix mutation applets software that she wrote. Highlights of her research results are (1) her Advances in Mathematics paper where she provides an explicit formula for the generating function enumerating totally positive Grassmann cells according to their dimension, (2) her Crelle’s Journal paper where she proves that K. Rietsch’s poset is shellable, in the spirit of the classical theorems on Bruhat orders due to A. Bjorner-M. Wachs and M. Dyer, 4 msri spring 2008: final report

(3) her work with Ardila and Klivans which characterizes the space of all positive points on the tropical space associated with an oriented matroid, and (4) her recent work with Corteel which found miraculous numerical correspondences be- tween the totally positive geometry and the convergence rates in the asymmetric ex- clusion process (ASEP). Since the ASEP is related to the Bethe ansatz in physics one might guess that this is only the tip of the iceberg in a fascinating new direction. • DARPA Langlands program postdoctoral fellow: Ghislain Fourier. Ghislain Fourier was appointed as a postdoc of the Combinatorial Representation Theory program through generous funding from the DARPA Langlands program grant. The Langlands Pro- gram, launched by in the late 60’s, ties together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic functions. The Langlands conjecture predicts that there is a correspondence between n-dimensional rep- resentations of the Galois group of a number field and automorphic representations of the group GL(n) over the ring of adeles of this field. This conjecture has an analogue when the number field is replaced by the field of functions on a smooth projective curve defined over a finite field. In this setting, this conjecture has a geometric version, called the geometric Langlands correspondence. There is a further generalization of this conjecture where the group GL(n) is replaced by a reductive G. The research of Ghislain Fourier focuses on the detailed structure of representations of the of the group of loops with values in a reductive algebraic group G. His research is particularly exciting because of the insight it gives into the way that finite dimen- sional representations of current algebras can live inside infinite dimensional highest weight representations of the affine Lie algebra. His work has extended known classifications of the finite dimensional representations of loop Lie algebras and the corresponding quantum groups to the twisted case. Fourier has worked with many of the main experts in this theory: with Peter Littelmann on tensor products of Demazure and Weyl modules, with V. Chari on Weyl modules in the twisted case and with Anne Schilling and Mark Shimozono, where they explained how to relate the crystals of Demazure modules and Kirillov-Reshetikhin modules.

Graduate student semester at MSRI There were 14 graduate students in residence at MSRI for the bulk of the Spring pro- grams (advisor listed in parentheses): Luca Moci (DeConcini), Brendon Rhoades (Reiner), Dan Swenson (Webb), Zajj Daugherty (Ram), Olcay Coskun (Boltje), Mbirika (Goodman), Jason Elliot (Robinson), David Nash (Kleshchev), Rahbar Virk (Ram), Martha Yip (Ram), Olivier Dudas (Bonnafe), Mike Hansen (Ram), John Graber (Goodman), Matt Davis (Ram). Arun Ram led a meeting of the graduate students each nonworkshop Friday 9:30-11:00. During these meetings we discussed mathematics, community, culture, teaching, job searches and many other topics. Primarily the discussions seemed to focus on explaining and dis- cussing mathematics terms that the students had heard “in the air” but didn’t know the meanings of. All in all the semester was a great success for the students. They came without expec- tations but with concept of a “gathering” of mathematicians in representation theory. Their maturation as mathematicians progressed at an astounding rate and they were very stimu- lated by the goings on. Though they came in “quite green” they became fluent with “the combinatorial representation theory 5 walk and the talk” of representation theory by the end. They claimed that it was easy to do so in this environment with so many “math plants all around”. The semester immersed the graduate students in a rich culture of the most exciting mathematics. The feeling that both the professionals and the students were doing the same kinds of work nurtured maturity and stimulated the students to discover, and be surprised by, their own mathematical potential. The semester was a particular success for the students in enabling them to meet the community and to extricate them from the bustle and distractions of their home institu- tions. They reported that “Everyone here is really focused, which makes it easy to remove distraction” and work on mathematics. They also felt that the natural time limitation of the program being only one semester was a good motivator for work – they had the “don’t waste the opportunity” feeling for their own research. For many of the students the semester was particularly valuable as a “job development” workshop. “We wanted a sense of what a job in mathematics is ... This was an intense dose.” This “career workshop” was an unplanned, supplementary, outcome which came from the community and the natural vertical integration all of the participants at MSRI for the semester. Several students felt bouyed by the realisation that mathematics has “a possible niche for me.” At the ouset mathematics research feels very imposing because “everyone knows so much” but the MSRI experience “has given us a view of how people make their way” in a career as a research mathematician. Several of the students reported on how effective the semester was for learning time management. They learned how to “partition the day: come in, sit down, work, stop and eat, work more, go home, take a break”. They learned concentration and how to take efficient breaks. When asked how MSRI made them learn this, the response was “There’s a natural emulation, everyone around us is doing itso it just, naturally, rubs off”. The students felt that the separation between faculty and graduate students is more blurred at MSRI than at their home dept. This may be a consequence of faculty/student ratio at MSRI–the graduate student community at MSRI exists but the “pack mentality” can’t take over. Several students reported positive progress and new results in their own research directly resulting from their interaction with senior members who were not their advisor. In general, they felt that the student/faculty ratio should not be changed and that the resources for research, particularly the non-circulating library, were very beneficial to their work. The primary gripes were the lack of after hours Hill buses and the crowding and lack of coordination at the MSRI/Evans lecture venue. There was some sentiment that there was a bit of an “excess of interesting talks to go to”.

Workshops

The Combinatorial Representation Theory program began with two workshops • Connections for Women: Introduction to the Spring, 2008 programs January 16, 2008 to January 18, 2008 Organized By: Bhama Srinivasan and Monica Vazirani • Introductory Workshop on Combinatorial Representation Theory January 22, 2008 to January 25, 2008 Organized By: Persi Diaconis, Arun Ram and Anne Schilling 6 msri spring 2008: final report

In addition, the Combinatorial Representation Theory program hosted two topical work- shops: • Lie Theory March 10, 2008 to March 14, 2008 Organized By: Alexander Kleshchev, Arun Ram, Richard Stanley and Bhama Srini- vasan • Topics in Combinatorial Representation Theory March 17, 2008 to March 21, 2008 Organized By: Sergey Fomin, Bernard Leclerc, Vic Reiner and Monica Vazirani

The workshop Lie Theory was a joint workshop between the Combinatorial Representation Theory program and the Representations of Finite groups program. In addition to MSRI workshop funding attached to programs, these workshops were supplemented by funds from National Security Agency. We had record attendance at these workshops – the S.S. Chern Hall at MSRI was packed to capacity for most lectures. By luck, the exciting • MSRI’s 25th Anniversary Celebration January 26, 2008 to January 30, 2008 Organized By: Alejandro Adem, Isadore Singer and Robert Bryant. immediately followed the introductory workshops of the Combinatorial Representation The- ory program. We were blessed with many exciting additional lectures related to the subject of our program. Among the talks at this celebration which were of relevance to our pro- gram: Persi Diaconis, an organizer of the Combinatorial Representation Theory program, gave the opening lecture of the 25th anniversary celebration; Andrei Okounkov gave the “fourth lecture in the series”, continuing the thread he had established in his lectures at the Introductory Workshop; Peter Ozsvath described the symmetric group combinatorics needed for computation of Heegard Floer homology invariants and Vaughan Jones gave a lively description of the history and recent results around the connections between operator algebras, knot invariants, diagram algebras and representation theory.

Connections for Women, January 16-18, 2008 Nearly 60 mathematicians, ranging from graduate student to emeritus professor, regis- tered for the workshop, and roughly that many attended, with the addition of several more MSRI members from both programs. This included roughly a dozen men and non-MSRI- member participants from as far as Australia, Uruguay, and . The scientific part of the workshop consisted of seven talks over the three days. The first day focused on Combinatorial Representation Theory, the second on Representation Theory of Finite Groups and Related Topics, and the third day on a historical overview of the field. The level of exposition in all talks was outstanding. The first day also included a poster session. Young researchers were encouraged to showcase their work. The formal session was preceded by a “poster preview” in which participants stood up and gave a 2-minute synopsis of their poster. This was very successful in both generating interest in later viewing of all the posters and in giving the younger participants a chance to advertise themselves to the whole group. The dozen or so 2-minute combinatorial representation theory 7 explanations were articulate and interesting and one of the posters was pictured in an article that the San Francisco Chronicle ran on January 25, 2008 about the events at MSRI. Participants also had the opportunity to advertise their research by submitting short (2 page) research abstracts. These were made available online on the workshop web page, and were copied and distributed during the workshop. Two panel discussions were held in an informal setting, with the participants and pan- elists alike sitting in a circle. Both panels were extremely popular and we received copious positive feedback. On the first day was a panel “From small colleges to large universities and everything in between.” The three panelists (moderated by Berkeley’s Jenny Harrison) represented three very different types of schools. Each gave 5-minute introductions to what it was like to be there and how their career paths led them there. The second day’s panel had the intriguing title “Three things I wish I knew then,” and consisted of a discussion of various issues that might arise in a young researcher’s early career. Some of the valuable advice given included being very honest about yourself and what you want while interviewing, how and when to bring up the 2-body problem, and to make connections to senior mathematicians and maintain those contacts. There were interesting discussions about how students’ perceptions are affected by the expectation that women should be more maternal and how to organize a local student AWM chapter or a “Noetherian Ring” at one’s home institution. Both panels elicited a lot of enthusiastic and lively discussion. After the scientific talks on the third day was a viewing of the film “Women and Math- ematics across Cultures” produced by the EWM in 1996. The video explores the impact of cultural differences on female professional lives. In the video, four women mathematicians who studied and worked in Europe and North and South America to tell their stories. In the discussion after the viewing European mathematicians in the audience uplifted spirits by refuting some of the negative experiences of some of the women in the film and noting how much things have improved over the past 10 years. The atmosphere at the conference banquet and other social activities was very cohesive. In addition to many members at MSRI, AWM president , former AWM presi- dent Bhama Srinivasan, and current president-elect Georgia Benkart were all in attendance.

Introductory workshop, January 22-25, 2008 The soul of Combinatorial Representation Theory lies in the interplay between combi- natorics and various branches of mathematics. Combinatorial methods are applied to solve problems in representation theory, Lie theory, geometry, and mathematical physics and, in symbiosis, deep combinatorial problems also arise from these areas. The goal of the Intro- ductory Workshop was to survey current and recent developments in the field, and set the stage for the focus of the program. There were over 150 registered participants from all over the world. Of these approximately 40 were women, 30 postdocs, and 50 graduate students. Lecture series: There were three lecture series: • Michel Brou´e, Complex reflection groups in representations of finite reflection groups, • Andrei Okounkov, Characters of symmetric groups • Arun Ram, Combinatorics of Lie type 8 msri spring 2008: final report

The lecture series of Brou´e,Okounkov and Ram were designed to provide the basic funda- mentals of the fields and explain the role of these topics in current research. The lecture series of M. Brou´eprovided a bridge between the Representations of Finite Groups program and the Combinatorial Representation Theory program. In these lectures he introduced finite reflection groups, unipotent characters, Deligne-Lusztig induction and restriction, and Harish-Chandra theory. The lectures of Arun Ram set out the fundamentals of the combinatorics of root systems and path models. In these lectures he gave definitions of Hecke algebras, double affine Hecke algebras, Macdonald polynomials, crystals, Chevalley groups, loop groups, flag varieties, loop Grassmanians and Mirkovic-Vilonen cycles. In total this formed a comprehensive overview of the combinatorics of Lie types. Andrei Okounkov (a Fields medallist) started with a survey of the character theory of the symmetric group. He proceeded to relate this to the Fock space realization, Gromov- Witten theory, the Hurwitz problem and finally the amazing recent results by Okounkov and Pandharipande on the relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves. Research talks: To complement the lecture series the postdoctoral fellows of the program each gave an hour long presentation introducing their research area. The topics were • Kevin Purbhoo, The Horn Inequalities and Their Generalizations • Syu Kato, Geometric representation theory of affine Hecke algebras • Lauren Williams, Total positivity for flag varieties: combinatorics, topology, and toric geometry • Ghislain Fourier, Finite dimensional modules for current and loop algebras • Sami Assaf, Applications of dual equivalence graphs These talks set the trend for the research activities pursued during the program. Sami Assaf gave a beautiful account on her recent work on dual equivalence graphs, which she used to give a combinatorial proof of the Macdonald-Kostka positvity, and related these graphs to crystal graphs. Ghislain Fourier presented the analysis of finite-dimensional modules for current and loop algebras. In particular he showed how to treat the twisted cases. This ties in with work of David Hernandez, another member of the program, on twisted q-characters for Kirillov-Reshetikhin modules for all types. 5-minute presentations: In addition, there were multiple sessions of 5-minute presenta- tions where the remaining participants were able to introduce themselves mathematically and gave a feel for their current research. These sessions were a great success (there were more volunteers than we could accommodate to speak)! The talks gave an overview of what our community is working on and enabled the participants of the program to make connections with each other. Arun Ram gave a sample 5 minute presentation about p-compact groups to stimulate the participation. Other highlights included Anne Schilling (affine Schubert calculus), David Hernandez (quantum affine algebras), Nat Thiem (supercharacters), Jason Bandlow (Macdonald polynomials), Brant Jones (Kazhdhan-Lustig polynomials), Monica Vazirani (crystals), Shona Yu (Brauer algebras), Mansaoru Koyama (discrete Fourier trans- form).

Lie Theory workshop, January 22-25, 2008 combinatorial representation theory 9

What the two Spring 2008 MSRI programs, Combinatorial Representation Theory and Representation Theory of Finite Groups, had in common was the central role played by Lie theory. In combinatorial representation theory the most important combinatorial objects used to model representations arise from Lie theory (tableaux, Littlewood-Richardson coef- ficients, Kazhdan-Lusztig polynomials, etc.). In finite groups there are several trends making Lie theory central. Some finite groups are either naturally a part of Lie theory (finite groups of Lie type) or are very closely connected to it on many levels (symmetric groups). Also, the theory of p-compact groups is a recent development providing new bridges between finite group theory, algebraic topology and complex reflection groups. So it was very natural for the two MSRI programs to run a joint workshop on Lie Theory. The workshop followed a standard MSRI 5 day workshop template. There were four 60 minute talks each day, except for Wednesday, when there were just two 60 minute talks in the morning. The structure seems to be optimal, as the uncrowded talk schedule left enough time for discussion. Each speaker was been given a short introduction by a ‘classic’ in that area (for example Robert Steinberg introduced George Lusztig, Charlie Curtis introduced Meinolf Geck, etc.). The new facilities at MSRI received positive comment: seating, visibility, blackboards, multimedia, etc. all seem outstanding. The general level of excitement of all participants was very high. Many participants gave very favorable reviews to the workshop, which has been one of the biggest such events in the history of MSRI. Significant research accomplishments were described in many talks. The auditorium was packed during all talks of the workshop. The following were some highlights of the conference: • Sergey Fomin gave an expository lecture on cluster algebras. This talk was part of the series of MSRI Evans talks—lecture series designed especially for students and faculty members of UC-Berkeley and given at the Evans Hall on campus. Cluster algebras arise in various algebraic and geometric contexts, with combinatorics providing a unifying framework. The presentation of the basic definitions and results of this emerging theory was guided by two sets of examples: coordinate rings of classical algebraic varieties, and cluster algebras associated with bordered oriented surfaces with marked points. The topic of cluster algebras has been continued in the talk of Bernard Leclerc who introduced the notion of monoidal categorification of a cluster algebra, and gave examples of such categorifications coming from the representation theory of quantum affine algebras. • Peter Fiebig gave a talk on Lusztig’s conjecture for characters of irreducible represen- tations of algebraic groups over a field of positive characteristic. Fiebig related sheaves of vector spaces on a complex affine flag variety to representations of the Lie algebra associated to Langlands dual root system. From this he extracted a new proof of Lusztig’s multiplicity conjecture for almost all characteristics. The main step in the construction of the above relation was a categorification of a natural map from the affine Hecke algebra to its periodic module via the theory of sheaves on moment graphs. This categorfication provides a non- topological proof of the multiplicity one case of Lusztig’s conjecture for all characteristics above the Coxeter number. • Edward Frenkel (UC-Berkeley) discussed his recent work with D. Gaitsgory, where the give conjectural description of the categories that the local geometric Langlands corre- spondence assigns to a local system on the punctured disc for the Langlands dual group of a complex G. These categories are given as categories of representations 10 msri spring 2008: final report of the corresponding affine Kac-Moody algebra of critical level. Sometimes these categories may also be realized as categories of D-modules or O-modules on some algebraic varieties or ind-schemes. Interrelations between these categories provide supporting evidence for the conjectures. In particular, the categories of Iwahori equivariant representations of critical level with fixed central character are equivalent to the categories of quasicoherent sheaves on the Springer fibers the Langlands dual group. • Meinolf Geck gave a spectacular talk on James conjecture for Hecke algebras of ex- ceptional type. The original James conjecture describes characters of irreducible modular representations of the symmetric group in terms of a corresponding theory for Hecke alge- bras at roots of unity. The original conjecture is still completely open, and, moreover, it is far from clear what the analogue for other types should look like. Recently Geck showed that Hecke algebras of finite type are cellular in the sense of Graham-Lehrer. This lead to a natural generalisation of the theory of Specht modules to Hecke algebras of any (finite) type and, in this framework, he has formulated a general version of James conjecture. In his talk, he described his recent joint work with Juergen Mueller, where they prove James’ conjecture for Hecke algebras of exceptional type. • George Lusztig gave a foundational talk on reductive algebraic groups. He explained how to use canonical bases theory to prove a result stated without proof by Kostant in 1966! The result is an explicit construction of the coordinate ring of a reductive algebraic group over the integers. Lusztig explained why such a proof could not be given in 1966. • Victor Ostrik presented a major development on Lusztig’s asymptotic Hecke ring. The explicit description of the asymptotic Hecke ring was laid out in a series of conjectures by Lusztig. In his talk, Ostrik described his recent joint work with R. Bezrukavnikov and M. Finkelberg in which they prove Lusztig’s conjectures in a very satisfactory way, using the theory of tensor categories. A remarkable use of classification results on monoidal tensor cat- egories provides an explanation of what goes wrong in the cases where Lusztig’s conjectures do not hold, providing a complete solution to the problem of determining the structure of the asymptotic Hecke algebras in these cases also. At the end of the talk, Ostrik presented an exciting and important new conjecture, due to him and Bezrukavnikov, which connects Lusztig’s asymptotic ring to representation theory of finite W -algebras. This provided a link to the number of talks on the conference which were dedicated to finite W -algebras, and generated a lot of discussion between the experts (Ostrik, Premet, Losev, Kleshchev, Goodwin, Brown). • Alexander Premet and Ivan Losev gave talks on the representation theory of finite W -algebras. One of the major problems in the theory is to classify finite dimensional rep- resentations. This problem is closely related to the theory of primitive ideals in universal enveloping algebras, deformations of singularities, quiver representations, and physics. In general, very little is known about representation theory of finite W -algebras outside of type A (which has been treated by Brundan and Kleshchev) and until recent work of Premet it was not known if general finite W algebras have any finite dimensional representations. A conjecture of Premet is that a 1-dimensional representation always exists. In his talk, Premet presented a very interesting positive solution to this problem for classical types using reduc- tion modulo p and lifting back, and results of Barbash and Vogan on primitive ideals. Ivan Losev presented a completely different approach to W -algebras based on the ideas of Fedosov quantization. He observes that a finite W -algebra is the invariant algebra for an action of a combinatorial representation theory 11 reductive group on a quantized symplectic affine variety. His results include an alternative definition of W , a relation between the sets of prime ideals of W and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation in classical types, and the separation of elements of the algebra by finite dimensional representations.

Topics in Combinatorial Representation Theory, March 17-21, 2008 20th century combinatorics taught us that representation theory is often the key to puz- zles involving our favorite combinatorial objects. In the reverse direction, answers to many central questions in representation theory required development of sophisticated combinato- rial techniques and constructions. This interplay, which has only intensified in recent years, was the focus of this workshop. It was aimed both at researchers and advanced graduate students working at this interface between representation theory and combinatorics. More specifically, the goal was to have speakers apprise them of the current status, problems and frontiers in the following hot areas of recent research: * quiver representations; * cluster algebras; * Macdonald and LLT polynomials; * representation-theoretic techniques in quantum/statistical mechanics; * crystal bases, graphs and Littelman path models; * affine Grassmannians, affine Schubert calculus, Mirkovic-Vilonen cycles; * dual canonical and semi-canonical bases; * combinatorial Hopf algebras. The workshop was organized following a standard MSRI weeklong research workshop template. There were four 50-minute talks each day, two each in the morning/afternoon, every day except Wednesday. The talks were grouped thematically on each day, e.g. Tuesday was the day for cluster algebras, Wednesday morning for path models, Thursday morning for affine Schubert calculus. The uncrowded talk schedule stimulated many questions after each talk as well as plenty of time for informal discussions during the breaks. Many participants gave favorable reviews to the new MSRI facilities. Even though the lecture hall was packed for many talks, it didn’t feel that way. Several talks announced “big news” research developments. • The very first talk at the workshop, given by John Stembridge, was simultaneously video- streamed to an Atlas of Lie Groups workshop (http://www.liegroups.org/papers/) so that, in a sense, he spoke at two workshops simultaneously. The MSRI computer people handled this challenge excellently. • Mark Haiman announced his recent work (joint with Ian Grojnowski) proving the conjec- ture that LLT polynomials expand positively in terms of Schur functions, and discussing its ramifications. In particular, the correct generalization of the LLT polynomials to other root systems/types was an important and illuminating part of the picture. • Arun Ram discussed his recent work ([arXiv:0803.1146], joint with Martha Yip) on the appropriate generalization of the type A Haglund-Haiman-Loehr formula for Macdonald polynomials to all types. The key tools were (folded) alcove walks parametrizing bases for Cherednik’s double affine Hecke algebra, and the connection to Schwer’s formula for 12 msri spring 2008: final report

Hall-Littlewood polynomials was made clear. There has already been follow-up work on this [arxiv:0804.4716] by another of the workshop’s speakers, Cristian Lenart, whose workshop talk was partly a ”preliminary report” for this work. • Jan Schroer explained work (joint with Christof Geiss and Bernard Leclerc) on how the representation theory of preprojective algebras can help in understanding a large class of cluster algebras arising in Lie theory, as coordinate rings of unipotent cells of Kac- Moody groups. In particular, this allows one to apply Lusztig’s theory of semicanonical bases and to obtain some semicanonical bases of these cluster algebras which contain all the cluster monomials. • Joel Kamnitzer gave his talk, joint with the Berkeley Colloquium, to a packed crowd in Evans Hall. He gave a very accessible account of his recent construction (together with Sabin Cautis) of a new categorification of the Jones and HOMFLY polynomials in knot theory. It was based on derived categories of equivariant coherent sheaves on the affine Grassmannian, and closely related with the talk in the workshop by Michael Kapovich on the Horn problem. Some ambience of the mathematical research activity It is not possible to describe all the exciting projects that were being worked on, all of the stimulating discussions that were going on, and all of the new connections that were being made this semester. Perhaps it is possible to give a feel for the energy in the build- ing throughout the semester by choosing a few representative topics which captured our attention. Interaction of geometry, representation theory and combinatorics The fascinating connection between geometry, representation theory and algebraic com- binatorics was a unifying topic of interest among MSRI members throughout the semester. Many formulas in algebraic combinatorics have a representation theoretic as well as an alge- braic geometric interpretation. This connection is very natural since many representations can be realised in a (algebraic) geometric context, and representation theoretic constructions have their geometric counterparts. The general philosophy is that a “really good” understanding of a formula can only be achieved once one has a good understanding of both aspects. One example for such a is formula is the n!-conjecture, another is the path model of a representation with its various interpretations in the combinatorial framework of models for crystals, its K-theoretic interpretation, its algebraic geometric interpretation in terms of galleries in the building etc. In February, Arun Ram and Martha Yip, both in residence at MSRI, discovered a new combinatorial formula for Macdonald polynomials. This new formula is valid for all root systems. One of the most exciting aspects of the Ram-Yip formula is the fact that it is in terms of the path model, which also has an algebraic geometric interpretation in terms of galleries in the building. The form of the new formula is the same as that of the formula of Haglund-Haiman-Loehr for type GL(n), but there is a fascinating, and not very well understood, collapsing of terms that relates the two formulas. Recent preprints of Cristian Lenart study this collapsing of terms. The connection between the path model combinatorics and the algebraic geometric interpretation was the centerpiece of of discussions at MSRI between Peter Littelmann and Cristian Lenart. The compression of terms seems to combinatorial representation theory 13 have an algebraic geometric background related to the interpretation of galleries (or alcove walks) in the framework of affine buildings an the affine grassmannian. The search for a better understanding of structure of flag varieties, Grassmanians, Schubert varieties and Demazure modules was an everywhere dense topic throughout the semester. Peter Littelmann and Alain Lascoux were able to use standard monomial the- ory to give geometric interpretations of some new decompositions of certain characters of Demazure modules for classical type groups discovered by Lascoux. Schubert varieties and Demazure modules were also the focus of many discussions (including Ghislain Fourier, David Hernandez, Bernard Leclerc, Anne Schilling and Peter Littelmann) about the pos- sible generalizations to the twisted setting of the geometric realization of special classes of finite dimensional representations of untwisted affine Kac-Moody algebras. One of the special families of finite dimensional modules of the affine quantum group is the family of Kirillov-Reshetikhin modules. Work of Littelmann and Fourier has confirmed that these arise as Demazure submodules (corresponding to a finite dimensional Lie alge- bra) inside a highest weight representation of the affine Lie algebra. For nonexceptional types, Okado and Schilling have recently proven that crystals for these modules exist. At MSRI Ghislain Fourier and Anne Schilling, in collaboration with Masato Okado, worked on combinatorial models for these Kirillov-Reshetikhin crystals. Their goal is to find a unique characterization for the Kirillov-Reshetikhin crystals Br,s thus enabling one to show that the combinatorial models for these crystals coincide with the crystals coming from the Uq(g)-modules. Of great interest, is not only the study of finite dimensional representations of affine Lie algebras and quantum groups but the relation between affine and finite type. In this vein, Jason Bandlow, Anne Schilling and Nicolas Thi´eryhave proved that there is a unique connected promotion operator which produces an affine crystal structure on a tensor product of two finite type crystals labeled by rectangles. Another development along these lines was the study of Hecke group algebras by Anne Schilling and Nicolas Thi´ery. Progress on this topic was stimulated by the lectures of Arun Ram during the Introductory Workshop and subsequent discusssions with him as well as many specialists at MSRI – such as Francesco Brenti, Mark Shimozono, John Stembridge, and Monica Vazirani. The Hecke group algebras are obtained by appropriately gluing the 0-Hecke algebra of a Coxeter group to its group algebra. Schilling and Thi´eryproved that the resulting algebra is a natural quotient of the affine Hecke algebra through its level 0 representation, explaining, in particular, why there are so many similarities between the representation theory of the classical 0-Hecke algebra and the affine Hecke algebra. One of the many classical appearances of Kostka-Foulkes polynomials is as (graded) de- composition numbers for the restriction of a finite dimensional representation of the affine Lie algebra to the subalgebra of finite type. The charge of a tableaux is a combinatorial statistic which gives a formula for the Kostka-Foulkes polynomials. Despite many attempts and ex- tensive discussions at MSRI between Pierre Baumann, Bernard Leclerc, Alain Lascoux, and Stephane Gaussent, the charge of a tableaux, defined by Lacoux and Schutzenberger, is, from the algebraic geometric point of view, still mysterious. Though the Kostka-Foulkes polyno- mials, have an important representation theoretic (q-weight multiplicities) and algebraic geometric underpinning (singularities of Schubert varieties) no satisfactory interpretation of the charge of a tableaux has been found. 14 msri spring 2008: final report Macdonald polynomials We have already mentioned the new combinatorial formula for Macdonald polynomials discovered by Arun Ram and Martha Yip. But this was only the tip of the iceberg for the MSRI activity around Macdonald polynomials. The stimulating stays of Adriano Garsia created great excitement. He arrived at the end of February equipped with an inspiring vision for a new, combinatorial, attack on the n! conjecture. In subsequent discussions, Sami Assaf learned the idea of “kicking” from Adriano Garsia and soon after created a shocking, conjectural, algorithm for producing for producing an explicit basis of the Garsia-Haiman modules. Proving that the new algorithm produces a basis with the desired properies is a challenge. Assaf and Garsia have succeeded in showing that the new method works for two column partitions and for hook shapes. The expectation is that the new algorithm will work in general. In a related development, there was significant interest in a new construction of Lee Chung, a student of Mark Haiman, which constructs a module which has graded character matching the k-Schur functions. This is analogous to the way that the graded character of the Garsia-Haiman modules match the Macdonald polynomials. Many MSRI members were interested in these new developments, since all indications are that the k-Schur functions can be viewed as building blocks for the Macdonald polynomials. The presence of Alain Lascoux at MSRI created a large amount of activity around the topic of Macdonald polynomials. One of the wonderful ideas he contributed was the idea that one can use “transition” to provide new combinatorial formulas for Macdonald polynomials. In his seminar talk during his stay at MSRI, Alain Lascoux followed up on the lectures of Phillipe DiFrancesco, and explained a miraculous way of connecting Macdonald polynomials to the remarkable combinatorics of alternating sign matrices, plane partitions and the Razumov-Stroganov conjecture. Extending this exciting idea, Alain Lascoux and Lauren Williams have begun investigating type B alternating sign matrices. During their stay at MSRI, Ole Warnaar and Alain Lascoux studied “interpolation” Macdonald polynomials and established combinatorial formulas for them. The Lascoux-Leclerc-Thibon (LLT) polynomials are building blocks for the Macdonald polynomials and the talk of Mark Haiman at the MSRI workshop Lie Theory focused on this connection. Arun Ram and Peter Tingley have begun a new collaboration to extend the known path model and crystal methods to give a combinatorial understanding of LLT polynomials and “level `” Fock spaces. Special cases of this combinatorics are known: the affine sl crystal based on the Young lattice and the Hecke algebra description of type A Fock space given by Leclerc and Thibon. Can these models be extended to more general settings? The approach for generalising these models is to combine a Hecke algebra construction of Fock space and a path model generalization of the Young lattice crystal. to design models for general type Fock space crystals. It is conjectured that the limit of the level ` Fock spaces gives a representation which realises the Macdonald polynomials.

Applications of combinatorial representation theory and computational ad- vances Several significant projects provided beautiful applications of combinatorial representa- tion theory. One of the striking results was the discovery by Lauren Williams, J.C. Novelli combinatorial representation theory 15 and J.Y. Thibon of a connection between the asymmetric exclusion process and combinato- rial Hopf algebras. Several important features of the stationary distribution of the process can be read directly from the combinatorial Hopf algebra perspective and there is further data available on the Hopf algebra side which, so far, is not yet understood in terms of the asymmetric exclusion process. Lauren was a postdoctoral fellow at MSRI for the whole semester and J.C. Novelli visited MSRI for a short period. Sami Assaf, Persi Diaconis and Kannan Soundararajan are completing a beautiful project on the study of random walks on cosets. A particular case of interest, where the group is the symmetric group and the subgroup is a Young subgroup), corresponds to the analysis of shuffles of a bicolored deck of cards. They show that logn shuffles are sufficient to mix up a deck with n cards which are half red and half black. The proof of these results uses representation theory (character formulas for the symmetric group evaluated at trans- positions), combinatorics (to get formulas for which decks are mostly likely to appear), and probabilistic and analytic methods (to get asymptotics for distance to uniformity). Computer exploration has been an invaluable research tool for suggesting and testing conjectures in combinatorics and combinatorial representation theory, requiring the imple- mentation of new features and algorithms in the field of symmetric function theory, geometry, Kac-Moody algebras and physical models. Several MSRI members are members of The Fo- cused Research group on ”Affine Schubert calculus” had a significant presence at MSRI with many members (Jason Bandlow, Francois Descouens, Anne Schilling, Mark Shimo- zono, Nicolas Thi´ery, Mike Zabrocki) being in residence for varying amounts of time during the semester. On of the goals of this research group is to share computational software development efforts between the participants, and at the end to make it freely available. Under the leadership of Florent Hivert and Nicolas Thi´ery, the open source algebraic combi- natorics package MuPAD-Combinat (http:://mupad-combinat.sf.net/) has been developed since 2001. The rapid growth of Sage (www.sagemath.org) makes it a much more viable alternative for a combinatorics package. Sage was started in 2005 by William Stein (now at the University of Washington) and it already consists of over two million lines of code. It incorporates several of the best free, open-source mathematics software packages available (GAP, Singular, Macaulay, GMP, MPFR just to name a few), as well as a huge original library, including several new algorithms not yet found elsewhere. About one year ago, Mike Hansen contacted the MuPAD-Combinat team. since then he has ported to Sage about thousands of lines of code of MuPAD-Combinat (about 1/3 of the total). With Mike Hansen and Nicolas Thi´eryin residence at MSRI for the semester, the coordination of the shift from MuPAD-Combinat to Sage-Combinat was greatly enhanced. It was also the occasion to start collaborating with Daniel Bump and Justin Walker (Stanford). Finally, during the last week of the program, the MSRI hosted a coding sprint where a large group of people came together to port further features from MuPAD-Combinat to Sage- Combinat (Mike Hansen, Nicolas Thi´ery, Jason Bandlow, Mark Shimozono, Brant Jones, Tom Denton, Nicolas Borie, Anne Schilling).

Cluster algebras, quivers, and quantum affine algebras Cluster algebras have been an important hot topic discussed by several members of the program from various points of view. One of the inventors of the theory, Sergey Fomin, spent 3 months at MSRI, during which he was always ready to share his knowledge and help 16 msri spring 2008: final report newcomers find their way in this quickly expanding new subject. In particular, he explained very convincingly to several of us the geometrical model for a large family of cluster algebras based on the combinatorics of triangulations of bordered hyperbolic surfaces with marked points, and he demonstrated its deep relations with Teichmuller theory, Penner coordinates, and Thurston’s laminations. Christof Geiss, Bernard Leclerc, and Jan Schr¨oer have completed their joint work on another geometric realization of yet another large class of cluster algebras, as coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. This generalizes to the Kac-Moody setting part of the foundational material developed by Fomin and Zelevinsky with various collaborators (Berenstein, Scott). But the approach is different, since GLS base their analysis of the cluster structure on a categorification given by appropriate Frobenius subcategories of the module category of the preprojective algebra attached to the Kac-Moody data. This allows them to relate these cluster algebras to previous work of Lusztig, and to construct in a uniform way semicanonical bases of the cluster algebras in question, which contain the set of cluster monomials. During his three week stay, Bernhard Keller lectured about his recent important result describing the mutation of quivers with potentials, due to Derksen, Weyman and Zelevinsky, in terms of equivalence of triangulated categories coming from Ginzburg DG-algebras. He discussed extensively with Leclerc about the particular examples of this setup coming from the above GLS results. Bernard Leclerc and David Hernandez have started exploring a new direction, linking cluster algebras and quantum affine algebras. They were motivated by a conjecture of Leclerc, stating that the Grothendieck rings of some apropriate monoidal categories M` of representations of Uq(bg) have a natural cluster algebra structure, whose cluster monomials are classes of simple objects of M`. To test the conjecture and gather some evidence, they explored a nice finite subset of the infinite set of (conjectural) clusters of M`, whose Drinfeld polynomials are expected to be labelled by certain roots of an infinite root system. This leads to a (conjectural) description of the general solution of a Y-system of type Xn × A`, where Xn is the Dynkin type of g, in terms of q-characters of Uq(bg). The conjecture has been proved in case Xn = An and arbitrary `, and also checked for D4,D5 when ` = 1. In another direction, Rinat Kedem and Philippe DiFrancesco have obtained an interpretation of the Q-systems attached to Uq(bg) in terms of cluster algebras (these are not the same cluster algebras as those studied by Leclerc and Hernandez). They proved for these cluster algebras a strenghtening of the Laurent phenomenon, namely, that if half of the 2n variables of a special cluster C are conveniently specialized, all the other cluster variables become polynomials in the remaining 4n variables of C. Other events and connections MSRI/Evans lectures • February 11, 2008, Alexander Kleshchev, Symmetric groups. The symmetric group is of course one of the most classical and basic mathematical objects. It is also known to be deep (no contradiction there!). This talk was about representation theory of symmetric groups—starting with Frobenius and ending quite recently with results of Ariki, Grojnowski, Rouquer and Chuang on representation theory of symmetric groups over a finite field. combinatorial representation theory 17

• February 25, 2008, Georgia Benkart, GL(n) representations and more – yeah Schur! Issai Schur’s dissertation on the representations of the general linear group GL(n) of invertible n by n matrices was the thesis that launched a thousand papers. It influenced work on symmetric functions, diagram algebras, knot and link invariants, and much more. • March 10, 2008, Sergey Fomin, Cluster algebras. Cluster algebras arise in various algebraic and geometric contexts, with combinatorics providing a unifying framework. My presentation of the basic definitions and results of this emerging theory will be guided by two sets of examples: coordinate rings of classical algebraic varieties, and cluster algebras associated with bordered oriented surfaces with marked points. • April 28, 2008, Lauren Williams, Combinatorics and statistical physics: a story of hopping particles. The asymmetric exclusion process (ASEP) is a simple but rich model from statistical physics concerning particles hopping on a 1-dimensional lattice: it serves as a primitive model for traffic flow and appears in a sequence alignment problem in computational biology. This talk will provide a gentle introduction to the ASEP followed by connections of the ASEP to combinatorics, including the totally non-negative part of the Grassmannian and combinatorial Hopf algebras. Running seminars during the semester An average week had about 8 seminar talks, not counting the talks down the hill at Evans Hall. There were two official seminars • Combinatorial Representation Theory seminar organised by Anne Schilling • Applied Representation Theory seminar organised by Persi Diaconis and many “unofficial” seminars. We encouraged the unofficial seminars since they were fo- cused on specialised topics and had the feel of “producing” results as opposed to “reporting” results. Many people in the program were actively participating in the RTG seminars • Seminar on representation theory of symmetric groups and closely related objects • Seminar on homological methods in representation theory • Seminar on representations of finite groups of Lie type and the complementary program seminar • Combinatorial Reciprocity Theorems organised by Matthias Beck. Perhaps the most exciting of all was the • Joint postdoctoral seminar organised by Sami Assaf which was certainly attended by a large cohort of nonpostdocs. The complete list of presenta- tions in seminars during the semester is available at http://www.msri.org/calendar/index seminars

NSF site visit One of the exciting weeks of the semester was the occasion of the NSF site visit, April 17-18, 2008. Many MSRI members contributed to the visit by meeting with the NSF team. There were special meetings with the program organisers, the workshop organisers, and the postdoctoral fellows and it was a welcome opportunity for the members to get to tell NSF a bit about the activities of the Combinatorial Representation Theory program. 18 msri spring 2008: final report Interaction with the nearby community There were a semi-infinite number of connections between the members at MSRI and activity in combinatorial representation theory at other institutions. Some examples are: – One of the highlights of the early part of the semester was a stimulating talk (a joint Combinatorial representation theory/Applied respresentation theory seminar) by Dan Bump from Stanford University entitled “Deformations of the Weyl Character formula and multiple Dirichlet series. This talk made connections between number theory, path models, crystals, character formulas and much of the research on these topics that was happening at MSRI this semester. In particular, the SAGE group began a new collaboration with Dan Bump and Justin Walker and stimulating relations were noticed between the work of Bump and collaborators and the new formula for Macdonald polynomials discovered by Ram and Yip. – There was no shortage of interaction between MSRI members and neighboring institu- tions. Among the speakers at the UC Davis Discrete Mathemtics and Represen- tation Theory seminar in Spring 2008 were MSRI members Brendon Rhoades, Nantel Bergeron, John Stembridge, Tom Halverson, Robert Guralnick, at the UC Riverside Lie theory seminar were MSRI members Rinat Kedem, David Hernandez, Michael Lau, Bernard Leclerc, Henning Andersen and Sergey Loktev, at the Algebra-Geometry- Combinatorics seminar at San Francisco State University were MSRI members Bren- don Rhoades, Adriano Garsia, Richard Stanley, Luis Serrano, H´el`eneBarcelo, Francesco Brenti, Kelli Talaska, Nantel Bergeron, and Lauren Williams, MSRI member Brendon Rhoades was a speaker in the UC Berkeley Representation Theory, Geometry and Combinatorics seminar and the Stanford combinatorics and geometry seminar saw MSRI members Mathias Beck, Arun Ram, Lauren Williams and Carlos Andr´eamong the list of speakers. – The Bay Area Discrete Math Day, Spring 2008 was held April 5, 2008 at MSRI. Sami Assaf was a main organiser of this workshop sponsored by MSRI and DE Shaw. Many MSRI members participated. MSRI members who were invited speakers included Rosa Orellana and Kevin Purbhoo. Rosa Orellana’s talk covered joint work done at MSRI with Andrew Mathas. – The 4th annual graduate student combinatorics conference was held at UC Davis, April 12-13, 2008. MSRI member Arun Ram was a keynote speaker, and many of the graduate students at MSRI attended and gave talks (two students of H´el`ene Barcelo, 4 students of Arun Ram, 2 students of Sergey Fomin, Mike Hansen, and one student of Peter Webb). – The 2008 Spring Western Section Meeting of the AMS in Claremont, CA, May 3-4, 2008 had a major MSRI presence. Anne Schilling was an invited plenary speaker and Mike Zabrocki and Anne Schilling organised a special session on Algebraic Combinatorics (see http://garsia.math.yorku.ca/claremont) Among the speakers were MSRI members Thomas Lam, J.C. Novelli, Nicolas Thi´ery, Adriano Garsia, and Mike Hansen. Several of these talks covered work done at MSRI, notably the talk of N. Thi´ery on Hecke group algebras, J.C. Novelli talked about joint work with Lauren Williams, and Jason Bandlow talked about joint work with Anne Schilling and Nicolas Thiery on promotion operators. The talk of Adriano Garsia outlined his ideas for giving a combinatorial representation theory 19

combinatorial proof of the n! along the lines of the techniques which are being pursued in joint work with Sami Assaf. – A special pleasure for the field of combinatorial representation theory was the occasion of a conference in honor of the career of MSRI member Georgia Benkart, held at UC San Diego, February 12-14, 2008. Many MSRI members attended the conference and MSRI members who gave invited talks were Edward Frenkel, Arun Ram and Tom Halverson.

Postdoctoral fellows For the senior members, one of the most stimulating aspects of the program was the presence, excitement, energy, and brilliance of the postdocs. The most exciting aspects of the programs often seemed to come from the postdocs. In particular, one thinks of the exciting work on affine Lie algebra representations of David Hernandez and Ghislain Fourier and their coauthors, of the exciting developments towards a combinatorial proof of the n! conjecture by Assaf, and the use of modular perverse sheaves for computing decomposition numbers as is being pioneered by the work of Daniel Juteau. Several new collaborations resulted from their stay at MSRI. Just to mention a few, David Hernandez has new collaborations with Bernard Leclerc and Edward Frenkel, Sami Assaf has new collaborations with Persi Diaconis and Kannan Soundararajan, and Ghislain Fourier continued his collaboration with Anne Schilling, and Arun Ram has begun a new collaboration with Lauren Williams. The best description of the MSRI postdoctoral experience comes from the postdocs themselves: • David Hernandez My visit to MSRI was very fruitful as I could present my work and results in several talks, and I could start several collaborations which I think will greatly influence my future research. As an example, in discussions with Bernard Leclerc we started to understand how to realize some cluster algebras in the Grothendieck ring of finite dimensional representation of quantum affine algebras. We hope representation theoretic interpretation of the these cluster algebras will lead to two new applications: a better understanding of the Grothendieck ring, new information on the structure of the corresponding cluster algebras. • Syu Kato The research I have done at MSRI is basically classified into three projects. All of them are more or less related to affine Hecke algebras and rational Cherednik algebras. During the semester at MSRI I was able to take two one-week trips to University of Utah. It was a consequence of these trip that I was able to begin an exciting new collaboration with P. Trapa and D. Ciubotaru. During the semester I gave two talks in MSRI and one talk at University of Utah. The first project is to understand the structure of some particular irreducible compo- nents of so-called ”exotic Springer fibers”, which is an analogue of Springer fibers which carries the action of affine Hecke algebras of type C with unequal parameters (at the level of its homology). The goal would have been to describe the explicit shape of such fibers in a way one can extract basically all information. I have obtained a conjectural characterization of the class of these fibers which is valid for n ≤ 3 or some specific case, but have not yet obtained the proof in general. 20 msri spring 2008: final report

In my previous work, we established a so-called ”exotic Deligne-Langlands correspon- dence”, which is an analogue of the Deligne-Langlands classification of irreducible of affine Hecke algebras of type C for unequal parameter case. A second project, which is on-going with Dan Ciubotaru at Utah, is to understand this in terms of the usual Langlands classi- fication. The result we obtained includes a description of the discrete series for the exotic Springer fiber geometry, the description of the Iwahori-Matsumoto involution for this discrete series, and the characterization of tempered modules which contain the sign representation as a finite Weyl group module. A branch of this second project is to study the compactification of our parameter space as in Zelevinsky and Ciubotaru-Trapa. My third project was to try to understand what the representation theory of rational Cherednik algebras looks like by analyzing the quiver varieties and using a Gordon-Stafford type construction. The result I obtained is somewhat similar to Ginzburg-Gordon-Stafford and I am now searching for a way to deepen it with Toshiro Kuwabara at Kyoto. • Jon Kujawa While a postdoctoral member at MSRI I had the paper “Cohomology and Support Varieties for Lie Superalgebras, II” (coauthored with Brian Boe and Daniel Nakano of the University of Georgia) accepted by the Proceedings of the London Mathematical Society. Irfan Bagci, Daniel Nakano, and I completed the writing of a paper entitled “Cohomology and Support Varieties for Lie Superalgebras of Type W (n)” and anticipate submitting it to a journal within the next few months. During visits by Brian Boe and Daniel Nakano to MSRI we continued with our collaboration. We are now considering several topics, including generalizing our earlier work to the modular and quantum cases. We also are considering ways to link our work in these papers to the parallel and independent work of Duflo and Serganova on associated varieties for Lie superalgebras. I presented aspects of my work with Bagci, Boe, and Nakano in the MSRI postdoctoral seminar and as an invited speaker in the “Homological Methods in Representation Theory” Conference hosted by MSRI. Additionally, I spoke on related results as an invited speaker in the “Special Session on Geometric and Combinatorial Representation Theory” at the AMS sectional meeting in Baton Rouge, LA on March 28-30, 2008. While at MSRI I initiated several new collaborations. The primary new project involves work with David Hill and Joshua Sussan (of UC Berkeley) to generalize work of Arakawa and Suzuki to construct a functor from the category O for the of type Q(n) to the degenerated affine Sergeev superalgebra, Hd. The ordinary degenerate affine Hecke algebra has proven to be a very interesting object of study considered by many researchers. One reason for this is because it surjects onto the group algebra of the symmetric group and, hence, plays an important role in studying the representations of the symmetric group — especially via the Lie theoretic approach of Ariki, Brundan, Grojnowski, Kleshchev, Vazirani, et. al. An analagous role is played by Hd for the spin representations of the symmetric group. As we continue our collaboration, I would expect that our functor will give interesting new results on the representation theory of Hd. This collaboration would not have begun if not for the conversations we began during the MSRI semester and it has been aided by numerous helpful conversations with other MSRI participants including Kleshchev, Leclerc, and Wang. Projects with Alexander Kleshchev (of the University of Oregon) and Nathan Geer (of Georgia Tech) also began during the semester. Beyond the specific projects discussed above, I have found my time at MSRI to be combinatorial representation theory 21 immensely enriching. Through formal and informal talks with a wide variety of people I have a much broader view of the current state of representation theory. Not only do I feel as if I have learned of many new results and interesting open questions, but I have a much better perspective on the direction the field is moving. • Sinead Lyle I have been working on two specific projects in collaboration with other mathematicians who visited MSRI during the last five months. The first is work with Andrew Mathas (Sydney) on a generalized q-analogue of the Carter–Payne theorem. The main idea behind our approach came from conversations with John Murray who was attending one of the earlier conferences. The work is still in progress; Andrew will be visiting Norwich in the Fall where we can hopefully tie together the outstanding details. The second project is some work with Matthew Fayers (Queen Mary’s, London / MIT) on reducible Specht modules. The reducible Specht modules for the Hecke algebras of type A have been classified except when the parameter q is equal to −1. In this case, there is a conjecture as to which Specht modules were reducible; we have been trying to prove the conjecture in one direction, namely to show that the modules that were conjectured to be reducible are in fact reducible. In characteristic 0, we have made a lot of progress using a variety of methods, and currently all but very small family of partitions have been dealt with. During our work, I also wrote some code for GAP to classify certain homomorphism spaces; while this did not turn out to be directly useful, it may well prove to be so in the future. In addition to these, I have been attending the talks given at the conferences and work- shops throughout the semester. I also found the PostDoc seminars to be very informative; since they were usually aimed at a non–expert audience, they did a good job of representing the different areas that are currently being worked on. Dave Hemmer and I organized a weekly seminar on representations of the symmetric groups, and I was encouraged to find it so well attended. My experience of MSRI has been entirely positive. It provides a perfect working en- viroment and operates with amazing efficiency. Over the last five months, I have had the opportunity to meet almost every mathematician in my area and to hear many of them speak. People from the highest level down have generally been approachable and happy to discuss their work. I feel privileged to have been part of the MSRI community. • Kevin Purbhoo During my time at MSRI, I completed the paper Compression of root systems and the E-sequence. This paper led to an interesting discovery that will eventually lead to a second, shorter, related paper. A second paper, A Littlewood-Richardson rule for Grassmannian permutations with Frank Sottile, was completed to the camera ready stage. A third paper is in preparation, tentatively titled Gr¨obnercycles in toric varieties via tropicalisation. In addition to these writing projects I began several new projects. * A study of the problem of computing Kronecker products for the symmetric group. A few weeks of computer explorations to test an idea produced mixed results and suggested that the idea may work, but will not be straightforward. * In a new collaboration with Lauren Williams, we are looking for a way to understand the totally non-negative part of a flag variety by degenerating the flag variety. 22 msri spring 2008: final report

One of the special benefits of residence at MSRI is the environment and the time to do a bit of ”side reading” and I took advantage of this to learn something about three main topics related to various ongoing or planned projects: quantum groups, intersection theory, and the KdV/KP heirarchy. In addition to several of the MSRI workshops, I attended the New York AMS sectional meeting, BAD Math day, and Lie Theory: The Mathematical Legacy of Bertram Kostant and gave five talks during the semester. • Nat Thiem My research activities at MSRI this spring primarily focused on the following two projects. (1) Develop a Hopf-like combinatorial understanding of the supercharacter theory asso- ciated with the finite group of unipotent upper-triangular matrices, via restriction, tensor products, and superinduction. (2) Understand the algebraic and combinatorial structure of the q-partition algebra. The supercharacter theory of the finite groups of unipotent upper-triangular matrices has a remarkably rich combinatorics, and can be viewed as a p-group analogue of the representation theory of the symmetric group, where the combinatorics of set partitions in supercharacter theory replaces the combinatorics of partitions in the representation theory of the symmetric group. It is therefore natural to expect a super-analogue to the ring of sym- metric functions. Over the course of the semester, it has become clear that the ring of symmetric functions in non-commuting variables is an appropriate analogue for the ring of supercharacters. This project has benefited greatly by conversations with two research groups throughout this semester: experts in supercharacter theory, including C. Andr´e,P. Diaconis, and M. Isaacs, and experts on the ring of symmetric functions in non-commuting variables, including N. Bergeron, B. Sagan, and M. Zabrocki. The second project is joint with T. Halverson and A. Ram. The partition algebra is the centralizer algebra of symmetric group acting on tensor powers of the permutation module. The q-partition algebra studies an analogous algebra where we replace the symmetric group with the finite general linear group. Halverson and I wrote a paper this semester that studies the combinatorics of the q-analogue of the tensor power module, leading to polynomials that interpolate between powers of the matrix dimensions and Bell numbers. In ongoing work with Halverson and Ram, we are studying the construction and relations for the q-partition algebra. There are other projects that also received attention, including an ongoing project with C.R. Vinroot on the representation theory of finite reductive groups, and an ongoing project with P. Diaconis and M. Isaacs on applications of supercharacter theory. • Lauren Williams I have had a stimulating mathematical experience during my stay at MSRI, continuing work on older projects and beginning work on new ones. In February I finished my paper with K. Rietsch, in which we use Lusztig’s canonical basis to prove that the non-negative part of G/P is a CW complex. I also finished my paper with J.C. Novelli and J.Y. Thibon about combinatorial Hopf algebras and the asymmetric exclusion process; Novelli visited MSRI so we were able to discuss the paper in person. A. Postnikov and D. Speyer and I finished our paper on the totally non-negative part of the combinatorial representation theory 23

Grassmannian, which we posted to the arXiv a year ago but did not submit until a few weeks ago (after drastically simplifying some of the proofs). I also continued working on my paper-in-preparation, which uses discrete Morse theory to prove that the closure of any cell of the non-negative part of G/P is contractible. I had many discussions with A. Lascoux, and we began a project investigating type B alternating sign matrices. When G. Musiker visited, we began thinking about f-polynomials and g-vectors for cluster algebras; we now have some “meta-conjectures.” F. Ardila and I continued working on a project investigating the connection between Bergman complexes of Coxeter arrangements and the tropical Grassmannian. I began working on a project with P. Pylvyavsky and T. Lam about parameterizations of cells in the totally non-negative part of a cominuscule flag variety. I discussed with A. Ram the connection of the asymmetric exclusion process to the XXZ model and certain Hecke algebras, and with K. Purbhoo we attempted to understand degenerations of flag varieties to toric varieties. In addition, I had interesting discussions with P. Diaconis, S. Fomin, B. Leclerc, and F. Brenti. In addition, being in the Bay Area gave me the opportunity to give a number of seminar talks: at MSRI in January, at Berkeley in April, and at San Francisco State and Stanford in May. I received valuable feedback from audience members at these talks. In summary, I had a extremely stimulating experience. I am very grateful to have been able to live in Berkeley and work in MSRI, with access to fantastic researchers and a top- notch library. I plan to spend this summer digesting what I’ve learned and making progress on the various new projects I began. Report on Representations of Finite Groups and Related Topics MSRI, spring 2008

1. Introduction The program on Representations of Finite Groups took place during the spring se- mester of 2008, from January 15 until May 23. It was organized by Jonathan Alperin, Michel Brou´e, Jon Carlson, Alexander Kleschev, Jeremy Rickard and Bhama Srini- vasan. Three of the organizers, Brou´e, Carlson and Kleshchev, were in residence for the entire program. Rickard was in residence at MSRI for all but the first month. Srinivasan participated in the program for two months. Alperin was the chairman of the organizing committee for the introductory workshop, and both Kleshchev and Srinivasan were on the organizing committee for the Lie theory workshop. The program on Representations of Finite Groups was coordinated with the con- currently running program on Combinatorial Representation Theory. The Connec- tions for Women workshop was joint with the other program and was organized by Monica Vazirani and Bhama Srinivasan. The workshop on Lie theory was also a joint effort of the two programs. Several members and posdocs were jointly funded by both programs and seminars and workshops associated to one of the programs were often attended by members of the other program.

2. Participants The following is a list of participants in the program. A few of the participants in the program were supported jointly with the program on Combinatorial Representa- tion Theory and should be considered members of both programs. These individuals are marked with an asterisk (∗). The follow mathematicians were members of the program for most of the entire semester (more than three months).

• David Benson (Symonds Professor), University of Aberdeen, • Cedric, Bonnaf´e, CNRS, Besan¸con, • Michel Brou´e (Organizer, Chancellors Professor), Institut Henri Poincar´e, • Robert Boltje, University of California, Santa Cruz, • Serge Bouc, Universit´e de Picardie - Jules Verne, • Jon Carlson (Organizer), University of Georgia, • Paul Fong, University of Illinois, Chicago, • David Hemmer, University of Buffalo, • Martin Isaacs, University of Wisconsin, • Nicolas Jacon∗, University of Besan¸con, • Vaughn Jones, University of California, Berkeley, • Radha Kessar, University of Aberdeen, • Alexander Kleshchev∗ (Organizer), University of Oregon, • Burkhard Kulshammer,¨ University of Jena, • Zongzhu Lin, Kansas State University, 1 2

• Markus Linckelmann, University of Aberdeen, • Gabriel Navarro, University of Valencia, • Julianne Rainboldt, Saint Louis University, • Jeremy Rickard (Organizer), University of Bristol, • Leonard Scott, University of Virginia, • Peter Symonds, University of Manchester, • Pham Huu Tiep, University of Florida, • Peter Webb, University of Minnesota. The following members of the program participated for a month or more. • Jonathan Brundan∗, University of Oregon, • Marc Cabannes, University of Paris, VII, • Francois Digne, Universit´e de Picardie - Jules Verne, • , University of Southern California, • Stephen Doty, Loyola University, • Jesper Grodal, University of Copenhagen, • Robert Guralnick, University of Southern California, • Gerhard Hiss, University of Aachen, • Henning Krause, University of Paderborn, • Gus Lehrer∗, University of Sydney, • Kay Magaard, Wayne State University, • Gunter Malle, University of Kaiserslautern, • Andrew Mathas∗, University of Sydney, • Jean Michel, University of Paris, VII, • Eamonn Obrien, University of Auckland, • Jorn Olsson, University of Copenhagen, • Daniel Nakano, University of Georgia, • Bhama Srinivasan (Organizer), University of Illinois, Chicago, • Alexandre Zalesskii∗, University of East Anglia, • Jiping Zhang, University of . The following members participated for less than a month but more than a week. • Meinolf Geck, University of Aberdeen, • Bernard Keller, University of Paris, VII, • George Lusztig∗, MIT, • Jan Saxl, Cambridge University. The postdoctoral fellows in the list that follows were supported and participated for the entire program. • Maria Chlouveraki∗, Ecole´ Polytechnique, Lausanne, • Daniel Juteau, CNRS, Caan, • Jonathan Kujawa∗, University of Oklahoma, • Sinead Lyle∗, University of East Anglia, • Atilla Maroti, University of Southern California, • Nadia Mazza, University of Aberdeen, • Julia Pevtsova, University of Washington, • Kari Ragnarsson, University of Illinois, Chicago, 3

• Nathaniel Thiem∗, University of Colorado. The following postdocs were supported as members for a month during the pro- gram. • David Craven, Oxford University, • Susanne Danz, University of Jena, • Felix Noeske, University of Aachen, • Britta Sp¨ath, University of Aachen, • Radu Stancu, University of Copenhagen.

3. Workshops There were four workshops associated to the program. Two of these were shared with the program on Combinatorial Representation theory. The Connections for Women workshop was organized by Monica Vazarani and Bhama Srinivasan. It was a joint project of the program on Representation of Finite Groups and the program on Combinatorial Representation Theory. The workshop was held for three days, January 16-18, 2008, during the first week of the program. In addition, to a series of lectures aimed at introducing the subjects covered by the program, the workshop featured a poster session for the participants to present their own results and two panel discussions on aspects of academic life. More details on the workshop can be found on the web page http://www.msri.org/calendar/workshops/WorkshopInfo/403/show workshop The Introductory Workshop on the Representation Theory of Finite Groups was organized by Jonathan Alperin (chairman), Robert Boltje and Markus Linckelmann, and was held February 4-8, 2008. The main part of the schedule consisted of four series of lectures on some of the very active aspects of modern group representation theory. The first series of lectures concentrated on character counting conjectures was given by Burkhard Kulshammer. The series on representations of finite groups of Lie type Jonathan Brundan and Cedric Bonnafe. In the third series Markus Linckelmann explored the connections between representation theory with algebraic topology. Joe Chuang’s three lectures on Brou´e’s abelian defect group conjecture, included an introduction to derived equivalences and an outlook on conjectures that might go beyond the abelian defect group case. Five other lectures on current work in representation theory were presented by Dave Benson, Serge Bouc, Paul Fong, Radha Kessar, and Gabriel Navarro. The workshop enjoyed additional support from a grant from NSA. More details on the workshop are available on the web page http://www.msri.org/calendar/workshops/WorkshopInfo/404/show workshop The workshop on Lie Theory was a joint effort of both the program on Represen- tation of Finite Groups and the program on Combinatorial Representation Theory. The subject matter was one of the most prominent areas of overlap in the interests of the two programs. The representation theory of finite groups and finite group theory in general have in many cases from insights coming from the theory of Lie algebras and Lie groups. Ideas have also flowed the other direction. Connections 4 between the representation theory of symmetric groups and representation of groups of Lie type have been the subject of intense study and development in the last few years. The workshop was organized by Alexander Kleshchev, Arun Ram, Richard Stan- ley (chair), Bhama Srinivasan. It featured lectures by Georgia Benkart, Victor Ginzburg, Viktor Ostrik, Sergey Fomin, George Lusztig, Peter Littelmann, Bernard Leclerc, Peter Fiebig, Alexander Premet, Cedric Bonnafe, Ivan Loseu, Edward Frenkel, Vyjayanthi Chari, Victor Kac, Meinolf Geck, Susumu Ariki, Gustav Lehrer and Gunter Malle. The workshop was held March 10-14, 2008 and was supported in part by a grant from NSA. Further details of the workshop can be found at the web site http://www.msri.org/calendar/workshops/WorkshopInfo/406/show workshop The workshop on Homological Methods in Representation Theory was organized by David Benson, Daniel Nakano (chairman) and Raphael Rouquier. It took place from March 31 to April 4 of 2008. Homological algebra and category theory have played an increasingly prominent role in group representation for the last several years and the development of homological techniques in representation theory has established deep connections with areas such as algebraic topology and commutative algebra. Some eighteen lectures were presented at the workshop. The light schedule allowed lots of time for interactions among the participants. The speakers were Markus Linckelmann, Cedric Bonnafe, Cornelius Pillen, Dave Benson, Paul Balmer, Srikanth Iyengar, David Hemmer, Brian Parshall, Bernhard Keller, Jesper Grodal, Henning Andersen, Jonathan Kujawa, Eric Friedlander, Radha Kessar Jon Carlson, Julia Pevtsova, Alison Parker and Henning Krause. Further details of the workshop can be found at the web site http://www.msri.org/calendar/workshops/WorkshopInfo/405/show workshop

4. Other Events Evans Lectures. Four individuals from the program present MSRI Evans Lec- tures at the University of California at Berkeley. The lectures and titles were

• Alexander Kleshchev, “Representations of Symmetric groups”, February 11, 2008. • David Benson “Classifying Spaces and Cohomology of Finite Groups”, March 31, 2008. • Julia Pevtsova, “Cohomology and Support Varieties”, April 14, 2008. • Radha Kessar, “Modular Representation Theory: A walk on the wild side”, May 12, 2008.

Course on Complex Reflection Groups. This course was presented by Michel Brou´e who was a member of the organizing committee for the program and Chan- cellor’s Professor at the University of California during the semester. The course was attended by several members in the program. 5

Postdoc Seminar. Once a week there was a seminar for the postdocs in the program to present their own work. This seminar was joint with the program on Combinatorial Representation Theory. Except for a few of those who were only at MSRI for a short term, every postdoc in the program presented a 45 minute lecture in the seminar. A total of 12 of the postdocs from the program lectured in the seminar.

5. Seminars and Working Groups The heart of the research part of the program was revealed in the several seminars that were held weekly during the program. Several individuals were encouraged to organize seminars by the organizers of the program. However, the organizers made no attempt to dictate the specific areas or topics to be considered. Modern representation theory of finite groups has two main streams which have both significant differences in methods and applications and significant overlaps in interest and motivation. The representation theories of and of symmetric groups uses specialized techniques and has connections with areas such as algebraic combinatorics, algebraic groups, Lie theory, and Hecke and Schur alge- bras. It was the area of maximal overlap with the concurrent program on Combina- torial Representation Theory at MSRI. The other main stream is general methods for representation theory which mostly involves representations in finite character- istics. There are several important conjectures by Alperin, Brou´e, Dade, McKay and others, whose study has been a driving force in the current research in the area. The area has strong connections with algebraic topology, homotopy theory and commutative ring theory. What follows is a list of the seminars and a sample of the activities and accom- plishments in the area of the seminar. The seminar on Representations of Groups of Lie Type was organized by Zongzhu Lin and his postdoc mentee Daniel Juteau. The seminar featured the works of many young people and include lectures by participants in the Combinatorial Representation Theory program such as Olivier Dudas, David Hernandez, Henning Andersen and Joel Kamnitzer. Other speakers included Lin, Doty, Bonnaf´e, Scott, Juteau and Tiep. One of the more spectacular developments of the last year has been the work of Pe- ter Fiebig in connection with the Lusztig conjecture. Fiebig reformulated Lusztig’s conjecture on the character formula of irreducible representations of algebraic groups in positive characteristics in terms of combinatorial intersection cohomology sheaves associated to the root systems. Fiebig attended the Lie Theory workshop and stayed for an additional two weeks even though he was not on the members list. During one of those weeks, formal discussions with a small group of members were held on the subject of this new work. In another direction there was quite a bit of interest in the work of one of the postdocs in the program. Daniel Juteau produced quite a few interesting results in applying his computation of the intersection cohomology of small nilpotent orbits to decomposing the Weyl modules in small characteristics. He produced a counter-example to a conjecture by Mirkovic and Vilonen. In addition, 6 his main idea, a geometric approach to modular representations involving modular character sheaves, expressed in his dissertation, attracted a lot of attention. The seminar on Representations of Symmetric Groups and Closely Re- lated Topics was organized by David Hemmer and his postdoctoral mentee Sinead Lyle. It also included lectures by some members of the program on Combinatorial Representation Theory, namely Anatoly Vershik, Hyohe Miyachi, Francesco Brenti, Olly Ruff. Another lecture was presented by David Hill from U.C., Berkeley. Speak- ers from the program on Representations of Finite groups included Juteau, Hemmer, Scott, Lyle, Kujawa and Mazza. One of the surprising results proved during the conference was Dave Hemmer’s stability theorem for symmetric group Specht module cohomology. Kleshchev and Brundan found new presentations of blocks of symmetric groups and cyclotomic Hecke algebras. These presentations establish an isomorphism between the blocks and cyclotomic Khovanov- Lauda algebras introduced three months ago. Srinivasan, Brou´e and Fong continued their work on global and local bijections in blocks of finite reductive groups. The seminar on Biset Functors was organized by Serge Bouc. All of the lectures in the seminar were presented by Bouc, Boltje, Ragnarsson and Webb. On the one hand this is a very technical area. But it has gained some prominence following Bouc’s classification of the endopermutation modules using these methods. It has been shown to be very important for several problems in block theory. One of the postdocs in the program, Kari Ragnarsson has made some progress defining Mackey functors and Burnside rings for fusion systems. During the program at MSRI, Bouc succeeded in proving one his own conjectures. He showed that for a group G, the cohomological Mackey functor for G over the base field k have projective resolutions with polynomial growth if and only if the Sylow p-subgroups of G are cyclic, in the case p > 2, or have sectional rank at most 2 if p = 2. Other notable results in block theory coming from the program included the joint work of David Craven, Charles Eaton, Radha Kessar and Markus Linckelmann. They have proved Puig’s finiteness conjecture for source algebras with a given defect group for blocks with Klein four defect groups. This also establishes Erdmann’s conjecture that all blocks with Klein four defect group have a simple module with trivial source. This work represents the first proof of Puig’s conjecture for defect groups other than cyclic p-groups. Linckelmann and Kessar have also demonstrated Alperin’s weight conjecture for blocks of defect 2 with a Brauer correspondent having a unique simple module. This was an application of Rouquier’s inductive result on the existence of local-global stable equivalences. The seminar on Homological Methods in Representation Theory was or- ganized by David Benson and Nadia Mazza. The seminar featured lectures by Vera Serganova from the University of California, Berkeley as well as lectures by members Ragnarsson, Webb, Nakano, Grodal, Lin, Webb, Symonds, Rickard and Carlson. There were a couple of notable advances to come from this area of the program. Dave Benson, partly with Julia Pevtsova, discovered methods for constructing vector 7 bundles over projective space in finite characteristics using the modules of constant Jordan type. The properties of these modules was developed by Carlson, Friedlan- der, Pevtsova and Suslin. Peter Symonds proved several theorems related to Castel- nuovo Mumford regularity. In particular, he settled some conjectures of Kemper and others on the regularity of rings of polynomial invariants and proved Benson’s conjecture on the regularity of cohomology rings. This last result has significant implications for the computation of cohomology. There was an informal working group on character theory led Martin Isaacs, Gabriel Navarro, Pham Huu Tiep and others. This is an area that has focused on important conjectures by Alperin, Dade, Isaacs, Navarro and others. One of the most striking results to come from the semester at MSRI is a proof Brauer’s height zero conjecture in the case of blocks of maximal defect in characteristic 2. The conjecture states that all complex irreducible characters in a p-block B of a finite group G have height zero if and only if the defect group of B is abelian. It was first proposed by Richard Brauer more than 50 years ago and has been confirmed for many specific groups. However, up until now there have been few results of any generality on the subject.

6. Postdocs There were nine postdoctoral fellow associated to the program who were fully funded for the entire program. Four of the postdocs were jointly funded with the program on Combinatorial Representation Theory. The program was blessed with a large number of applications for the few postdoctoral positions. In the end. several of the applicants who did not receive any of the full semester offers were funded for shorter periods as a part of the program. There were at least two of these who were able to participate for the entire program by obtaining other funding. Each postdoc was assigned a mentor who assured his or her continuing activity in the program. Almost all of the postdocs spoke at the postdoc seminar on Friday mornings and many also presented at least one lecture in a seminar. The following is description of some of the activities and accomplishments of the postdocs during the semester at MSRI. The first nine are the ones who had regular postdoc positions.

Maria Chlouveraki has postdoctoral position at the Ecole´ Polytechnique in Lausanne. While at MSRI she began and finished the calculation of the Rouquier blocks for the complex reflection groups of all of the infinite series, thus completing their determination for all complex reflection groups. She has written two papers on the subject: “Rouquier blocks of the cyclotomic Ariki-Koike algebras” and “Rouquier blocks of the cyclotomic Hecke algebras of G”. Both of the papers have been submitted and posted on the archiv. During the time at MSRI she served as coordinator for the seminars for both programs.

Daniel Juteau is a recent PhD from Paris. In the fall of 2007 he had a postdoctoral position at the Ecole´ Polytechnique in Lausanne (EPFL). 8

Beginning in October he will hold a research position (Charge de Recherche) in Caen. While at MSRI, he presented two lectures in seminars in addition to lecturing in the postdoctoral seminar. With Zongzhu Lin, he organized the seminar of representations of groups of Lie type. He submitted a second article based on his PhD thesis, ”Decomposition numbers for perverse sheaves”, to the Annales de l’Institut Fourier and com- pleted a third article ”Modular Springer correspondence and decomposition matrices” also based on his thesis. This last should be submitted soon. An- other article, ”Modular representations of reductive groups and geometry of affine Grassmannians”, was submitted to the Duke Mathematical Journal. Juteau worked with Geordie Williamson on project, which aimed to make a link between equivariant multiplicities and p-smoothness of Schubert vari- eties. This work should lead to another article to be submitted soon. Juteau reports learning a lot about representations of reductive groups working with Pierre Baumann, Cidric Lecouvey, Zongzhu Lin and particu- larly Leonard Scott. He was able completed a part of his research program, concerning representations of reductive groups and the minimal degeneration singularities in the affine Grassmannian.

Jonathan Kujawa is currently an assistant professor at the University of Oklahoma. He started at this position in the Fall of 2007. At MSRI, he spoke in the seminar on Representations of Symmetric Groups as well as in the Postdoctoral Seminar. He was also one of the invited speakers at the workshop on Homological Methods in Representation Theory at MSRI in early April. He expects at least three papers to come out of work initiated while at MSRI: He continued his work with Brian Boe and Daniel Nakano during their visits to MSRI. They are now considering several topics involving support varieties for Lie superalgebras, including generalizing earlier work to the modular and quantum cases. They also are considering ways to link their work to the parallel and independent results of Duflo and Serganova on associated varieties for Lie superalgebras. While at MSRI Kujawa initiated several new collaborations. The pri- mary new project involves work with David Hill and Joshua Sussan (of UC Berkeley) to generalize work of Arakawa and Suzuki to construct a functor from the category O for the Lie superalgebra of type Q(n) to the degener- ated affine Sergeev superalgebra, Hd. Some of this plays an important role in studying the representations of the symmetric group, especially the Lie theoretic approach of Ariki, Brundan, Grojnowski, Kleshchev, Vazirani and others. Kujawa also started projects with Alexander Kleshchev and Nathan Geer during the semester.

Sinead Lyle continues her position as a lecturer at the University of East Anglia. While at MSRI she worked on a project on reducible Specht modules with Matthew Fayers and with Andrew Mathas on Carter-Payne homomor- phisms. The latter is an attempt to find a generalized q-analogue of the 9

Carter–Payne theorem. The main idea behind our approach came from con- versations with John Murray who attending one of the workshops. The work is still in progress though they expect to produce a paper soon. A paper on the results with Fayers has been submitted. Dr. Lyle served as one of the organizers of the seminar on Representations of Symmetric Groups and Closely Related Topics. She presented a lecture in that seminar and in the postdoc seminar.

Attila Maroti has a position at the University of Southern California as an Adjunct Research Assistant Professor until the end of 2008. After that he will be at the Renyi Institute in Budapest. During the semester he worked mostly on two projects. For the first he is working with Bob Guralnick on the non-coprime k(GV) problem. Classi- cally, the k(GV )-problem is to show that if V is a finite faithful G-module and (|G|, |V |) = 1, then the number of conjugacy classes k(GV ) of the semidi- rect product GV is at most |V |. The current question is what is the best possible upper bound for k(GV ) if the co-primeness condition is dropped and it is assumed that V is a completely reducible G-module? Maroti and Guralnick have made significant progress on the question and a paper ”On the extra-special case of the non-coprime k(GV ) problem” should be written soon. The other project with Andreas Lucchini is on two invariants of the generating graph of a finite group. Define a graph Γ on the elements of G by connecting two vertices by an edge if and only if they generate G. Determine the clique and chromatic numbers of this graph. A paper on this subject is nearly finished.

Nadia Mazza is currently holding a postdoctoral position at the University of Aberdeen. While at MSRI, she spoke in the seminar on Representa- tions of Symmetric Groups as well as the Postdoctoral Seminar. She was a coorganizer of the seminar on Homological Methods which met weekly. In collaboration with Jon Carlson and Dave Hemmer, Mazza determined the group of endotrivial modules for the symmetric and alternating groups in odd characteristic. This finished earlier joint work of hers with Carlson and Dan Nakano. This research was entirely done within the semester program at the MSRI and a paper describing the results has been submitted. Other work with Diaz, Glesser and Park considered the problem of control of trans- fer and weak closure for fusion systems, generalizing results by Glauberman on the K-infinities functors in finite groups. The research started at the be- ginning of the program and the paper was submitted shortly after the end of the program. Meanwhile, a similar previous collaboration of the same four which was submitted at the beginning of the program, has been accepted for publication. In collaboration with Serge Bouc, Mazza started an investiga- tion of the functorial properties of the Dade group for fusion systems. They define also a functor of endotrivial modules and reduce the problem of glu- ing of endo-permutation modules to the question of determining the higher 10

limits of these two functors and a related one. She has also been investigat- ing properties of the category of elementary abelian p-subgroups of rank at least two. The goals are twofold: first, determine how morphisms given by a fusion system on the considered p-group act on this category, and second, investigate possible relationships with the category of centric subgroups of a p-group or a fusion system on a p-group.

Julia Pevtsova begins an assistant professorship at the University of Wash- ington in the fall of 2008. She presented one of the Evan’s Lectures during the semester at MSRI as well as giving plenary lectures at the Connections for Women workshops and the workshop on Homological Methods in Repre- sentation Theory. During the semester at MSRI, she completed a paper on ”Spectrum of the tensor triangulated category of perfect complexes over a stack, with P. Smith and continued work on a project on finite dimensional pointed Hopf algebras with with M. Mastnak, P. Schauenburg, and S. Witherspoon. She completed another paper on ”Constructions for Infinitesimal group schemes”, with E. Friedlander. The results were proved for a paper with D. Benson on ”Vec- tor bundles and modules of constant Jordan type”. In addition, she worked with Jon Carlson and Friedlander on a project new to develop invariants for modular representations. A paper on on this subject with the title ”Higher rank varieties and generic kernels” should be written soon. There may be at least one more paper to come from this project.

Kari Ragnarsson had a visiting position at the University of Illinois at Chicago in the fall of 2007. He will have a visiting position at DePaul Uni- versity, also in Chicago, in the fall of 2008. His main focus during the semes- ter at MSRI was an ongoing project to define Burnside rings and Mackey functors for fusion systems. The object was to generalize specific proper- ties exhibited by classical Mackey functors when examined at a prime, and much of his time was spent on a preliminary paper illustrating these p-local properties. He benefited in this regard from conversations with his postdoc mentor Peter Webb and with Serge Bouc. The preliminary paper is in the advanced stages of writing, while the fusion version is still work in progress. Two papers were completed and submitted during his time at MSRI. Ragnarsson spoke in the seminar on Homological Methods in Represen- tation Theory and the seminar on Biset Functors as well as the Postdoc Seminar. Beyond his work on Mackey functors and fusion systems, he has submitted two paper ”Obtainable sizes of finite topologies” and ”Homotopy type of the boolean complex of a Coxeter system” both written with Bridget Eileen Tenner. Most of the work for these papers was done during the semes- ter at MSRI. Another paper on ”Fusion data in the Burnside ring” concerns results obtained at MSRI in work with Radu Stancu. 11

Nat Thiem is an assistant professor at the University of Colorado at Boul- der. While at MSRI he worked on projects to develop a Hopf-like com- binatorial understanding of the supercharacter theory associated with the finite group of unipotent upper-triangular matrices, and to understand the algebraic and combinatorial structure of the q-partition algebra. The first project project has benefited greatly by conversations with two research groups throughout this semester: experts in supercharacter theory, including C. Andr´e, P. Diaconis, and M. Isaacs, and experts on the ring of symmetric functions in non-commuting variables, including N. Bergeron, B. Sagan, and M. Zabrocki. The second project is joint with T. Halverson and A. Ram. During the semester Thiem wrote a paper that studies the combinatorics of the q-analogue of the tensor power module, leading to polynomials that interpolate between powers of the matrix dimensions and Bell numbers. He continued his work on a project with C.R. Vinroot on the representation theory of finite reductive groups, and on a project with P. Diaconis and M. Isaacs on applications of supercharacter theory. The following is a description of the activities of the postdoctoral fellows that participated in the program for less than the entire semester or were not fully funded by MSRI. David Craven completed his PhD. at Oxford University in the early part of 2008. In the coming year, he will remain at Oxford becoming a Junior Research Fellow of Christ Church College. Caven participated in the pro- gram at MSRI for one month, mostly in March of 2008. During that month he and his collaborators had a remarkable success. It was joint work with Charles Eaton, Radha Kessar and Markus Linckelmann, all of whom were at MSRI. They proved a conjecture of Karin Erdmann on the structure of blocks of finite groups whose defect group is the Klein four group. The con- jecture basically says that all simple modules in such a block are periodic or have trivial source. They show that there are only three isomorphism types of source algebra of a block with Klein four defect group, confirming a wide-ranging conjecture of Puig for these blocks. The conjecture of Puig had previously only been verified in the case of blocks of cyclic defect group. In addition to the above results, Craven wrote a paper extending the work in his thesis while at MSRI.

Susanne Danz has been an assistant at the University of Jena. Beginning in August, she will be at Oxford University for eight months. Danz was supported by a fellowship from the Deutsche Forschungsgemeinschaft and was at MSRI for the entire program. She presented a lecture in the postdoc seminar. At MSRI she was mostly working on the problem of determining vertices of Specht modules and simple modules for the symmetric and alternating groups. This was the topic of her DFG project. In joint work with Burkhard Kuelshammer she proved part of a conjecture on vertices of the basic spin 12

module in characteristic 2 which had been set up in joint work with Rene Zimmermann in an earlier paper. She also investigated vertices, sources and Green correspondents of the simple modules for the large Mathieu groups and their covering groups. Her work at MSRI resulted in three paper, one written with Kuelshammer on the vertices of the basic spin module for the symmetric group in characteristic 2 and two on the vertices and sources of module of the simple modules for the Mathieu groups, one of which was written with Kuelshammer.

Adam Glesser is a Research Fellow at the University of Aberdeen. In the coming academic year he will hold a visiting Assistant Professor position at Suffolk University in Boston. He was in residence for most of the semester at MSRI. While at MSRI he worked on a project on fusion systems with Anto- nio Diaz, Nadia Mazza and Sejong Park. A paper describing this work has been submitted for publication. Another project with Markus Linckelmann considered the commuting category of a fusion system. Some preliminary re- sults were obtained. In addition, he made significant progress on solo fusion systems and will submit a paper on this subject in the near future.

Felix Noeske currently holds a position as an assistant professor, not tenure track, at the University in Aachen, Germany. He spent approximately a month at MSRI mostly in March. He spoke in the postdoc seminar. His main work has been on the modular atlas project, meaning the computation of Brauer characters for simple and almost simple groups. While at MSRI he also began a project with Eamonn Obrien to implement an efficient Schreier- Sims method for matrix groups in the computer algebra systems GAP and MAGMA. He also spent time working on fine tuning the condensation tech- niques to calculate the modular characters of the Fischer Group F24 in char- acteristic 2 and the Thompson group in characteristic 5. The latter project is a collaboration with Gerhard Hiss and Jon Carlson.

Britta Sp¨ath has a position in Kaiserslautern, but will assume a fellowship in Paris in the fall of 2008. Her work has concentrated on the McKay conjec- ture which can be stated as follows. If H is a finite group, ` a prime, Irrl0 (H) the set of irreducible characters of H whose degree is not divisible by ` and P a Sylow `-subgroup of H, then |Irrl0 (H)| = |Irrl0 (NH (P ))|. Some recent work of Isaacs, Malle and Navarro shows that in order to prove the McKay Conjecture for all finite groups one can concentrate on simple groups of Lie type and prove that bijections as above satisfying also some equivariance conditions exist. Sp¨ath has been working on proving that such bijections exist; in particular she is verifying that certain characters of Levi subgroups can be extended to their stabilizers in the normalizers of the Levi subgroups. This involves a good knowledge of the properties of these groups and consists of many technical details. Some of her work will also be useful in the work 13

of Srinivasan, Brou´e and Fong.

Radu Stancu has a postdoctoral fellowship at the University of Copen- hagen, which he has occupied since the fall of 2007. He participated in the program at MSRI for approximately one month in late February until late March. While at MSRI he completed one older project with Markus Linck- elmann on the graded center of the stable module category of 2-groups in characteristic 2. In particular they proved that in a lot of the cases this center is infinite dimensional as a k-vector space and give the k-basis in the case of a Klein four group. He started and completed a project with Silvia Onofrei on Stelmacher’s 2-version of the Glauberman’s ZJ-Theorem in the context of fusion systems, and began a project with Peter Symonds on pro-fusion sys- tems. He began a project with Kari Ragnarsson on the characteristic biset for fusion systems. They proved that a fusion system is saturated if and only if it admits a characteristic biset. The works with Linckelmann and Onofrei should submitted for publication soon.

7. Diversity and Human Resource Development The theory of finite group representations has always been a very geographically diverse subject with a large European component. This fact was certainly confirmed in the composition of the membership of the program. From the early planning, the organizers intended to emphasize the role of the best young researchers in the program. The program had nine regular post doctoral fellows, four of whom were jointly supported with the program on Combinatorial Representation Theory. In the original selection process, the top three choices for the postdoc positions were women. Of the nine in the final selection, four were women. In addition, the program offered shorter term support to four postdocs and two of these were women. On the whole, the program had an extremely strong group of young participants. The number of excellent young female members bodes well for the future of an area in which women have traditionally been under represented.

8. Summary and impact While it may in general be difficut to predict the overall impact of a program such as the MSRI on the future of a research area, the signs are very positive. The program featured a large diverse group of young researchers. The research in the area has expanded into some new and unexpected directions of study. At the same time, some significant progress was made on a few of the old questions that have been driving the research. Examples of the more significant advances made during the semester of the program include

• The verification by Navarro and Tiep of the Brauer height-zero conjecture for block of maximal defect in characteristic 2, • The proof by Symonds of the conjecture on the Csstelnuovo-Munford regu- larity of cohomology rings and other rings of invariants, 14

• The proof of the Puig conjecture for blocks with noncylic defect groups of order four, by Craven, Eaton, Kessar and Linckelmann. • The development by Benson, following ideas of Suslin and others, of con- structions of bundles using modules of constant Jordan type. Some of the results build bridges to other areas of mathematics. Some have compu- tational implications. On the whole, it was a very successful program that promises a strong future for the subject. Report on Connections for Women Workshop, Teichmuller Theory and Kleinian Group program, MSRI, August 16--17, 2007

Organizers: Moon Duchin, Caroline Series

Our conception of the Connections for Women workshop was to showcase female speakers, mainly around the postdoc stage of their careers, from areas of specialization adjoining Teichmuller theory and Kleinian groups. Graduate students were heavily represented in the audience, but senior mathematicians attended as well; in the end, the audience was about sixty people, with roughly half women.

The workshop featured 12 speakers in all, and drew speakers and participants from as far as Japan and Spain. The format was two one-hour expository talks each morning, followed each afternoon by four half-hour talks on research themes. The workshop built in several hours of conversation time, providing lunch and coffee for the participants, so that interactions could build on the introductions provided by the short talks.

We elected not to include any programming explicitly devoted to women's issues, and to welcome men to participate in every aspect of the workshop. The thinking was that by giving young women the floor to showcase their expertise in front of an audience representative of the wider community, we would give them an opportunity to make valuable professional connections while situating the conference in the mainstream of the broader MSRI programs. Furthermore, the mandate for the 30-minute talks was to focus not on the theorems in one's most recent work, but on the themes and points of view that frame one's research program. This mandate went toward the workshop's overarching goal of highlighting connections and overlaps, even between areas that do not normally share the same conference space, from Ulcigrai's Hamiltonian flows to Delp's Hilbert metrics to Yaman's convergence groups.

We rate the format of more, shorter talks to have been a great success. In addition, the placement of the workshop at the beginning of the semester-long program was appealing, as it maximized the potential for the connections to mature into professional relationships, and even collaborations. We have some anecdotal evidence to suggest that the exposure that the speakers got in this workshop (in person, and through the video lectures on the MSRI website) will be parlayed into future speaking invitations, and even into greater employment success.

Teichmuller theory and Kleinian groups CONNECTIONS FOR WOMEN August 16-17 2007

Thursday August 16 Friday August 17

9:30- Ania Lenzhen: Teichmuller space and its Genevieve Walsh: Surfaces in 3- 10:30 metrics manifolds

11:00- Moon Duchin: The curve complex and Jane Gilman: A survey of Schottky 12:00 its relatives groups

12:00- Lunch and conversation 2:00

2:00- Elmas Irmak: Mapping class groups, Pallavi Dani: Measuring sets in 2:30 curve and arc complexes on surfaces infinite groups

2:40- Corinna Ulcigrai: Ergodic properties of Kelly Delp: Convex projective 3:10 flows on surfaces structures on 2-orbifolds

3:10- Coffee and conversation 4:00

Ege Fujikawa: Teichmuller space and 4:00- Alexandra Pettet: Teichmuller moduli space for Riemann surfaces of 4:30 spaces of n-tori infinite type

Asli Yaman: Kleinian groups from 4:40- Sarah Koch: Teichmuller theory and the geometric group theory 5:10 endomorphisms of Pn viewpoint

Currently Available Videos

• Anna Lenzhen , Teichmuller Theory and its Metrics August 16,2007, 09:30 AM to 10:30 AM

• Moon Duchin , The Curve Complex and its Relatives August 16,2007, 11:00 AM to 12:00 PM

• Elmas Irmak , Mapping Class Groups, Curve and Arc Complexes on Surfaces August 16,2007, 02:00 PM to 02:30 PM

• Corinna Ulcigrai , Ergodic Properties of Flows on Surfaces August 16,2007, 02:30 PM to 03:10 PM

• Ege Fujikawa , Teichmuller Space and Moduli Space for Riemann Surfaces of Infinite Type August 16,2007, 04:00 PM to 04:30 PM

• Sarah Koch , Teichmuller Theory and Endomorphisms of Pn August 16,2007, 04:30 PM to 05:10 PM

• Genevieve Walsh , Surfaces in 3-Manifolds August 17,2007, 09:30 AM to 10:30 AM

• Jane Gilman , A Survey of Schottky Groups August 17,2007, 11:00 AM to 12:00 PM

• Pallavi Dani , Measuring Sets in Infinite Groups August 17,2007, 02:00 PM to 02:30 PM

• Kelly Delp , Convex Projectives Structures on 2-orbifolds August 17,2007, 02:30 PM to 03:10 PM

• Alexandra Pettet , Teichmuller Spaces of n-tori August 17,2007, 04:00 PM to 04:30 PM

• Asli Yaman , Kleinian Groups from the Geometric Group Theory Viewpoint August 17,2007, 04:30 PM to 05:10 PM

Participant List

Name Role Institution Baba, Shinpei Participant UC Davis Barnhill, Angela Kubena Participant Northwestern University Barrera, Carlos Participant UC Davis Bonfert-Taylor, Petra Participant Wesleyan University Brendle, Tara E Participant Louisiana State University Calderin, Ivo Janier Participant Florida State University Cavendish, William Palmer Notetaker N/A Choi, YoungEun Participant Penn State Altoona Dani, Pallavi Participant University of Oklahoma Delp, Kelly Annette Participant Buffalo State College Drutu, Cornelia Participant N/A Duchin, Moon Organizer UC Davis Fernos, Talia Participant University of California, Los Angeles University of California at Santa Finegold, Brie Participant Barbara Fujikawa, Ege Participant Sophia University Gilman, Jane Participant Rutgers University, Newark Gokturk, Ali Participant Brown University Gouvea, Ezra Participant UC Davis Greenberg, Michael Participant Brown University Irmak, Elmas Participant Bowling Green State University Kim, Inkang Participant Seoul National University Kim, Youngju Participant The City University of New York Koch, Sarah Participant Cornell University Koshlap, Marilyn Margaret Participant Laney College Lamb, Evelyn James Participant Rice University Lee, Jaejeong Participant UC Davis Lee, Juhyun Participant Seoul National University University of Illinois, Urbana- Leininger, Christopher J Participant Champaign Lenzhen, Anna B. Participant University of Michigan Lotay, Jason Participant University College Oxford Mangahas, Johanna Participant University of Michigan Manning, Jason Participant University of Buffalo McCammond, Jon Participant University of California, Santa Barbara Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Peng, Irine Participant University of Chicago Pettet, Alexandra Participant Stanford University Pfaff, Catherine Participant Rutgers University Putman, Andrew Participant University of Chicago Rafi, Kasra Participant University of Chicago Sabalka, Lucas Participant UC Davis Université de Strasbourg (Louis Salepci, Nermin Participant Pasteur) SanAgustin, Keefe L Participant Brandeis University Schleimer, Saul Participant University of Warwick Series, Caroline Organizer University of Warwick Smillie, John Participant Cornell University Smirnova-Nagnibeda, Tatiana Participant University of Geneva Suh, Chan-Ho Participant University of California, Davis Tao, Jing Participant University of Illinois at Chicago Ulcigrai, Corinna Participant Princeton University Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Participant Cornell University Walsh, Genevieve Participant Tufts University Wilde, Natalie Participant Brigham Young University Yaman, Asli Participant Centre de Recerca Matematica

CONFERENCE REPORT

TOPICSIN TEICHMULLER¨ THEORY AND KLEINIANGROUPS

ORGANIZING COMMITTEE: Jeffrey Brock (Brown), Richard Canary (Michigan), Howard Masur (UIC), Maryam Mirzakhani (Princeton), and Alan Reid (Texas)

From November August 20 to August 24, 2007, we held a in introductory workshop entitled “Introduction to Teichmuller Theory and Kleinian Groups.” This workshop consisted of five speakers giving minicourses coordinated to vary- ing degrees. The minicourses were given by Jeff Brock, Ken Bromberg, Alex Eskin, Yair Minsky, and John Smillie. Roughly speaking, the minicourses given by Brock, Bromberg and Minsky were coordinated with the intent to introduce the participants to the central tools used in the proof of the ending lamination conjec- ture, and the minicourses given by Eskin and Smillie were intented to at once introduce the participants to the basic questions in the theory of dynamics of bil- liards and translation surfaces and to recent developments in counting problems in Teichm¨ller space. Careful notes were taken and typeset with detailed illustrations by William Cavendish, a graduate student of Jeff Brock. These are now available on the MSRI website.

1 Talks in Kleinian groups

The coordinated minicourses given by Brock, Bromberg and Minsky were in- tended to introduce students to basics in the theory of Kleinian groups and dis- crete, faithful PSL2(C) representations of surface groups π1(S). Yair Minsky’s lead-off talk described the basics of hyperbolic space, the elements of the isom- etry group, and the Margulis decomposition of a hyperbolic 3-manifold into its “thick” and “thin” parts, where the injectivity radius is greater than some univer- sal constant and less than that constant. Bromberg introduced participants to the shape of a Margulis tube, the embedded tubular neighborhood of a short geodesic, and the different shapes it can take. Brock introduced the notion of algebraic and geometric limits of Kleinian groups, in particular their difference in the setting of quasi-Fuchsian groups. This was made prominent by an introduction of the Kerckhoff-Thurston example when a rank-2 cusp emerges in the geometric limit of a quasi-Fuchsian groups under the iteration of a Dehn twist on one side.

1 The next topic discussed was the notion of interpolation of surfaces in hy- perbolic 3-manifolds, a central tool in all discussions of ends in of hyperbolic 3- manifolds. Bromberg introduced the somewhat more general notion of interpola- tion of Lipschitz surfaces, which encapsulates the other types of surfaces (pleated and simplicial-hyperbolic) that have been used in this context. Minsky then intro- duced the model theorem for Kleinian “punctured-torus groups”, and a succinct proof using the interpolation methods Bromberg had described. Because of the connection with continued fraction expansions for irrational numbers, the use of the Farey graph makes explicit how the geometry of hyperbolic 3-manifolds in this case connects with classical notions from number theory. In the third day’s lectures, Yair Minsky introduced students to the fundamen- tals of the curve complex, its Gromov hyperbolicity, and its use in studying prop- erties of the mapping class group and Teichmuller¨ space, and length estimates in the setting of Kleinian surface groups. These methods generalize the puncutred torus case, so the connection with the Farey graph served as a useful motivat- ing device. Bromberg then used his interpolation technique to give a variant of Minsky’s theorem that the curves with a given length bound in a hyperbolic 3- manifold always form a uniformly quasi-convex set. The minicourse concluded with Brock’s lecture on the construction of the model manifold explicitly in the case of the five holed sphere, using the (somewhat simpler in this setting) notion of hierarchies of tight geodesics in the curve complex. As a gratifying indication of the success of this sequence of lectures, there were a number of days following the conference where a group of five or six graduate and postdocs, and senior including William Cavendish, Ali Gokturk, Corinna Ulicgrai, Saul Schleimer, Aaron Magid and other students could be found at the main blackboard in the coffee area of the MSRI loungue, reconstruct- ing the combinatorial structure of the model manifold in the five holed sphere case. With the considerable ramp up to the ultimate conclusion, it was clear that enough time had been given for them to digest and develop interest in the key points of the proof of the ending lamination conjecture in the low complexity cases.

2 Talks in Teichmuller¨ theory

Two of the main series of lectures given in the Introductory workshop were given by John Smillie and Alex Eskin. A major theme in Teichmuller¨ theory for the last twenty-five years has been the study of translation surfaces and the SL2(R) action on the moduli space of translation surfaces. This subject arises naturally

2 in Teichmuller¨ theory in studying Teichmuller¨ maps, and also quite prominently in the study of billiards in rational angled polygons. John Smillie gave a series of three lectures introducing this important subject to the graduate students and postdocs. A second major subject in Teichmuller¨ theory has been the study of the map- ping class group and its action on Teichmuller¨ space. A recurring theme is to examine to what extent the group has properties similar to those of a lattice in a Lie group, and to what extent Teichmuller¨ space resembles a manifold of nonpos- itive curvature. Alex Eskin gave two lectures where he introduced the action of the mapping class group on Teichmuller¨ space. In the first lecture he discussed his recent beautiful work with Athreya, Bufetov, and Mirzakhani on lattice counting problems associated with this action. In a second lecture he discussed his work with Mirzakhani on asymptotics of the number of pseudo-Anosov elements whose translation length is smaller than a given number. Both of these subjects have their roots in ideas from Lie groups and manifold theory. Smillie’s lectures were widely praised by participants as providing a solid ba- sis from which to begin understanding fundamental questions in the area of trans- lation surfaces, and Eskin’s lectures provided a clear and illuminating window into one of the major technical advances in the connection between the study of Teichmuller¨ theory and the study of Lie groups and their ergodic and number theoretic aspects.

3

Introduction to Teichmuller Theory and Kleinian Groups August 20, 2007 to August 24, 2007

Monday August 20

09:30AM - 10:30AM Yair Minsky "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 10:30AM - 11:00AM Morning Break 11:00AM - 12:00PM John Smillie "Introduction to rational billiards and translation surfaces" 12:00PM - 02:00PM Lunch at MSRI 02:00PM - 03:00PM Kenneth Bromberg "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 03:00PM - 03:30PM Afternoon Tea 03:30PM - 04:30PM Jeffrey Brock "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 04:30PM - 05:30PM Discussion Session

Tuesday August 21

09:30AM - 10:30AM Kenneth Bromberg "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 10:30AM - 11:00AM Morning Break 11:00AM - 12:00PM John Smillie "Introduction to Rational Billiards and Translation Surfaces" 12:00PM - 02:00PM Lunch at MSRI 02:00PM - 03:00PM Jeffrey Brock "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 03:00PM - 03:30PM Afternoon Tea 03:30PM - 04:30PM Yair Minsky "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 04:30PM - 05:30PM Discussion Sessions

Wednesday August 22

09:30AM - 10:30AM Yair Minsky "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 10:30AM - 11:00AM Morning Break 11:00AM - 12:00PM John Smillie "Introduction to Rational Billiards and Translation Surfaces" 12:00PM - 02:00PM Lunch at MSRI 02:00PM - 03:00PM Kenneth Bromberg "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 03:00PM - 03:30PM Afternoon Tea 03:30PM - 04:30PM Jeffrey Brock "Introductory topics in Kleinian groups and hyperbolic 3-manifolds" 04:30PM - 04:30PM Discussion Sessions

Thursday August 23

10:00AM - 11:00AM Alex Eskin "Counting problems in Teichmüller space" 11:00AM - 12:00PM Problem Session: "Kleinian groups and hyperbolic 3-manifolds" 12:00PM - 01:30PM Lunch at MSRI 01:30PM - 02:30PM Open Hour 02:30PM - 03:30PM Discussion Session: "Counting problems" 03:30PM - 04:00PM Afternoon Tea 04:00PM - 05:00PM Special Topics: "Teichmüller theory and Kleinian groups"

Friday August 24

10:00AM - 11:00AM Alex Eskin "Counting problems in Teichmüller space" 11:00AM - 12:00PM Problem Session: "Kleinian groups and hyperbolic 3-manifolds" 12:00PM - 01:30PM Lunch at MSRI 01:30PM - 02:30PM Open Hour 02:30PM - 03:30PM Problem Session: "Counting problems" 03:30PM - 04:00PM Afternoon Tea 04:00PM - 05:00PM Special Topics: "Teichmüller theory and Kleinian groups"

Currently Available Videos

• Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 20,2007, 09:30 AM to 10:30 AM

• John Smillie , Introduction to rational billiards and translation surfaces August 20,2007, 11:00 AM to 12:00 PM

• Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 20,2007, 02:00 PM to 03:00 PM

• Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 20,2007, 03:30 PM to 04:30 PM

• Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 21,2007, 09:30 AM to 10:30 AM

• John Smillie , Introduction to rational billiards and translation surfaces August 21,2007, 11:00 AM to 12:00 PM

• Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 21,2007, 02:00 PM to 03:00 PM

• Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 21,2007, 03:30 PM to 04:30 PM

• Yair Minsky , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 22,2007, 09:30 AM to 10:30 AM

• John Smillie , Introduction to rational billiards and translation surfaces August 22,2007, 11:00 AM to 12:00 PM

• Kenneth Bromberg , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 22,2007, 02:00 PM to 03:00 PM

• Jeffrey Brock , Introductory topics in Kleinian groups and hyperbolic 3-manifolds August 22,2007, 03:30 PM to 04:30 PM

• Alex Eskin , Counting Problems in Teichmüller Space August 23,2007, 10:00 AM to 10:50 AM

• Kasra Rafi , Flat structures and hyperbolic structures on surfaces August 23,2007, 04:00 PM to 05:00 PM

• Alex Eskin , Counting Problems in Teichmüller Space August 24,2007, 10:00 AM to 10:50 AM

• Christopher Leininger , Kleinian aspects of the mapping class group August 24,2007, 04:00 PM to 05:00 PM

Participant List

Name Role Institution Algom Kfir, Yael Participant University of Utah Aramayona, Javier Participant University of Illinois at Urbana-Champaign Atkinson, Christopher Participant University of Illinois at Chicago Baba, Shinpei Participant UC Davis Babson, Eric Participant University of California Barnhill, Angela Kubena Participant Northwestern University Barrera, Carlos Participant UC Davis Behrstock, Jason Participant Columbia University Bonfert-Taylor, Petra Participant Wesleyan University Brendle, Tara E Participant Louisiana State University Brock, Jeffrey F. Organizer Brown University Bromberg, Kenneth W. Participant University of Utah Calderin, Ivo Janier Participant Florida State University Calta, Kariane Participant Cornell University Canary, Richard Douglas Organizer University of Michigan Cashen, Christopher Henry Participant University of Illinois, Chicago Cavendish, William Palmer Participant N/A Chatterji, Indira Lara Participant OSU cheraghi, davoud Participant SUNY at Stony Brook Cheung, Yitwah Participant San Francisco State University Childers, Leah Participant Louisiana State University Korea Advanced Institute of Science and Choudhury, Dhrubajit Participant Technology Cruz-Cota, Aldo-Hilario Participant University of California, Santa Barbara Danciger, Jeffrey E Participant Stanford University Davis, Michael W. Participant Ohio State University Delp, Kelly Annette Participant Buffalo State College Drutu, Cornelia Participant N/A Duchin, Moon Participant UC Davis Dumas, David Participant Brown University Dymarz, Tullia Maria Participant Yale University Edwards, Robert Participant UCLA Eskin, Alex Participant University of Chicago Fera, Joseph Louis Participant Wesleyan University Fernos, Talia Participant University of California, Los Angeles Fujikawa, Ege Participant Sophia University Futer, David Participant Michigan State University Gabai, David Participant Princeton University Gadre, Vaibhav S Participant Caltech Gilman, Jane Participant Rutgers University, Newark Gokturk, Ali Participant Brown University Gouvea, Ezra Participant UC Davis Greenberg, Michael Participant Brown University Groves, Daniel Peter Participant University of Illinois Guo, Ren Participant Rutgers University Gupta, Subhojoy Participant Yale University Korea Advanced Institute of Science and Ha, Jeasoon Participant Technology Hass, Joel Participant UC Davis Hoban, Ryan Francis Participant University of Maryland Huang, Zheng Participant University of Michigan Irmak, Elmas Participant Bowling Green State University Ji, Lizhen Participant univ of Michigan Kent, Richard Peabody Participant N/A Kerckhoff, Steven Participant Stanford University Kim, Inkang Participant Seoul National University Kim, Sam Participant Kyungpook National University Kim, Sang-hyun Participant N/A Kim, Youngju Participant The City University of New York Koshlap, Marilyn Margaret Participant Laney College Lamb, Evelyn James Participant Rice University Landes, Emily Rose Participant University of Texas, Austin Lazowski, Andrew Participant Wesleyan University Leary, Ian Participant Ohio State University Ledrappier, François Participant Korea Advanced Institute of Science and Lee, Gye-Seon Participant Technology Lee, Jaejeong Participant UC Davis Lee, Juhyun Participant Seoul National University Leininger, Christopher J Participant University of Illinois, Urbana-Champaign Lelievre, Samuel L Participant University of Warwick Lenzhen, Anna B. Participant University of Michigan Lim, Seonhee Participant Cornell Univeristy Lotay, Jason Participant University College Oxford Louder, Larsen Edmund Participant University of Utah Louwsma, Joel Participant California Institute of Technology Magid, Aaron Daniel Participant University of Michigan Malone, William Participant University of Utah Mangahas, Johanna Participant University of Michigan Manning, Jason Participant University of Buffalo Masur, Howard A. Organizer University of Illinois, Chicago McCammond, Jon Participant University of California, Santa Barbara Minsky, Yair Nathan Participant Yale University Mirzakhani, Maryam Organizer Princeton University Modami, Babak Participant Yale University Nakamura, Kei Participant University of California, Davis Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Papazoglou, Panagiotis Participant University of Athens Peng, Irine Participant University of Chicago Pfaff, Catherine Participant Rutgers University Putman, Thomas Andrew Participant Massachusetts Institute of Technology Rafalski, Shawn Thomas Participant University of Illinois at Chicago Rafi, Kasra Participant University of Chicago Reid, Alan William Organizer University of Texas Sabalka, Lucas Participant UC Davis Salepci, Nermin Participant Université de Strasbourg (Louis Pasteur) SanAgustin, Keefe L Participant Brandeis University Schleimer, Saul Participant University of Warwick Smillie, John Participant Cornell University Suh, Chan-Ho Participant University of California, Davis Tao, Jing Participant University of Illinois at Chicago Tavakoli, Kourosh Participant City University of New York Thomas, Anne Caroline Mary Participant Cornell University Thompson, Josh Jerome Participant N/A Tsai, Chia-yen Participant University of Illinois at Urbana-Champaign Ulcigrai, Corinna Participant Princeton University Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Participant Cornell University Walsh, Genevieve Participant Tufts University Wiest, Bert Participant N/A Yaman, Asli Participant Centre de Recerca Matematica CONFERENCE REPORT

TOPICSIN TEICHMULLER¨ THEORY AND KLEINIANGROUPS

ORGANIZING COMMITTEE: Jeffrey Brock (Brown), Kenneth Bromberg (Utah), Richard Canary (Michigan), Howard Masur (UIC), Maryam Mirzakhani (Princeton), Alan Reid (Texas), and John Smillie (Cornell)

From November 12 to November 16 we held a workshop entitled “Topics in Teichmuller Theory and Kleinian Groups.” It was exciting to see such a broad range of researchers speak on many disparate themes ranging from classical 3- manifold topology to counting problems in Teichmuller¨ space. The conference made clear that more unites these fields than divides them. Here are some of the highlights.

1 Talks in 3-manifolds and Kleinian groups

Recent advances in the area, such as Perelman’s solution of Thurston’s Geometriza- tion Conjecture, have given rise to the expectation that, given Perelman’s work, a great deal of the focus of 3-manifold topology will be on understanding the ge- ometry and topology of finite volume hyperbolic 3-manifolds. To this end, one of the exciting new developments has been the application of technology used in the proof of the Ending Lamination Conjecture to understanding the geometry and topology of compact 3-manifolds. Two of the talks (by Hossein Namazi and Juan Souto) are directly related to this. The theme of these talks could be described as “the hyperbolic geometry of Heegaard splittings”. Heegaard splittings have long been a basic tool in the theory of 3-manifolds, and they have enjoyed somewhat of a renaissance of late. The talks of Namazi and Souto described recent advances on understanding Heegaard splittings geometrically. For example, Souto’s talk was focused on the problem on the rank of π1 versus Heegaard genus and described how for every k and ε there is a g such that every non-Haken hyperbolic 3-manifold with at least injectivity radius ε and whose fundamental group is generated by k elements admits a genus g Heegaard splitting. Namazi’s talk described ongoing work on how combinato- rial information in the curve complex can result in information about Heegaard splittings about 3-manifolds.

1 Dave Gabai described the proof that the minimal volume hyperbolic 3-manifold is the so-called Weeks manifold. The proof of this relies on rigourous computer programming together with an analysis of basically a special handle decomposi- tion called a Mom-structure. This structure was well-suited to enumerating cusped hyperbolic 3-manifolds of low-complexity which proved an important step. This work also relied on recent work of Agol-Storm-Thurston which used some of Perelman’s work. It is worth remarking that the “Mom technology” described in the talk has proved useful recently in work of Lackenby and Meyerhoff on proving that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. gave a talk whose title may seem far removed from the general theme of the semester, however, the proof of their main result, relies heavily on many different aspects of the geometry and topology of hyperbolic 3-manifolds. For example, a key part of their proof is the construction of a certain complexity function which as part of the data for a hyperbolic 3-manifold involves the volume and length spectrum. Chris Leininger’s talk described his work with Saul Schleimer regarding the connectivity of the space of ending laminations, and his new work joint with Mitra and Schleimer on the existence of Cannon-Thurston maps for surface groups into the curve complex. This work has recently inspired a new proof by Gabai that treats connectivity of the space of ending laminations in general. Teruhiko Soma gave an account of his recent work with Kenichi Ohshika giving a complete description of geometric limtis of hyperbolic 3-manifolds homeomor- phic to S × R. This result relies strongly on the model theorem of Brock-Canary- Minsky for hyperbolic 3-manifolds with the homotopy type of a surface, and an analysis of the behavior of the hierarchical combinatorial models on which the Minsky model used in their proof of the ending lamination conjecture is based. In particular their use of so-called bug-infested hierarchies, which allow for holes to appear in the standard hierarchy model, was a key development.

2 Talks in Teichmuller¨ theory

Many of the talks were directly connected to Teichmuller¨ theory. These speakers were Yitwah Cheung, Patrick Hooper, Kariane Calta, Matthew Bainbridge, Corinna Ulcigrai and Miriam Mirzakhani who was one of the program orga- nizers. This is a young filed and the talks were filled with new ideas and new

2 directions. The Teichmuller¨ theory in question is the investigation of the Teichmuller¨ flow on spaces of flat surfaces. One way to motivate these questions is to look at the problem of the billiard flow on a polygon. Consider a polygonal region in the plane. Consider a billiard trajectory, that is to say, a path starting in the interior of the region which moves with constant velocity until it hits a side at which point it bounces so that the angle of incidence is equal to the angle of reflection. One fundamental question is how the trajectories are distributed. In the particular case of the square there is a dichotomy: in a given direction the trajectories are either all closed or uniformly distributed. The billiard flow on the square can be analyzed as a family of constant slope flows on the torus. When the region is a more general polygon for which the angles are rational multiples of π then the behavior can be analyzed in terms of geodesic flows on flat surfaces of higher genus with singular points. As in the case of the torus these geodesic flows break up into a family of directional flows. A fundamental tool in addressing this problem is the study of the Teichmuller¨ flow on the space of flat surfaces. Masur’s theorem connects the behavior of the distribution of orbits on flat surface with the divergence of the flat surface under the Teichmuller¨ flow. Yitwah Cheung spoke on joint work with Alex Eskin. In this work they show that with the hypothesis of boundedness of the trajectory can be replaced with the hypothesis of slow divergence. A novel feature of their work is the use of Delaunay triangulations of the flat surface to obtain geometric information about the degeneration of the surface. Much recent work in the field has focused on Veech’s discovery of non-arithmetic lattice examples. These lead to billiard tables which posses a dichotomy like that for the square. Unlike the case of the square the flow on these surfaces cannot be analyzed in terms of flows on the torus or even flows on branched covers of the torus. Veech showed that the billiard flow on the regular n-gon with n > 4 gives a family of such examples. Patrick Hooper showed in his talk that in some sense one could take a limit of Veech’s examples and an infinite genus lattice example. This surface displays a weak form of the Veech dichotomy. This is very surprising work and got confer- ence participants wondering what else could be done for infinite genus surfaces. Kariane Calta spoke at the conference on a deeper investigation of some of the techniques that she had used in her discovery of interesting new families of non-arithmetic lattice examples, discovered independently by Curt McMullen. In her talk Kariane investigated the connection between the J-invariant and algebraic

3 properties of the holonomy map for the surface. She made a connection with the work of Kenyon and Smillie where they used the J-invariant to determine which acute rational triangles had the lattice property. Matthew Bainbridge considered a second important question in the field of polyg- onal billiards involves the existence of closed trajectories and how the number of trajectories of length less than L grows as L increases. In the case of the square the number of closed geodesics of length less than L is asymptotic to cL2. One case where these problems can be solved is the lattice case. In his talk, he dis- cussed rates of growth of closed geodesics on flat surfaces which were one step more complicated than these lattice examples. In this work Matt used techniques from algebraic geometry to calculate the volumes of moduli spaces of genus two translation surfaces. Corinna Ulcigrai discussed the question of properties for multi-valued Hamiltonians on surfaces. These flows can be viewed as reparametrized direc- tional flows on flat surfaces. The fact that the time reparametrization blows up near singular points of the surfaces makes these questions delicate. Understand- ing mixing properties of the Hamiltonian flows involves precise estimates on the near returns of the orbits to the singular points. Getting these estimates was a mix of standard techniques and completely new techniques which Corinna explained very well. Maryam Mirzakhani spoke about joint work with Alex Eskin where they related the problem of finding closed geodesics of the Teichmuller¨ flow on the space of quadratic differentials to the problem of counting the number of pseudo-Anosov on a surface with a fixed bound on the expansion constant. This work applies the Teichmuller¨ theory techniques to a problem in surface topology. Francois Labourie talked about the geometry of the space of representations of surface groups in SLn(R) where many of the features of hyperbolic geometry translate (via the notion of geodesic currents and cross ratios) to the setting of one of the connected component of the representation variety (called n-Hitchin repre- sentations), providing a glimpse of higher Teichmuller¨ theory that was otherwise absent from the conference.

3 Other comments

These talks are some of the highlights of the conference, which was marked by considerable activity outside of the lecture hall. The presence of many participants

4 from the Geometric Group Theory program led to many lively discussions during breaks and after hours. The list of speakers and participants reflects an emphasis on early career math- ematicians: over half of the speakers were pre-tenure. Furthermore, five out of the twenty-two speakers were women, and twenty of the roughly one-hundred regis- tered participants were women.

5

Topics in Geometric Group Theory November 05, 2007 to November 09, 2007

Schedule

Monday November 5

08:45AM - 09:00AM "Welcome to MSRI" talk Coarse differentiation and the geometry 09:00AM - 09:50AM Alex Eskin of polycyclic groups. 10:00AM - 10:30AM Tea 10:30AM - 11:20AM Yehuda Shalom TBD Alexander On asymptotic dimension of Coxeter 11:30AM - 12:20PM Dranishnikov groups 12:30PM - 02:00PM Lunch 02:00PM - 02:50PM Indira Chatterji Median spaces and applications 03:00PM - 03:30PM Tea Discrete groups and Morse theory 04:10PM - 05:00PM Mladen Bestvina (in 610 Evans Hall, UC-Berkeley)

Tuesday November 6

Automorphism groups of right-angled 09:00AM - 09:50AM Karen Vogtmann Artin group 10:00AM - 10:30AM Tea Higher isoperimetric inequalities for 10:30AM - 11:20AM Panagiotis Papazoglou complexes and groups 11:30AM - 12:20PM Anne Thomas Lattices acting on polyhedral complexes 12:30PM - 02:00PM Lunch Normal automorphisms of relatively 02:00PM - 02:50PM Denis Osin hyperbolic groups 03:00PM - 03:30PM Tea Rank-1 isometries on CAT(0) spaces and 03:30PM - 04:20PM Koji Fujiwara quasi-homomorphisms 04:30PM – 06:30PM Reception

Wednesday November 7

A family of simple groups acting on 09:00AM - 09:50AM Bertrand REMY buildings 10:00AM - 10:30AM Tea 10:30AM - 11:20AM Dan Margalit Dimension of Torelli groups On the homology of finite index 11:30AM - 12:20PM Thomas Putman subgroups of the mapping class group 12:30PM - 02:00PM Lunch Quasi-isometric rigidity of the mapping 02:00PM - 02:50PM Jason Behrstock class groups 03:00PM - 03:30PM Tea 03:30PM - 04:20PM Jan Dymara L^2 cohomology of buildings

Thursday November 8

Geometric Makanin algorithm for solving 09:00AM - 09:50AM Vincent Guirardel equations in virtually free groups 10:00AM - 10:30AM Tea Global fixed points for centralizers and 10:30AM - 11:20AM Mike Handel Morita's Theorem (joint work with John Franks) 11:30AM - 12:20PM Larsen Louder Krull dimension for limit groups 12:30PM - 02:00PM Lunch Volodymyr Space of marked groups and non-uniform 02:00PM - 02:50PM Nekrashevych exponential growth 03:00PM - 03:30PM Tea

Friday November 9

09:00AM - 09:50AM Martin Bridson Finitely presented, residually-free groups 10:00AM - 10:30AM Tea 10:30AM - 11:30AM Tadeusz Januszkiewicz Groups with fixed point properties Residual finiteness and separability of 11:30AM - 12:20PM Jason Manning quasi-convex subgroups 12:30PM - 02:00PM Lunch Kleiner's proof of the polynomial growth 02:00PM - 02:50PM David Fisher theorem 03:00PM - 03:30PM Tea

Currently Available Videos

• Alex Eskin , Coarse Differentiation and the Geometry of Polycyclic Groups November 5,2007, 09:00 AM to 09:50 AM

• Yehuda Shalom , Y. Shalom Lecture November 5,2007, 10:30 AM to 11:20 AM

• Alexander Dranishnikov , On Asymptotic Dimension of Coxeter Groups November 5,2007, 11:30 AM to 12:20 PM

• Indira Chatterji , Median Spaces and Applications November 5,2007, 02:00 PM to 02:50 PM

• Karen Vogtmann , Automorphism Groups of Right-angled Artin Groups November 6,2007, 09:00 AM to 09:50 AM

• Panagiotis Papazoglou , Higher Isoperimetric Inequalities for Complexes and Groups November 6,2007, 10:30 AM to 11:20 AM

• Anne Thomas , Lattices Acting on Polyhedral Complexes November 6,2007, 11:30 AM to 12:20 PM

• Denis Osin , Normal Automorphisms of Relatively Hyperbolic Groups November 6,2007, 02:00 PM to 02:50 PM

• Koji Fujiwara , Rank-1 Isometries on CAT(0) Spaces and Quasi-homomorphisms November 6,2007, 03:30 PM to 04:20 PM

• Bertrand REMY , A Family of Simple Groups Acting on Buildings November 7,2007, 09:00 AM to 09:50 AM

• Dan Margalit , Dimension of Torelli Groups November 7,2007, 10:30 AM to 11:20 AM

• Thomas Putman , On the Homology of Finite Index Subgroups of the Mapping Class Group November 7,2007, 11:30 AM to 12:20 PM

• Jason Behrstock , Quasi-isometric Rigidity of the Mapping Class Groups November 7,2007, 02:00 PM to 02:50 PM

• Jan Dymara , L^2 Cohomology of Buildings November 7,2007, 03:30 PM to 04:20 PM

• Vincent Guirardel , Geometric Makanin Algorithm for Solving Equations in Virtually Free Groups November 8,2007, 09:00 AM to 09:50 AM

• Michael Handel , Global fixed points for centralizers and Morita's Theorem (joint work with John Franks) November 8,2007, 10:30 AM to 11:20 AM

• Larsen Louder , Krull Dimension for Limit Groups November 8,2007, 11:30 AM to 12:20 PM

• Volodymyr Nekrashevych , Space of Marked Groups and Non-uniform Exponential Growth November 8,2007, 02:00 PM to 02:50 PM

• Martin Bridson , Finitely Presented, Residually-free Groups November 9,2007, 09:00 AM to 09:50 AM

• Tadeusz Januszkiewicz , Groups with Fixed Point Properties November 9,2007, 10:30 AM to 11:30 AM

• Jason Manning , Residual Finiteness and Separability of Quasi-convex Subgroups November 9,2007, 11:30 AM to 12:20 PM

• David Fisher , Kleiner's Proof of the Polynomial Growth Theorem November 9,2007, 02:00 PM to 02:50 PM

Participant List

Name Role Institution Algom Kfir, Yael Notetaker University of Utah Alperin, Roger C. Participant San Jose State University Amram Blei, Meirav Participant Bar Ilan University Aramayona, Javier Participant University of Illinois at Urbana-Champaign Arzhantseva, Goulnara Participant University of Geneva Atanasov, Risto Participant Western Carolina University Badus, Alina Participant University of Pennsylvania Barnard, Josh Participant University of South Alabama Barnhill, Angela Kubena Participant Northwestern University Behrstock, Jason Speaker Columbia University Bell, Robert W Participant Michigan State University Bennett, Hanna Notetaker University of Chicago Berkove, Ethan J Participant Lafayette College Bestvina, Mladen Speaker University of Utah Bleak, Collin Participant University of Nebraska at Lincoln Brady, Noel Organizer University of Oklahoma Bridson, Martin Speaker University of Oxford Broaddus, Nathan Participant University of Chicago Bumagin, Inna Participant Carleton University Calderin, Ivo Janier Participant Florida State University Caprace, Pierre-Emmanuel Participant Institut des Hautes Études Scientifiques (IHES) Carette, Mathieu Marc Participant Université Libre de Bruxelles Cashen, Christopher Henry Participant University of Illinois, Chicago Casson, Andrew John Participant Yale University Cavendish, William Palmer Participant N/A Chatterji, Indira Lara Speaker OSU Chaynikov, Vladimir Participant Vanderbilt University Childers, Leah Participant Louisiana State University Clair, Bryan Participant Saint Louis University Cleary, Sean Participant CUNY, City College Connell, Chris G. Participant Indiana University Coulbois, Thierry Participant Universite Aix-Marseille III (France) Culler, Marc Participant University of Illinois, Chicago Dani, Pallavi Participant University of Oklahoma Davidson, Peter John Participant University of Glasgow Davis, Michael W. Organizer Ohio State University Denham, Graham Campbell Participant University of Western Ontario Dranishnikov, Alexander Speaker University of Florida Dymara, Jan Speaker Uniwersytet Wroclawski Edwards, Robert Participant UCLA Elder, Murray Participant Stevens Institute of Technology Eskin, Alex Speaker University of Chicago Falk, Michael J. Participant Northern Arizona University Feighn, Mark Organizer Rutgers University Fel'shtyn, Alexander Participant Boise State University and Szczecin Univesrsity Fernos, Talia Participant University of California, Los Angeles Fisher, David M Participant Indiana University Forester, Max Participant University of Oklahoma Francaviglia, Stefano Participant University of Pisa Freden, Eric Participant Southern Utah University Fujiwara, Koji Speaker Tôhoku University Futer, David Participant Michigan State University Gadre, Vaibhav S Participant Caltech Geller, William Participant Indiana U.-Purdue U. Indianapolis Geoghegan, Ross Participant SUNY, Binghamton Goldman, Bill Participant University of Maryland Groves, Daniel Peter Participant University of Illinois Guilbault, Craig Robert Participant UW-Milwaukee Guirardel, Vincent Speaker Universite Paul Sabatier (Toulouse) Guralnik, Dan Participant Vanderbilt University Guth, Larry Participant Massachusetts Institute of Technology Hambleton, Ian Participant McMaster University Handel, Michael Speaker Lehman College Hass, Joel Participant UC Davis Henderson, Jim Participant U. of Colorado-Colorado Springs Hensel, Sebastian Participant University of Bonn Hermiller, Susan Participant University of Nebraska Hsu, Timothy M. Participant San Jose State University Huang, Zheng Participant University of Michigan James, Justin Amery Participant Minnesota State University - Moorhead Januszkiewicz, Tadeusz Speaker Ohio State University Kaminker, Jerry Participant IUPUI Kapovich, Ilya Participant University of Illinois at Urbana-Champaign Kapovich, Michael Participant University of California Kar, Aditi Participant Ohio State University Katerman, Eric Participant University of Texas-Austin Kim, Sam Participant Kyungpook National University Kim, Sang-hyun Participant N/A Klein, Tom Participant McMaster University Kramer, Linus Karl Heinz Participant Universitaet Muenster Lackenby, Marc Speaker University of Oxford Lai, Yvonne (Yuan-Juang) Participant UC Davis Leary, Ian Participant Ohio State University Levitt, Gilbert Participant universite de Caen Lim, Seonhee Notetaker Cornell Univeristy Lotay, Jason Participant University College Oxford Louder, Larsen Edmund Speaker University of Utah Louwsma, Joel Participant California Institute of Technology Lustig, Martin Participant Universite P. Cezanne - Aix Marseille III Mackay, John Malcolm Participant University of Michigan Macura, Natasa Participant Trinity University Malone, William Notetaker University of Utah Mangahas, Johanna Participant University of Michigan Manning, Jason Speaker University of Buffalo Marden, Albert Participant University of Minnesota Margalit, Dan Speaker University of Utah Martinez-Pedroza, Eduardo Participant University of Oklahoma Institute of Mathematics "Simion Stoilow" of the Romanian Matei, Daniel Participant Academy McCammond, Jon Participant University of California, Santa Barbara Meier, John Participant Lafayette College Min, Honglin Participant Rutgers University-Newark Minasyan, Ashot Participant University of Geneva Mislin, Guido Participant ETH Mitra, Mahan Participant THE INSTITUTE OF MATHEMATICAL SCIENCES Mooney, Christopher Paul Participant University of Wisconsin-Milwaukee Nekrashevych, Volodymyr Speaker Texas A & M University Nipper, Emanuel Participant Universität Bonn Nucinkis, Brita Participant University of Southampton Okun, Boris L Participant University of Wisconsin-Milwaukee Osajda, Damian Longin Participant University of Wroclawski Osin, Denis V Speaker The City College of CUNY Otera, Daniele Ettore Participant Universitè de Neuchatel Papazoglou, Panagiotis Speaker University of Athens Pejic, Michael Participant N/A Peng, Irine Participant University of Chicago Pershell, Karoline Patricia Participant Rice University Peterson, Valerie Participant University of Illinois at Urbana-Champaign Pfaff, Catherine Notetaker Rutgers University Przytycki, Piotr Participant Polish Academy of Sciences Putman, Thomas Andrew Speaker Massachusetts Institute of Technology REMY, Bertrand Speaker N/A Riley, Tim Participant Bristol University Rinker, Mark Participant University of San Francisco Sabalka, Lucas Participant UC Davis Schroeder, Timothy Alan Participant University of Wisconsin-Milwaukee Scott, Richard Allan Participant Santa Clara University Shalom, Yehuda - Speaker Tel-Aviv University Shwartz, Robert Participant Bar Ilan University Sonkin, Dmitriy M Participant University of Virginia Storm, Peter Allen Participant University of Pennsylvania Swenson, Eric Participant Brigham Young University Tamura, Makoto Participant Osaka Sangyo University Tessera, Romain Participant Vanderbilt University Thomas, Anne Caroline Mary Speaker Cornell University Tran, Quan Thua Participant university of oklahoma Tsai, Chia-yen Participant University of Illinois at Urbana-Champaign Tsemo, Aristide Participant N/A Valette, Alain Participant Universite de Neuchatel Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Speaker Cornell University Wilton, Henry Participant University of Texas Wise, Daniel Speaker McGill University Wortman, Kevin Participant University of Utah Xie, Xiangdong Participant Geogia Souther University Yaman, Asli Participant Centre de Recerca Matematica

REPORT ON GGT CONNECTIONS FOR WOMEN WORKSHOP

The Connections for Women workshop for the Geometric Group Theory program was held Thursday and Friday, August 23-24, 2007, prior to the Introductory workshop for the program, which was held the following week. The organizers were Ruth Charney, Indira Chatterji and Karen Vogtmann. The organizers chose four theorems which are fundamental in the subject: Mostow rigidity, Gromov’s polynomial growth theorem, the Bass-Serre and Rips theory of groups acting on trees, and Thurston’s classification of surface mapping classes. We then chose four speakers and asked each to give a two-hour introductory minicourse on the proofs and applications of these classical theorems. Speakers were selected for the quality of their current research, the extent to which they use the particular theorem in their own research, and their expository ability. The quality of the lectures was in fact excellent (including the lectures on Gromov’s theorem, which were prepared by one of the organizers at the last minute due to the fact that an invited speaker was unexpectedly denied a visa.) The format of each minicourse was a one-hour lecture followed by a one-hour discussion session and then the second hour of lecture. The discussion section was conducted by arranging participants randomly into groups of four; each group was asked to go over the first lecture and come up with questions. The lecturer then came in for the last 15-20 minutes of this hour to answer questions, before giving the second lecture. The lectures were held in the auditorium, but the discussion groups met in the lunchroom, decks and halls of the building. Prior to the workshop, participants were offered the opportunity to send a one- or two- page PDF file of a poster explaining their current research interests. The word “poster” was deliberately used instead of “abstract” in order to encourage graphics and attention to presentation. We received 24 posters in all. The deadline for submitting posters was several days before the workshop; the posters were then copied, bound and distributed to all participants at the beginning of the workshop. MSRI recruited a notetaker for the lectures, who was paid a small stipend. The resulting notes, together with videos of the lectures, are posted on the MSRI web site. The notes are also available at http://www.math.cornell.edu/˜vogtmann/MSRI, which contains lecture notes or links to lecture notes from all of the GGT and TK workshops and short courses this semester at MSRI. There were approximately 60 officially registered participants, including about 10 men. Attendance at the lectures was signifiantly higher than this; the audience included general MSRI members in residence for the GGT semester, participants in the TK Introductory workshop and people arriving early for the GGT introductory workshop. Participation in the discussion sessions was smaller, ranging from 20 to 40; this was due at least partly to 1 2 REPORT ON GGT CONNECTIONS FOR WOMEN WORKSHOP the fact that there were very interesting lectures in the TK introductory workshop which conflicted with the discussion sessions. MSRI also arranged a dinner on Thursday evening in a local Chinese restaurant, which was attended by about 40 participants.

Scheduling comment: The conflict with the TK introductory workshop was very unfortunate, since the TK workshop was of interest to almost all Connections participants and vice versa. The TK organizers were very cooperative in trying to avoid conflicts, but this made for an exhaust- ing schedule as well as some unavoidable conflicts. It occurred because we were told that was impossible to hold the Connections workshop over the weekend, although the week- end between the introductory workshops for the two programs would in fact have been ideal. We are happy to see that this policy has been reversed for upcoming Connections workshops.

Connections for Women: Geometric Group Theory August 203, 2007 to August 24, 2007

Thursday August 23

08:45AM - 09:00 AM Welcome from MSRI 09:00AM - 09:50 AM Anna Wienhard “Mostow rigidity part I” 09:50AM - 10:00AM Break 10:00AM - 10:30 AM Small group discussion of Part I 10:30AM - 10:50 AM Discussion of questions from small groups 10:50AM - 11:00 AM Break 11:00AM - 11:50 AM Anna Wienhard “Mostow rigidity part II” 11:50AM - 12:00 PM Questions 12:00PM - 01:30 PM Lunch 01:30PM - 02:20 PM Genevieve Walsh “Thurston's classification of surface automorphisms Part I” 02:20PM - 02:30PM Break 02:30PM - 03:00 PM Small group discussion of Part I 03:00PM - 03:20 PM Discussion of questions from small groups 03:20PM - 04:00 PM Afternoon tea 04:00PM - 04:50 PM Genevieve Walsh “Thurston's classification of surface automorphisms Part II” 04:50PM - 05:00PM Questions

Friday August 24

09:00AM - 09:50 AM Lisa Carbone “Trees and Group Actions Part I” 09:50AM - 10:00 AM Break 10:00AM - 10:30 AM Small group discussion of Part I 10:30AM - 10:50 AM Discussion of questions from small groups 10:50AM - 11:00 AM Break 11:00AM - 11:50 AM Lisa Carbone “Trees and Group Actions Part II” 11:50AM - 12:00 PM Questions 12:00PM - 01:30 PM Lunch 01:30PM - 02:20 PM Indira Chatterji “Gromov's polynomial growth theorem Part I” 02:20PM - 02:30 PM Break 02:30PM - 03:00 PM Small group discussion of Part I 03:00PM - 03:20 PM Discussion of questions from small groups 03:20PM - 04:00 PM Afternoon tea 04:00PM - 04:50 PM Indira Chatterji “Gromov's polynomial growth theorem Part II” 04:50PM - 05:00 PM Questions

Currently Available Videos

• Anna Wienhard , Mostow rigidity part I August 23,2007, 09:00 AM to 09:50 AM

• Anna Wienhard , Mostow rigidity part II August 23,2007, 11:00 AM to 11:50 AM

• Genevieve Walsh , Thurston's classification of surface automorphisms Part I August 23,2007, 01:30 PM to 02:20 PM

• Genevieve Walsh , Thurston's classification of surface automorphisms Part II August 23,2007, 04:00 PM to 04:50 PM

• Lisa Carbone , Trees and Group Actions Part I August 24,2007, 09:00 AM to 09:50 AM

• Lisa Carbone , Trees and Group Actions Part I I August 24,2007, 11:00 AM to 11:50 AM

• Indira Chatterji , Gromov's polynomial growth theorem Part I August 24,2007, 01:30 PM to 02:20 PM

• Indira Chatterji , Gromov's polynomial growth theorem Part II August 24,2007, 04:00 PM to 04:50 PM

Participant List

Name Role Institution Abrams, Aaron David Participant Emory University Algom Kfir, Yael Participant University of Utah University of Illinois at Urbana- Aramayona, Javier Participant Champaign Arzhantseva, Goulnara Speaker Université de Genève Badus, Alina Participant University of Pennsylvania Banu, Letitia Mihaela Participant University of Western Ontario Barnhill, Angela Kubena Participant Northwestern University Bennett, Hanna Participant University of Chicago Brendle, Tara E Participant Louisiana State University Calderin, Ivo Janier Participant Florida State University Carbone, Lisa Speaker Rutgers University Cavendish, William Palmer Participant N/A Charney, Ruth Organizer Brandeis University Chatterji, Indira Lara Organizer OSU Childers, Leah Participant Louisiana State University Clancy, Maura Participant National University of Ireland, Galway cobbs, leigh Participant Rutgers Davie, Emille Kennae Participant University of California Santa Barbara Delp, Kelly Annette Participant Buffalo State College Duchin, Moon Participant University of California, Davis Dymarz, Tullia Maria Participant Yale University Fadnavis, Sukhada Sharad Participant Caltech Fein, Gregory MacLean Participant Rutgers University-Newark Fernos, Talia Participant University of California, Los Angeles University of California at Santa Finegold, Brie Participant Barbara Fujikawa, Ege Participant Sophia University Futer, David Participant Michigan State University Gouvea, Ezra Participant UC Davis Iftime, Mihaela D. Participant Boston University Irmak, Elmas Participant Bowling Green State University Kahobaei, Delaram Participant College of Technology Kalanidhi, Sharada Participant Bank of America Kar, Aditi Participant Ohio State University Kenny, Aisling Participant Dublin City University Komlos, Hanna Participant Rutgers University Leary, Ian Participant Ohio State University Lim, Seonhee Participant Cornell Univeristy Malone, William Participant University of Utah Mangahas, Johanna Participant University of Michigan McGathey, Natalie Joy Participant University of Illinois at Chicago MECHAM, TARALEE Participant UNIVERSITY OF OKLAHOMA Mihaila, Ioana Notetaker Cal Poly Pomona Min, Honglin Participant Rutgers University-Newark Moon, Soyoung Participant Institut de Mathematiques Mukherjee, Antara Participant Univ. of Oklahoma Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Peng, Irine Participant University of Chicago Perin, Chloé Participant Université de Caen University of Illinois at Urbana- Peterson, Valerie Participant Champaign Pettet, Alexandra Participant Stanford University Pfaff, Catherine Participant Rutgers University Pfaff, Catherine Participant Rutgers University Putman, Andrew Participant University of Chicago Rafi, Kasra Participant University of Chicago Riley, Tim Participant Bristol University SanAgustin, Keefe L Participant Brandeis University Schenk, Candace Participant Binghamton University Talelli, Olympia Participant University of Athens Tao, Jing Participant University of Illinois at Chicago Thomas, Anne Caroline Mary Participant Cornell University University of Illinois at Urbana- Tsai, Chia-yen Participant Champaign Ulcigrai, Corinna Participant Princeton University Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Organizer Cornell University Walsh, Genevieve Speaker Tufts University Wassink, Bronlyn Participant Binghamton University Wienhard, Anna Katharina Speaker University of Chicago Yaman, Asli Participant Centre de Recerca Matematica

Introduction to Geometric Group Theory

August 20, 2007 to August 24, 2007

Monday August 27

09:00 AM - 09:50 AM James Cannon: Geometric Group Theory by Example 10:30 AM - 11:20 AM Ruth Charney: Coxeter and Artin groups 01:00 PM - 01:50 PM Benson Farb: A crash course on mapping class groups, I: Algebra versus topology 02:00 PM - 02:50PM Michael Davis: Cohomology of Coxeter groups and buildings 03:45 PM - 04:35 PM Daniel Groves: Homomorphisms to mapping class groups

Tuesday August 28

09:00 AM - 09:50 AM Bruce Kleiner: Quasi-isometric rigidity 10:30 AM - 11:20 AM Ian Leary: Finiteness properties and Bestvina-Brady Morse theory 01:00 PM - 01:50 PM James Cannon: Curvature in Group Theory 02:00 PM - 02:50PM Kim Ruane: Automorphisms of Graph Products 03:45 PM - 04:35 PM Jennifer Taback: An introduction to Thompson's group F

Wednesday August 29

09:00 AM - 09:50 AM Ruth Charney: Coxeter and Artin groups 10:30 AM - 11:20 AM Benson Farb: A crash course in mapping class groups, II: Moduli space and the Thurston Classification 11:30 AM - 12:20 PM Bruce Kleiner: Quasi-isometric rigidity

Thursday August 30

09:00 AM - 09:50 AM Ian Leary: Finiteness properties and Bestvina-Brady Morse theory 10:30 AM - 11:20 AM James Cannon: Curvature and Group Algorithms 01:00 PM - 01:50 PM Ruth Charney: Coxeter and Artin groups 02:00 PM - 02:50 PM Alain Valette: Groups embedding quasi-isometrically into Hilbert spaces 03:45 PM - 04:35 PM Kevin Wortman: SL(n,Z[t]) is not FP_(n-1)

Friday September 1

09:00 AM - 09:50 AM Bruce Kleiner: Quasi-isometric rigidity 10:30 AM - 11:20 AM Benson Farb: A crash course on mapping class groups, II: The Torelli group 01:00PM - 01:50 PM Ian Leary: Finiteness properties and Bestvina-Brady Morse theory 02:00 PM - 02:50 PM Jon McCammond: Triangle, Squares and Biautomaticity 03:45 PM - 04:35 PM Henry Wilton: Subgroup separability in residually free groups

Currently Available Videos

• James Cannon , Geometric Group Theory by Example August 27,2007, 09:00 AM to 09:50 AM

• Ruth Charney , Coxeter and Artin groups August 27,2007, 10:30 AM to 11:20 AM

• Benson Farb , A crash course on mapping class groups, I: Algebra versus topology August 27,2007, 01:00 PM to 01:50 PM

• Michael Davis , Cohomology of Coxeter groups and buildings August 27,2007, 02:00 PM to 02:50 PM

• Daniel Groves , Homomorphisms to mapping class groups August 27,2007, 03:45 PM to 04:45 PM

• Bruce Kleiner , Quasi-isometric rigidity August 28,2007, 09:00 AM to 09:50 AM

• Ian Leary , Finiteness properties and Bestvina-Brady Morse theory August 28,2007, 10:30 AM to 11:20 AM

• James Cannon , Curvature in Group Theory August 28,2007, 01:00 PM to 01:50 PM

• Kim Ruane , Automorphisms of Graph Products August 28,2007, 02:00 PM to 02:50 PM

• Jennifer Taback , An introduction to Thompson's group F August 28,2007, 03:45 PM to 04:45 PM

• Ruth Charney , Coxeter and Artin groups August 29,2007, 09:00 AM to 09:50 AM

• Benson Farb , A crash course in mapping class groups, II: Moduli space and the Thurston Classification August 29,2007, 10:30 AM to 11:20 AM

• Bruce Kleiner , Quasi-isometric rigidity August 29,2007, 11:30 AM to 12:20 PM

• Ian Leary , Finiteness properties and Bestvina-Brady Morse theory August 30,2007, 09:00 AM to 09:50 AM

• James Cannon , Curvature and Group Algorithms August 30,2007, 10:30 AM to 11:20 AM

• Ruth Charney , Coxeter and Artin groups August 30,2007, 01:00 PM to 01:50 PM

• Kevin Wortman , SL(n,Z[t]) is not FP_(n-1) August 30,2007, 03:45 PM to 04:35 PM

• Bruce Kleiner , Quasi-isometric rigidity August 31,2007, 09:00 AM to 09:50 AM

• Benson Farb , A crash course on mapping class groups, II: The Torelli group August 31,2007, 10:30 AM to 11:20 AM

• Ian Leary , Finiteness properties and Bestvina-Brady Morse theory August 31,2007, 01:00 PM to 01:50 PM

• Jon McCammond , Triangle, Squares and Biautomaticity August 31,2007, 02:00 PM to 02:50 PM

• Henry Wilton , Subgroup separability in residually free groups August 31,2007, 03:45 PM to 04:35 PM

Participant List

Name Role Institution Abrams, Aaron David Participant Emory University Alperin, Roger C. Participant San Jose State University Amos, Gabriel Participant University of California Antolin Pichel, Yago Participant Universitat Autonoma de Barcelona Aramayona, Javier Participant University of Illinois at Urbana-Champaign Ardila, Federico Participant Microsoft Research Arenas, Ruben Participant University of California, San Diego Atkinson, Christopher Participant University of Illinois at Chicago Baba, Shinpei Participant UC Davis Bacardit, Lluis Participant Univirsitat Autonoma de Barcelona Badus, Alina Participant University of Pennsylvania bagci, irfan Participant University of Georgia Baker, Owen Participant Cornell Barnhill, Angela Kubena Participant Northwestern University Barrera, Carlos Participant UC Davis Behrstock, Jason Participant Columbia University Bennett, Hanna Participant University of Chicago Berkove, Ethan J Member Lafayette College Bestvina, Mladen Organizer University of Utah Birman, Joan Participant Barnard-Columbia Brendle, Tara E Participant Louisiana State University Calderin, Ivo Janier Participant Florida State University Cannon, James Speaker Brigham Young University Cashen, Christopher Henry Participant University of Illinois, Chicago Cavendish, William Palmer Participant N/A Charney, Ruth Speaker Brandeis University Chatterji, Indira Lara Participant OSU Chaynikov, Vladimir Participant Vanderbilt University Childers, Leah Participant Louisiana State University Korea Advanced Institute of Science and Choudhury, Dhrubajit Participant Technology Clancy, Maura Participant National University of Ireland, Galway cobbs, leigh Participant Rutgers Coulbois, Thierry Participant Universite Aix-Marseille III (France) Cruz Morales, John Alexander Participant National University Danciger, Jeffrey E Participant Stanford University Dani, Pallavi Participant University of Oklahoma Das, Manav Participant University of Louisville Davie, Emille Kennae Participant University of California Santa Barbara Davis, Michael W. Speaker Ohio State University de Balle Pigem, Borja Participant UPC Delucchi, Emanuele Participant Binghamton University (SUNY) Deshpande, Priyavrat C. Participant The University of Western Ontario Dison, William Participant Imperial College Dymarz, Tullia Maria Participant Yale University Edwards, Robert Participant UCLA Farb, Benson Speaker University of Chicago Fein, Gregory MacLean Participant Rutgers University-Newark Fernandes, Praphat Xavier Participant Emory University Fernos, Talia Participant University of California, Los Angeles Frankel, Steven Participant The Cooper Union Friedman, Michael Participant Bar-Ilan University Fujikawa, Ege Participant Sophia University Futer, David Participant Michigan State University Geoghegan, Ross Participant SUNY, Binghamton Gongopadhyay, Krishnendu Participant Indian Institute of Technology Gouvea, Ezra Participant UC Davis Groves, Daniel Peter Speaker University of Illinois Korea Advanced Institute of Science and Ha, Jaesoon Participant Technology Hass, Joel Participant UC Davis Hensel, Sebastian Participant University of Bonn Hoban, Ryan Francis Participant University of Maryland Hsu, Timothy M. Participant San Jose State University Irmak, Elmas Participant Bowling Green State University Ji, Lizhen Participant univ of Michigan Jones, Keith Michael Participant SUNY Binghamton Juhász, Arye Participant Technion (Israel Institute of Technology) Kalanidhi, Sharada Participant Bank of America Kar, Aditi Participant Ohio State University Kenny, Aisling Participant Dublin City University Kerckhoff, Steven Participant Stanford University Kim, Sam Participant Kyungpook National University Kleiner, Bruce A. Speaker Yale University Komlos, Hanna Participant Rutgers University Kostyuk, Victor Participant Cornell - Mathematics Le Donne, Enrico Participant Yale University Leary, Ian Speaker Ohio State University Korea Advanced Institute of Science and Lee, Gye-Seon Participant Technology Lee, Jaejeong Participant UC Davis Lim, Seonhee Participant Cornell Univeristy Lotay, Jason Participant University College Oxford Louder, Larsen Edmund Participant University of Utah Louwsma, Joel Participant California Institute of Technology Macdonald, Jeremy Participant McGill University Mackay, John Malcolm Participant University of Michigan Madani, Farid Participant Ecole doctorale de Mathématiques Paris centre Malestein, Justin Participant University of Chicago Malone, William Participant University of Utah Malone, William Participant University of Utah Mangahas, Johanna Participant University of Michigan Margolis, Max Balsam Participant Brandeis University Martinez-Pedroza, Eduardo Participant University of Oklahoma Matsuzaki, Katsuhiko Participant Okayama University McCammond, Jon Organizer University of California, Santa Barbara McCune, David Participant University of Nebraska-Lincoln Meakin, John C. Participant University of Nebraska MECHAM, TARALEE Participant UNIVERSITY OF OKLAHOMA Mese, Chikako Participant Johns Hopkins University Mihaila, Ioana Notetaker Cal Poly Pomona Moon, Soyoung Participant Institut de Mathematiques Mukherjee, Antara Participant Univ. of Oklahoma Nakamura, Kei Participant University of California, Davis Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Okun, Boris L Participant University of Wisconsin-Milwaukee Osajda, Damian Longin Member N/A Ottman, Ryan Joseph Participant University of California- Santa Barbara Papazoglou, Panagiotis Participant University of Athens Peng, Irine Participant University of Chicago Perin, Chloé Participant Université de Caen Peterson, Valerie Participant University of Illinois at Urbana-Champaign Pettet, Alexandra Participant Stanford University Pfaff, Catherine Participant Rutgers University Pommersheim, Jamie Participant Reed College Putman, Andrew Participant University of Chicago Riley, Tim Participant Bristol University Rinker, Mark Participant University of San Francisco Ruane, Kim Speaker Tufts University Ruiz, Amanda Participant San Francisco State University Sabalka, Lucas Participant UC Davis Sageev, Michah Organizer Technion (Israel Institute of Technology) Sahattchieve, Jordan Antonov Participant University of Michigan Schenk, Candace Participant Binghamton University Scott, Richard Allan Participant Santa Clara University Sohrabi, Mahmood Participant Carleton Univertsity Staley, Daniel Todd Participant Rutgers University Stallings, John R. Participant UCB - University of California, Berkeley Sulway, Robert John Participant University of California, Santa Barbara Sunic, Zoran Participant Texas A&M University Swenson, Eric Participant Brigham Young University Taback, Jennifer Speaker Bowdoin College Talelli, Olympia Participant University of Athens Tao, Jing Participant University of Illinois at Chicago Thomas, Anne Caroline Mary Participant Cornell University Tsai, Chia-yen Participant University of Illinois at Urbana-Champaign Ulcigrai, Corinna Participant Princeton University Upadhyay, Ashish Kumar Participant Birla Institute of Technology and Science, Pilani Valette, Alain Speaker Universite de Neuchatel Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Organizer Cornell University Wall, Liam Participant Oxford University Walsh, Genevieve Participant Tufts University Wassink, Bronlyn Participant Binghamton University Wiest, Bert Participant N/A Wilton, Henry Speaker University of Texas Wortman, Kevin Speaker University of Utah Yaman, Asli Participant Centre de Recerca Matematica Zawislak, Pawel Participant University of Wroclaw

Topics in Geometric Group Theory November 5–9, 2007 Organizers: Noel Brady, Mike Davis, Mark Feighn.

Topics in Geometric Group Theory was the third of the three major workshops of the semester-long parent program in Geometric Group Theory, which ran from August 20 to December 14, 2007 at the Mathematical Sciences Research Institute, Berkeley, CA. The first two workshops, Connections for Women: Geometric Group Theory and Introduction to Geometric Group Theory, were held in August 2007. These consisted of several series of mini-courses, which introduced young researchers to various themes and directions in the field. The aim of the Topics in Geometric Group Theory workshop was to present, in a conference format, cutting-edge developments across the board in geometric group theory. The workshop consisted of 22 invited speakers, each giving a one hour presenta- tion, according to the following daily schedule. Monday Nov. 5 Indira Chatterji, Alexander Dranishnikov, Alex Eskin, Yehuda Shalom. Tuesday Nov. 6 Koji Fujiwara, Denis Osin, Panos Papasoglu, Anne Thomas, Karen Vogtmann. Wednesday Nov. 7 Jason Behrstock, Jan Dymara, Dan Margalit, Thomas Putman, Bertrand Remy. Thursday Nov. 8 Vincent Guirardel, Michael Handel, Lars Louder, Volodymyr Nekrashevych. Friday Nov. 9 Martin Bridson, David Fisher, Tadeusz Januszkiewicz, Jason Manning. There were over 130 registered participants. In addition, members of the parallel program on Teichmuller Theory and Kleinian Groups attended many of the talks. The conference room was full, and there were very lively discussions between the talks. The speaker demographics clearly reflect the fact that much of the energy and momentum of the field is generated by the steady influx of young researchers. Thir- teen of the twenty-two speakers received PhDs within the last 10 years; five of these were post-doctoral associates, three of whom had just received PhDs in 2007. The workshop organizers, in consultation with the program organizers, decided to only to invite people who had not already presented at one of the two earlier workshops in August.

1 The talks covered a broad spectrum of topics in geometric group theory: coarse geometry of groups, asymptotic dimension, CAT(0) groups, CAT(0) cube complexes and generalizations, automorphism groups of free groups, of surface groups and of right-angled Artin groups, L2-cohomology, buildings and polyhedral complexes, map- ping class groups and special subgroups, relatively hyperbolic groups and applications of group theoretical hyperbolic Dehn filling, equations over free groups, limit groups, fixed point properties, and higher filling invariants of groups. The workshop was flanked by two talks which addressed one of the central prob- lems in the field; namely, Gromov’s program on the quasi-isometric classification of infinite discrete groups. The opening talk, by Alex Eskin, described the new idea of coarse differentiation which has been used by Eskin-Fisher-Whyte to prove the quasi- isometric rigidity of Sol. This is one of the most significant tools in quasi-isometry classification in recent years. The closing talk, by David Fisher, gave an account of the recent proof by Bruce Kleiner of Gromov’s polynomial growth theorem. Unlike Gromov’s proof, Kleiner’s proof makes no use of the structure theory of locally com- pact groups. Kleiner’s preprint had just appeared a few weeks before the workshop, so Fisher’s presentation generated a lot of excitement and discussion. These two talks demonstrated how Gromov’s foundational program of quasi-isometric classification still dominates research in the field on the one hand, and gave a glimpse at the level of sophistication to which tools for dealing with quasi-isometric rigidity have evolved on the other hand. The development of a dictionary between automorphism groups of free groups and automorphism groups of surface groups (mapping class groups) has had a long and rich history, including joint contributions from group theorists, Teichmuller theorists and low-dimensional topologists. This research was well represented at the confer- ence, with talks by Jason Behrstock, Koji Fujiwara, Michael Handel, Dan Margalit, Thomas Putman and Karen Vogtmann. It was interesting to see that techniques from CAT(0) geometry, Morse theory, and quasi-isometric rigidity are increasingly being applied to mapping class groups and their special subgroups. These topics were of particular interest to participants from the parallel program on Teichmuller Theory and Kleinian Groups. The pioneering work of Zlil Sela and (independently) of Olga Kharlampovich and Alexei Miasnikov on the elementary theory of free groups has had huge impact on the field. Some of this impact was seen in the talks of Martin Bridson, Vincent Guirardel and Lars Louder, who described properties of limit groups and of equations over free groups. Another exciting development in recent years was the generalization of Thurston’s hyperbolic Dehn filling theory to the context of relatively hyperbolic groups. This

2 was carried out by Groves-Manning and independently by Osin. At the workshop participants got to learn about the first waves of applications of hyperbolic Dehn filling in group theory. Denis Osin presented applications to automorphisms of rela- tively hyperbolic groups, and Jason Manning described how he, Ian Agol and Daniel Groves used Dehn filling to relate residual finiteness and subgroup separability for hyperbolic groups.

3

Topics in Geometric Group Theory November 05, 2007 to November 09, 2007

Schedule

Monday November 5

08:45AM - 09:00AM "Welcome to MSRI" talk Coarse differentiation and the geometry 09:00AM - 09:50AM Alex Eskin of polycyclic groups. 10:00AM - 10:30AM Tea 10:30AM - 11:20AM Yehuda Shalom TBD Alexander On asymptotic dimension of Coxeter 11:30AM - 12:20PM Dranishnikov groups 12:30PM - 02:00PM Lunch 02:00PM - 02:50PM Indira Chatterji Median spaces and applications 03:00PM - 03:30PM Tea Discrete groups and Morse theory 04:10PM - 05:00PM Mladen Bestvina (in 610 Evans Hall, UC-Berkeley)

Tuesday November 6

Automorphism groups of right-angled 09:00AM - 09:50AM Karen Vogtmann Artin group 10:00AM - 10:30AM Tea Higher isoperimetric inequalities for 10:30AM - 11:20AM Panagiotis Papazoglou complexes and groups 11:30AM - 12:20PM Anne Thomas Lattices acting on polyhedral complexes 12:30PM - 02:00PM Lunch Normal automorphisms of relatively 02:00PM - 02:50PM Denis Osin hyperbolic groups 03:00PM - 03:30PM Tea Rank-1 isometries on CAT(0) spaces and 03:30PM - 04:20PM Koji Fujiwara quasi-homomorphisms 04:30PM – 06:30PM Reception

Wednesday November 7

A family of simple groups acting on 09:00AM - 09:50AM Bertrand REMY buildings 10:00AM - 10:30AM Tea 10:30AM - 11:20AM Dan Margalit Dimension of Torelli groups On the homology of finite index 11:30AM - 12:20PM Thomas Putman subgroups of the mapping class group 12:30PM - 02:00PM Lunch Quasi-isometric rigidity of the mapping 02:00PM - 02:50PM Jason Behrstock class groups 03:00PM - 03:30PM Tea 03:30PM - 04:20PM Jan Dymara L^2 cohomology of buildings

Thursday November 8

Geometric Makanin algorithm for solving 09:00AM - 09:50AM Vincent Guirardel equations in virtually free groups 10:00AM - 10:30AM Tea Global fixed points for centralizers and 10:30AM - 11:20AM Mike Handel Morita's Theorem (joint work with John Franks) 11:30AM - 12:20PM Larsen Louder Krull dimension for limit groups 12:30PM - 02:00PM Lunch Volodymyr Space of marked groups and non-uniform 02:00PM - 02:50PM Nekrashevych exponential growth 03:00PM - 03:30PM Tea

Friday November 9

09:00AM - 09:50AM Martin Bridson Finitely presented, residually-free groups 10:00AM - 10:30AM Tea 10:30AM - 11:30AM Tadeusz Januszkiewicz Groups with fixed point properties Residual finiteness and separability of 11:30AM - 12:20PM Jason Manning quasi-convex subgroups 12:30PM - 02:00PM Lunch Kleiner's proof of the polynomial growth 02:00PM - 02:50PM David Fisher theorem 03:00PM - 03:30PM Tea

Currently Available Videos

• Alex Eskin , Coarse Differentiation and the Geometry of Polycyclic Groups November 5,2007, 09:00 AM to 09:50 AM

• Yehuda Shalom , Y. Shalom Lecture November 5,2007, 10:30 AM to 11:20 AM

• Alexander Dranishnikov , On Asymptotic Dimension of Coxeter Groups November 5,2007, 11:30 AM to 12:20 PM

• Indira Chatterji , Median Spaces and Applications November 5,2007, 02:00 PM to 02:50 PM

• Karen Vogtmann , Automorphism Groups of Right-angled Artin Groups November 6,2007, 09:00 AM to 09:50 AM

• Panagiotis Papazoglou , Higher Isoperimetric Inequalities for Complexes and Groups November 6,2007, 10:30 AM to 11:20 AM

• Anne Thomas , Lattices Acting on Polyhedral Complexes November 6,2007, 11:30 AM to 12:20 PM

• Denis Osin , Normal Automorphisms of Relatively Hyperbolic Groups November 6,2007, 02:00 PM to 02:50 PM

• Koji Fujiwara , Rank-1 Isometries on CAT(0) Spaces and Quasi-homomorphisms November 6,2007, 03:30 PM to 04:20 PM

• Bertrand REMY , A Family of Simple Groups Acting on Buildings November 7,2007, 09:00 AM to 09:50 AM

• Dan Margalit , Dimension of Torelli Groups November 7,2007, 10:30 AM to 11:20 AM

• Thomas Putman , On the Homology of Finite Index Subgroups of the Mapping Class Group November 7,2007, 11:30 AM to 12:20 PM

• Jason Behrstock , Quasi-isometric Rigidity of the Mapping Class Groups November 7,2007, 02:00 PM to 02:50 PM

• Jan Dymara , L^2 Cohomology of Buildings November 7,2007, 03:30 PM to 04:20 PM

• Vincent Guirardel , Geometric Makanin Algorithm for Solving Equations in Virtually Free Groups November 8,2007, 09:00 AM to 09:50 AM

• Michael Handel , Global fixed points for centralizers and Morita's Theorem (joint work with John Franks) November 8,2007, 10:30 AM to 11:20 AM

• Larsen Louder , Krull Dimension for Limit Groups November 8,2007, 11:30 AM to 12:20 PM

• Volodymyr Nekrashevych , Space of Marked Groups and Non-uniform Exponential Growth November 8,2007, 02:00 PM to 02:50 PM

• Martin Bridson , Finitely Presented, Residually-free Groups November 9,2007, 09:00 AM to 09:50 AM

• Tadeusz Januszkiewicz , Groups with Fixed Point Properties November 9,2007, 10:30 AM to 11:30 AM

• Jason Manning , Residual Finiteness and Separability of Quasi-convex Subgroups November 9,2007, 11:30 AM to 12:20 PM

• David Fisher , Kleiner's Proof of the Polynomial Growth Theorem November 9,2007, 02:00 PM to 02:50 PM

Participant List

Name Role Institution Algom Kfir, Yael Notetaker University of Utah Alperin, Roger C. Participant San Jose State University Amram Blei, Meirav Participant Bar Ilan University Aramayona, Javier Participant University of Illinois at Urbana-Champaign Arzhantseva, Goulnara Participant University of Geneva Atanasov, Risto Participant Western Carolina University Badus, Alina Participant University of Pennsylvania Barnard, Josh Participant University of South Alabama Barnhill, Angela Kubena Participant Northwestern University Behrstock, Jason Speaker Columbia University Bell, Robert W Participant Michigan State University Bennett, Hanna Notetaker University of Chicago Berkove, Ethan J Participant Lafayette College Bestvina, Mladen Speaker University of Utah Bleak, Collin Participant University of Nebraska at Lincoln Brady, Noel Organizer University of Oklahoma Bridson, Martin Speaker University of Oxford Broaddus, Nathan Participant University of Chicago Bumagin, Inna Participant Carleton University Calderin, Ivo Janier Participant Florida State University Caprace, Pierre-Emmanuel Participant Institut des Hautes Études Scientifiques (IHES) Carette, Mathieu Marc Participant Université Libre de Bruxelles Cashen, Christopher Henry Participant University of Illinois, Chicago Casson, Andrew John Participant Yale University Cavendish, William Palmer Participant N/A Chatterji, Indira Lara Speaker OSU Chaynikov, Vladimir Participant Vanderbilt University Childers, Leah Participant Louisiana State University Clair, Bryan Participant Saint Louis University Cleary, Sean Participant CUNY, City College Connell, Chris G. Participant Indiana University Coulbois, Thierry Participant Universite Aix-Marseille III (France) Culler, Marc Participant University of Illinois, Chicago Dani, Pallavi Participant University of Oklahoma Davidson, Peter John Participant University of Glasgow Davis, Michael W. Organizer Ohio State University Denham, Graham Campbell Participant University of Western Ontario Dranishnikov, Alexander Speaker University of Florida Dymara, Jan Speaker Uniwersytet Wroclawski Edwards, Robert Participant UCLA Elder, Murray Participant Stevens Institute of Technology Eskin, Alex Speaker University of Chicago Falk, Michael J. Participant Northern Arizona University Feighn, Mark Organizer Rutgers University Fel'shtyn, Alexander Participant Boise State University and Szczecin Univesrsity Fernos, Talia Participant University of California, Los Angeles Fisher, David M Participant Indiana University Forester, Max Participant University of Oklahoma Francaviglia, Stefano Participant University of Pisa Freden, Eric Participant Southern Utah University Fujiwara, Koji Speaker Tôhoku University Futer, David Participant Michigan State University Gadre, Vaibhav S Participant Caltech Geller, William Participant Indiana U.-Purdue U. Indianapolis Geoghegan, Ross Participant SUNY, Binghamton Goldman, Bill Participant University of Maryland Groves, Daniel Peter Participant University of Illinois Guilbault, Craig Robert Participant UW-Milwaukee Guirardel, Vincent Speaker Universite Paul Sabatier (Toulouse) Guralnik, Dan Participant Vanderbilt University Guth, Larry Participant Massachusetts Institute of Technology Hambleton, Ian Participant McMaster University Handel, Michael Speaker Lehman College Hass, Joel Participant UC Davis Henderson, Jim Participant U. of Colorado-Colorado Springs Hensel, Sebastian Participant University of Bonn Hermiller, Susan Participant University of Nebraska Hsu, Timothy M. Participant San Jose State University Huang, Zheng Participant University of Michigan James, Justin Amery Participant Minnesota State University - Moorhead Januszkiewicz, Tadeusz Speaker Ohio State University Kaminker, Jerry Participant IUPUI Kapovich, Ilya Participant University of Illinois at Urbana-Champaign Kapovich, Michael Participant University of California Kar, Aditi Participant Ohio State University Katerman, Eric Participant University of Texas-Austin Kim, Sam Participant Kyungpook National University Kim, Sang-hyun Participant N/A Klein, Tom Participant McMaster University Kramer, Linus Karl Heinz Participant Universitaet Muenster Lackenby, Marc Speaker University of Oxford Lai, Yvonne (Yuan-Juang) Participant UC Davis Leary, Ian Participant Ohio State University Levitt, Gilbert Participant universite de Caen Lim, Seonhee Notetaker Cornell Univeristy Lotay, Jason Participant University College Oxford Louder, Larsen Edmund Speaker University of Utah Louwsma, Joel Participant California Institute of Technology Lustig, Martin Participant Universite P. Cezanne - Aix Marseille III Mackay, John Malcolm Participant University of Michigan Macura, Natasa Participant Trinity University Malone, William Notetaker University of Utah Mangahas, Johanna Participant University of Michigan Manning, Jason Speaker University of Buffalo Marden, Albert Participant University of Minnesota Margalit, Dan Speaker University of Utah Martinez-Pedroza, Eduardo Participant University of Oklahoma Institute of Mathematics "Simion Stoilow" of the Romanian Matei, Daniel Participant Academy McCammond, Jon Participant University of California, Santa Barbara Meier, John Participant Lafayette College Min, Honglin Participant Rutgers University-Newark Minasyan, Ashot Participant University of Geneva Mislin, Guido Participant ETH Mitra, Mahan Participant THE INSTITUTE OF MATHEMATICAL SCIENCES Mooney, Christopher Paul Participant University of Wisconsin-Milwaukee Nekrashevych, Volodymyr Speaker Texas A & M University Nipper, Emanuel Participant Universität Bonn Nucinkis, Brita Participant University of Southampton Okun, Boris L Participant University of Wisconsin-Milwaukee Osajda, Damian Longin Participant University of Wroclawski Osin, Denis V Speaker The City College of CUNY Otera, Daniele Ettore Participant Universitè de Neuchatel Papazoglou, Panagiotis Speaker University of Athens Pejic, Michael Participant N/A Peng, Irine Participant University of Chicago Pershell, Karoline Patricia Participant Rice University Peterson, Valerie Participant University of Illinois at Urbana-Champaign Pfaff, Catherine Notetaker Rutgers University Przytycki, Piotr Participant Polish Academy of Sciences Putman, Thomas Andrew Speaker Massachusetts Institute of Technology REMY, Bertrand Speaker N/A Riley, Tim Participant Bristol University Rinker, Mark Participant University of San Francisco Sabalka, Lucas Participant UC Davis Schroeder, Timothy Alan Participant University of Wisconsin-Milwaukee Scott, Richard Allan Participant Santa Clara University Shalom, Yehuda - Speaker Tel-Aviv University Shwartz, Robert Participant Bar Ilan University Sonkin, Dmitriy M Participant University of Virginia Storm, Peter Allen Participant University of Pennsylvania Swenson, Eric Participant Brigham Young University Tamura, Makoto Participant Osaka Sangyo University Tessera, Romain Participant Vanderbilt University Thomas, Anne Caroline Mary Speaker Cornell University Tran, Quan Thua Participant university of oklahoma Tsai, Chia-yen Participant University of Illinois at Urbana-Champaign Tsemo, Aristide Participant N/A Valette, Alain Participant Universite de Neuchatel Vavrichek, Diane Participant University of Michigan Vogtmann, Karen Speaker Cornell University Wilton, Henry Participant University of Texas Wise, Daniel Speaker McGill University Wortman, Kevin Participant University of Utah Xie, Xiangdong Participant Geogia Souther University Yaman, Asli Participant Centre de Recerca Matematica

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¿ Connections for Women: Introduction to the Spring 2008 Programs

January 16 - 18, 2008

Wednesday January 16, 2008

9:00AM - 9:30AM Opening Remarks 9:30AM - 10:30AM Monica Vazirani “Type A combinatorics” 10:30AM - 11:00AM Morning Break 11:00AM - 12:00PM Poster Preview 12:00PM – 2:00PM Lunch 2:00PM – 3:00PM Anne Schilling “Crystal bases -- what it is all about?” 3:00PM - 3:30PM Afternoon tea 3:30PM - 4:30PM Panel -- From small colleges to large universities and everything in between 4:30PM - 6:00PM Poster Session 6:30PM – 9:00PM Banquet: “Great China Restaurant”, 2115 Kittredge St, Berkeley

Thursday January 17, 2008

“l-modular representations of finite groups of Lie 9:30AM - 10:30AM Bhama Srinivasan type: past, present, future” 10:30AM - 11:00AM Morning Break “Modular Representations of Algebraic Groups: Or To Characteristic Zero and Back Again, with 11:00AM - 12:00PM Terrell Hodge Applications to Representations of Finite Groups of Lie Type in the Defining Characteristic” 12:00PM – 1:30PM Lunch 1:30PM - 2:30PM Julia Pevtsova “Cohomology and support varieties” 3:00PM - 3:30PM Afternoon tea 4:00PM - 5:00PM Panel -- 3 things I wish I knew then

Friday January 18, 2008

“Combinatorial Representation Theory - Old and 9:30AM - 10:30AM Georgia Benkart New” 10:30AM - 11:00AM Morning Break 11:00AM - 12:00PM T. Y. Lam “Early History of Finite Group Representations “ 12:00PM - 2:00PM Lunch 2:00PM - 3:00PM Film (EWM): Women and Mathematics across Cultures 3:00PM - 3:30PM Afternoon tea Currently Available Videos

• Monica Vazirani , Type A Combinatorics January 16,2008, 09:30 AM to 10:30 AM

• Anne Schilling , Crystal BAses--What is it all about? January 16,2008, 02:00 PM to 03:00 PM

• Bhama Srinivasan , l-Modular Representations of Finite Groups of Lie Type: Past, Present, Future January 17,2008, 09:30 AM to 10:30 AM

• Terrell Hodge , Modular Representations of Algebraic Groups: Or To Characteristic Zero and Back Again, with Applications to Representations of Finite Groups of Lie Type in the Defining Characteristic January 17,2008, 11:00 AM to 12:00 PM

• Julia Pevtsova , Cohomology and Support Varieties January 17,2008, 01:30 PM to 02:30 PM

• Georgia Benkart , Combinatorial Representation Theory - Old and New January 18,2008, 09:30 AM to 10:30 AM

• T. Y. Lam , Early History of Finite Group Representations January 18,2008, 11:00 AM to 12:00 PM

Participant List

Name Role Institution Alghamdi, Ahmad M. Participant Umm Alqura University Arslan, Ogul Participant University of Florida Assaf, Sami H Participant MIT Barcelo, Hélène Participant MSRI - Mathematical Sciences Research Institute Beier, Julie C Participant North Carolina State University Benkart, Georgia M. Participant University of Wisconsin bidkhori, hoda Participant Massachusetts Institute of Technology Brichard, Joelle Participant Columbia University École Polytechnique Fédérale de Lausanne Chlouveraki, Maria Participant (EPFL) Christodoulopoulou, Konstantina Participant University of Windsor Daugherty, Zajj Participant University of Wisconsin Davis, Darnice Marie Participant N/A Dobria, Eunice Voichita Participant Florida Atlantic University Du, Ruoxia Participant MIT - Massachusetts Institute of Technology Endelman, Robin Participant University College of the Fraser Valley Goodman, Frederick Participant University of Iowa Graber, John Participant University of Iowa Hersh, Patricia Lynn Participant Indiana University Hodge, Terrell L. Speaker Western Michigan University Holmes, Susan Participant Stanford University Juteau, Daniel Pierre Participant Jussieu Karaali, Gizem Participant Pomona College Lam, Tsit-Yuen Participant UC Berkeley Li, Huilan Participant UQAM Liu, Fu Participant University of California, Davis Lyle, Sinead Participant University of East Anglia Maroti, Attila Participant University of Southern California Mbirika, Abukuse (Aba) Participant University of Iowa Mihnea, Amalya Participant Florida Atlantic University Mogel, Jennifer Participant University of California, Santa Cruz Moreno, Jose A Participant N/A Morier-Genoud, Sophie Participant University of Michigan Murphy, Aileen Participant Saint Louis University Orellana, Rosa Participant Dartmouth College Facultad de Ciencias, Universidad de la Pereira, Mariana Participant Republica Perkins, D. Kala Participant Infinitee Pevtsova, Julia A Participant University of Washington Qing, Yulan Participant Massachusetts Institute of Technology Rainbolt, Julianne Geering Participant St. Louis University Ram, Arun Participant University of Melbourne Ruiz, Amanda Participant San Francisco State University Sanus, Lucia Participant Universitat de Valencia Sazdanovic, Radmila Tomislav Participant The George Washington University Schilling, Anne Participant University of California, Davis Späth, Britta Kerstin Participant N/A Srinivasan, Bhama Organizer University of Illinois, Chicago Sun, Jie Participant University of Alberta Talaska, Kelli Participant University of Michigan van Willigenburg, Steph Participant UBC Vazirani, Monica Joy Organizer University of California, Davis Vigneras, Marie-France Participant Université de Paris 7 (Diderot) Vuletic, Mirjana Participant California Institute of Technology Wiesner, Emilie Participant Ithaca College Williams, Lauren Kiyomi Participant Harvard University Yin, Jingbin Participant MIT Yip, Martha Participant University of Wisconsin Yoo, Meesue Participant University of Pennsylvania Yu, Josephine T. Participant Massachusetts Institute of Technology Yu, Shona Huimin Participant The University of Sydney Zárate, Alma Leticia Participant CINVESTAV IPN

Introductory workshop, January 22-25, 2008 The soul of Combinatorial Representation Theory (CRT) lies in the interplay between combinatorics and various branches of mathematics. Combinatorial meth- ods are applied to solve problems in representation theory, Lie theory, geometry, and mathematical physics and, in symbiosis, deep combinatorial problems also arise from these areas. The goal of the Introductory Workshop was to survey current and recent developments in the field, and set the stage for the focus of the program. Lecture series: There were three lecture series:

•• Michel Brou´e, Complex reflection groups in representations of finite re- flection groups,

•• Andrei Okounkov, Characters of symmetric groups

•• Arun Ram, Combinatorics of Lie type

The lecture series of M. Brou´eprovided a bridge between the RTF program and the CRT program. The lecture series of Brou´e,Okounkov and Ram were designed to provide the basic fundamentals of the fields and explain the role of these topics in current research. Brou´eintroduced finite reflection groups, unipotent characters, Deligne-Lusztig induction and restriction, and Harish-Chandra theory. The lectures of Arun Ram set out the fundamentals of the combinatorics of root systems and path models. In these lectures he gave definitions of Hecke algebras, double affine Hecke algebras, Macdonald polynomials, crystals, Chevalley groups, loop groups, flag varieties, loop Grassmanians and Mirkovic-Vilonen cycles. In total this formed a comprehensive overview of the combinatorics of Lie types. These lectures greatly inspired a project on Hecke group algebras completed at MSRI by Anne Schilling and Nicolas Thiery (preprint arXiv:0804.3781v1 [math.RT]). Andrei Okounkov (a Fields medallist) started with a survey of the character theory of the symmetric group. He proceeded to relate this to the Fock space realization, Gromov-Witten theory, the Hurwitz problem and finally the amazing recent results by Okounkov and Pandharipande on the relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson- Thomas correspondence for local curves. Research talks: To complement the lecture series the postdoctoral fellows of the program each gave an hour long presentation introducing their research area. The topics were

•• Kevin Purbhoo, The Horn Inequalities and Their Generalizations

•• Syu Kato, Geometric representation theory of affine Hecke algebras

•• Lauren Williams, Total positivity for flag varieties: combinatorics, topol- ogy, and toric geometry

•• Ghislain Fourier, Finite dimensional modules for current and loop algebras

1 •• Sami Assaf, Applications of dual equivalence graphs

These talks set the trend for the research activities pursued during the program. Sami Assaf gave a beautiful account on her recent work on dual equivalence graphs, which she used to give a combinatorial proof of the Macdonald-Kostka positvity, and related these graphs to crystal graphs. Ghislain Fourier presented the analysis of finite-dimensional modules for current and loop algebras. In particular he showed how to treat the twisted cases. This ties in with work of David Hernandez, another member of the program, on twisted q-characters for Kirillov-Reshetikhin modules for all types. 5-minute presentations: In addition, there were multiple sessions of 5-minute presentations where the remaining participants were able to introduce themselves mathematically and gave a feel for their current research. These sessions were a great success (there were more volunteers than we could accommodate to speak)! The talks gave an overview of what our community is working on and enabled the participants of the program to make connections with each other. Arun Ram gave a sample 5 minute presen- tation about p-compact groups to stimulate the participation. Other highlights in- cluded Anne Schilling (affine Schubert calculus), David Hernandez (quantum affine algebras), Nat Thiem (supercharacters), Jason Bandlow (Macdonald polynomials), Brant Jones (Kazhdhan-Lustig polynomials), Monica Vazirani (crystals), Shona Yu (Brauer algebras), Mansaoru Koyama (discrete Fourier transform). Funding: Andrei Okounkov, who was one of the mini course speakers, was funded. The other main speakers were in residence at the program. Our funding included support for 12 postgraduate participants and 13 graduate students to attend the workshop for a total budget of $30,000. The total number of registered participants was 154, with participants from the USA, , UK, Korea, Brazil, Japan, Ireland, Spain, Israel, Germany, France, Nigeria, China, Australia, Uruguay, Mexico. Of these approximately 40 were women, 30 postdocs, and 50 graduate students.

2 Introductory Workshop on Combinatorial Representation Theory January 22-25, 2008

Tuesday January 22, 2008

9:00AM - 10:00AM Arun Ram Combinatorics of Lie type 10:00AM - 11:00AM Morning Break Complex reflection groups in representations of finite 11:00AM - 12:00PM Michel Broué reductive groups 12:00PM - 2:00PM Lunch at MSRI 2:00PM - 3:00PM Andrei Okounkov Characters of symmetric groups 3:00PM - 4:00PM Afternoon Tea 4:00PM - 5:00PM 5 Minute Presentations 5:00PM - 6:00PM Reception Wednesday January 23, 2008

9:00AM - 10:00AM Arun Ram Combinatorics of Lie type 10:00AM - 11:00AM Morning Break 11:00AM - 12:00PM Andrei Okounkov Characters of symmetric groups 12:00PM - 2:00PM Lunch at MSRI 2:00PM - 3:00PM Kevin Purbhoo The Horn inequalities and their generalizations 3:00PM - 4:00PM Afternoon Tea 4:00PM - 5:00PM Syu Kato Geometric representation theory of affine Hecke algebras Thursday January 24, 2008

Complex reflection groups in representations of finite 9:00AM - 10:00AM Michel Broué reflective groups 10:00AM - 11:00AM Morning Break 11:00AM - 12:00PM 5 Minute Presentations 12:00PM - 2:00PM Lunch at MSRI 2:00PM - 3:00PM Andrei Okounkov Characters of symmetric groups 3:00PM - 4:00PM Afternoon Tea Total positivity for flag varieties: combinatorics, topology, 4:00PM - 5:00PM Lauren Williams and toric geometry Friday January 25, 2008

9:00AM - 10:00AM Arun Ram Combinatorics of Lie type 10:00AM - 11:00AM Morning Break 11:00AM - 12:00PM Ghislain Fourier Finite dimensional modules for current and loop algebras 12:00PM - 2:00PM Lunch at MSRI 2:00PM - 3:00PM Sami Assaf Applications of dual equivalence graphs 3:00PM - 4:00PM Afternoon Tea Complex reflection groups in representations of finite 4:00PM - 5:00PM Michel Broué reflective groups

Currently Available Videos

• Arun Ram , Combinatorics of Lie type January 22,2008, 09:00 AM to 10:00 AM

• Michel Broué , Complex Reflection Groups in Representations of Finite Reductive Groups January 22,2008, 11:00 AM to 12:00 PM

• Andrei Okounkov , Characters of Symmetric Groups January 22,2008, 02:00 PM to 03:00 PM

• Arun Ram , Combinatorics of Lie Type January 23,2008, 09:00 AM to 10:00 AM

• Andrei Okounkov , Characters of Symmetric Groups January 23,2008, 11:00 AM to 12:00 PM

• Kevin Purbhoo , The Horn Inequalities and Their Generalizations January 23,2008, 02:00 PM to 03:00 PM

• Syu Kato , Geometric representation theory of affine Hecke algebras January 23,2008, 04:00 PM to 05:00 PM

• Michel Broué , Complex Reflection Groups in Representations of Finite Reductive Groups January 24,2008, 09:00 AM to 10:00 AM

• Andrei Okounkov , Characters of Symmetric Groups January 24,2008, 02:00 PM to 03:00 PM

• Lauren Williams , Total positivity for flag varieties: combinatorics, topology, and toric geometry January 24,2008, 04:00 PM to 05:00 PM

• Arun Ram , Combinatorics of Lie Type January 25,2008, 09:00 AM to 10:00 AM

• Ghislain Fourier , Finite dimensional modules for current and loop algebras January 25,2008, 11:00 AM to 12:00 PM

• Sami Assaf , Applications of dual equivalence graphs January 25,2008, 02:00 PM to 03:00 PM

• Michel Broué , Complex Reflection Groups in Representations of Finite Reductive Groups January 25,2008, 04:00 PM to 05:00 PM

Participant List

Name Role Institution Alghamdi, Ahmad M. Speaker Umm Alqura University Amdeberhan, Tewodros Participant Massachusetts Institute of Technology Andre, Carlos Participant University of Lisbon Ardila, Federico Participant Microsoft Research Arslan, Ogul Participant University of Florida Assaf, Sami H Participant MIT Bandlow, Jason Participant UC Davis Barcelo, Hélène Participant MSRI - Mathematical Sciences Research Institute Barry, Michael J.J. Participant Allegheny College Beck, Matthias Participant San Francisco State University Benkart, Georgia M. Participant University of Wisconsin Berg, Chris Participant UC Davis Berget, Andrew Schaffer Participant University of Minnesota Beyene, Kumsa Abraham Participant Jimma university ,Ambo college bidkhori, hoda Participant Massachusetts Institute of Technology Boltje, Robert Participant University of California, Santa Cruz Braun, Ben Participant Washington University in St. Louis Brenti, Francesco Participant N/A Brichard, Joelle Participant Columbia University Broué, Michel Participant Institut Henri Poincaré Chlouveraki, Maria Participant École Polytechnique Fédérale de Lausanne (EPFL) Coskun, Olcay Participant Bilkent University Crites, Andrew Participant University Of Washington Croitoru, Dorian Participant Massachusetts Institute of Technology Culbertson, Jared Participant Louisiana State University Daugherty, Zajj Participant University of Wisconsin Davis, James William Participant NC State University Davis, Matt Participant University of Wisconsin Denton, Tom Participant University of California di Francesco, Philippe Participant Service de Physique Theorique Diaconis, Persi Organizer Stanford University Ding, Kequan Participant Chinese Academy of Sciences Dobria, Eunice Voichita Participant Florida Atlantic University Doker, Jeff Participant UC Berkeley Dolbin, RJ Participant UC Riverside Du, Ruoxia Participant MIT - Massachusetts Institute of Technology Edwards, Robert Participant UCLA Elias, Benjamin Seth Participant Columbia University Elizalde, Sergi Participant Dartmouth College Elliot, Jason Walter Participant University of Illinois Eu, Sen-Peng Participant U of Minnesota Ferreira, Jeff Participant UC Davis Fong, Paul Participant University of Illinois, Chicago Fourier, Ghislain Participant Universität zu Köln Freedman, Michael H Participant Microsoft Gao, Shanzhen Participant Florida Atlantic University Garrousian, Mehdi Participant University of Western Ontario Glesser, Adam Marc Participant N/A Goodman, Frederick Participant University of Iowa Graber, John Participant University of Iowa Gramain, Jean-Baptiste B Participant EPFL Grant, Joseph Steven Participant University of Bristol Greene, Curtis Participant Haverford College Haiman, Mark David Participant UCB - University of California, Berkeley Hales, Alfred W. Participant Institute for Defense Analyses (CCR-LJ) Halverson, Tom Participant Macalester University Hansen, Mike Participant N/A Hemmer, David J. Participant State University at Buffalo, SUNY Hernandez, David Participant N/A Hill, David Edward Participant University of California, Berkeley Hodge, Terrell L. Participant Western Michigan University Johnson, Paul Participant University of Michigan Jones, Brant Participant University of California, Davis Jung, Ji-Hye Participant Seoul National University Juteau, Daniel Pierre Participant Jussieu University Kang, Seok-Jin Participant Seoul National University Kantor, William U. Participant University of Oregon Kashuba, Iryna Participant University of Sao Paulo Kato, Syu Participant Kyoto University Kedem, Rinat Participant University of Illinois, Urbana-Champaign Kedlaya, Kiran Sridhara Participant Massachusetts Institute of Technology Kim, Jeong-Ah Participant University of Seoul Kim, Myungho Participant Seoul National University Konvalinka, Matjaz Participant MIT Koyama, Masanori Participant University of Wisconsin Madison Kujawa, Jonathan R Participant University of Oklahoma Kumsa, Abraham Beyene Participant Jimma university ,Ambo college Lin, Zongzhu Participant Kansas State University Liu, Fu Participant University of California, Davis Liu, Zhihua Participant Florida Atlantic University Lyle, Sinead Participant University of East Anglia MacQuarrie, John William Participant The University of Manchester Maeno, Toshiaki Participant Kyoto University Maisch, Filix Portocarrero Participant UCSC Maroti, Attila Participant University of Southern California Mathas, Andrew Participant University of Sydney Mazza, Nadia Participant University of Aberdeen Mbirika, Abukuse (Aba) Participant University of Iowa Mihnea, Amalya Participant Florida Atlantic University Miller, Michael Gatewood Participant UC Santa Cruz Moci, Luca Participant Roma Tre Mogel, Jennifer Participant University of California, Santa Cruz Morales, Alejandro Henry Participant Massachusetts Institute of Technology Morier-Genoud, Sophie Participant University of Michigan Murray, John Cyril Participant National University of Ireland, Maynooth Musiker, Gregg Participant Massachusetts Institute of Technology Nash, David Notetaker University of Oregon Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Noonan, John Participant Mount Vernon Nazarene University Oh, SuHo Participant Massachusetts Institute of Technology Okada, Soichi Participant Nagoya University Okounkov, Andrei Speaker Princeton University Orellana, Rosa Participant Dartmouth College Orrison, Michael Participant Harvey Mudd College Ovchinnikov, Sergei Participant San Francisco State University Panova, Greta Participant Harvard University Park, Euiyong Participant Seoul National University Park, Euiyong Participant Seoul National University Pereira, Mariana Participant Facultad de Ciencias, Universidad de la Republica Pon, Steven Participant UC Davis Poonen, Bjorn Participant UCB - University of California, Berkeley Purbhoo, Kevin Participant University of Waterloo Qing, Yulan Participant Massachusetts Institute of Technology Ragnarsson, Kari Participant University of Illinois, Chicago Rainbolt, Julianne Geering Participant St. Louis University Ram, Arun Organizer University of Melbourne University of California-Riverside Mathematics Ridenour, Timothy Blake Participant Department Ruiz, Amanda Participant San Francisco State University Sanus, Lucia Participant Universitat de Valencia Schilling, Anne Organizer University of California, Davis Serrano, Luis Participant University of Michigan Shalile, Armin Participant Oxford University Shin, Dong-Uy Participant Hanyang University Skyner, Tony Participant Bristol University Solomon, Louis Participant University of Wisconsin Späth, Britta Kerstin Participant N/A Srinivasan, Bhama Participant University of Illinois, Chicago Suh, Uhi Rinn Participant Seoul National University Sutherland, Andrew V. Participant MIT Swenson, Daniel Participant University of Minnesota Symonds, Peter Participant University of Manchester Talaska, Kelli Participant University of Michigan Tefera, Akalu Participant Grand Valley State University Tenner, Bridget Eileen Participant DePaul University tevlin, lenny Participant yeshiva university Thiem, Nat Participant University of Colorado Thiéry, Nicolas M Participant Faculté d'Orsay, Université Paris Sud Tiep, Pham Huu Participant University of Florida Tijjani, Manir Participant Federal Airport Authority of Nigeria (FAAN) Tingley, Peter Participant UC Berkeley Vazirani, Monica Joy Participant University of California, Davis Virk, Rahbar Participant University of Wisconsin Vuletic, Mirjana Participant California Institute of Technology Walter, Marty Participant University of Colorado Webb, Peter Participant University of Minnesota Williams, Lauren Kiyomi Participant Harvard University Wilson, Benjamin John Participant University of Sydney Wolff, Tom Participant Ohio University Wong Kew, Rich Participant Postdoc Research Fellows Xu, Zhe Participant ucsc Yacobi, Oded Participant UCSD Yang, Shih-Wei Participant Northeastern University Yin, Jingbin Participant MIT Yip, Martha Participant University of Wisconsin Yoo, Meesue Participant University of Pennsylvania Yu, Josephine T. Participant Massachusetts Institute of Technology Yu, Shona Huimin Participant The University of Sydney Zárate, Alma Leticia Participant CINVESTAV IPN

1 SCIENTIFIC GOALS

In the Spring of 2008 two half-year programs took place at MSRI: Combinatorial Representation Theory and Representation Theory of Finite Groups. What the two programs had in common was the central role played by Lie theory. In combinato- rial representation theory the most important combinatorial objects used to model representations arise from Lie theory (tableaux, Littlewood-Richardson coefficients, Kazhdan-Lusztig polynomials, etc.). In finite groups there are several trends making Lie theory central. Some finite groups are either naturally a part of Lie theory (finite groups of Lie type) or are very closely connected to it on many levels (symmetric groups). Central general conjectures in representation theory of finite groups can often be tested on these important classes of groups and sometimes, using the Clas- sification Theorem, might even be reduced to questions about these groups. On the other hand, the theory of p-compact groups provides a bridge between a finite group theory, algebraic topology and complex reflection groups. So it was very natural for the two MSRI programs to run a joint workshop on Lie Theory.

2 ORGANIZATIONAL STRUCTURE

The workshop has been organized following a standard MSRI 5 day long research workshop template. There were four 60 minute talks each day, except for Wednesday, when there were just two 60 minute talks in the morning. The structure seems to be optimal, as the uncrowded talk schedule left enough time for discussion. Each speaker has been given a short introduction by a ‘classic’ in that area (for example Robert Steinberg presented George Lusztig, Charley Curtis presented Meinolf Geck, etc.). The new facilities at MSRI were positively commented on by many participants. Seating, visibility, blackboards, multimedia, etc. all seem outstanding.

3 SCIENTIFIC DEVELOPMENTS

Let us mention in the beginning that a general level of excitement of all participants was very high. Many participants and some UC-Berkeley faculty members gave very favorable reviews to the workshop, which has been one of the biggest such events in the history of MSRI. Significant research accomplishments were described in many talks. The auditorium was packed during all talks of the workshop. We now describe some of the most significant talks of the conference. Complete list of talks is given in an appendix below. Victor Ostrik (University of Oregon) presented a major development on Lusztig’s asymptotic Hecke ring (joint work with Bezrukavnikov and Finkelberg). As explained by Lusztig, the asymptotic Hecke ring plays a significant role in represen-

1 tation theory of finite groups of Lie type and in the theory of character sheaves. The explicit description of the asymptotic Hecke ring for each two-sided cell has been con- jectured by Lusztig. The authors prove the Lusztig’s conjecture in a very satisfactory way, using the theory of tensor categories. It turns out that the conjecture holds true for all non-exceptional cells, of which there is only one in type E7 and two in type E8. What is remarkable, the classification results on monoidal tensor categories provide an explanation of what goes wrong in the exceptional cases, and provide a complete solution in these cases also. In the end of the talk, Ostrik presented an exciting and important new conjecture, due to him and Bezrukavnikov, which con- nects Lusztig’s asymptotic ring to representation theory of finite W -algebras. This has provided a link to the number of talks on the conference which were dedicated to finite W -algebras, and generated a lot of discussion between the experts (Ostrik, Premet, Losev, Kleshchev, Goodwin, Brown). Alexander Premet (University of Manchester) and Ivan Losev (Belarus- sian State University) presented two significant talks on representation theory of finite W -algebras. One of the major problems of the theory is classification of finite dimensional representations of these algebras. This is closely related to the theory of primitive ideals in universal enveloping algebras, deformations of singularities, quiver representations, and physics, to name just a few connections. On the other hand, very little is known about representation theory of finite W -algebras outside of type A (which has been treated by Brundan and Kleshchev). Until recent work of Premet it has not even been known if in general finite W algebras have any finite dimen- sional representations. Even more importantly is a result conjectured by Premet that 1-dimensional representation always exists. In his talk Premet presented a very in- teresting solution of this problem for classical types using reduction modulo p and lifting back, as well as results of Barbash and Vogan on primitive ideals. On the other hand, Ivan Losev presented a completely different approach to W -algebras based on the ideas of Fedosov quantization. He observe that the finite W -algebras W is the in- variant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and used this observation to study W . The results include an alternative definition of W , a relation between the sets of prime ideals of W and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of W in the case of classical Lie algebra, and the separation of elements of W by finite dimensional representations. Meinolf Geck (University of Aberdeen) gave a spectacular talk on James conjecture for Hecke algebras of exceptional type. Originally, James conjecture is concerned with modular representation theory of symmetric groups. Roughly speak- ing it describes characters of irreducible representations under certain non-obvious restrictions in terms of the corresponding theory for Hecke algebras at roots of unity. It can also be described as a (non-obvious) symmetric group analogue of Lusztig’s conjecture for algebraic groups. It needs to be said that the conjecture is completely

2 open, and, moreover, it is far from clear what the analogue for other types should look like. Recently Geck showed that Hecke algebras of finite type are cellular in the sense of Graham-Lehrer. This lead to a natural generalisation of the theory of Specht modules to Hecke algebras of any (finite) type. In this framework, he also formulated a general version of James conjecture. In a joint work with Juergen Mueller, he now proved James’ conjecture for Hecke algebras of exceptional type. George Lusztig (MIT) made a foundational talk on reductive algebraic groups. He explained how to use canonical bases theory to prove the result announced but left unproved by Kostant in 1966! The result is an explicit construction of the coordinate ring of a reductive algebraic group over the integers. Lusztig explained why such a proof could not be found in 1966. Sergei Fomin (University of Michigan) gave an expository lecture on cluster algeras. This talk was part of the series of MSRI Evans talks—lecture series designed especially for students and faculty members of UC-Berkeley and given at the Evans Hall on campus. Cluster algebras arise in various algebraic and geometric contexts, with combinatorics providing a unifying framework. The presentation of the basic definitions and results of this emerging theory was guided by two sets of examples: coordinate rings of classical algebraic varieties, and cluster algebras associated with bordered oriented surfaces with marked points. The topic of cluster algebras has been continued in the talk of Bernard Leclerc (University of Caen) who introduced the notion of monoidal categorification of a cluster algebra, and gave examples of such categorifications coming from the representation theory of quantum affine algebras. Peter Fiebig () gave a talk on Lusztig’s conjecture on characters of irreducible representation of algebraic groups over an in positive char- acteristic. Fiebig related sheaves of vector spaces on a complex affine flag variety to representations of the Lie algebra associated to Langlands dual root system. From this he extracted a new proof of Lusztig’s multiplicity conjecture for almost all character- istics. The main step in the construction of the above relation was a categorification of a natural map from the affine Hecke algebra to its periodic module via the theory of sheaves on moment graphs. Using this categorification a non-topological proof of the multiplicity one case of Lusztig’s conjecture for all characteristics above the Coxeter number was presented. Edward Frenkel (UC-Berkeley) discussed a conjectural description of the cat- egories assigned by the local geometric Langlands correspondence assigns to a local system on the punctured disc for the Langlands dual group of a complex reductive group G. The categories are given as categories of representations of the correspond- ing affine Kac-Moody algebra of critical level. Sometimes these categories may also be realized as categories of D-modules or O-modules on some algebraic varieties or ind-schemes. Interrelations between these categories provide supporting evidence for the conjectures. In particular, the categories of Iwahori equivariant representations of critical level with fixed central character are equivalent to the categories of quasi-

3 coherent sheaves on the Springer fibers the Langlands dual group.

4

Lie Theory March 10, 2008 to March 14, 2008

Schedule

Monday March 10, 2008

Georgia Bases, Lattices, Hyperbolic Kac-Moody 9:30AM - 10:30AM Benkart Algebras Mirabolic affine Grassmannian and 11:00AM 12:00PM Victor Ginzburg character sheaves 12:00PM - 2:00PM Lunch Tensor categories attached to cells in 2:00PM - 3:00PM Viktor Ostrik Weyl groups. 3:00PM - 4:00PM Tea Evans talk (60 Evans Hall): Cluster 4:10PM - 5:00PM Sergey Fomin algebras

Tuesday March 11, 2008

Canonical bases and construction of 9:30AM - 10:30AM George Lusztig Chevalley groups over Z Peter Superbosonization of invariant matrix 11:00AM - 2:00PM Littelmann ensembles 12:00PM - 2:00PM Lunch Monoidal categorifications of cluster 2:00PM - 3:00PM Bernard Leclerc algebras 3:00PM - 3:30PM Tea Sheaves on affine flag varieties, modular 3:30PM - 4:30PM Peter Fiebig representations and Lusztig's conjecture

Wednesday March 12, 2008

Alexander Small nonrestricted representations and 9:30AM - 10:30AM Premet completely prime primitive ideals. Semicontinuity properties of Kazhdan- 11:00AM - 2:00PM Cedric Bonnafe Lusztig cells

Thursday March 13, 2008

Quantized symplectic actions and W- 9:30AM - 10:30AM Ivan Loseu algebras Langlands correspondence for loop 11:00AM 12:00PM Edward Frenkel groups 12:00PM - 2:00PM Lunch Vyjayanthi Abelian Ideals, KR--modules and Koszul 2:00PM - 3:00PM Chari algebras 3:00PM - 4:00PM Tea Poisson vertex algebras and integrable 4:00PM - 5:00PM Victor Kac systems (!at EVANS HALL 60!)

Friday March 14, 2008

James' conjecture for Hecke algebras of 9:30AM - 10:30AM Meinolf Geck exceptional type Two applications of Littelmann's path 11:00AM - 2:00PM Susumu Ariki model 12:00PM - 1:30PM Lunch Coxeter group actions on the 1:30PM - 2:30PM Gustav Lehrer cohomology of toric varieties 2:30PM - 3:00PM Tea 3:00PM - 4:00PM Gunter Malle TBD

Currently Available Videos

• Georgia Benkart , Bases, Lattices, Hyperbolic Kac-Moody Algebras March 10,2008, 09:30 AM to 10:30 AM

• Victor Ginzburg , Mirabolic affine Grassmannian and character sheaves March 10,2008, 11:00 AM to 12:00 PM

• Viktor Ostrik , Tensor categories attached to cells in Weyl groups March 10,2008, 02:00 PM to 03:00 PM

• Sergey Fomin , Cluster algebras March 10,2008, 04:10 PM to 05:00 PM

• George Lusztig , Canonical bases and construction of Chevalley groups over Z March 11,2008, 09:30 AM to 10:30 AM

• Peter Littelmann , Superbosonization of invariant matrix ensembles March 11,2008, 11:00 AM to 12:00 PM

• Bernard Leclerc , Monoidal categorifications of cluster algebras March 11,2008, 02:00 PM to 03:00 PM

• Peter Fiebig , Sheaves on affine flag varieties, modular representations and Lusztig's conjecture March 11,2008, 03:30 PM to 04:30 PM

• Alexander Premet , Small nonrestricted representations and completely prime primitive ideals. March 12,2008, 09:30 AM to 10:30 AM

• Cedric Bonnafe , Semicontinuity properties of Kazhdan-Lusztig cells March 12,2008, 11:00 AM to 12:00 PM

• Ivan Loseu , Quantized symplectic actions and W-algebras March 13,2008, 09:30 AM to 10:30 AM

• Edward Frenkel , Langlands correspondence for loop groups March 13,2008, 11:00 AM to 12:00 PM

• Vyjayanthi Chari , Abelian Ideals, KR--modules and Koszul algebras March 13,2008, 02:00 PM to 03:00 PM

• Victor Kac , Poisson vertex algebras and integrable systems March 13,2008, 04:00 PM to 05:00 PM

• Meinolf Geck , James' conjecture for Hecke algebras of exceptional type March 14,2008, 09:30 AM to 10:30 AM

• Susumu Ariki , Two applications of Littelmann's path model March 14,2008, 11:00 AM to 12:00 PM

• Gustav Lehrer , Coxeter group actions on the cohomology of toric varieties March 14,2008, 01:30 PM to 02:30 PM

• Gunter Malle , Counting Characters of p'-degree March 14,2008, 03:00 PM to 04:00 PM

Participant List

Name Role Institution Alghamdi, Ahmad M. Participant Umm Alqura University Andersen, Henning Haahr Participant Aarhus Universitet Andikfar, Hossein Participant University of Toledo Andre, Carlos Participant University of Lisbon Andruskiewitsch, Nicolas Participant Universidad Nacional de Cordoba Ariki, Susumu Speaker Kyoto University Armstrong, Drew Participant University of Minnesota Assaf, Sami H Participant MIT Banu, Letitia Mihaela Participant University of Western Ontario Barcelo, Hélène Participant MSRI - Mathematical Sciences Research Institute Barnet-Lamb, Thomas James Participant Harvard Unviersity Batra, Punita Participant Harish Chandra Research Institute Baumann, Pierre Participant N/A Beck, Matthias Participant San Francisco State University Benkart, Georgia M. Speaker University of Wisconsin Benson, Dave J. Participant University of Aberdeen Bessenrodt, Christine Participant Leibniz Universitaet Hannover bidkhori, hoda Participant Massachusetts Institute of Technology Boltje, Robert Participant University of California, Santa Cruz Bonnafe, Cedric Speaker Centre National de la Recherche Scientifique Bouc, Serge Participant CNRS - Université de Picardie Broto, Carles Speaker N/A Broué, Michel Speaker Institut Henri Poincaré Brown, Jonathan Participant University of Oregon Bunke, Thomas Participant Universidade de Sao Paulo Bystron, Jakub Participant Charles University, Prague Campbell, Peter Participant University of Bristol Can, Mahir Bilen Participant university of western ontario Carlson, Jon F. Participant University of Georgia Carrell, James B. Participant University of British Columbia Chari, Vyjayanthi Speaker University of California, Riverside Chebolu, Sunil Kumar Participant University of Western Ontario Cheng, Shun-Jen Participant Academia Sinica Cherniavsky, Yonah Participant Technion -- Israel Institute of Technology, Haifa, Israel coleman, paula Participant cima phamarcuticals Cramer, Tim Participant Yale University Craven, David A Participant University of Oxford Curtis, Charles W. Participant University of Oregon Danz, Susanne Participant University of Jena Daugherty, Zajj Participant University of Wisconsin Davis, Matt Participant Unversity of Wisconsin Denton, Tom Participant University of California Dobria, Eunice Voichita Participant Florida Atlantic University Doty, Stephen Participant N/A Douglass, J. Matthew Participant University of North Texas Laboratoire de Mathematiques de Besançon CNRS (UMR 6623) UFR - Dudas, Olivier Participant ST Edwards, Robert Participant UCLA Ehrig, Michael Participant University of Cologne Elliot, Jason Walter Participant University of Illinois Fayers, Matthew Participant Massachusetts Institute of Technology Feldvoss, Joerg Participant University of South Alabama Fiebig, Peter Participant Universität Freiberg Fishel, Susanna Dodds Participant Arizona State University Fomin, Sergey Speaker University of Michigan Fong, Paul Participant University of Illinois, Chicago Fourier, Ghislain Participant Universität zu Köln Frenkel, Edward Speaker UCB - University of California, Berkeley Futorny, Vyacheslav Participant Universidade de Sao Paulo Gan, Wee Teck Participant Princeton University Gaussent, Stephane Participant Universite de Nancy Geck, Meinolf J Speaker King's College, Aberdeen University Geiss, Christof Participant UNAM - Universidad Nacional Autonoma de Mexico Ghanam, Ryad Participant University of Pittsburgh-Greensburg Ginzburg, Victor A. Speaker University of Chicago Glesser, Adam Marc Participant N/A Gonzales, Richard Participant NPR Goodman, Frederick Participant University of Iowa Goodwin, Simon Participant University of Birmingham Gorelik, Maria Participant Weizmann Institute of Science Graber, John Participant University of Iowa Greene, Curtis Participant Haverford College Guilhot, Jeremie Participant University of Aberdeen Guralnick, Robert M. Participant USC Haglund, James B. Participant Univ. of Pennsylvania Haiman, Mark David Participant UCB - University of California, Berkeley Hansen, Mike Participant N/A He, Xuhua Participant Stony Brook University Helminck, Aloysius Gerardus Participant NC State University Hemmer, David J. Participant State University at Buffalo, SUNY Hernandez, David Participant N/A Hill, David Edward Participant University of California, Berkeley Himstedt, Frank Participant Technische Universität Munchen Hiss, Gerhard Richard Participant RWTH Aachen Hoshino, Ayumu Participant University of Tokyo Hoyt, Crystal Faye Participant Weizmann Institute of Science Ip, Ivan Participant Yale University jacon, Nicolas Participant Universite de Franche comte Jakelic, Dijana Organizer University of Illinois at Chicago Johnson, Garrett Participant University of California at Santa Barbara Jones, Benjamin F. Participant University of Georgia Jones, Brant Participant University of California, Davis Juan, Lourdes Participant Texas Tech University Juteau, Daniel Pierre Participant Jussieu University Kac, Victor Speaker MIT - Massachusetts Institute of Technology Kasatani, Masahiro Participant Kyoto University Kashiwara, Masaki Speaker L'Institut de Mathématiques de Jussieu Kashuba, Iryna Participant University of Sao Paulo Kedem, Rinat Participant University of Illinois, Urbana-Champaign Khare, Apoorva Participant University of California at Riverside Khongsap, Totrakool Participant University of Virginia KIM, SungSoon Yj Participant University of Paris 7 Kivran-Swaine, Terence Joseph Participant The Graduate Center of CUNY Kleshchev, Alexander Organizer University of Oregon Konvalinka, Matjaz Participant MIT Krause, Henning Participant University of Paderborn Kuelshammer, Burkhard Participant University of Jena Kujawa, Jonathan R Participant University of Oklahoma Kuwabara, Toshiro Participant Kyoto University Kwon, Namhee Participant University of Toledo Lam, Thomas Participant Harvard University Lascoux, Alain Participant N/A Lasy, Trafim Participant Paris 7 Lau, Michael Participant University of Windsor Lauda, Aaron D Participant Columbia University Lauve, Aaron Participant Texas A&M University Le, Tung Thien Participant Wayne State University Leclerc, Bernard Speaker Université de Caen Lehrer, Gustav I. Speaker University of Sydney Li, Yiqiang Participant Yale University Lien, TseChing Participant University of Wisconsin, Madison Lin, Zongzhu Participant Kansas State University Littelmann, Peter Speaker N/A Loktev, Sergei Participant N/A Loseu, Ivan Speaker N/A Lu, Dan Participant Yale University Lusztig, George Speaker MIT Lyle, Sinead Participant University of East Anglia Lynd, Justin Participant The Ohio State University Malle, Gunter Participant TU Kaiserslautern Marko, Frantisek Participant PennState University Hazleton Maroti, Attila Participant University of Southern California Mazza, Nadia Participant University of Aberdeen Mbirika, Abukuse (Aba) Participant University of Iowa Milas, Antun Participant Rutgers University, New Brunswick Miyachi, Hyohe Participant Nagoya University Moci, Luca Participant Roma Tre Moreau, Anne Participant ETH Zürich Morier-Genoud, Sophie Participant University of Michigan Musiker, Gregg Participant Massachusetts Institute of Technology Nakanishi, Tomoki Participant Nagoya University Nakano, Daniel Participant University of Georgia Nakashima, Toshiki Participant Sophia University Nash, David Participant University of Oregon Nevins, Monica Participant University of Ottawa Nguyen, Hung Ngoc Participant University of Florida Noeske, Felix Participant RWTH Aachen University O'Brien, Eamonn A Participant University of Auckland Ohn, Christian Participant Universite de Valenciennes Okada, Soichi Participant Nagoya University Orellana, Rosa Participant Dartmouth College Ostrik, Viktor Speaker University of Oregon Pal, Tanusree Participant Harish-Chandra Research Institute Park, Sejong Participant University of Aberdeen Premet, Alexander Speaker University of Manchester Purbhoo, Kevin Participant University of Waterloo Ragnarsson, Kari Participant University of Illinois, Chicago Rainbolt, Julianne Geering Participant St. Louis University Ram, Arun Organizer University of Melbourne Reif, Shifra Participant Weizmann institute of Science Roehrle, Gerhard Erich Participant University of Bochum Roichman, Yuval Participant Bar-Ilan University Ronan, John T. Participant N/A Roth, Ilan Participant UC Berkeley SanAgustin, Keefe L Participant Brandeis University Saxl, Jan Participant University of Cambridge Schilling, Anne Participant University of California, Davis Scott, Leonard Participant University of Virginia Serrano, Luis Guillermo Participant University of Michigan Shchigolev, Vladimir Participant Lomonosov State University Shinkado, Takuya Participant Kyoto university Shtukaturov, Konstantin Yurievich Participant Institute of Power Transportation Skoda, Zoran Participant Institute Ruder Boskovic Smirnov, Evgeny Participant Universitaet Bonn Solomon, Louis Participant University of Wisconsin Späth, Britta Kerstin Participant N/A Srinivasan, Bhama Participant University of Illinois, Chicago Stanley, Richard P. Organizer MIT - Massachusetts Institute of Technology Steinberg, Robert Participant UCLA Stokke, Anna Participant University of Winnipeg Stroppel, Catharina Speaker N/A Stump, Christian Participant University of Vienna Sun, Jie Participant University of Alberta Swenson, Daniel Participant University of Minnesota Symonds, Peter Participant University of Manchester Tan, Kai Meng Participant National University of Singapore Taskin, Muge Participant York University Thiem, Nat Participant University of Colorado Tiep, Pham Huu Participant University of Florida Tikaradze, Akaki Participant The University of Chicago Tingley, Peter Participant UC Berkeley Townsley, Lisa Participant Benedictine University Tsuchioka, Shunsuke Participant Research Institute for Mathematical Sciences Kyoto University Vazirani, Monica Joy Participant University of California, Davis Virk, Rahbar Notetaker University of Wisconsin Walia, Rajeev Participant University of California Riverside Walter, Marty Participant University of Colorado Wan, Jinkui Participant University of Virginia Wang, Weiqiang Participant University of Virginia Watanabe, Hidekazu Participant University of Tokyo Watson, Holly Lynn Participant University of South Carolina Webb, Peter Participant University of Minnesota Wiesner, Emilie Participant Ithaca College Williams, Lauren Kiyomi Participant Harvard University Williamson, Geordie Participant Universität Freiburg Witherspoon, Sarah Participant Texas A&M University Yamamoto, Makoto Participant University of California Yip, Martha Notetaker University of Wisconsin Yoo, Meesue Participant University of Pennsylvania Zalesski, Alexandre Participant N/A Zhu, Minxian Participant Yale University Zorrilla Masías, Henry Participant Universidad Nacional Federico Villarreal

Topics in Combinatorial Representation Theory MSRI Workshop March 17-21, 2008

SCIENTIFIC GOALS ------

20th century combinatorics taught us that representation theory is often the key to puzzles involving our favorite combinatorial objects. In the reverse direction, answers to many central questions in representation theory required development of sophisticated combinatorial techniques and constructions.

This interplay, which has only intensified in recent years, was the focus of this workshop. It was aimed both at researchers and advanced graduate students working at this interface between representation theory and combinatorics. More specifically, the goal was to have speakers apprise them of the current status, problems and frontiers in the following hot areas of recent research:

* quiver representations; * cluster algebras; * Macdonald and LLT polynomials; * representation-theoretic techniques in quantum/statistical mechanics; * crystal bases, graphs and Littelman path models; * affine Grassmannians, affine Schubert calculus, Mirkovic-Vilonen cycles; * dual canonical and semi-canonical bases; * combinatorial Hopf algebras.

ORGANIZATIONAL STRUCTURE ------

The workshop was organized following a standard MSRI weeklong research workshop template. There were four 50-minute talks each day, two each in the morning/afternoon, every day except Wednesday. The talks were grouped thematically on each day, e.g. Tuesday was the day for cluster algebras, Wednesday morning for path models, Thursday morning for affine Schubert calculus.

The structure appeared to work well, in that the uncrowded talk schedule left plenty of time for questions after each talk (of which there were many after most talks), as well as plenty of time for informal discussions during the various breaks.

Many participants gave favorable reviews to the relatively new MSRI facilities, e.g. in terms of the seating and visibility in the auditorium, the availability of blackboards, even outdoors! Even though the lecture hall was packed for many talks, it didn't feel that way.

We should mention that the very first talk at the workshop, given by John Stembridge, was simultaneously video-streamed to an Atlas of Lie Groups workshop so that, in a sense, he spoke at two workshops simultaneously. The MSRI computer people handled this very well.

BIG DEVELOPMENTS ------

Many of the talks announced "big news" research developments.

Mark Haiman announced his recent work (joint with Ian Grojnowski) proving the conjecture that LLT polynomials expand positively in terms of Schur functions, and discussing its ramifications. In particular, the correct generalization of the LLT polynomials to other root systems/types was an important and illuminating part of the picture.

Arun Ram discussed his recent work (, joint with Martha Yip) on the appropriate generalization of the type A Haglund-Haiman-Loehr formula for Macdonald polynomials to all types. The key tools were (folded) alcove walks parametrizing bases for Cherednik's double affine Hecke algebra, and the connection to Schwer's formula for Hall-Littlewood polynomials was made clear. There has already been follow-up work on this by another of the workshop's speakers, Cristian Lenart, whose workshop talk was partly a "preliminary report" for this work.

Jan Schroer explained work (joint with Christof Geiss and Bernard Leclerc) on how the representation theory of preprojective algebras can help in understanding a large class of cluster algebras arising in Lie theory, as coordinate rings of unipotent cells of Kac-Moody groups. In particular, this allows one to apply Lusztig's theory of semicanonical bases and to obtain some semicanonical bases of these cluster algebras which contain all the cluster monomials.

We were also very pleased with the Berkeley Colloquium given by our invited speaker, Joel Kamnitzer, to a packed crowd in Evans Hall. He gave a very accessible account of his recent construction (together with Sabin Cautis) of a new categorification of the Jones and HOMFLY polynomials in knot theory. It was based on derived categories of equivariant coherent sheaves on the affine Grassmannian, and closely related with the talk in the workshop by Michael Kapovich on the Horn problem.

COLLABORATIONS/INTRODUCTIONS ------

There appeared to be a great deal of mixing/consultation between people on the combinatorial side and those more expert in representation or group theory.

The workshop also gave the opportunity for some disparate groups of researchers within combinatorial representation theory to consult each other, and gave other groups the opportunity to meet in person and continue their ongoing collaboration. It appears that some new collaborations were begun during the workshop.

In addition, there were many introductions of grad students, postdocs and young faculty to each other and to more senior researchers.

Topics in Combinatorial Representation Theory

March 17, 2008 to March 21, 2008

Schedule

Monday March 17, 2008

9:15AM - 9:30AM Welcome and Opening Remarks 9:30AM - 10:30AM John Stembridge Admissible W-graphs 10:30AM - 11:00AM Morning tea Hopf algebras, Towers of Algebras and Dual 11:00AM - 12:00PM Thomas Lam graded graphs 12:00PM - 2:00PM Lunch Restrictions of irreducible representations to 2:00PM - 3:00PM Michael Kapovich Levi subgroups and ideal triangles in Euclidean buildings 3:00PM - 4:00PM Afternoon tea Unipotent classes and special Weyl group 4:00PM - 5:00PM George Lusztig representations

Tuesday March 18, 2008

Combinatorics of Coxeter elements and cluster 9:30AM - 10:30AM Andrei Zelevinsky algebras of finite type 10:30AM - 11:00AM Morning tea Dual semicanonical bases and cluster algebras 11:00AM - 12:00PM Jan Schroeer

12:00PM - 2:00PM Lunch Alexander 2:00PM - 3:00PM Total positivity, matroids, and clusters Postnikov 3:00PM - 4:00PM Afternoon tea 4:00PM - 5:00PM David Speyer Sortable elements -- beyond finite type 5:00PM - 6:00PM Reception

Wednesday March 19, 2008

9:30AM - 10:30AM Arun Ram Path models 10:30AM - 11:00AM Morning tea Cristian-Paul Hall-Littlewood polynomials, alcove walks, 11:00AM - 12:00PM Lenart and the Macdonald polynomial inv statistic

Thursday March 20, 2008

Schubert Polynomials for the affine 9:30AM - 10:30AM Anne Schilling Grassmannian of the 10:30AM - 11:00AM Morning tea k-Schur functions via multidegrees of Matrix 11:00AM - 12:00PM Mark Shimozono Affine Schubert Varieties 12:00PM - 2:00PM Lunch 2:00PM - 3:00PM Mark Haiman LLT polynomials 3:00PM - 3:55PM Afternoon tea (UC Berkeley Math Colloquium, 60 Evans) 4:10PM - 5:00PM Joel Kamnitzer Knot homology, geometric representation theory, and derived categories

Friday March 21, 2008

The Razumov-Stroganov conjecture: loop gas, Philippe di 9:30AM - 10:30AM alternating sign matrices, plane partitions and Francesco orbital varieties 10:30AM - 11:00AM Morning tea Quantum algebra characters and cluster 11:00AM - 12:00PM Rinat Kedem algebras 12:00PM - 2:00PM Lunch 2:00PM - 3:00PM Harm Derksen Quivers with Potentials 3:00PM -4:00PM Afternoon tea Some Problems of Asymptotic Representation 4:00PM - 5:00PM Anatoly Vershik Theory (ART)

Currently Available Videos

• Georgia Benkart , Bases, Lattices, Hyperbolic Kac-Moody Algebras March 10,2008, 09:30 AM to 10:30 AM

• Victor Ginzburg , Mirabolic affine Grassmannian and character sheaves March 10,2008, 11:00 AM to 12:00 PM

• Viktor Ostrik , Tensor categories attached to cells in Weyl groups March 10,2008, 02:00 PM to 03:00 PM

• Sergey Fomin , Cluster algebras March 10,2008, 04:10 PM to 05:00 PM

• George Lusztig , Canonical bases and construction of Chevalley groups over Z March 11,2008, 09:30 AM to 10:30 AM

• Peter Littelmann , Superbosonization of invariant matrix ensembles March 11,2008, 11:00 AM to 12:00 PM

• Bernard Leclerc , Monoidal categorifications of cluster algebras March 11,2008, 02:00 PM to 03:00 PM

• Peter Fiebig , Sheaves on affine flag varieties, modular representations and Lusztig's conjecture March 11,2008, 03:30 PM to 04:30 PM

• Alexander Premet , Small nonrestricted representations and completely prime primitive ideals. March 12,2008, 09:30 AM to 10:30 AM

• Cedric Bonnafe , Semicontinuity properties of Kazhdan-Lusztig cells March 12,2008, 11:00 AM to 12:00 PM

• Ivan Loseu , Quantized symplectic actions and W-algebras March 13,2008, 09:30 AM to 10:30 AM

• Edward Frenkel , Langlands correspondence for loop groups March 13,2008, 11:00 AM to 12:00 PM

• Vyjayanthi Chari , Abelian Ideals, KR--modules and Koszul algebras March 13,2008, 02:00 PM to 03:00 PM

• Victor Kac , Poisson vertex algebras and integrable systems March 13,2008, 04:00 PM to 05:00 PM

• Meinolf Geck , James' conjecture for Hecke algebras of exceptional type March 14,2008, 09:30 AM to 10:30 AM

• Susumu Ariki , Two applications of Littelmann's path model March 14,2008, 11:00 AM to 12:00 PM

• Gustav Lehrer , Coxeter group actions on the cohomology of toric varieties March 14,2008, 01:30 PM to 02:30 PM

• Gunter Malle , Counting Characters of p'-degree March 14,2008, 03:00 PM to 04:00 PM

Participant List

Name Role Institution Andersen, Henning Haahr Participant Aarhus Universitet Ardila, Federico Participant Microsoft Research Armstrong, Drew Participant University of Minnesota Assaf, Sami H Participant MIT Bandlow, Jason Participant UC Davis Barcelo, Hélène Participant MSRI - Mathematical Sciences Research Institute Baumann, Pierre Participant N/A Baur, Karin U Participant ETH Zurich Beazley, Elizabeth Participant University of Chicago Beck, Matthias Participant San Francisco State University Benkart, Georgia M. Participant University of Wisconsin Berenstein, Arkady Participant University of Oregon Bergeron-Brlek, Anouk Participant York University bidkhori, hoda Participant Massachusetts Institute of Technology Billey, Sara C. Participant University of Washington Bouc, Serge Participant CNRS - Université de Picardie Braun, Ben Participant Washington University in St. Louis Brenti, Francesco Participant N/A Brunson, Jason Cornelius Participant Virginia Tech Bunke, Thomas Participant Universidade de Sao Paulo Bystron, Jakub Participant Charles University, Prague Carrell, James B. Participant University of British Columbia Chen, Li-Chung Participant UC Berkeley Cheung, Rex Participant Yale University Cramer, Tim Participant Yale University Craven, David A Participant University of Oxford Danz, Susanne Participant University of Jena Daugherty, Zajj Participant University of Wisconsin Davis, Matt Participant University of Wisconsin Derksen, Harm Speaker University of Michigan Descouens, Francois Participant Fields Institute di Francesco, Philippe Speaker Service de Physique Theorique Ding, Kequan Participant Chinese Academy of Sciences Doty, Stephen Participant N/A Eaton, Charles William Participant University of Manchester Ehrig, Michael Participant University of Cologne Elliot, Jason Walter Participant University of Illinois Eu, Sen-Peng Participant U of Minnesota Fayers, Matthew Participant Massachusetts Institute of Technology Fiebig, Peter Participant Universität Freiberg Fomin, Sergey Organizer University of Michigan Fourier, Ghislain Participant Universität zu Köln Gaussent, Stephane Participant Universite de Nancy Geiss, Christof Participant UNAM - Universidad Nacional Autonoma de Mexico Goodman, Frederick Participant University of Iowa Graber, John Participant University of Iowa Greene, Curtis Participant Haverford College Greenstein, Jacob Participant UC, Riverside Guralnick, Robert M. Participant USC Haglund, James B. Participant Univ. of Pennsylvania Haiman, Mark David Speaker UCB - University of California, Berkeley Hansen, Mike Participant N/A Hemmer, David J. Participant State University at Buffalo, SUNY Hernandez, David Participant N/A Hill, David Edward Participant University of California, Berkeley Hoshino, Ayumu Participant University of Tokyo Hoyt, Crystal Faye Participant Weizmann Institute of Science Jones, Brant Participant University of California, Davis Jung, Ji-Hye Participant Seoul National University Juteau, Daniel Pierre Participant Jussieu University Kamnitzer, Joel Speaker American Institute of Mathematics Kang, Seok-Jin Participant Seoul National University Kapovich, Michael Speaker University of California Karaali, Gizem Participant Pomona College Kasatani, Masahiro Participant Kyoto University Kedem, Rinat Speaker University of Illinois, Urbana-Champaign Kim, Jeong-Ah Participant University of Seoul Kim, Myungho Participant Seoul National University Kinser, Ryan Participant University of Michigan Knutson, Allen Speaker University of California, San Diego Krause, Henning Participant University of Paderborn Kuelshammer, Burkhard Participant University of Jena Kuwabara, Toshiro Participant Kyoto University Kwon, Jae-Hoon Participant University of Seoul Lam, Thomas Speaker Harvard University Lascoux, Alain Participant N/A Lau, Michael Participant University of Windsor Lauve, Aaron Participant Texas A&M University Leclerc, Bernard Organizer Université de Caen Lecouvey, Cedric Participant N/A Lee, Dong-il Participant Seoul National University Lehrer, Gustav I. Participant University of Sydney Lenart, Cristian-Paul Speaker SUNY, Albany Li, Huilan Participant UQAM Lien, TseChing Participant University of Wisconsin, Madison Lin, Zongzhu Participant Kansas State University Littelmann, Peter Participant N/A Liu, Fu Participant University of California, Davis Loktev, Sergei Participant N/A Lusztig, George Speaker MIT Lyle, Sinead Participant University of East Anglia Maisch, Filix Portocarrero Participant UCSC Maroti, Attila Participant USC Mbirika, Abukuse (Aba) Notetaker University of Iowa Miyachi, Hyohe Participant Nagoya University Moci, Luca Participant Roma Tre Moreau, Anne Participant ETH Zürich Morier-Genoud, Sophie Participant University of Michigan Musiker, Gregg Participant Massachusetts Institute of Technology Nakanishi, Tomoki Participant Nagoya University Nash, David Participant University of Oregon Ngo Dac, Tuan Participant Institute for Advanced Study Ochoa Daza, Maicol Arley Participant Cornell University Oh, Young-Tak Participant Sogang university Ohn, Christian Participant Universite de Valenciennes Okada, Soichi Participant Nagoya University Opocenska, Katerina Participant Charles University, Prague Orellana, Rosa Participant Dartmouth College Ovchinnikov, Sergei Participant San Francisco State University Park, Euiyong Participant Seoul National University Pevtsova, Julia A Participant University of Washington Postnikov, Alexander Speaker MIT Purbhoo, Kevin Participant University of Waterloo Rainbolt, Julianne Geering Participant St. Louis University Ram, Arun Speaker University of Melbourne Reading, Nathan Participant North Carolina State University Reif, Shifra Participant Weizmann institute of Science Reiner, Vic Organizer University of Minnesota Roehrle, Gerhard Erich Participant University of Bochum Sagan, Bruce E. Participant National Science Foundation Sakamoto, Reiho Participant University of Tokyo Saleh, Ibrahim Abdou Participant Kansas State University SanAgustin, Keefe L Participant Brandeis University Savage, Alistair Rowland John Participant University of Ottawa Saxl, Jan Participant University of Cambridge Schedler, Travis Jeremy Participant University of Chicago Schilling, Anne Speaker University of California, Davis Schlosser, Michael J Participant University of Vienna Schroeer, Jan Hendrik Speaker University of Bonn Scott, Jeanne Participant University of Leeds Serrano, Luis Guillermo Participant University of Michigan Seven, Ahmet Participant Middle East Technical University Shchigolev, Vladimir Participant Lomonosov Moscow State University Shimozono, Mark Speaker Virginia Polytechnic Institute & State University Shin, Dong-Uy Participant Hanyang University Shinkado, Takuya Participant Kyoto university Siemons, Johannes Participant University of East Anglis Smirnov, Evgeny Participant Universitaet Bonn Solomon, Louis Participant University of Wisconsin Sottile, Frank Participant Texas A&M University Speyer, David Speaker UCB - University of California, Berkeley Stanley, Richard Participant Chevron Information Technology Co Stembridge, John Speaker University of Michigan Stump, Christian Participant University of Vienna Subrahmanyam, Kambhampati Venkata Participant Chennai Mathematical Institute Symonds, Peter Participant University of Manchester Talaska, Kelli Participant University of Michigan Tan, Kai Meng Participant National University of Singapore Taskin, Muge Participant York University Tenner, Bridget Eileen Participant DePaul University tevlin, lenny Participant yeshiva university Thiem, Nat Participant University of Colorado Thiéry, Nicolas M Participant Faculté d'Orsay, Université Paris Sud Thind, Jaimie Participant SUNY - Stony Brook Tiep, Pham Huu Participant University of Florida Tikaradze, Akaki Participant The University of Chicago Tingley, Peter Participant UC Berkeley Tran, Thao Participant Northeastern University Research Institute for Mathematical Sciences Kyoto Tsuchioka, Shunsuke Participant University tyert, jfgui Participant N/A Tyler, Eiko Nakayama Participant N/A Vazirani, Monica Joy Organizer University of California, Davis Vershik, Anatoly M. Speaker Russian Academy of Science Vinroot, Ryan Participant University of Arizona Warnaar, Ole Participant The University of Melbourne Watanabe, Hidekazu Participant University of Tokyo Webb, Peter Participant University of Minnesota Webster, Ben Participant institute for advanced study White, Jacob Anthony Participant Arizona State University Willenbring, Jeb Participant University of Wisconsin-Milwaukee Williamson, Geordie Participant Universität Freiburg Woo, Alexander Participant UC Davis Yamada, Hiro-Fumi Participant Okayama University Yang, Shih-Wei Participant Northeastern University Yip, Martha Participant University of Wisconsin Yoo, Meesue Participant University of Pennsylvania Yvonne, Xavier Participant University of Strasbourg Zalesski, Alexandre Participant N/A Zelevinsky, Andrei Speaker Northeastern University Zhao, Lei Participant University of Virginia Zwicknagl, Sebastian Participant UC Riverside

Report on "Introductory Workshop on the Representation Theory of Finite Groups"

Held at MSRI, February 4, 2008 – February 8, 2008

Organizers: Jonathan Alperin (Chair), Robert Boltje, Markus Linckelmann

1. Goals: The goal of this workshop was to introduce the participants to the representation theory of finite groups. Scheduled were four series of lectures and five single session lectures. The four series were meant to give introductions to four particularly active fields: counting conjectures, representation theory of finite groups of Lie type, representation theory and topology, and Broue’s abelian defect group conjecture. The first of these series also included a thorough introduction into the general terminology used in the representation theory of finite groups. The five single talks at the end of the workshop reported on more specialized new results. The idea was to introduce both graduate students and participants of the parallel program on “Combinatorial Representation Theory” to representations of finite groups in order to make upcoming workshops and seminars more accessible for these audiences and to foster interaction between the two programs.

2. Description of Lectures: The first series of lectures on “Block Theory and Counting Conjectures” was held by Burkhard Kulshammer. The aim was to introduce terminology used in block theory and use this terminology to present counting conjectures as the Alperin-McKay conjecture, Alperin’s weight conjecture and Dade’s refinements.

The second series on representations of finite groups of Lie type was split into two parts. Jonathan Brundan gave two talks on the theory in the defining characteristic and Cedric Bonnafe gave a two-lecture introduction into the representation theory in non-defining characteristic, the emphasis being on Deligne-Lusztig theory. Cedric Bonnafe used an almost axiomatic setup for Deligne-Lusztig theory and applied that to a very thorough treatment of the of degree 2.

The third series on connections between representation theory and topology was held by Markus Linckelmann. He thoroughly presented the basic constructions from simplicial topology and applied them to small categories like fusion systems associated to blocks.

Joe Chuang gave a series of three lectures on Broue’s abelian defect group conjecture. This included an introduction to derived equivalences and an outlook on conjectures that could go beyond the abelian defect group case.

Radha Kessar gave a lecture on new work studying sources of simple modules, in particular in finite classical groups. Gabriel Navarro gave a detailed overview of the McKay conjecture, and also his joint work with Isaacs and Malle which reduces the conjecture to quasi-simple groups. Paul Fong reported on recent joint work with Broue and Srinivasan related to Alperin’s weight conjecture for finite reductive groups in non- defining characteristic. Serge Bouc gave an overview of his work on biset functors and applications to the classification of endo-permutation modules. Finally, Dave Benson’ talk on “squeezed resolutions” introduced a representation theoretic way of computing the homology of the loop space on the p-completion of the classifying space of a finite group.

3. Achievements: All four lecture series gave exceptionally high quality introductions to various topics in the representation theory of finite groups, even surpassing the high expectations we had when selecting the speakers. The workshop attracted many graduate students. The ones we talked to were full of praise for the introductory lecture series. MSRI posted videos of all the lectures on their web site. This will be a valuable source not only for the participants of the workshop but also for anybody who wants to become acquainted with the subject. The seminars that took place later in the program always attracted a number of people from the parallel program on “Combinatorial Representation Theory”. We would like to think that the successful introductory workshop was one of the reasons for this.

The five more specialized talks made the workshop well rounded. With the reduction of the McKay conjecture to quasi-simple groups and the classification of endo-trivial modules these lectures contained highlights of the research achievements of the last years. Graduate students and post-docs made comments that it felt reassuring that the main open questions and conjectures in the field are not only fascinating but also approachable and that they lead to interesting new connections, for example with algebraic topology.

All of this was made possible by the structure and support given by MSRI!

Introductory Workshop on the Representation Theory of Finite Groups February 4 -8, 2008

Schedule Monday February 4, 2008 09:15AM - 09:30AM Opening Remarks 09:30AM - 10:30AM Burkhard Kuelshammer Blocks and Counting Conjectures 11:00AM - 12:00PM Jonathan Brundan Representations of Algebraic Groups in Characteristic p 12:00PM - 02:00PM Lunch at MSRI 02:45PM - 03:45PM Markus Linckelmann Representation Theory and Topology 03:45PM - 04:15PM Afternoon Tea 04:15PM - 05:15PM Joe Chuang Broué's Abelian Defect Group Conjecture Tuesday February 5, 2008 09:30AM - 10:30AM Burkhard Kuelshammer Blocks and Counting Conjectures 11:00AM - 12:00PM Jonathan Brundan Representations of Algebraic Groups in Characteristic p 12:00PM - 02:00PM Lunch at MSRI 02:45PM - 03:45PM Markus Linckelmann Representation Theory and Topology 03:45PM - 04:15PM Afternoon Tea 04:15PM - 05:15PM Joe Chuang Broué's Abelian Defect Group Conjecture 05:15PM - 06:30PM Reception Wednesday February 6, 2008 09:30AM - 10:30AM Burkhard Kuelshammer Blocks and Counting Conjectures 11:00AM - 12:00PM Cedric BONNAFE Introduction to Deligne-Lusztig Theory Thursday February 7, 2008 09:30AM - 10:30AM Cedric BONNAFE Introduction to Deligne-Lusztig Theory 11:00AM - 12:00PM Markus Linckelmann Representation Theory and Topology 12:00PM - 02:00PM Lunch at MSRI 02:45PM - 03:45PM Joe Chuang Broué's Abelian Defect Group Conjecture 03:45PM - 04:15PM Afternoon Tea On Duality Inducing Automorphisms and Sources of 04:15PM - 05:15PM Radha Kessar Simple Modules in Finite Classical Groups Friday February 8, 2008 09:30AM - 10:30AM Gabriel Navarro Around the McKay Conjecture A Bijection towards Counting Conjectures in Finite 11:00AM - 12:00PM Paul Fong Reductive Groups 12:00PM - 02:00PM Lunch at MSRI 02:45PM - 03:45PM Serge Bouc Biset Functors 03:45PM - 04:15PM Afternoon Tea Squeezed Resolutions over the Group Algebra of a Finite 04:15PM - 05:15PM Dave Benson Group

Currently Available Videos

• Burkhard Kuelshammer , Blocks and Counting Conjectures February 4,2008, 09:30 AM to 10:30 AM

• Jonathan Brundan , Representations of Algebriac Groups in Characteristic p February 4,2008, 11:00 AM to 12:00 PM

• Markus Linckelmann , Representation Theory and Topology February 4,2008, 02:45 PM to 03:45 PM

• Joe Chuang , Broué's Abelian Defect Group Conjecture February 4,2008, 04:15 PM to 05:15 PM

• Burkhard Kuelshammer , Blocks and Counting Conjectures February 5,2008, 09:30 AM to 10:30 AM

• Jonathan Brundan , Representations of Algebriac Groups in Characteristic p February 5,2008, 11:00 AM to 12:00 PM

• Markus Linckelmann , Representation Theory and Topology February 5,2008, 02:45 PM to 03:45 PM

• Joe Chuang , Broué's Abelian Defect Group Conjecture February 5,2008, 04:15 PM to 05:15 PM

• Burkhard Kuelshammer , Blocks and Counting Conjectures February 6,2008, 09:30 AM to 10:30 AM

• Cedric Bonnafe , Introduction to Deligne-Lusztig Theory February 6,2008, 11:00 AM to 12:00 PM

• Cedric Bonnafe , Introduction to Deligne-Lusztig Theory February 7,2008, 09:30 AM to 10:30 AM

• Markus Linckelmann , Representation Theory and Topology February 7,2008, 11:00 AM to 12:00 PM

• Joe Chuang , Broué's Abelian Defect Group Conjecture February 7,2008, 02:45 PM to 03:45 PM

• Radha Kessar , On Duality Inducing Automorphisms and Sources of Simple Modules in Finite Classical Groups February 7,2008, 04:15 PM to 05:15 PM

• Gabriel Navarro , Around the McKay Conjecture February 8,2008, 09:30 AM to 10:30 AM

• Paul Fong , A Bijection towards Counting Conjectures in Finite Reductive Groups February 8,2008, 11:00 AM to 12:00 PM

• Serge Bouc , Biset Functors February 8,2008, 02:45 PM to 03:45 PM

• Dave Benson , Squeezed Resolutions over the Group Algebra of a Finite Group February 8,2008, 04:15 PM to 05:15 PM

Participant List

Name Role Institution Al Sharo, Khaled Ahmad Participant UCSC Alghamdi, Ahmad M. Participant Umm Alqura University Alperin, Jon Organizer University of Chicago Andre, Carlos Participant University of Lisbon Arslan, Ogul Participant University of Florida Assaf, Sami H Member MIT Kwame Nkrumah University of Science and AYEKPLE, YAO ELIKEM Participant Technology Balmer, Paul Participant UCLA Benkart, Georgia M. Participant University of Wisconsin Benson, Dave J. Member University of Aberdeen Berg, Chris Participant UC Davis Bhatt, Sheela Participant Eritrean Institute of Technology Bird, Katherine Anne Participant University of Illinois at Chicago Blomgren, Martin Participant KTH Bogdanic, Dusko Participant University of Oxford Boltje, Robert Organizer University of California, Santa Cruz Bonnafe, Cedric Member Centre National de la Recherche Scientifique Bouc, Serge Member CNRS - Université de Picardie Brichard, Joelle Participant Columbia University Brundan, Jonathan Speaker University of Oregon Carlson, Jon F. Participant University of Georgia Chebolu, Sunil Kumar Participant University of Western Ontario Chlouveraki, Maria Participant École Polytechnique Fédérale de Lausanne (EPFL) Chuang, Joe Participant Universityof Bristol Coskun, Olcay Participant Bilkent University Dabbaghian-Abdoly, Vahid Participant Simon Fraser University Danz, Susanne Participant University of Jena Deshpande, Priyavrat C. Participant The University of Western Ontario Dexter, Kathleen D Participant University of Illinois at Chicago Doty, Stephen Participant N/A Elliot, Jason Walter Participant University of Illinois Eu, Sen-Peng Participant U of Minnesota Fel'shtyn, Alexander Participant Boise State University and Szczecin University Fong, Paul Member University of Illinois, Chicago Futorny, Vyacheslav Participant Universidade de Sao Paulo Geline, Michael Aaron Participant U. of Chicago Glesser, Adam Marc Participant N/A Goodman, Frederick Participant University of Iowa Graber, John Participant Universityof Iowa Gramain, Jean-Baptiste B Participant EPFL Grant, Joseph Steven Participant University of Bristol Grime, Matthew Participant University of Bristol Halverson, Tom Participant University of Macalester Hemmer, David J. Participant State University at Buffalo, SUNY Hendrickson, Anders O.F. Participant University of Wisconsin at Madison Herman, Allen Participant University of Regina Hernandez, David Participant N/A Hill, David Edward Participant University of California, Berkeley Isaacs, I. Martin Participant University of Wisconsin Jones, Brant Participant University of California, Davis Juteau, Daniel Pierre Notetaker Jussieu University Kamnitzer, Joel Member American Institute of Mathematics Kantor, William U. Participant University of Oregon Kashuba, Iryna Participant University of Sao Paulo Kato, Syu Participant Kyoto University Kessar, Radha Member University of Aberdeen Kivran-Swaine, Terence Joseph Participant The Graduate Center of CUNY Koshitani, Shigeo Participant Chiba University Kuelshammer, Burkhard Participant University of Jena Kujawa, Jonathan R Member University of Oklahoma Kunugi, Naoko Participant Tokyo University of Science Le, Tung Thien Participant Wayne State University Lin, Zongzhu Participant Kansas State University Linckelmann, Markus Organizer N/A Loukaki, Maria Participant University of Crete Lyle, Sinead Participant University of East Anglia Lynd, Justin Participant The Ohio State University MacQuarrie, John William Participant The University of Manchester Maisch, Filix Portocarrero Participant UCSC Maroti, Attila Participant USC Mathas, Andrew Participant University of Sydney Mazza, Nadia Member University of Aberdeen Mbirika, Abukuse (Aba) Participant University of Iowa Miller, Michael Gatewood Participant UC Santa Cruz Mitra, Dipra Ranjan Participant University of Regina Moci, Luca Participant Roma Tre Mogel, Jennifer Participant University of California, Santa Cruz Naehrig, Michael Participant RWTH Aachen University Naehrig, Natalie Participant RWTH Aachen University Narasaki, Ryo Participant Osaka university Nash, David Participant University of Oregon Navarro, Gabriel Member Universitat de Valencia Nenciu, Adriana Participant University of Wisconsin Nguyen, Hung Ngoc Participant University of Florida Nikolova-Popova, Daniela Borislavova Participant Florida Atllantic University Noeske, Felix Participant RWTH Aachen University Onofrei, Sylvia Elena Participant Kansas State University Orellana, Rosa Participant Dartmouth College Orrison, Michael Participant Harvey Mudd College Otto, Benjamin Allen Participant University of Wisconsin - Madison Park, Sejong Participant University of Aberdeen Pevtsova, Julia A Participant University of Washington Pforte, Lars Participant NUI Maynooth Poonen, Bjorn Participant UCB - University of California, Berkeley Ragnarsson, Kari Participant University of Illinois, Chicago Rainbolt, Julianne Geering Member St. Louis University Salminen, Adam Participant University of Evansville Sangroniz, Josu Participant Universidad del Pais Vasco Sanus, Lucia Participant Universitat de Valencia Saunders, Neil John Participant University of Sydney Schaps, Mary Participant Bar_Ilan University Shalile, Armin Participant Oxford University Skyner, Tony Participant Bristol University Späth, Britta Kerstin Participant N/A Srinivasan, Bhama Participant University of Illinois, Chicago Stancu, Radu Participant University of Kansas Sun, Jie Participant University of Alberta Swenson, Daniel Participant University of Minnesota Symonds, Peter Participant University of Manchester Thiem, Nat Participant University of Colorado Tiep, Pham Huu Participant University of Florida Turull, Alexandre Participant University of Florida Vazirani, Monica Joy Member University of California, Davis Wakefield, Thomas Philip Participant Kent State University Walter, Marty Participant University of Colorado Webb, Peter Participant University of Minnesota Williams, Adrian Leonard Participant Mahasarakham University Wolff, Tom Participant Ohio University Yamamoto, Makoto Participant University of California Zárate, Alma Leticia Participant CINVESTAV IPN Zorrilla Masías, Henry Participant Universidad Nacional Federico Villarreal

Homological Methods in Representation Theory Held on March 31, 2008 to April 04, 2008 at MSRI Organized By: David Benson, Daniel Nakano (Chair), Raphael Rouquier

Over the last century, homological algebra has been a fundamental tool in studying properties of topological spaces. Group cohomology can be defined using an algebraic definition (as given by Brauer, Eilenberg and MacLane) and also topologically via classifying spaces. The nature of this subject has led to many deep interactions between mathematicians from diverse backgrounds who apply group cohomology to representation theory along with those who study algebraic topology via homotopy theory. More recently we have seen new advances in the subject through applications of commutative algebra, group actions and representation theory.

In the last 40 years, cohomology and other homological methods have been used in other ways to study algebraic objects by introducing geometry (i.e., algebraic varieties, derived categories) that captures information about the algebras and their representations. This workshop was aimed at exploring the latest developments in the connections between representations and their underlying geometry. The talks were presented in a manner accessible to young researchers in the field. The main topics included:

Varieties for Modules: Support varieties were first developed 25 years ago by Alperin and Carlson to study complexes of modules for finite groups from a geometric viewpoint via the cohomology. Most recently this theory has been introduced in more axiomatic settings using tensor triangulated categories. In the talks of Balmer and Pevtsova we saw how this setting was useful for applications in both representation theory and geometry. A refinement of the theory has been introduced by Carlson, Friedlander and Pevtsova. This entails studying the Jordan type of modules. The talks by Carlson and Friedlander presented recent work in this direction. Iyengar discussed support varieties in the context of infinitely generated modules and their connections with localizing subcategories of the stable module category.

Support varieties also have various applications in more concrete representation theoretic settings. Kujawa demonstrated how module varieties can be used to study the combinatorics and representation theory for Lie superalgebras.

Derived Categories: Several of the lectures presented new results on the structure of triang- gulated categories. Equivalences of derived categories and the stable module categories have played a prominent role in representation theory. Keller’s lecture demonstrated how mutations of quivers generated equivalences between derived categories of DG algebras that were introduced by Ginzburg. Linckelmann’s lecture first focused on defining the notion of the center of stable categories then presented some applications to modular representations of finite groups. Bonnafe discussed consequences of how the Deligne- Lusztig functors behave as functors on the level of derived categories of a finite reductive group and a fixed Levi subgroup. Krause presented a survey on a homological invariant called the representation dimension of an algebra. This invariant has renewed interest especially with Rouquier’s computation of the representation dimension of the exterior algebra which involved using derived categories

Representations and Cohomology of Specific Groups and Algebras: The theoretical aspects of the subject can be used in a myriad of applications to specific groups and algebras. These include symmetric groups, finite Chevalley groups, reductive algebraic groups, Frobenius kernels, and quantum groups. For reductive groups and quantum groups, homological criteria are important for constructing canonical filtrations of modules. Parshall presented new results with Cline and Scott on filtration questions of (reduced) standard/costandard modules and their homological properties. Andersen described how tensoring the Steinberg module with a restricted simple module leads to constructing tilting modules for the reductive group and the quantum group. This has connections with the construction of p-filtrations.

Several talks were devoted to providing methods to compute cohomology. Parker presented an overview and some new results on cohomological calculations for Schur algebras, symmetric groups and Hecke algebras via the Schur and inverse Schur functor. Hemmer lectured on topological techniques which led to recent advances in computing the cohomology for Young modules for symmetric groups. Pillen addressed an old problem initiated by Quillen which is to determine the initial vanishing range of cohomology for finite Chevalley groups. This question was reintroduced by Friedlander two years ago in a meeting in Oberwolfach.

Connections with Topology: In the Evans Lecture, Benson presented a lecture for a general mathematical audience which explained how cohomology for groups arises from an purely algebraic definition and also from topology through the use of classifying spaces. The interactions between algebra and topology are rich because of these two manifestations. Grodal discussed how homotopical approaches to p-local finite groups (or p-completed classifying spaces) can be used to give global information for finite groups. In the same spirit, Kessar showed how homological methods pertinent to fusion systems and p-completed classifying spaces can be used to provide information about p-fusion in finite groups.

There were 18 talks at the workshop. Five of the talks were given by semi-recent Ph.Ds (post 2000). These speakers included Grodal, Hemmer, Kujawa, Parker and Pevtsova. With at most four talks per day (and only 2 talks on Wednesday) there was ample time for informal mathematical interaction between participants.

List of speakers:

Henning Andersen Aarhus University Paul Balmer UCLA David Benson University of Aberdeen Cedric Bonnafe University of Besancon Jon Carlson University of Georgia Eric Friedlander University of Southern California Jesper Grodal University of Copenhagen David Hemmer SUNY, Buffalo Srikanth Iyengar University of Nebraska Bernhard Keller University of Paris 6 Radha Kessar University of Aberdeen Henning Krause University of Paderborn Jonathan Kujawa University of Oklahoma Markus Linckelmann University of Aberdeen Alison Parker University of Leeds Brian Parshall University of Virginia Julia Pevtsova University of Washington Cornelius Pillen University of South Alabama

Conference schedule:

Monday March 31, 2008

9:30AM - 10:30AM Markus Linckelmann

Title: On graded centers of stable and derived module categories

Abstract: The center of a category C, introduced by P. Gabriel, consists of all natural transformations on the identity functor on C. The terminology is motivated by the fact that if C is the module category of a ring A then the center of C is isomorphic to the center of the ring A. If C is an additive category equipped with a self equivalence - for example, a triangulated category - one can refine this concept to get a graded version, the graded center of C. This is a graded ring which in some ways behaves like a cohomology ring. Examples include the derived module category and the stable module category of a self-injective algebra. We consider in particular graded centers of stable categories and some applications in modular representation theory.

11:00AM - 12:00PM Cedric Bonnafe

Title: Deligne-Lusztig restriction of modular Gelfand-Graev representations

Abstract: The "modular" Deligne-Lusztig functors (induction and restriction=) are functors between the derived categories of a finite reductive group and one of its Levi subgroups. Very few informations are known about them. The aim of this talk is to give some results about the action of Deligne-Lusztig restriction on the Gelfand-Graev projective modules. We shall then discuss some consequences and some related results.

2:00PM - 3:00PM Cornelius Pillen

Title: Cohomology of finite groups of Lie type

Abstract: Let G be a reductive algebraic group over a field k of prime characteristic p r which is split over the prime field Fp. Set q=p and Fq be the finite field with q elements. Let Fr :G →G denote the Frobenius map. Then the fixed points of the rth iterate of the Frobenius map, denoted G(Fq ), is a finite Chevalley group. The question of interest in i this talk is to determine the least i > 0 such that the cohomology group H (G(Fq ), k) ≠ 0.

i Quillen showed that H (GLn(Fq ), k) = 0$ for all 0 < i < r(p - 1) for all n. In that work, he noted that for all finite Chevalley groups there exist constants C depending on the root i system and the prime such that H (G(Fq ), k)=0 for 0 < i < Cr. However, no explicit value of C is given except for G = SL2 (and p odd) in which case one can take C = (p-1)/2. Further, it was not shown whether these vanishing ranges were sharp. Indeed, in the case of SL2, one can see that these bounds are not sharp in general. Later work by Friedlander and by Hiller extended Quillen's results and found vanishing ranges for groups of all types. Since then, few if any results have been obtained in this direction.

The goal here is to exploit techniques developed by Bendel, Nakano and the presenter i which relate H (G(Fq ), k) to extensions over G. In particular, for a group G of classical type and r=1, under the assumption that p > h (the Coxeter number), we improve on the vanishing ranges of Hiller and in many cases find sharp bounds.

4:10PM - 5:00PM David Benson

Title: Classifying spaces and cohomology of finite groups ( Evans Lecture )

Abstract: I shall give a gentle introduction to the cohomology of finite groups from the point of view of algebra, topology, group actions and number theory.

Tuesday April 1, 2008

9:30AM - 10:30AM Paul Balmer

Title: The geometry of tensor triangulated categories of representations

Abstract: We'll explain how to do geometry in tensor triangulated categories, with emphasis on the derived and stable categories of kG-modules. We'll see how basic geometric ideas can be exported to this framework and then applied to modular representation theory. In particular, we'll discuss the construction of endotrivial modules by means of gluing techniques.

11:00AM - 12:00PM Srikanth Iyengar

Title: Localizing subcategories of the stable module category of a finite group

Abstract: This talk concerns the support, as defined by Benson, Carlson, and Rickard, of a (possibly infinite-dimensional) representation of finite group. I will discuss a proof of the following result: Two representations have the same support if and only if they generate the same localizing subcategory of the stable module category of infinite dimensional representations. As a corollary one obtains that the localizing subcategories of stable module category are parameterized by subsets of non-maximal prime ideals in the cohomology ring of the group.

These results are part of on-going work, in collaboration with Benson and Krause.

2:00PM - 3:00PM David Hemmer

Title: An application of topology to computing the cohomology of Young modules for the symmetric group

Abstract: We will discuss recent joint work with F. Cohen and D. Nakano which allows us to compute cohomology of Young modules for the symmetric group, often in all possible degrees. The methods are strong enough to determine, in any characteristic, which Young modules have vanishing cohomology in all degrees, and there are many of them! We can also give explicit formulas for low degree cohomology and arbitrary symmetric groups.

We demonstrate a stability result for cohomology which is reminiscent of generic cohomology for algebraic groups. We believe this is the first theorem for the symmetric group involving multiplying a partition by "p", something analogous to a Frobenius twist for symmetric groups! Finally we show that knowledge just the space of homomorphisms between two Young modules is enough to determine the cohomology in all degrees.

3:30PM - 4:30PM Brian Parshall

Title: Reduced standard modules, filtrations, and cohomology

Abstract: Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. A reduced standard module ∆(λ) is a rational G-module obtained by reduction from a minimal lattice for the irreducible representation of high weight λ of the corresponding quantum enveloping algebra at a pth root of unity. The dual notion of a reduced costandard ∇(λ) module ∇red(λ) can also be defined. Thus, these modules are, in some sense, very similar to standard modules ∆(λ) and costandard modules ∇(λ) for G obtained from the complex semisimple Lie algebra (as in Steinberg's notes). And, like ∆(λ) and ∇(λ), there are several equivalent definitions of ∆red(λ) and ∇red(λ), one due to Z. Lin. We discuss the homological properties of these modules, filtration questions involving them and corresponding interesting highest weight categories, and applications to cohomology of finite groups. This is joint work with E. Cline and L. Scott.

Wednesday April 2, 2008

9:30AM - 10:30AM Bernhard Keller

Title: Mutations of quivers with potentials and derived equivalences

Abstract: Mutations of quivers with potentials were recently introduced by Derksen- Weyman-Zelevinsky with motivations coming from superpotentials in physics, Calabi- Yau algebras and cluster algebras. Mutations can be thought of as a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflections. We will show how mutations give rise to equivalences between the derived categories of certain differential graded algebras introduced by Ginzburg. This is joint work with Dong Yang.

11:00AM - 12:00PM Jesper Grodal

Title: Local-to-global principles for groups and p-local finite groups (See Abstract)

Abstract: A p-local finite group is an algebraic structure which mimics the p-local structure in a finite group, and topologically corresponds to the p-completed classifying space. A central question in group theory is to what extend the local structure determines the global structure. In this talk I will present a homotopical approach to this question via p-local finite groups, and give some concrete local-to-global results for specific classes of finite groups. This talk is joint work with Bob Oliver.

Thursday April 3, 2008

9:30AM - 10:30AM Henning Andersen

Title: Tensoring with the Steinberg module

Abstract: Let G be a semisimple algebraic group over a field of characteristic p > 0 and denote by Uq the corresponding quantum group at a root of unity. In both cases we have a Steinberg module, Stp and Stq, respectively. This module is both simple, induced from a Borel subgroup/subalgebra, and tilting. We study the effect of tensoring finite dimensional modules for G or Uq by the Steinberg module. In particular, we prove that if

L is a restricted module (for G or Uq) then the L⊗Stp, respectively L⊗Stq is tilting.

11:00AM - 12:00PM Jonathan Kujawa

Title: Cohomology and Varieties for Lie Superalgebras

Abstract: In this talk we will discuss ongoing work to use relative cohomology to study the representation theory of Lie superalgebras over the complex numbers. In particular, one sees that the cohomology ring is finitely generated (in fact, a polynomial ring), and one is naturally led to the theory of support varieties for Lie superalgebras. Our investigations have shown that this approach has close connections to both the representation theory and combinatorics of Lie superalgebras. In particular, we will discuss our efforts to calculate specific examples and to extend the theory to the Cartan- type simple Lie superalgebras. This work is joint with Brian Boe, Daniel Nakano, and Irfan Bagci.

2:00PM - 3:00PM Eric Friedlander

Title: p-nilpotent operators and vector bundles

Abstract: Recent joint work with Julia Pevtsova investigates aspects of the representation theory of infinitesimal group schemes. The universal p-nilpotent operator is introduced and shown to determine the local Jordan type of modules. This operator also leads to vector bundles on the projectivization of the spectrum of the cohomology which can differentiate modules with the same local Jordan type.

3:30PM - 04:30PM Radha Kessar

Title: Controlling fusion in saturated fusion systems

Abstract: Saturated fusion systems were introduced by L. Puig in order to distill the common essence of p-fusion in finite groups and the fusion of Brauer pairs in modular representation theory. The work of C.Broto, R.Levi and R.Oliver studying p-completed classifying spaces of finite groups through the ambient fusion systems has generated much interest in these categories. In my talk, I will explain how some of the homological machinery developed around fusion systems can be used to carry over classical theorems of Glauberman and others on control of p-fusion in finite groups to the realm of saturated fusion systems.

Friday April 4, 2008

9:30AM - 10:30AM Jon Carlson

Title: Generic kernels and other constructions of modules over group (scheme) algebras

Abstract: Together with Eric Friedlander, Julia Pevtsova and , we have examined some constructions of canonical submodules of kG-modules in the case that G is a finite group scheme and k is a field of characteristic p. The constructions are based on the rank varieties of the modules.

11:00AM - 12:00PM Julia Pevtsova

Title: Spectra of tensor triangulated categories

Abstract: I shall start by discussing a general framework of studying tensor triangulated categories via geometric ideas as introduced by P. Balmer and then concentrate on two specific applications: to the category of modular representations of a finite group scheme and to the bounded derived category of a quotient stack. This is joint work with S. Paul Smith.

2:00PM - 3:00PM Alison Parker

Title: Homomorphisms and higher extensions for Schur algebras and symmetric groups

Abstract: In 2005 Anton Cox and I wrote a general article which surveyed (and in some cases mildly generalised) many of the more recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. In this talk I will update this survey and give an overview of various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit values for these Ext groups have been determined.

3:30PM - 4:30PM Henning Krause

Title: The representation dimension of a finite dimensional algebra

Abstract: The representation dimension is a homological invariant which Auslander introduced in 1971. For many years, only some basic properties were known. For instance, Auslander proved that representation dimension less or equal 2 characterizes finite representation type. My talk provides a survey of some recent work which sheds some new light on this dimension. In particular, its geometric nature will be discussed.

All the talks are available on Currently Available Videos

Homological Methods in Representation Theory March 31, 2008 to April 4, 2008 Schedule Monday March 31, 2008 9:30AM - 9:45AM Welcome 9:45AM - 10:30AM Markus Linckelmann On graded centers of stable and derived module categories Deligne-Lusztig restriction of modular Gelfand-Graev 11:00AM - 12:00PM Cedric Bonnafe representations 12:00AM - 2:00PM Lunch 2:00PM - 3:00PM Cornelius Pillen Cohomology of finite groups of Lie type 3:00PM - 3:30PM Tea Classifying spaces and cohomology of finite groups 4:10PM - 5:00PM Dave Benson ( Lecture will be held at Evans Hall ) Tuesday April 1, 2008 9:30AM - 10:30AM Paul Balmer The geometry of tensor triangulated categories of representations Localizing subcategories of the stable module category of a finite 11:00AM - 12:00PM Srikanth Iyengar group 12:00PM - 2:00PM Lunch An application of topology to computing the cohomology of Young 2:00PM - 3:00PM David Hemmer modules for the symmetric group 3:00PM - 3:30PM Tea 3:30PM - 4:30PM Brian Parshall Reduced standard modules, filtrations, and cohomology 4:30PM - 5:30PM Reception Wednesday April 2, 2008 9:30AM - 10:30AM Bernhard Keller Mutations of quivers with potentials and derived equivalences 11:00AM - 12:00PM Jesper Grodal Local-to-global principles for groups and p-local finite groups Thursday April 3, 2008 9:30AM - 10:30AM Henning Andersen Tensoring with the Steinberg module 11:00AM - 12:00PM Jonathan Kujawa Cohomology and Varieties for Lie Superalgebras 12:00PM - 2:00PM Lunch 2:00PM - 3:00PM Eric Friedlander P-nilpotent operators and vector bundles 3:00PM - 3:30PM Tea 3:30PM - 4:30PM Radha Kessar Controlling fusion in saturated fusion systems Friday April 4, 2008 Generic kernels and other constructions of modules over group 9:30AM - 10:30AM Jon Carlson (scheme) algebras 11:00AM - 12:00PM Julia Pevtsova Spectra of tensor triangulated categories 12:00PM - 2:00PM Lunch 2:00PM - 3:00PM Alison Parker TBA 3:00PM - 3:30PM Tea 3:30PM - 4:30PM Henning Krause The representation dimension of a finite dimensional algebra

Currently Available Videos

• Markus Linckelmann , On graded centers of stable and derived module categories March 31,2008, 09:30 AM to 10:30 AM

• Cedric Bonnafe , Deligne-Lusztig restriction of modular Gelfand-Graev representations March 31,2008, 11:00 AM to 12:00 PM

• Cornelius Pillen , Cohomology of finite groups of Lie type March 31,2008, 02:00 PM to 03:00 PM

• Dave Benson , Classifying spaces and cohomology of finite groups March 31,2008, 04:10 PM to 05:00 PM

• Paul Balmer , The geometry of tensor triangulated categories of representations April 1,2008, 09:30 AM to 10:30 AM

• Srikanth Iyengar , Localizing subcategories of the stable module category of a finite group April 1,2008, 11:00 AM to 12:00 PM

• David Hemmer , An application of topology to computing the cohomology of Young modules for the symmetric group April 1,2008, 02:00 PM to 03:00 PM

• Brian Parshall , Reduced standard modules, filtrations, and cohomology April 1,2008, 03:30 PM to 04:30 PM

• Bernhard Keller , Mutations of quivers with potentials and derived equivalences April 2,2008, 09:30 AM to 10:30 AM

• Jesper Grodal , Local-to-global principles for groups and p-local finite groups April 2,2008, 11:00 AM to 12:00 PM

• Henning Andersen , Tensoring with the Steinberg module April 3,2008, 09:30 AM to 10:30 AM

• Jonathan Kujawa , Cohomology and Varieties for Lie Superalgebras April 3,2008, 11:00 AM to 12:00 PM

• Eric Friedlander , p-nilpotent operators and vector bundles April 3,2008, 02:00 PM to 03:00 PM

• Radha Kessar , Controlling fusion in saturated fusion systems April 3,2008, 03:30 PM to 04:30 PM

• Jon Carlson , Generic kernels and other constructions of modules over group (scheme) algebras April 4,2008, 09:30 AM to 10:30 AM

• Julia Pevtsova , Spectra of tensor triangulated categories April 4,2008, 11:00 AM to 12:00 PM

• Alison Parker , Homomorphisms and higher extensions for Schur algebras and symmetric groups April 4,2008, 02:00 PM to 03:00 PM

• Henning Krause , The representation dimension of a finite dimensional algebra April 4,2008, 03:30 PM to 04:30 PM

Participant List

Name Role Institution Alghamdi, Ahmad M. Participant Umm Alqura University Alperin, Jon Participant University of Chicago Andersen, Henning Haahr Participant Aarhus Universitet Apostolou, Stavros George Participant Aarhus Universitet Ariki, Susumu Participant Kyoto University Assaf, Sami H Participant MIT Babson, Eric Participant University of California bagci, irfan Participant University of Georgia Balmer, Paul Speaker UCLA Barcelo, Hélène Participant MSRI - Mathematical Sciences Research Institute Baumann, Pierre Participant N/A Beil, Charles Participant UC Santa Barbara Bendel, Christopher Participant University of Wisconsin-Stout Benson, Dave J. Organizer University of Aberdeen Bleher, Frauke Participant University of Iowa Boe, Brian D Participant University of Georgia Bonnafe, Cedric Participant Centre National de la Recherche Scientifique Bouc, Serge Participant CNRS - Université de Picardie Bunke, Thomas Participant Universidade de Sao Paulo Cabanes, Marc Participant University of Jussieu Carlson, Jon F. Speaker University of Georgia Chebolu, Sunil Kumar Participant University of Western Ontario Chen, Xueqing Participant University of Wisconsin--Whitewater Chlouveraki, Maria Participant École Polytechnique Fédérale de Lausanne (EPFL) Collins, Michael J. Participant Oxford University Cooper, Bobbe Participant University of Georgia Craven, David A Participant University of Oxford Danz, Susanne Participant University of Jena Doty, Stephen Participant N/A Drupieski, Christopher Martin Participant University of Virginia Laboratoire de Mathematiques de Besançon CNRS (UMR 6623) UFR - Dudas, Olivier Participant ST Eaton, Charles William Participant University of Manchester Elliot, Jason Walter Participant University of Illinois Emerton, Matthew James Participant Northwestern University Erdmann, Karin Participant Oxford University Fiebig, Peter Participant Universität Freiberg Fong, Paul Participant University of Illinois, Chicago Friedlander, Eric Speaker Northwestern University Futorny, Vyacheslav Participant Universidade de Sao Paulo Gill, Christopher Participant University of Oxford Glesser, Adam Marc Participant N/A Gonzales, Richard Participant NPR Graber, John Participant University of Iowa Green, Edward Lewis Participant Virginia Tech Grime, Matthew Participant University of Bristol Grodal, Jesper Participant University of Copenhagen Halverson, Tom Participant University of Macalester Harris, Morton E. Participant University of Illinois at Chicago Hemmer, David J. Participant State University at Buffalo, SUNY Hernandez, David Participant N/A Hill, David Edward Participant University of California, Berkeley Hodge, Terrell L. Participant Western Michigan University Howard, Thomas T Participant UC Santa Barbara Iyengar, Srikanth B. Speaker University of Nebraska Juteau, Daniel Pierre Participant University of Jussieu Kaptanoglu, Semra Ozturk Participant Middle East Technical University Karuppuchamy, Paramasamy Participant University of Virginia Keller, Bernhard Participant University Paris 7 Kessar, Radha Participant University of Aberdeen Koshitani, Shigeo Participant Chiba University Krause, Henning Participant University of Paderborn Kuelshammer, Burkhard Participant University of Jena Kujawa, Jonathan R Participant University of Oklahoma Lau, Michael Participant University of Windsor Lin, Zongzhu Participant Kansas State University Linckelmann, Markus Participant N/A Lynd, Justin Participant The Ohio State University MacQuarrie, John William Participant The University of Manchester Mantese, Francesca Participant University of Verona Marcus, Andrei Participant Babes-Bolyai University Mazza, Nadia Participant University of Aberdeen Mbirika, Abukuse (Aba) Participant University of Iowa Minac, Jan Participant University of Western Ontario Moci, Luca Participant Roma Tre Nakano, Daniel Organizer University of Georgia Nath, Rishi Participant CUNY-York College Onofrei, Sylvia Elena Participant Kansas State University Park, Sejong Participant University of Aberdeen Parker, Alison Elizabeth Participant University of Leeds Parshall, Brian Participant University of Virginia Pevtsova, Julia A Participant University of Washington Pillen, Cornelius Speaker University of South Alabama Pop, Flaviu Vasile Participant Faculty of Mathematics and Computer Science Ragnarsson, Kari Participant University of Illinois, Chicago Rahman, Abdul Participant Howard University Rainbolt, Julianne Geering Participant St. Louis University Ribet, Kenneth A. Participant UC Berkeley Rose, Andrew James Participant University of Warwick Rouquier, Raphael Organizer Institut de Mathématiques de Jussieu SanAgustin, Keefe L Participant Brandeis University Schroll, Sibylle Participant University of Oxford Scott, Leonard Participant University of Virginia Shchigolev, Vladimir Participant Lomonosov Moscow State University Stancu, Mihai Radu Participant University of Copenhagen Swenson, Daniel Participant University of Minnesota Symonds, Peter Participant University of Manchester Talelli, Olympia Participant University of Athens Torres Giese, Enrique Participant University of Rochester Townsley, Lisa Participant Benedictine University Turner, Will Participant Oxford University Vazirani, Monica Joy Participant University of California, Davis Vejdemo-Johansson, Mikael Participant Fakultät für Mathe. und Info. Vershik, Anatoly M. Participant Russian Academy of Science Vitória, Jorge Participant University of Warwick Walter, Marty Participant University of Colorado Webb, Peter Participant University of Minnesota Witherspoon, Sarah Participant Texas A&M University Zacharia, Daniel Participant Syracuse University Zalesski, Alexandre Participant N/A Zárate, Alma Leticia Participant CINVESTAV IPN Zhang, Jiping Participant Peking Univiversity

NSF-CDI Workshop: Computation and Complex Systems October 12 2007 MSRI, Berkeley, California Organized By: Robert Bryant (MSRI) and Masoud Nikravesh (CITRIS-UC Berkeley and CS-LBNL) http://www.msri.org/calendar/workshops/WorkshopInfo/448/show_workshop

The National Science Foundation (NSF) is funding a major new initiative, beginning in 2008, on Cyber-enabled Discovery and Innovation (CDI). This initiative is intended to foster American competitiveness through research contributing to "a new generation of computationally based discovery concepts and tools to deal with complex, data-rich, and interacting systems." The NSF notes several examples of themes in this research category: data mining of large sets, interacting complex systems, high-performance computational experimentation, virtual environments, and educating researchers and students in computational discovery.

It is clear that mathematics must play a key role in research in the areas outlined for this initiative. The Division of Mathematical Sciences at the National Science Foundation is strongly supportive of the involvement of mathematical and cross-disciplinary groups in research projects connected with CDI. As a first step in promoting mathematics involvement, the NSF Mathematical Sciences Institutes have begun a coordinated effort to inform the mathematical community about this initiative and to facilitate the development of research proposals.

On October 12, 2007, MSRI will host a one-day workshop at the Mathemeatical Sciences Research Institute (MSRI) in Berkeley directed at assisting mathematical scientists in responding to this initiative. The plan is for this workshop is to include key lectures about large scale interdisciplinary problems, round table discussions about mathematical challenges in these and related areas, and Q & A sessions about the structure of the CDI initiative and the NSF's expectations for proposals.

All interested researchers are welcome to attend. There is no charge for this workshop but registration is required and can be done on-line at http://www.msri.org/calendar/workshops/WorkshopInfo/448/show_workshop . Lunch will be provided to all registrants. .

The plan for MSRI workshop is to include key lectures about large scale interdisciplinary problems, round table discussions about mathematical challenges in these and related areas, and Q & A sessions about the structure of the CDI initiative and the NSF's expectations for proposals.

Speakers: Large Scale Interdisciplinary Systems

Climate Modeling: Understanding the effect of human-induced changes to the composition of the atmosphere on the climate system is one of the most compelling scientific missions of our time. Largely a result of energy production, these compositional changes have been demonstrated to have a detectable influence on many aspects of the climate system. Computer modeling is an important tool in the detection and attribution of recent climate change and is the principal vehicle to make predictions regarding future climate change. Current state-of-the-art climate models are a complex coupling of models of the major climate subsystems. Present-day submodels describe the atmosphere, ocean, sea ice, and land systems. The large computational burden required to adequately describe any of these subsystems forces compromises to be made both in sophistication and fidelity. Advances in computing technology permit better simulation of the climate system by allowing the inclusion of more processes relevant to climate behavior, as well as more highly resolved discretizations of the climate subsystems. Trends in both of these aspects of climate modeling increase our understanding of human-induced effects on the recent past climate and reduce the uncertainty in our predictions of future climate change.

Astrophysics:

The study of astronomy — the study of the Universe as a whole and of its component parts, past, present, and future — is surely one of the earliest sciences pursued by mankind. Today this evolution is also marked by the words used to describe the field: astronomy tends to refer to the more descriptive aspects of the subject, while astrophysics is used to describe activities related to the use of physical sciences (including both physics and chemistry) as explanatory tools for what astronomers observe. Furthermore, astronomy is now intimately linked to virtually all other sciences. For example, physicists study the nature of fundamental interactions by looking at the evolution of the very early Universe and by studying the properties of highly evolved stars — exploding stars (e.g., supernovae), white dwarfs, and neutron stars. Biologists and chemists are examining the origins of life by considering the organic chemistry of the interstellar medium. Geoscientists interested in the origins of the planets are collaborating with astronomers who are finding numerous planetary systems orbiting nearby stars.

Speakers: Computing tools and Infrastructure for Large Scale Interdisciplinary Research

Building a Science Data Server, Berkeley Water Center-UCB Effective water management is not purely a scientific problem, a political problem, a technological problem, a computer science problem nor a socioeconomic problem; it is a complex, 21st Century problem that demands collaborative coordination between all of these disciplines. The Berkeley Water Center has been developed to integrate expertise across disciplines in support of a new research mode for water investigations

Scientific Data Management Managing scientific data has been identified as one of the most important emerging needs by the scientific community because of the sheer volume and increasing complexity of data being collected. Effectively generating, managing, and analyzing this information requires a comprehensive, end-to-end approach to data management that encompasses all of the stages from the initial data acquisition to the final analysis of the data. Based on the community input, we have identified three significant requirements. First, more efficient access to storage systems is needed. In particular, parallel file system improvements are needed to write and read large volumes of data without slowing a simulation, analysis, or visualization engine. Second, scientists require technologies to facilitate better understanding of their data, in particular the ability to effectively perform complex data analysis and searches over large data sets. Specialized feature discovery, parallel statistical analysis, and efficient indexing are needed before the data can be understood or visualized. Finally, generating the data, collecting and storing the results, data post- processing, and analysis of results is a tedious, fragmented process. Workflow tools for automation of this process in a robust, tractable, and recoverable fashion are required to enhance scientific exploration.

Scientific Data Visualization and Analysis

The goal of scientific visualization is to help scientists view and better understand their data. This data can come from experiments or numerical simulations. Often the size and complexity of the data makes it difficult to understand by direct inspection. Also, the data may be generated at several times during an experiment or simulation and understanding how the data varies with time may be difficult.

Scientific visualization can help with these difficulties by representing the data so that it may be viewed in its entirety. In the case of time varying data, animations can be created that show this variation in a natural way. Using virtual reality techniques, the data can be viewed and manipulated naturally in a true three dimensional environment (e.g. depth is explicitly perceived and not just implied).

All these techniques can allow scientists to better understand their data. Viewing the data in this way can quickly draw the scientist's attention to interesting and/or anomalous portions of the data. Because of this, we encourage scientists to use scientific visualization from the beginning of their experiments and simulations and not just when they think they have everything operating correctly. This also allows the scientists to develop a set of visualization tools and techniques that will help them understand their data as their research matures.

Citizen Cyber Science

Citizen cyber science model for scientific and social systems to enhance scientific discovery and innovation by bringing people and resources together across geographical and cultural boundaries.

Panelist: Scientific Panel

Richard Karp (Berkeley), University Professor Department of Electrical Engineering and Computer Sciences with additional appointments in Mathematics, Bioengineering and Operations Research, University of California at Berkeley and ICSI.

Research Interests: Theoretical computer science, combinatorial algorithms, discrete probability, computational biology, internet algorithms

Richard M. Karp was born in Boston, Massachusetts on January 3, 1935. He attended Boston Latin School and Harvard University, receiving the Ph.D. in 1959. From 1959 to 1968 he was a member of the Mathematical Sciences Department at IBM Research. From 1968 to 1994 and from 1999 to the present he has been a Professor at the University of California, Berkeley, where he held the Class of 1939 Chair and is currently a University Professor. From 1988 to 1995 and 1999 to the present he has been a Research Scientist at the International Computer Science Institute in Berkeley. From 1995 to 1999 he was a Professor at the University of Washington. During the 1985-86 academic year he was the co-organizer of a Computational Complexity Year at the Mathematical sciences research Institute in Berkeley. During the 1999-2000 academic year he was the Hewlett-Packard Visiting Professor at the Mathematical Sciences Research Institute. The unifying theme in Karp's work has been the study of combinatorial algorithms. His 1972 paper, "Reducibility Among Combinatorial Problems," showed that many of the most commonly studied combinatorial problems are NP-complete, and hence likely to be intractable. Much of his work has concerned parallel algorithms, the probabilistic analysis of combinatorial optimization algorithms and the construction of randomized algorithms for combinatorial problems.

His current activities center around algorithmic methods in genomics and computer networking. He has supervised thirty-six Ph.D. dissertations. His honors and awards include: U.S. National Medal of Science, Turing Award, Fulkerson Prize, Harvey Prize (Technion), Centennial Medal (Harvard), Lanchester Prize, Von Neumann Theory Prize, Von Neumann Lectureship, Distinguished Teaching Award (Berkeley), Faculty Research Lecturer (Berkeley), Miller Research Professor (Berkeley), Babbage Prize and eight honorary degrees. He is a member of the U.S. National Academies of Sciences and Engineering, the American Philosophical Society and the French Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science, the Association for Computing Machinery and the Institute for Operations Research and Management Science.

Stuart Russell (Berkeley): Professor, EECS Associate Chair, CS Chair Director, Center for Intelligent Systems (CIS)

Research Areas Artificial Intelligence (AI), Biosystems (BIO), Control, Intelligent Systems, and Robotics (CIR), machine learning; real-time decision-making; algorithms; probabilistic reasoning; computational biology

Biography He received his B.A. with first-class honours in Physics from Oxford University, 1982, and a Ph.D. in Computer Science from Stanford, 1986. He then joined the faculty of the UC Berkeley where he is currently the Associate Chair of EECS, a professor of computer science, director of the Center for Intelligent Systems, and holder of the Smith-Zadeh Chair in Engineering.

In 1990, he received the Presidential Young Investigator Award of the National Science Foundation, and in 1995 he was co-winner of the Computers and Thought Award. He was a 1996 Miller Professor of the University of California and was appointed to a Chancellor's Professorship in 2000. In 1998, he gave the Forsythe Memorial Lectures at Stanford University. He is a Fellow and former Executive Council member of the American Association for Artificial Intelligence and a Fellow of the Association for Computing Machinery.

He has published over 100 papers on a wide range of topics in artificial intelligence. His books include "The Use of Knowledge in Analogy and Induction" (Pitman, 1989), "Do the Right Thing: Studies in Limited Rationality" (with Eric Wefald, MIT Press, 1991), and "Artificial Intelligence: A Modern Approach" (with Peter Norvig, Prentice Hall, 1995, 2003).

Horst Simon (LBL) Dr. Horst Simon was named Associate Laboratory Director (ALD) for Computing Sciences at Berkeley Lab in 2004. In his role as the ALD for Computing Sciences, Horst represents the interests of the Lab’s scientific computing divisions, NERSC and Computational Research, in the formulation of Laboratory policy, and leads the overall direction of the two divisions. He also coordinates constructive interactions within the computing sciences divisions to seek coupling with other scientific programs. Horst joined LBNL in early 1996 as director of the newly formed NERSC Division, and was one of the key architects in establishing NERSC at its new location in Berkeley. The NERSC Center is DOE’s flagship supercomputing facility for unclassified research funded by DOE’s Office of Science and is currently supports 2,677 users at more than 300 institutions. Under Horst’s leadership, NERSC has enabled important discoveries in fields ranging from global climate modeling to combustion to astrophysics. Horst is also the founding director of Berkeley Lab's Computational Research Division, which conducts applied research and development in computer science, computational science, and applied mathematics. His research interests are in the development of sparse matrix algorithms, algorithms for large-scale eigenvalue problems, and domain decomposition algorithms for unstructured domains for parallel processing. Horst's recursive spectral bisection algorithm is regarded as a breakthrough in parallel algorithms for unstructured computations, and his algorithm research efforts were honored with the 1988 Gordon Bell Prize for parallel processing research. Horst was member of the NASA team that developed the NAS Parallel Benchmarks, a widely used standard for evaluating the performance of massively parallel systems. He is also one of four editors of the twice-yearly “TOP500” list of the world’s most powerful computing systems.

Kathy Yelick (Berkeley), Leader of Berkeley Lab's New Berkeley Institute for Performance Studies (BIPS) Kathy Yelick, a professor of computer science at UC Berkeley with a joint appointment in Berkeley Lab's Computational Research Division, has been named the new lead for CRD's Future Technologies Group (FTG). She will also be leading the newly established Berkeley Institute for Performance Studies (BIPS), which will bring together a number of projects in computer performance and evaluation. BIPS is the umbrella organization encompassing several research activities at LBNL and UC Berkeley: • The Performance Evaluation Research Center (PERC) • The Berkeley Benchmarking and Optimization Group (BeBOP) • LAPACK/ScaLAPACK project • Architectural evaluation research project • Benchmarking and performance optimization project The main goal of Kathy's research is to develop techniques for obtaining high performance on a wide range of computational platforms and to ease the programming effort required to obtain performance. Kathy is perhaps best known for her efforts in global address space (GAS) languages, which attempt to present the programmer with a shared memory model for parallel programming. These efforts have led to the design of Unified Parallel C (UPC), which merged some of the ideas from three shared address space dialects of C: Split-C, AC (from IDA), and PCP (from LLNL). In recent years, UPC has gained recognition as an alternative to message passing programming for large- scale machines. Compaq, Sun, Cray, HP, and SGI are implementing UPC, and Kathy is currently leading a large effort at LBNL to implement UPC on Linux clusters and IBM machines and to develop new optimizations. The language provides a uniform programming model for both shared and distributed memory hardware. She has also worked on other global address space languages such as Titanium, which is based on Java. Kathy has also done some notable work on single processor optimizations including techniques for automatically optimizing sparse matrix algorithms for memory hierarchies. These efforts are part of an NSF funded project called BeBOP (Berkeley Benchmarking and Optimization) that is working on methods to take advantage of special structure such as symmetry and triangular solves.

Jeff Grossman (Berkeley), Executive Director Center of Integrated Nanomechanical Systems Berkeley Nanosciences and Nanoengineering Institute

The computational nanoscience group at UC Berkeley, headed by Jeffrey C. Grossman, is actively engaged in a number of research areas relating to the simulation of nanoscale materials and interfaces. Our focus is on the application and development of cutting-edge classical and quantum simulation tools to understand, predict, and design novel nanoscale materials with applications to:

• build new sensing and collection devices; • design highly efficient “3 rd generation” photovoltaics; • understand water in confined conditions; • control self-assembly and synthesis of nanoscale objects; • predict new thermoelectric materials.

Particular emphasis is given to the chemistry and physics at the nanoscale interfaces which play a crucial role in both underlying physical as well as system-level device properties. In addition to using standard classical and ab initio approaches to compute, e.g., structural, optical, and electronic properties, we are also developing a new class of simulation tools that bridge the gap between these traditional methods and the needs of experimentalists and engineers.

Organizers:

Robert Bryant, MSRI Director Bryant received his Ph.D. in mathematics in 1979 at the University of North Carolina at Chapel Hill, working under the late Robert B. Gardner, who had been a UC Berkeley mathematics student of Professor Shiing-Shen Chern. This makes Bryant an "academic grandchild" and mathematical disciple of S.-S. Chern, co-founder and first director of MSRI. Bryant's recent work on Finsler geometry was directly inspired by conversations with Chern. "Chern made MSRI into an important mathematical center, practically overnight," says Bryant.

Beginning in 1979, Robert Bryant served on the faculty at Rice University for seven years, and then moved to Duke. He has held numerous visiting positions at universities and research institutes around the world. He visited MSRI during the 2001-02 academic year as the Clay Mathematics Visiting Professor while on sabbatical from Duke University. Bryant returned to MSRI during the fall 2003 term as co-organizer of the Institute's semester-long research program in Differential Geometry.

Robert Bryant was appointed by President Bush to serve on the Board of the Vietnam Education Foundation (2002-05). He was elected to the American Academy of Arts and Sciences in 2002. Just last week he was elected a Fellow of the prestigious National Academy of Sciences.

Robert Bryant is a distinguished differential geometer, with an extensive list of publications. Bryant's research interests center on exterior differential systems and the geometry of differential equations as well as their applications to Riemannian geometry, special holonomy, and mathematical physics. In addition to his teaching and research in differential geometry, he is Vice President of the American Math Society (AMS), and serves as Director of The Institute for Advanced Study (IAS)/Park City Mathematics Institute (PCMI). Bryant has also been active in the Chamber Arts Society of Durham, NC, and became its director in 2000.

Robert Bryant's appointment as director of MSRI follows a year-long, nationwide search process led by a committee of the Institute's trustees chaired by Julius Krevans, which included Deborah Loewenberg Ball, Charles Fefferman, Dusa McDuff, Donald Saari, Roger Strauch, and Craig Evans, who represented UC Berkeley's Math Department. After assessing a group of outstanding candidates, the committee presented its recommendation to MSRI's Board, which made the final selection.

Robert Bryant succeeds David Eisenbud, who has served as MSRI director since 1997 and is also a tenured professor on the faculty at UC Berkeley. In August Eisenbud will leave MSRI to become a full-time member of its Department of Mathematics. "Bryant is a superb researcher, internationally renowned in the mathematical community, one whose scientific interests and knowledge are very broad," commented Eisenbud. "He is loved as a lecturer and teacher because he thinks so clearly, and cares so visibly and effectively about his audiences and students. He possesses a rare quality of selflessness," noted Eisenbud. "I believe that these characteristics, combined with Bryant's organizational skill and love of culture will make him a great director. I look forward very much to watching MSRI's progress under his leadership."

Masoud Nikravesh (CITRIS Berkeley and Life Science-LBNL), CITRIS Executive Director for Computational Science and Engineering

The Executive Director of Computational Science and Engineering provides high level executive support to the CITRIS Director and LBNL Associate Laboratory Director. Responsibilities of the Executive Director for Computational Science and Engineering include program development and management of all aspects of the program’s operations, including budget matters. The Executive Director will also provide intellectual leadership in curriculum development and judgment in defining outreach services. The Executive Director will develop and maintain liaison with the advisory board and coordinate fund raising in consultation with the program’s constituent Schools and Lawrence Berkeley National Lab (LBNL). With direct reporting responsibility to the Director of CITRIS and the Associate Lab Director of Computing Sciences at LBNL, the incumbent will interact with highly multidisciplinary team of researchers, identify potential synergy among them, leverage user’s facility at Lawrence Berkeley National Laboratory, advocate CITRIS and DOE-SciDac funded programs to a broader community. CITRIS is supported by a large number of industrial and government sponsors, and the Executive Director of Computational Science and Engineering (CSE) will interact with their respective representatives. The Executive Director will interact with these sponsors, extend existing research programs or develop new ones, and oversee execution of these programs once funded. In this capacity, the Executive Director of CSE will interact significantly with CITRIS Director, who takes the lead on industrial and government interactions and negotiations.

Prior to joining CITRIS, Dr. Nikravesh was Executive Director of BISC (Berkeley Initiative in Soft Computing) of the Computer Science Division at the University of California, Berkeley and the BT Senior Research Fellow since 2000 and hold non-Senate academic appointment from 2001 to 2007 prior to joining the CITRIS. BISC is a world leading center for basic and applied research in soft computing (Computational Intelligence and Machine Learning) with over 5000 members worldwide. Dr. Nikravesh is also Research Scientist in the Imaging and Informatics Group at Life Sciences and previously at NERSC (National Energy Research Scientific Computing Division) at the Lawrence Berkeley National Laboratory. Dr. Nikravesh is a member of the Industrial Relations Advisory Board at EECS, UC Berkeley, the LBNL-NERSC representative to the DiMI- UC Discovery Program (appointed by the NERSC Director) and he is a member of the Executive Committee and member of the Research Council-UC Discovery program (appointed by UC Provost and Senior Vice President). In addition, he has been an invited lecturer throughout the world including China, Germany, Hong Kong, Finland, Canada, UK, Turkey, New Zealand, and Mexico as well as at many conferences and scientific events, IFSA, NAFIPS, FuzzIEEE, IEEEGrc, INDIN, GEECO, ICSCCW, AICE, ACM, ACC, ACS, SEG and SPE. He has been a member of IEEE, AICHE, SPE, AAPG, SEG, ACS, NAFIPS, IFSA, and other scientific scholarly societies.

Computation and Complex Systems Friday October 12, 2007

8:30AM - 8:45AM Registration

8:45AM - 9:00AM Introduction: Robert Bryant (Director,MSRI), Peter March (Director,NSF-DMS)

9:00AM - 9:45AM Climate Modelling: William Collins and Inez Fung (UCB, Earth & Planetary Science)

9:50AM - 10:35AM Astrophysics: Josh Bloom, Geoffrey Bower, Eliot Quataert (UCB, Astrophysics), and Dave Schlegel (LBNL)

10:35AM - 11:00AM Coffee Break

11:00AM - 12:15PM NSF Panel: Peter March (NSF-DMS), Eduardo Misawa (NSF-DCMS)

12:15PM - 1:30PM Lunch

1:30PM - 1:55PM Building a Science Data Server: Deb Agarwal (Berkeley Water Center - Berkeley)

2:00PM - 2:25PM Scientific Data Management: Arie Shoshani (CS-LBNL)

2:30PM - 2:55PM Scientific Data Visualization and Analysis: Wes Bethel (CS-LBNL)

3:00PM - 3:25PM Citizen Cyber Science: Dave Anderson (Berkeley-SSL)

3:30PM - 4:00PM Afternoon Tea

4:00PM - 5:30PM Panel -Enabling Cooperation with Mathematical Scientists: Robert Bryant (MSRI-Moderator), Daryl Chrzan (Berkeley), Jeffrey Grossman (Berkeley), Richard Karp (Berkeley), Stuart Russell (Berkeley), Horst Simon (CS-LBNL), and Kathy Yelick (Berkeley)

Currently Available Videos

• William Collins, Inez Fung , Climate Modelling October 12,2007, 09:00 AM to 09:45 AM

• William Collins, Inez Fung , Climate Modelling October 12,2007, 09:00 AM to 09:45 AM

• Joshua Bloom , Astrophysics October 12,2007, 09:50 AM to 10:35 AM

• Peter March, Eduardo Misawa , NSF Panel October 12,2007, 11:00 AM to 12:15 PM

• Peter March, Eduardo Misawa , NSF Panel October 12,2007, 11:00 AM to 12:15 PM

• Deb Agarwal , Building a Science Data Server October 12,2007, 01:30 PM to 01:55 PM

• Arie Shoshani , Scientific Data Management October 12,2007, 02:00 PM to 02:25 PM

• Wes Bethel , Scientific Data Visualization and Analysis October 12,2007, 02:30 PM to 02:55 PM

• David Anderson , Citizen Cyber Science October 12,2007, 03:00 PM to 03:25 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

• Robert Bryant, Daryl Chrzan, Robert Grossman, Richard Karp, Stuart Russell, Horst Simon, Katherine Yelick , Panel -Enabling Cooperation with Mathematical Scientists October 12,2007, 04:00 PM to 05:30 PM

Participant List

Name Role Institution Agarwal, Deb Speaker UCB and LBL Anderson, David Participant U.C. Berkeley Assadi, Amir Participant UW Madison Bartlett, Peter L. Participant UC Berkeley Bayer, David Allen CAS Rep Columbia University Bethel, Wes Participant Lawrence Berkeley National Laboratory Bloom, Joshua Participant UC Berkeley Borrelli, Francesco Participant UCB Bower, Geoffrey Speaker University of California Bryant, Robert Leamon Participant MSRI Cadieu, Charles Participant Redwood Center Cayco, Bem E. Participant San Jose State University Chrzan, Daryl Participant University of California, Berkeley Collins, William Speaker UC Berkeley Crutchfield, James Participant University of California at Davis de Figueiredo, rui Participant UC Irvine Denning, John Participant Kaiser Permanente Ding, Kequan Participant Chinese Academy of Sciences El Karoui, Noureddine Participant Stanford University Eswaran, Krish Participant UC Berkeley Evans, Craig Participant UC, Berkeley Fateman, Richard Participant UCB - University of California, Berkeley Feeley, Ryan Patrick Participant UCB - University of California, Berkeley Universidad de la República Oriental del Ferrer Santos, Walter Participant Uruguay Fung, Inez Speaker UC, Berkeley Gerber, Richard Participant LBNL Grossman, Jeffrey Participant University of California, Berkeley Gunter, Dan Participant LBNL Hahnfeldt, Philip Participant Tufts University School of Medicine Harrison, Jenny Participant UCB - University of California, Berkeley Hesse, Marc Participant Stanford University Hlatky, Lynn Participant CSEMC, Tufts Univ. School of Medicine Hobbs, Allen Participant Kaiserp Permanante Horowitz, Roberto Participant University of California, berkeley Iftime, Mihaela D. Participant Boston University Imanyuel, Larens Participant Universal System of Natural Units Inoue, Takahide Participant University of California Ivanov, Paul Participant UC Berkeley Kafadar, Karen Participant Indiana University Kalczynski, Pawel J Participant California State University, Fullerton Kane, Abdoul Participant University of Toronto Karp, Richard Participant University of California, Berkeley Kazanci, Caner Participant University of Georgia Koepsell, Kilian Participant UC Berkeley Korsan, Bob Participant Decisions, Decisions! Krener, Arthur J. Participant Naval Postgraduate School Lauritzen, Thomas Zaccarin Participant Redwood Neuroscience Institute Ligocki, Terry Participant Lawrence Berkeley National Lab (LBNL) Lim, Lek-Heng Participant University of California, Berkeley Lin, En-Bing Participant Central Michigan University Lin, Florence J. Participant University of Southern California Lipkin, Efrem Participant CoDesign Madera, Martin Participant UC Santa Cruz Mahoney, Michael Participant Yale University NSF March, Peter representative National Science Foundation Marcus, Bob Participant N/A McCorquodale, Peter Participant Lawrence Berkeley National Laboratory McMains, Sara Participant ucb Meyer, Francois Participant University of Colorado Millman, Jarrod Participant UC Berkeley NSF Misawa, Eduardo representative National Science Foundation Mislove, Michael Participant Tulane University Mittelmann, Hans CAS Rep Arizona State University Montanari, Andrea Participant Stanford University Morton, Jason Participant University of California, Berkeley Newsam, Shawn Participant UC Merced Ng, Esmond G. Participant Lawrence Berkeley National Laboratory Nicolau, Monica Participant Stanford Nikravesh, Masoud Organizer University of California, Berkeley Nolan, Deborah Participant UCB - University of California, Berkeley Otoo, Ekow Participant LBNL Packard, Andrew Participant UC Berkeley Pejic, Michael Participant N/A Pitman, Jim Participant UCB - University of California, Berkeley Purdy, David Gregory Participant UCB - University of California, Berkeley Quataert, Eliot Participant UC Berkeley Rajagopal, Ram Participant N/A Rajapakse, Indika Participant Fred Hutchinson Cancer Research Center Ravindran, Sivaguru S Participant University of Alabama in Huntsville Rice, John A. Participant UCB - University of California, Berkeley Rosenberg, David S. Participant UC Berkeley Roth, Ilan Participant UC Berkeley Russell, Stuart J. Speaker UC Berkeley Sachs, Rainer K. Participant University of California at Berkeley Sachs, Sonia Participant IBM Research Saito, Naoki Participant University of California, Davis Sakuma, Thais Harumi Participant Federal State University of Rio de Janeiro Sarwate, Anand Dilip Participant UC Berkeley Scapolla, Terenzio Participant N/A Schlegel, David Participant LBL Sethian, James A. Participant UCB - University of California, Berkeley Shahbazova, Shahnaz Nadir Participant UC Berkeley Shan, Eric Participant N/A Shankar, Natarajan Participant SRI International Shapiro, Vadim Participant University of Wisconsin, Madison Shoshani, Arie Participant LBNL Simon, Horst D. Participant LBNL Skinner, David E Participant LBNL Tomizuka, Masayoshi Participant University of California, Berkeley Topcu, Ufuk Participant UC, BERKELEY Trinkle, David Participant UC Berkeley Velasco, Mauricio Fernando Participant UC Berkeley Wang, Yue Participant UC Berkeley Warnock, Robert Lee Participant Stanford Linear Accelerator Center Wong Kew, Rich Participant Postdoc Research Fellows Wu, John Participant LBNL Yan, Ning Participant Stanford University Yelick, Katherine Participant UCB - University of California, Berkeley Yu, Bin Participant UCB - University of California, Berkeley

REPORT ON MSRI HOT TOPIC WORKSHOP ”Contact structures, dynamics and the Seiberg-Witten equations in dimension 3” Held at MSRI, June 9th–June 13th

Organizer: Helmut Hofer, Michael Hutchings, Peter Kronheimer, Tom Mrowka and Cliff Taubes

INTRODUCTION

In 2007 Cliff Taubes proved the Weinstein conjecture in dimension three. The proof exploits a novel relationships between Seiberg-Witten theory and contact geometry on three-dimensional manifolds. Further developments of the ideas lead to an isomorphism between Seiberg-Witten Floer cohomology (SWF) and embedded contact homology (ECH). The latter is a form of Floer homology that was defined by Michael Hutchings. The workshop gave a his- toric perspective of the Weinstein conjecture, introduced the salient features of both the contact geometry side and the Seiberg-Witten side, and explained how they are related. Further some recent applications were discussed. ORGANIZATION

The workshop consisted of several mini-courses providing an introduction to the Weinstein conjecture, the embedded contact homology (ECH) and Seiberg-Witten Floer homology (SWF) and Taubes lecture series on the proof of the Weinstein conjecture in dimension three and the equivalence of ECH and SWF. In addition several one hour lectures were devoted to related topics.

SCHEDULE

Monday June 9, 2008

09:00AM - 10:00AM Helmut Hofer The History of the Weinstein Conjecture 11:00AM - 12:00PM Helmut Hofer What do we know about the Weinstein Conjecture in higher dimensions? 01:30PM - 02:30PM Introduction to the Seiberg-Witten equations

1 03:00PM - 04:00PM Richard Siefring The asymptotic behavior of punctured pseudoholomorphic curves

Tuesday June 10, 2008

09:00AM - 10:00AM Michael Hutchings Background for embedded contact homology 11:00AM - 12:00PM Peter Kronheimer Introduction to the Seiberg-Witten-Floer homologies I 01:30PM - 02:30PM Clifford Taubes Outline of the relationship between Seiberg-Witten and the Weinstein conjecture 03:00PM - 04:00PM Robert Lipshitz Towards Heegaard Floer homology for 3-manifolds with boundary

Wednesday June 11, 2008

09:00AM - 10:00AM Michael Hutchings Introduction to embedded contact homology I 11:00AM - 12:00PM Tomasz Mrowka Introduction to the Seiberg-Witten-Floer homologies II 01:30PM - 02:30PM Clifford Taubes Seiberg-Witten and Weinstein Conjecture I 03:00PM - 04:00PM Clifford Taubes Seiberg-Witten and Weinstein Conjecture II 04:30PM - 05:30PM Ko Honda Relative Contact Homologies

Thursday June 12, 2008

09:00AM - 10:00AM Michael Hutchings Introduction to embedded contact homology II 11:00AM - 12:00PM Peter Kronheimer Existence theorems for Seiberg-Witten-Floer homology 01:30PM - 02:30PM Clifford Taubes Seiberg-Witten and Weinstein Conjecture III 03:00PM - 04:00PM Tim Perutz A symplectic Gysin sequence

2 Friday June 13, 2008

09:00AM - 10:00AM Clifford Taubes Seiberg-Witten and embedded contact homology I 11:00AM - 12:00PM Clifford Taubes Seiberg-Witten and embedded contact homology II 01:30PM - 02:30PM Cagatay Kutluhan Seiberg-Witten Floer homology and symplectic forms on S1 × M 3 03:00PM - 04:00PM Robion Kirby Broken Lefschetz fibrations for all smooth oriented 4-manifolds

ABSTRACTS

Helmut Hofer gave two talks. He described how Rabinowitz’s existence proof for periodic orbits on star-shaped energy surfaces in R2n led Weinstein to his famous conjectures. He explained the relationship of the Weinstein conjecture to a pseudoholomorphic curve theory, symplectic field theory and Gromov-Witten theory.

Ko Honda defined in his talk a version of contact homology and embed- ded contact homology for contact 3-manifolds with boundary, which we call sutured contact homology and sutured embedded contact homology. Gave an outline for proving a criterion for determining whether a 3-manifold fibers over the circle with a given fiber, in the framework of embedded contact ho- mology. Conjectured the equality of the sutured Heegaard Floer homology and sutured embedded contact homology groups, and presented some evi- dence in the easiest cases.

Michael Hutchings gave a three-lecture minicourse on embedded con- tact homology (ECH). The first lecture gave a detailed discussion of the background from topology, contact geometry, and holomorphic curve theory that is required for the definition of ECH. The second lecture defined ECH and explained the fundamental index inequality which lies at its foundations. The third lecture introduced some additional structures on ECH and the re- lation of ECH with Seiberg-Witten Floer homology, showed how to compute examples of ECH, and discussed several important open problems for the further development of ECH .

3 Rob Kirby defined broken achiral Lefschetz fibration, and then stated that Lekili had removed the word ”achiral” from the theorem by showing, us- ing singularity theory of maps from 4-space to 2-space, that an anti-Lefschetz singularity could be replaced by three Lefschetz singularities and one round handle. The neatest description uses an improvement of Baykur. He sketched some possible applications of this theorem to homotopy 4-spheres.

Peter Kronheimer gave two lectures. The first lecture described two aspects of finite-dimensional Morse theory, with a view to seeing how these come into play in Seiberg-Witten Floer homology: these were, first, the study of Morse theory on manifolds with boundary, in the case that the gradient flow is tangent to the boundary; and second, the study of Morse theory in the presence of semi-free circle actions. One can pass from the latter to the former by blowing up. The second lecture described the calculation of the ”bar” version of Seiberg-Witten Floer homology. This calculation can be reduced to algebraic topology; it leads to a non-vanishing theorem for all variants of Seiberg-Witten Floer homology for spin-c structures with torsion first Chern class. These two lectures, together with the lectures by Tom Mrowka, formed a series of four.

Cagatay Kutluhan talked about a joint work with Clifford H. Taubes in which it is proved that monopole Floer homology of a closed, connected, orientable 3-manifold M resembles that of a manifold which fibers over the circle if the 4-manifold S1 × M admits a symplectic form. The proof in- volves techniques and ideas from Taubes’ proof of the Weinstein Conjecture. Although the work was initially motivated with the intention of completely characterizing when 4-manifolds of the form S1 ×M admit symplectic forms, the techniques in the proof might yield new ideas in the study of dynamical problems in dimension three as well.

Robert Lipshitz spoke about an extension of one group from Heegaard Floer homology – a family of invariants conjecturally equivalent to monopole Floer homology and embedded contact homology – to three-manifolds with parametrized boundary. The extension of Lipshitz-Ozsvath-Thurston assigns a differential graded algebra to a surface and dg modules over that algebra to a 3-manifold with boundary parametrized by the surface. The talk relates to another approach to proving equivalence of different Floer theories: by finding axioms characterizing them.

4 Tom Mrowka presented in a first lecture the basics of the Seiberg-Witten equations, their structure and compactness properties of the moduli spaces of solutions. The second lecture discussed how the finite dimensional construc- tions (blowing up the fixed point set of a semi-free circle action) play out in the infinite dimensional Seiberg-Witten context and calculated the three version of Floer homology for the basic example of S3 from the definitions.

Tim Perutz presented a new exact triangle in symplectic Floer homol- ogy, closely related to the Gysin sequence for the homology of a sphere- bundle. The maps in the sequence invoke ‘pseudo-holomorphic quilts’ in the sense of Wehrheim-Woodward. A motivating class of examples arise in what is conjecturally a symplectic model for Seiberg-Witten Floer homology of 3-manifolds. In these cases, the symplectic Gysin sequence is precisely anal- ogous to an exact triangle for the Floer homology of a connected sum.

Richard Siefring discussed the asymptotic behavior of punctured pseu- doholomorphic curves. The main result was a formula for the relative asymp- totic behavior of two pseudoholomorphic curves that approach the same peri- odic orbit or the Reeb vector field. This asymptotic formula is important for controlling intersections and embeddedness of punctured pseudoholomorphic curves, and is needed for the proof of Hutchings’ index inequality in the most general case.

Clifford Taubes gave six talks on his work that relates the Seiberg- Witten Floer cohomology of a compact, 3-dimensional contact manifold with the latter’s embedded contact homology as defined by Michael Hutchings. The first talk was an outline to the proof by Taubes of the 3-dimensional We- instein conjecture which posited that the Reeb orbit associated to any such contact structure must have at least one closed, integral curve. The second talk went through the basic PDE estimates that are used in the proof, and then explained how these estimates lead to the existence of a closed orbit if a certain energy bound is obeyed. The third lecture described a dichotomy that asserts either the energy bound, or a particular rate of divergence of both the energy and an associated Chern-Simons function. The fourth talk explained why the divergence of the Chern-Simons function can not occur. This utilizes a novel result about the spectral flow for certain families of Dirac operators on odd dimensional manifolds. The last two talks outlined

5 the author’s proof that there is an isomorphism between the Seiberg-Witten Floer cohomology of the 3-manifold and the embedded contact homology.

AVAILABLE VIDEOS

• Helmut Hofer , The History of Weinstein Conjecture. June 9, 2008, 09:00 AM to 10:00 AM • Helmut Hofer , What do we know about the Weinstein Conjecture in higher dimensions? June 9, 2008, 11:00 AM to 12:00 PM • Tomasz Mrowka , Introduction to the Seiberg-Witten equations. June 9, 2008, 01:30 PM to 02:30 PM • Richard Siefring , The asymptotic behavior of punctured pseudoholomor- phic curves. June 9, 2008, 03:00 PM to 04:00 PM • Michael Hutchings , Background for embedded contact homology. June 10, 2008, 09:00 AM to 10:00 AM • Peter Kronheimer , Introduction to the Seiberg-Witten-Floer homologies I. June 10, 2008, 11:00 AM to 12:00 PM • Clifford Taubes , Outline of the relationship between Seiberg-Witten and the Weinstein conjecture. June 10, 2008, 01:30 PM to 02:30 PM • Robert Lipshitz , Towards Heegard Floer homology for 3-manifolds with boundary. June 10, 2008, 03:00 PM to 04:00 PM • Michael Hutchings , Introduction to embedded contact homology I. June 11, 2008, 09:00 AM to 10:00 AM • Tomasz Mrowka , Introduction to the Seiberg-Witten-Floer homologies II. June 11, 2008, 11:00 AM to 12:00 PM • Clifford Taubes , Seiberg-Witten and Weinstein Conjecture I. June 11, 2008, 01:30 PM to 02:30 PM • Clifford Taubes , Seiberg-Witten and Weinstein Conjecture II. June 11,2 008, 03:00 PM to 04:00 PM • Ko Honda , Relative Contact Homologies. June 11,2008, 04:30 PM to 05:30 PM • Michael Hutchings , Introduction to embedded contact homology II. June 12, 2008, 09:00 AM to 10:00 AM • Peter Kronheimer , Existence theorems for Seiberg-Witten-Floer homol- ogy. June 12, 2008, 11:00 AM to 12:00 PM • Clifford Taubes , Seiberg-Witten and Weinstein Conjecture III. June 12, 2008, 01:30 PM to 02:30 PM

6 • Tim Perutz , A symplectic Gysin sequence June 12, 2008, 03:00 PM to 04:00 PM • Clifford Taubes , Seiberg-Witten and embedded contact homology I. June 13, 2008, 09:00 AM to 10:00 AM • Clifford Taubes , Seiberg-Witten and embedded contact homology II. June 13, 2008, 11:00 AM to 12:00 PM • Cagatay Kutluhan Seiberg-Witten Floer homology and symplectic forms on S1 × M 3. June 13, 2008, 01:30 PM to 02:30 PM • Robion Kirby , Broken Lefschetz fibrations for all smooth oriented 4- manifolds. June 13, 2008, 03:00 PM to 04:00 PM

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Hot Topics: Contact structures, dynamics and the Seiberg-Witten equations in dimension 3

June 9 – June 13, 2008

Schedule

Monday June 9, 2008

9:00 AM - Helmut Hofer The History of the Weinstein Conjecture 10:00 AM 10:00 AM – Morning Break 11:00 AM 11:00 AM– What do we know about the Weinstein Conjecture in higher Helmut Hofer 12:00 PM dimensions? 12:00 PM- Lunch 1:30 PM 1:30 PM – Tomasz Mrowka Introduction to the Seiberg-Witten equations 2:30 PM 2:30 PM – Tea 3:00 PM 3:00 PM – The asymptotic behavior of punctured pseudoholomorphic Richard Siefring 4:00 PM curves

Tuesday June 10, 2008

9:00 AM – Michael Hutchings Background for embedded contact homology 10:00 AM 10:00 AM – Morning Break 11:00 AM 11:00 AM – Peter Kronheimer Introduction to the Seiberg-Witten-Floer homologies I 12:00 PM 12:00 PM- Lunch 1:30 PM 1:30 PM – Outline of the relationship between Seiberg-Witten and the 2:30 PM Weinstein conjecture 2:30 PM – Tea 3:00 PM 3:00 PM – Robert Lipshitz Towards Heegard Floer homology for 3-manifolds with boundary 4:00 PM 4:00 PM – Reception 5:00 PM

Wednesday June 11, 2008

9:00 AM – Michael Hutchings Introduction to embedded contact homology I 10:00 AM 10:00 AM – Morning Break 11:00 AM 11:00 AM – Tomasz Mrowka Introduction to the Seiberg-Witten-Floer homologies II 12:00 PM 12:00 PM- Lunch 1:30 PM 1:30 PM – Clifford Taubes Seiberg-Witten and Weinstein Conjecture I 2:30 PM 2:30 PM – Tea 3:00 PM 3:00 PM – Clifford Taubes Seiberg-Witten and Weinstein Conjecture II 4:00 PM 4:30 PM – Ko Honda TBA 5:30 PM

Thursday June 12, 2008

9:00 AM – Michael Hutchings Introduction to embedded contact homology II 10:00 AM 10:00 AM – Morning Break 11:00 AM 11:00 AM – Peter Kronheimer Existence theorems for Seiberg-Witten-Floer homology 12:00 PM 12:00 PM- Lunch 1:30 PM 1:30 PM – Clifford Taubes Seiberg-Witten and Weinstein Conjecture III 2:30 PM 2:30 PM – Tea 3:00 PM 3:00 PM – Tim Perutz A symplectic Gysin sequence 4:00 PM

Friday June 13, 2008

9:00 AM – Clifford Taubes Seiberg-Witten and embedded contact homology I 10:00 AM 10:00 AM – Morning Break 11:00 AM 11:00 AM – Clifford Taubes Seiberg-Witten and embedded contact homology II 12:00 PM 12:00 PM- Lunch 1:30 PM 1:30 PM – Seiberg-Witten Floer homology and symplectic forms on S^1 x Cagatay Kutluhan 2:30 PM M^3 2:30 PM – Tea 3:00 PM 3:00 PM – Robion Kirby Broken Lefschetz fibrations for all smooth oriented 4-manifolds 4:00 PM

Currently Available Videos

• Helmut Hofer , The History of Weinstein Conjecture. June 9,2008, 09:00 AM to 10:00 AM

• Helmut Hofer , What do we know about the Weinstein Conjecture in higher dimensions? June 9,2008, 11:00 AM to 12:00 PM

• Tomasz Mrowka , Introduction to the Seiberg-Witten equations June 9,2008, 01:30 PM to 02:30 PM

• Richard Siefring , The asymptotic behavior of punctured pseudoholomorphic curves June 9,2008, 03:00 PM to 04:00 PM

• Michael Hutchings , Background for embedded contact homology June 10,2008, 09:00 AM to 10:00 AM

• Peter Kronheimer , Introduction to the Seiberg-Witten-Floer homologies I June 10,2008, 11:00 AM to 12:00 PM

• Clifford Taubes , Outline of the relationship between Seiberg-Witten and the Weinstein conjecture June 10,2008, 01:30 PM to 02:30 PM

• Robert Lipshitz , Towards Heegard Floer homology for 3-manifolds with boundary June 10,2008, 03:00 PM to 04:00 PM

• Michael Hutchings , Introduction to embedded contact homology I June 11,2008, 09:00 AM to 10:00 AM

• Tomasz Mrowka , Introduction to the Seiberg-Witten-Floer homologies II June 11,2008, 11:00 AM to 12:00 PM

• Clifford Taubes , Seiberg-Witten and Weinstein Conjecture I June 11,2008, 01:30 PM to 02:30 PM

• Clifford Taubes , Seiberg-Witten and Weinstein Conjecture II June 11,2008, 03:00 PM to 04:00 PM

• Ko Honda , Relative Contact Homologies June 11,2008, 04:30 PM to 05:30 PM

• Michael Hutchings , Introduction to embedded contact homology II June 12,2008, 09:00 AM to 10:00 AM

• Peter Kronheimer , Existence theorems for Seiberg-Witten-Floer homology June 12,2008, 11:00 AM to 12:00 PM

• Clifford Taubes , Seiberg-Witten and Weinstein Conjecture III June 12,2008, 01:30 PM to 02:30 PM

• Tim Perutz , A symplectic Gysin sequence June 12,2008, 03:00 PM to 04:00 PM

• Clifford Taubes , Seiberg-Witten and embedded contact homology II June 13,2008, 11:00 AM to 12:00 PM

• Clifford Taubes , Seiberg-Witten and embedded contact homology II June 13,2008, 11:00 AM to 12:00 PM

• Cagatay Kutluhan , Seiberg-Witten Floer homology and symplectic forms on S^1 x M^3 June 13,2008, 01:30 PM to 02:30 PM

• Robion Kirby , Broken Lefschetz fibrations for all smooth oriented 4-manifolds June 13,2008, 03:00 PM to 04:00 PM

Participant List

Name Role Institution Abbas, Casim Participant Michigan State University Agol, Ian Participant UCB Albers, Peter Participant New York University Auroux, Denis Participant MIT Bramham, Barney Participant University of Leipzig Burghelea, Dan Participant OSU Canez, Santiago Participant UC Berkeley Chapin, Jeff Scott Participant Michigan State University Choi, Ka Participant N/A Choi, Keon Participant UC Berkeley Colin, Vincent Participant Universite de Nantes Cotton, Andrew Participant UC Berkeley Crombecque, David Participant Bryn Mawr College Diogo, Luis Miguel Participant Stanford University Doria, Celso Melchiades Participant Michigan State University Dotterrer, Dominic Participant N/A Early, Nicholas Joseph Participant Louisiana State University Etgu, Tolga Participant Koc University Farris, David Participant Berkeley Fel'shtyn, Alexander Participant Boise State University Fish, Joel Participant University of Michigan Frohman, Charles Participant University of Iowa Ghiggini, Paolo Participant Caltech Golovko, Roman Participant University of Southern California Hays, Chris Participant Michigan State University He, Zhenqi Participant MIT Hecht, Michael Participant Universität Leipzig Hofer, Helmut H. Organizer New York University Honda, Ko Speaker Univ. of Southern California Hong, Yoon hi Participant N/A Hutchings, Michael Organizer University of California Hwang, Cheuk-Man Participant Michigan State University Kim, Yon-Seo Participant University of California, Los Angeles UCB - University of California, Kirby, Robion C. Speaker Berkeley Kronheimer, Peter Benedict Organizer Harvard University Kuperberg, Krystyna Participant Auburn University Kuperberg, Wlodzimierz Participant University of Auburn Kusner, Robert Barnard Participant University of Massachusetts Kutluhan, Cagatay Participant University of Michigan Lai, Hsin-Hong Participant Brandeis University Lee, Yi-Jen Participant Purdue University Lekili, Yanki Participant MIT Leness, Thomas Participant Florida International University Lipshitz, Robert Speaker Columbia University Lipyanskiy, Max Participant MIT Lisi, Samuel Thomas Participant Stanford University Liu, Yang Participant University of Georgia Lotay, Jason Participant University College Oxford mathews, Daniel Participant N/A Maydanskiy, Maksim I Participant MIT Mazzucchelli, Marco Participant Università di Pisa Momin, Al S Participant Courant Institute Montgomery, Richard W. Participant UCSC Mrowka, Tomasz S. Organizer MIT Noetzel, Gregor Participant Universitaet Leipzig Otterson, James Joaquim de Almeida Participant Imperial College Perkins, Kala Participant Infiniti-Ed Perutz, Tim Speaker Columbia University Rezazadegan, Reza Participant Rutgers University Roth, Ilan Participant UC Berkeley Scharlemann, Martin G. Participant UC Santa Barbara Siefring, Richard Speaker Stanford University Sivek, Steven Participant MIT Srikrishnan, Vivek Participant Penn State University Takakura, Tatsuru Participant N/A Massachusetts Institute of Tanaka, Yuuji Participant Technology Taubes, Clifford Organizer Harvard University Thakkar, Pragnesh Laxmiram Participant Gandhinagar Institute Of Technology Tsai, Chung-Jun Participant Harvard University van erp, johannes Participant univeristy of pennsylvania Walker, Kevin M Participant Microsoft Wendl, Chris M Participant ETH Zurich Wysocki, Krzysztof Participant Penn State University Xu, Feng Participant Duke University Yang, Dingyu Participant NYU Zehmisch, Kai Participant Universität Leipzig

Table of Contents 1. Introduction ...... 1 2. Funding Information ...... 2 3. Recruitment, Application and Admissions Procedures ...... 2 4. Summary of Participant Demographics ...... 3 5. Housing and Lodging for the Students ...... 3 Table 1. Student Data ………………………………………………………………… 4 6. Pre-Research Seminar ...... 4 7. Research Projects and Student Presentations ...... 5 8. Evaluation of Student Work ...... 6 9. Colloquia Series ...... 7 10. Graduate School and Funding Workshops...... 7 11. Additional Workshops ...... 8 12. Recreational/Cultural Activities ...... 8 13. Program Evaluation During MSRI-UP ...... 8 14. End-of-Program Evaluation ...... 9 15. Long Term Summative Evaluation ……………………………………………………. 9 16. Conclusion ...... 10

Appendix A: Sample Student Technical Report Appendix B: Results of the Student Pre-Evaluation Appendix C: Results of the End-of-Program Student Evaluation

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2008 Mathematical Sciences Research Institute – Undergraduate Program (MSRI-UP) Final Report

1. Introduction The second Mathematical Sciences Research Institute – Undergraduate Program (MSRI-UP) took place from June 14 to July 26, 2008 at the Mathematical Sciences Research Institute, Berkeley, CA under the leadership of the On-Site Director Ivelisse Rubio from the University of Puerto Rico – Río Piedras and Co-Directors Duane Cooper, from Morehouse College, Ricardo Cortez from Tulane University, Herbert Medina from Loyola Marymount University, and Suzanne Weekes. Although the main part of MSRI-UP happens during summer, as is explained below, significant parts of the program happens after the summer. The 2008 MSRI-UP was funded by grants from the National Security Agency (NSA) and the National Science Foundation (NSF), the Mathematical Sciences Research Institute (MSRI) and a donation from the Gauss Research Foundation in Puerto Rico. The program was designed for undergraduates who are majoring in mathematics or related science. Seventeen students from universities in Texas, California, Louisiana, Massachusetts, New York, Puerto Rico, Missouri and Georgia participated in the six- week research program in mathematics. Eight of the students were women. Participants in MSRI-UP received round-trip travel to Berkeley, CA, room and board for the duration of the program, and a $3,000 stipend. Five MSRI-UP faculty and staff from the University of Puerto Rico – Río Piedras, Tulane University, the University of Iowa, and the Rutgers University were responsible for assisting the students in creating an academic and research environment that would help to achieve the main objective of the program:

2 To identify talented students, especially those from underrepresented groups, who are interested in mathematics and make available to them meaningful research opportunities, the necessary skills and knowledge to participate in successful collaborations, and academic peers and mentors who can advise, encourage and support them through a successful graduate program.

The objective is designed to contribute significantly toward meeting the program goal of increasing the number of graduate degrees in the mathematical sciences, especially doctorates, earned by U.S. citizens and permanent residents by cultivating heretofore untapped mathematical talent within the U.S. Black, Hispanic/Latino and Native American communities.

2. Funding Information The funding for MSRI-UP can be summarized as follows: 1. National Security Agency $155,0001 2. The National Science Foundation $76,4002 3. Gauss Research Foundation $2,000 4. Mathematical Sciences Research Institute

In addition, the MSRI provided much additional support by allowing MSRI-UP to use classrooms, offices, computers, by facilitating transportation, and providing administrative assistance.

3. Recruitment, Application and Admissions Procedures

1 Grant number H98230-08-1-0063. 2 Grant number 0754872.

3 The Co-Directors began recruiting for the 2008 MSRI-UP at the Annual SACNAS Conference in Kansas City, MO in October, 2007. The Co-Directors passed out applications, brochures and posters and talked to dozens of students and faculty about the program. The MSRI-UP home page also provided information about and applications for the program. The Co-Directors e-mailed flyers to hundreds of mathematicians, to SACNAS members who belong to the mathematics community and to professors who had sent letters of recommendations for students to previous summer programs. The on-line application had to be completed by February 27, 2008 and consisted of four items: a completed student application form, transcripts, a statement of interest, and a letter of recommendation.3 The 2008 MSRI-UP received seventy five applications. The Co-Directors Cooper, Cortez, Medina, Rubio and Weekes reviewed each application and evaluated it using four criteria: 1.) the student’s grades in mathematics courses; 2.) the student’s mathematical background; 3.) the statement of interest; and 4.) the letter of recommendation. Based on these four criterions, each of the Co-Directors gave an applicant a score between 1 and 9. The scores were summed and averaged and this score served as the initial measure for evaluating each applicant. The Co-Directors proceeded to discuss individual applications and eventually reached a consensus on fourteen of the admits, the other three students were brought and financed from grants of the research advisor, Prof. Víctor Moll.

4. Summary of Participant Demographics Table 1 details the demographics of the seventeen MSRI-UP students. The diversity of the group across type of university and geography was achieved accidentally. The Co-Directors paid some special attention to gender during the selection process. Achieving this type of diversity and gender balance is important to creating the academic and research environment explained below and to achieving one of the MSRI-UP objectives.

5. Housing and Lodging for the Students

3 Please see http://www.msri.org/up/intropage for the MSRI-UP application.

4 Sixteen of the seventeen students were housed Stern Hall Dormitories at the University of California, Berkeley; the other lived in a private residency. On weekdays, the students’ lunch was served at MSRI. The students had dinner at the dinning facilities of Stern Hall Dormitories. All MSRI-UP students ate their meals together. On weekends, breakfast and dinner was served at the dinning facilities of Stern Hall Dormitories. On Saturdays the students the program had organized outings and the students had lunch there. On Sundays the students had lunch on their own. Sharing meals with their MSRI-UP peers promoted mathematical discussions and enhanced the collaborative and intellectual environment of MSRI-UP.

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Table 1 2008 Mathematical Sciences Research Institute (MSRI-UP) Student Data University Gender University of Puerto Rico – Río Piedras 3 Male 9 Morehouse College, GA 2 Female 8 University of California – Berkeley 2 Tulane University, LA 2 University of Puerto Rico – Mayagüez 1 Major University of Texas, Austin 1 Mathematics 13 University of Massachusetts, Boston 1 Applied Mathematics 1 Columbia University, NY 1 Systems Engineering 1 California State University, Pomona 1 Computer Science 1 State University of New York, Bufalo 1 Physics 1 Washington University, MO 1 California Institute of Technology 1 Graduation Date Fall 2008 2 State in which Students Study 2009 6 California 4 2010 7 Puerto Rico 4 2011 1 New York 2 Did not specify 1 Louisiana 2 Georgia 2 Texas 1 Massachusetts 1 Missouri 1

Ethnicity Latino 8 African American 3 Native American 1 Pacific Islander 1 Asian 1

6 White/ Caucasian 3

6. Pre-Research Seminar The seventeen students, led by Professor Victor Moll, from Tulane University, participated in the conducted research in the area of Experimental Mathematics. Professor Moll was assisted by his former graduate student Luis Medina who is now a Posdoc at Rutgers University and two graduate students: Erin Beyerstedt from Tulane University and Candice Price from The university of Iowa. During the first eight days of MSRI-UP, students participated in a pre-research seminar and a computational laboratory linked to their seminar. Professor Moll planned the seminar and laboratory so that he would familiarize students with the fundamental concepts and theories and the main tools of their research topic. He not only introduced students to the mathematical field, but also provided them with sufficient background and introduced them to their undergraduate-research project so that students could complete their project during the rest of the program. The computational laboratory was dedicated to activities that supplemented the mathematics from the pre-research seminar and prepared students to tackle the research projects. The laboratories gave students the opportunity to use the computer algebra system Mathematica to solve problems, develop and test conjectures, and were exposed to the tools that allowed them to complete their research project. The pre-research phase was conducted in Baker Board Room. Twenty desktop computers were placed in the room. The students continued working on their dorms during the night and the staff was available for consultation until almost midnight.

7. Research Projects and Student Presentations

7 The focus of MSRI-UP is undergraduate research. After the first eight days of the program all students worked exclusively on an undergraduate research project carefully designed by their seminar leader. There were two groups of four students and three groups of three students. Students wrote technical reports and presented the results of their research at the MSRI-UP Student Colloquium the day of the program. By the beginning of the second week of the program, students had a description of their possible research projects. The students did preliminary reading and literature searches on their projects and soon thereafter made a presentation to the other students and staff in which they gave an overview of their research project and the methods and techniques that they hoped to use to tackle it. During the last four weeks of MSRI-UP the students worked in the offices assigned to the at MSRI and used the computers available to them to run Mathematica programs related to their projects. Each one of the graduate students assisted two of the research groups, the posdoc assisted the other group and helped the graduate students with their four groups; Professor Moll supervised also supervised all the groups. The students met with the staff to discuss their projects; they worked on preparing their technical reports and oral presentation; and worked tirelessly to complete their projects. The staff also worked tirelessly to help students with their work. Indeed, all the MSRI-UP staff was busy consulting with students during day and also at night. During the program, MSRI-UP participants were introduced to some of the techniques that are used while conducting successful research in the mathematical sciences. Indeed, many MSRI-UP students learned to work as part of a research team, to develop an effective faculty advisor-student relationship, to use the computer as a tool, to use the Internet as a resource, to prepare and give oral presentations, and to write technical reports using LaTeX. The outcome of all the students’ and staff’s work and dedication resulted in five technical reports and an equal number of oral presentations in the Student Colloquium Series.4 A collection of all the technical reports is available on a CD and all technical reports can be downloaded from the MSRI-UP web site. Fifteen of the seventeen MSRI-UP students will present their research at the 2008 SACNAS Conference Salt Lake City, Utah in October, 2008. We expect that many of the students will present their research at the

8 Joint Mathematics Meetings in Washington, DC in January of 2009 or the NCUR Conference in La Crosse, Wisconsin in April of 2009. The program’s objectives could be achieved only within a rich and intense academic and research environment. The research environment in MSRI-UP created the richness and intensity and thus drove the program. The program was a success largely because of the outstanding staff that worked so hard to make it a success.

8. Evaluation of Student Work Close interaction with students allowed the staff to give individuals feedback on their work throughout the program. Indeed, seminar leaders and associates gave students written feedback on drafts of their technical reports, and transparencies for their oral presentation. Professor Moll also wrote an end-of-program evaluation for each student in their seminar. Each evaluation contains a section with valuable feedback and advice that can help the student to improve her/his work and this part of the evaluation was sent to the student a few weeks after the program ended. The evaluation also contains a confidential part (similar to a confidential letter of recommendation) that will be kept by MSRI-UP for purposes of long-term evaluation of program effectiveness.

9. Colloquia Series The 2008 MSRI-UP hosted six mathematicians for a colloquia series5: Luis Melara, University of Colorado; Rosa Orellana, Dartmouth College; Javier Arsuaga, San Francisco State University; Matthias Beck, San Francisco State University; Robin Wilson, California State Polytechnic University; Pamela Williams, Sandia National Laboratories. The colloquia series stimulated the mathematical interests of the students and gave them a glimpse of current mathematical research. In addition to this, they provided the

4 Please see Appendix ### for a complete collection of the abstracts for the students’ research projects and one of the technical reports. The technical reports can be accessed in http://www.msri.org/up/resproj/2008. 5 Please see http://www.msri.org/up/colloquia/2008 for more details on the colloquia.

9 students with additional role models and expanded their network of mentors. Some of the colloquium speakers also disseminated information about the graduate programs in the mathematical sciences at their institution.

MSRI-UP students also had the opportunity to attend several of the other seminars that were presented at MSRI during the summer.

10. Graduate School Workshops and Individual Academic Advising of MSRI-UP Students Dr. Colette Patt, Director of Diversity Programs in the Physical Sciences at the University of California, Berkeley visited MSRI-UP and gave a workshop on applying for graduate school and another on finding funding for graduate school. The workshops addressed questions/issues such as the significant differences between a master’s and a doctoral program, the funding opportunities available for most graduate programs, and the benefits of obtaining a graduate degree. In addition to this basic information, Dr. Patt also presented successful techniques for applying to graduate school. She discussed the elements that constitute a good statement of purpose, the types of professors from whom one should seek letters of recommendation, and successful techniques for addressing not-so-stellar semesters. Dr. Patt also discussed the successful strategies for compiling a winning national fellowship application. She also provided the students with written related material.

11. Additional Workshops The other three workshops were devoted to the development of skills that are important to every mathematician. The first was devoted to learning LaTeX, the typesetting program most widely used by mathematicians. The second and third workshop familiarized students with good practices in preparing and delivering a mathematics oral and poster presentation. The workshops were designed and run by Professor Ivelisse Rubio. All three of these workshops were necessary as MSRI-UP students prepared their

10 technical report and transparencies using LaTeX, gave an end-of-program oral presentation, and will present posters at the conferences mentioned above.

12. Recreational/Cultural Activities In addition to all the academic activities described above, MSRI-UP students were treated to several recreational activities. These included a picnic at Tilden Park, a tour of San Francisco Bay, a trip to Monterrey and a visit to Monterey Aquarium, attending a San Francisco Giants baseball game, a trip to Muir Woods and a kayaking trip. These carefully-planned recreational and cultural activities were essential to MSRI-UP’s success as they gave students the opportunity to put mathematics aside for a few hours so that they could come back later to their work with renewed vigor; they also helped to build the MSRI-UP mentored community as all staff participated in the activities with the students.

13. Program Evaluation During MSRI-UP Formal evaluation started the first day of the program when the students completed an online pre-evaluation prepared by Professor Rubio using the Student Assessment of Learning Gains (http://www.salgsite.org/instructor/signin) web site. The results of this pre- evaluation are included in Appendix B. Also, informal formative program evaluation took place from the beginning of MSRI-UP through conversations with students and staff. Formal formative evaluation continued the first Friday of the program when Professor Rubio met with the students in order to solicit feedback on the program. The aim of the meeting was to listen to student concerns, complaints and methods for improving the program. The students’ feedback was communicated to the staff so that adjustments could be made in order to improve the program. A similar evaluation meeting was held the second Friday of the program. During the fourth week of the program Professor Rubio met individually with each one of the students and staff of the program. During these meetings the Director had the opportunity to have more close contact with the students and staff, to listen to individual concerns and provide

11 individual mentoring to thye students. In addition to these forums, the staff’s close interaction with the students enabled them to gather informal feedback that also led to adjustments to improve the program. To obtain additional input from the staff, the Directors called several staff meetings in which they discussed program improvement.

14. End-of-Program Evaluation Each MSRI-UP student was required to complete a comprehensive, end-of-program, written evaluation. Appendix C contains a sample of the evaluation form. The evaluation form had both year-to-year formative evaluation questions designed for soliciting feedback in order to improve future institutes and summative-evaluation questions to measure the effectiveness of MSRI-UP in accomplishing the Comment [D1]: En este parrafo hay program objectives. The results of the end of program evaluation are provided in Appendix C. que insertar las estadisticas. Post-program conversations between the MSRI-UP staff and the Directors indicated that the staff felt that the institute was successful in accomplishing its objectives.

15. Long-Term Summative Evaluation Measuring the program’s effectiveness in achieving the MSRI-UP goal will take several years. In order to do so, the Directors will follow the educational progress of each MSRI-UP student for several years. To facilitate this process, each year MSRI-UP students will provide the Directors contact information that is valid for at least one year. The On-Site Director will contact each student who has participated in MSRI-UP during the year that he/she was the On-Site Director. The Director will ask him/her to fill out a questionnaire to ascertain their educational progress and also to obtain new contact information.

12 16. Conclusion Students, staff and visitors have communicated to the Director that the 2008 MSRI-UP was very successful in achieving its objectives. The Director also saw the mathematical maturing, blossoming and development of many MSRI-UP students during the six weeks of the program---transformations that undoubtedly will contribute towards the academic careers of these students. By the end of the program many students had produced technical reports and given oral presentations that were at a very high undergraduate level and many were clearly on their way to a first-rate graduate program in mathematics. The data that will verify that the MSRI-UP objectives contribute towards the goal of increasing the number of Latinos/Chicanos, African American and Native Americans earning graduate degrees in the mathematical sciences will not be available for several years. The Directors are committed to continuing to do this in the years to come.

13 Numerical Integration Techniques with Rational Landen Transformations

Natasha Cayco Gaji´c Nathan Kallus California Institute of Technology University of California at Berkeley Pasadena, CA Berkeley, CA Jess Stigile Washington University in St. Louis St. Louis, MO

July 25, 2008

Abstract

Landen transformations are maps on an integrand that preserve the value of the integral. These originally appeared in the context of elliptic integrals. We present an implementation and numerical improvement of an integration technique based on Landen transformations of rational integrands developed by Boros, Manna, and Moll. The method presented deals with the problem of coefficient growth while preserving the fast order of convergence of the original procedure. We discuss the success of the method. Additionally, we investi- gate the possibility of extending the technique to non-rational functions using rational-function approximations.

1 Background

The indefinite integrals of rational functions were studied by Bernoulli, Hermite, Risch, and others beginning in the 18th century. More recently, a technique of definite integration of rational functions over the real line was developed by Boros, Manna, and Moll [?]. Their technique is based on a class of transformations studied by 18th- century surveyor and amateur mathematician John Landen. A Landen transformation is a transformation on the parameters of an integrand which preserves the value of the integral. Gauss studied one such transformation. He observed that the elliptic integral

π 2 √ dθ G(a, b) = 2 2 2 2 Z0 a cos θ + b sin θ

1 is invariant under the transformation a + b √ E :(a, b) 7→ , ab .  2 

This is known as the elliptic Landen transformation. Because the value of G(a, b) is invariant under E, taking the limit as E is iterated yields a result that is useful in calculating π to high precision [?]: π = G(a, b) 2AGM (a, b) where AGM (a, b) is the arithmetic-geometric mean of a and b, that is, the common limit as n → ∞ of an + bn an+1 = and bn+1 = anbn. 2 p The study of Landen transformations has led to the discovery of a Landen trans- formation L for rational integrands [?]. For an example of a rational Landen trans- cx4+dx2+e ∞ formation, let R(x) = x6+ax4+bx2+1 and consider the integral I = −∞ R(x)dx. Us- ing trigonometric identities, one can find rational functions Rm suchR that cot mθ = 2 (cot θ) −1 x2−1 2 2 Rm(cot θ). For m = 2, cot 2θ = 2 cot θ so that R (x) = 2x . Letting y = R (x) 2 and solving for x in terms of y yields x±(y) = y ± y + 1. The original integral can be written as 0 ∞ p I = R(x)dx + R(x′)dx′. Z−∞ Z0 ′ Under the change of variables x 7→ x+(y), x 7→ x−(y),

∞ I = (R+(y) + R−(y))dy Z−∞ where

2 2 R+(y) = R(y + y + 1) + R(y − y + 1) y p 2 p 2 − − − R (y) = 2 R(y + y + 1) R(y y + 1) . y + 1  p p  p Simplification leads to the equations

∞ cy˜ 4 + dy˜ 2 +e ˜ I = 6 4 2 dy Z−∞ y +ay ˜ + ˜by + 1 where ab + 5a + 5b + 9 a + b + 6 a˜ = , ˜b = , (a + b + 2)4/3 (a + b + 2)2/3 c + d + e (b + 3)c + 2d + (a + 3)e c + e c˜ = a + b + 2, d˜= , e˜ = . (a + b + 2)2/3 a + b + 2 (a + b + 2)1/3

2 The transformation L :(a, b, c, d, e) 7→ (˜a,˜b, c,˜ d,˜ e˜) is a rational Landen transforma- tion, as it preserves the value of I. To see this, suppose a = 2, b = 1, c = 4, d = 2, e = 26 ˜ 9 × 1/3 ˜ 4. Then the given transformation results ina ˜ = 5×51/3 , b = 52/3 , c˜ = 2 5 , d = 8 8, e˜ = 51/3 and a direct calculation shows ∞ 4 2 ∞ 4 ˜ 2 cx + dx + e cy˜ + dy +e ˜ ≈ I = 6 4 2 dx = 6 4 2 dy 12.5881 Z−∞ x + ax + bx + 1 Z−∞ y +ay ˜ + ˜by + 1 We investigate the use of a class of rational Landen transformations in a technique of numerical integration of rational integrals over the real line, as well as the imple- mentation of this technique on a computer and a possible extension of the technique to non-rational integrals.

2 The Rational Landen Transformation

We are interested in the numerical integration of rational functions R(x) over the real line. Let Rp be the set of rational functions of degree at most (p − 2, p). For any m ∈ N p ∈ 2N, there exists a Landen transformation Lm,p : Rp → Rp and a rational function φ in the coefficients of rational functions such that for any R ∈ Rp, ∞ ◦ Ln 1 lim φ m,p (R) = R n→∞ π Z−∞  with order of convergence m [?]. By that, we mean that for any n

∞ ∞ m ◦ Ln − 1 ◦ Ln−1 − 1 φ m,p (R) R < C φ m,p (R) R π Z−∞ π Z−∞   with some constant C. Furthermore, Lm,p can be expressed as a tuple of rational expressions in the coefficients, one for each transformed coefficient; and φ is the ratio of the constant terms of the numerator and denominator. As an example, let m = 10 and p = 4. Under L10,4, the transformed leading coefficient of the denominator is a homogeneous polynomial in the coefficients with a total of 110 distinct terms:

8 7 2 6 a4 7→ 33000a4b0a0 + 580800a2a4b0a0 − 184800a1a4b0a0 + ···

x2+x+1 ∞ 6π Take R(x) = x4−2x3+3x2−2x+2 . By the residue theorem, −∞ R = 5 . Upon applying L10,4, R 2930046x2 − 1199x + 2929328 L10 4(R) = , 2441288x4 + 3116x3 + 4882813x2 + 3116x + 2441525 1.00x2 + 4.09 × 10−4x + 1 ≈ 1.1998 × 1.00x4 + 1.28 × 10−3x3 + 2.00x2 + 1.28 × 10−3x + 1 2 −35 2 x − 1.58 × 10 x + 1 L10 4(R) ≈ 1.2 × , x4 + 3.10 × 10−35x3 + 2x2 + 3.10 × 10−35x + 1

3 As L10,4 is iterated, the transformed coefficients approach the coefficients in the bi- nomial expansions of 1 + x2 in the numerator and (1 + x2)2 in the denominator. The 6 → ∞ factor in front of the rational expression is φn, which approaches 5 = 1.2 as n . Below are the values of φ to 50 digits through three iterations:

φ0 = 1/2 φ1 = 1.1997943908008314475583907598734397558902734970971 φ2 = 1.2000000000000000000000000000000000230367994760222 φ3 = 1.2000000000000000000000000000000000000000000000000

2 n 6 x +1 6 1 L → × × L10 4 Continuing this process we see that 10,4(R) 5 x4+2x2+1 = 5 x2+1 . Since , is integral-perserving, ∞ ∞ 6 × dx 6π R = 2 = . Z−∞ 5 Z−∞ x + 1 5

Thus by applying Lm,p repeatedly, it is possible to numerically compute a rational function’s integral over the real line without ever evaluating the integrand or its antiderivative. Unfortunately, this transformation is that it is limited to rational functions over the real line. In attempting to extend the technique to non-rational functions, we consider Pad´eapproximants.

3 Pad´eApproximants

The Pad´eapproximant of order (L, M) of any function f is the rational function with numerator of degree L and denominator of degree M which is the unique solution to the following: ∞ ··· L i a0 + a1z + + aLz L+M+1 ciz = + O x 1 + b1z + ··· + b zM Xi=0 M  ∞ i where i=0 ciz is the power series expansion of f around zero and ai, bi can be found by solvingP a system of L + M + 1 equations. In this paper the Pad´eapproximant of degree (L, M) will be denoted by [L/M]. The power series of [L/M] agrees with the power series of f up to the (M + L + 1)th term. For example, consider the function ex. Then, 4 1 2 1 3 1 4 1 + 9 x + 12 x + 126 x + 3024 x [4/5] = − 5 5 2 − 5 3 5 4 − 1 5 . 1 9 x + 36 x 252 x + 3024 x 15120 x Moreover, note that the series expansion centered at x = 0 of ex is given by 1 1 1 1 1, ex =1 + x + x2 + x3 + x4 + x5 + x6 2 6 24 120 720 1 1 1 1 + x7 + x8 + x9 + x10 + O(x11) 5040 40320 362880 3628800

4 y 1.0015

1.0010

1.0005

1.0000

0.9995

x -0.01 0.00 0.01 0.02 0.03

Figure 1: The Pad´eapproximant [1/1] of f(z) = 1 + ǫz + z2 has a defect near z = ǫ. Plotted are y = f(x) (solid) and y = [1/1](x) (dashed), with ǫ = 0.01. and the series expansion centered at x = 0 of [4/5] is given by

1 2 1 3 1 4 1 5 1, 6 [4/5] =1 + x + x + x + x + x + x 2 6 24 120 720 1 1 1 127 + x7 + x8 + x9 + x10 + O(x11). 5040 40320 362880 457228800 Hence, the power series expansions agree up to the tenth term (10 = M + L + 1). We want to use Pad´eapproximants to extend the Landen integration algorithm to non-rational functions. Ideally, the Pad´eapproximant of a non-rational function will be an accurate rational approximation to which the Landen technique can be applied: ∞ ∞ × ◦ Ln − − ≈ π lim φ m,p ([p 2/p]) = [p 2/p] f. n→∞ Z−∞ Z−∞  However, some Pad´eapproximants contain defects: extraneous poles and nearby zeros that do not correspond to poles or zeros in the original function. Defects behave erratically in that they may appear and disappear unpredictably as the degree of approximation is varied. Consider, as an example, the function f(z) = 1 + ǫz + z2 with ǫ ≪ 1. Then, 1 + (ǫ + ǫ−1)z [1/1] = 1 − z/ǫ is the Pad´eapproximant of f of order (1, 1) centered around zero. Moreover, [1/1] has a defect around z = ǫ; this is very close to the center of the approximation as ǫ approaches zero(Fig. 1). It is difficult to produce a Pad´eapproximant that is guaranteed to be free of artificial poles. The presence of a defect hinders the accurate ∞ evaluation of −∞[L/M]dx. Later in this section, we try to find a class of functions R 5 cosh t Table 1: For g(t) = 1+(sinh t)2n , the percent error of the Pad´eapproximant [2n + 18/2n + 20] is small. n % Error n % Error 11 9.13944×10−9 2 2.93663×10−6 12 1.06776×10−8 3 1.91479 ×10−7 13 3.25114×10−9 4 3.54743×10−7 14 3.50755×10−9 5 8.15335×10−8 15 1.17916×10−9 6 2.3177×10−7 16 1.21993 ×10−9 7 5.20637×10−8 17 4.46229 ×10−10 8 1.01022×10−7 18 4.51199 ×10−10 9 2.44876×10−8 19 1.77544 ×10−10 10 3.36035×10−8 20 1.77875 ×10−10 whose Pad´eapproximants do not have defects so that we may apply the Landen technique to non-rational functions as well. We still have not been able to produce a class of functions whose Pad´eapproximants do not contain defects. For a rational function R of degree (l, m), if L ≥ l and M ≥ m, then the Pad´e approximant [L/M] = R. In other words, the Pad´eapproximant of sufficient order of a rational function has no defects. We define a rational function in disguise (RFID) to be a function of the form g (x) = R (µ (x)) µ′ (x) with R rational, g non-rational, and µ an increasing function such that µ (−∞) = −∞ and µ (∞) = ∞. The following relationship is satisfied:

∞ ∞ R = g. Z−∞ Z−∞ We originally thought that much like the Pad´eapproximants of rational functions, the Pad´eapproximants of RFIDs would not have defects. To investigate, we first 1 ∈ Z+ chose R(x) = 1+x2n for n and made the substitution µ(t) = sinh t. This yields the RFID cosh t g(t) = . 1 + (sinh t)2n We numerically evaluated the integral of the Pad´eapproximant of g over the real ∞ line. This approximates −∞ g(t)dt. In our numerical experiments we noticed that the percent error does notR increase as n increases if we choose the order (m, m + 2) of the Pad´eapproximant to be linear in n. In particular, the choice of m = 2n + 18 generated small error (Table 1). Since the integrals converge, we assume that the Pad´eapproximant has no defects. This is also a good approximation because the percent error is small. We followed a similar process to investigate other RFIDs (Tables 2, 3). From this data, we conclude that the Pad´eapproximant of a RFID need not be free of defects. For instance, consider the RFID that results from the substitution µ(t) = t2 sinh t

6 1 Table 2: Percent error for the Pad´eapproximant R(x) = 1+x2n Substitution Order of Pad´e Maximum Percent Error (2 ≤ n ≤ 20) sinh t [2n + 18/2n + 20] 2.93663 × 10−6 t2 sinh t [8n + 20/8n + 22] 1.44445630 × 10−9 t cosh t [4n + 20/4n + 22] 3.271 × 10−8 t3 cosh t [8n + 20/8n + 22] 4.915 × 10−14 t cosh t + sinh t [4n + 20/4n + 22] 1.965 × 10−8

1 Table 3: Percent error for the Pad´eapproximant of R(x) = 1+x2n+x4n Substitution Order of Pad´e Maximum Percent Error (2 ≤ n ≤ 10) sinh t [4n + 20/4n + 22] 2.588 × 10−6 t2 sinh t [8n + 20/8n + 22] Defects for 7 ≤ n ≤ 10 t cosh t [8n + 20/8n + 22] 4.00009 × 10−11 t3 cosh t [8n + 20/8n + 22] Defects for 9 ≤ n ≤ 10 t cosh t + sinh t [8n + 20/8n + 22] Defects for n = 3

1 into the rational function R(x) = 1+x14+x28 : 2t sinh t + t2 cosh t g(t) = . 1 + (t2 sinh t)14 + (t2 sinh t)28 We computed the [76/78] Pad´eapproximant of g and subsequently tried to integrate it over the real line using Mathematica. We were unsuccessful as the value of the integral could not be computed. This led us to believe that there was a defect. By plotting [76/78] (Fig. 2), we see that there is a pole. We know that there are no poles in our RFID, so we conclude that the Pad´eapproximant has defects. Our investigations of specific examples indicate that there may exist a subclass of RFIDs whose Pad´eapproximants have no defects. We were then concerned with the intervals where the Pad´eapproximant of a function f is “good,” that is, where the Pad´eapproximant has no defects and accurately approximates f. We noticed that

1. ´ 1016

5. ´ 1015

-1.84 -1.83 -1.82 -1.81 -1.80

-5. ´ 1015

-1. ´ 1016

2t sinh t+t2 cosh t Figure 2: A defect in the Pad´eapproximant [76/78] of 1+(t2 sinh t)14+(t2 sinh t)28 .

7 cos x − for some functions, such as sinh x and 1+x2 , the Pad´eapproximant of order (m 2, m) 2 is good on the interval (−m, m) (Fig. 3). For other functions, including e−x , the Pad´eapproximant had no defects on (−m, m) but was a poor approximation (Fig. 4). We conjecture that our (−m, m) rule of thumb works for a specific class of functions which we hope to identify more completely in future research and we would also like to find rules for other classes of functions.

100 2. ´ 107

7 50 1. ´ 10

-20 -10 10 20 -6 -4 -2 2 4 6

- ´ 7 1. 10 -50

-2. ´ 107 (a) (b) -100

7 4000 2. ´ 10

2000 1. ´ 107

-10 -5 5 10 -20 -10 10 20

- 2000 -1. ´ 107

-4000 -2. ´ 107 (c) (d) Figure 3: An example with f(x) = sinh x (thick line), plotted in (a). The Pad´e approximant [m − 2/m] (thin line) has no defects and closely approximates f on (−m, m) for (b) m = 4, (c) m = 10, and (d) m = 20. Note that in (b) and (d), artificial poles lie just outside this interval.

4 Generating Rational Landen Transforms

We are interested in realizing the technique described in [?] as a computationally feasible method of numerical integration. Certain obstacles need to be surmounted in order to accomplish this. First, we must pre-generate the rational expressions of the Landen transformation for various (p, m) prior to runtime. Generating this transfor- mation is computationally costly and a tradeoff between space and time is necessary to make this application practical. Additionally, we need to compute the iterations of Lm,p in a reasonable amount of time. As Lm,p is iterated, the coefficients of the intermediate rational functions converge to the binomial coefficients in the expansion

8 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

(a) -4 -2 2 4 (b)-6 -4 -2 2 4 6 0.04

0.06

0.03 0.04

0.02 0.02

-10 -5 5 10 0.01

-0.02 (c) (d) -20 -10 10 20 Figure 4: An example where the (−m, m) rule fails for f(x) = e( − x2) (thick line), plotted in (a). The Pad´eapproximant [m − 2/m] (thin line) has no poles on (−m, m) but is a bad approximation for (b) m = 4, (c) m = 10, and (d) m = 20.

2 p−2 2 p of (1+x ) 2 for the numerator and (1+x ) 2 for the denominator. As the coefficients approach integers, an increasing number of digits are needed to encode the coefficients as exact rational numbers; and as the number of digits increases, the evaluation of rational expressions in the coefficients becomes more time-intensive. With increasing m, the rational expressions describing Lm,p also increase in complexity. We began by devising a Mathematica procedure for generating the rational ex- pressions of Lm,p. Although this procedure is not a part of runtime, we needed it to run in a reasonable timescale. We tabulate below the time in seconds needed to generate Lm,p for each (p, m) and are interested in their relative growth (Table 4). The growth of computational time is nearly exponential in either p or m. This is unfortunate but not detrimental as this is not part of runtime. We then consider the growth of the size of the rational expression of Lm,p. Size is measured as the total number of operands in the fully simplified expression. This has ramifications about the time needed to iterate the expression in runtime. We tabulate the size for each (p, m) (Table 5). Again, we see nearly exponential growth in either p or m. Finally, we study the growth of the size of the coefficients of the iterations of Lm,p. Here we measure size as the maximum number of digits in the numerators and

9 Table 4: Time in seconds to generate Lm,p m = 2 3 4 5 6 p = 2 22 48 72 140 186 4 81 299 739 1704 3233 6 218 1180 4056 11475 27462 8 464 3346 14966 51877 153110 10 857 7732 42535 175755 614909

Table 5: Size of the rational expression of Lm,p m = 2 3 4 5 6 p = 2 22 48 72 140 186 4 81 299 739 1704 3233 6 218 1180 4056 11475 27462 8 464 3346 14966 51877 153110 10 857 7732 42535 175755 614909 denominators of each coefficient in its rational form. Suppose

7 + 2x2 + 4x3 + x7 R(x) = . 15707 + 18861x2 + 9432x4 + 2513x6 + 376x8 + 30x10 + x12

As L3,12 is iterated on R, the size of the coefficients increases exponentially (Fig. 5). This rapid growth is a universal trend for all m and all rational functions R. Coupled with the rapid growth of the rational expression of Lm,p, this presents a bottleneck for repeated iteration. That is, as the coefficients under Lm,p grow in size, the trans- formation becomes much more computationally intensive when iterated. In fact, the → 1 coefficients (once normalized) are close to binomial coefficients so that R(x) L 1+x2 . A reduction in the complexity, albeit at the cost of error, is needed. A desirable so- lution should minimize error as well as computational time. We investigated the use of continued fractions to re-encode the coefficients with fewer digits.

5 Continued Fractions as Best Approximants

Any real x can be expressed as a continued fraction: 1 x = a0 + 1 , a0 ∈ Z, aj ∈ N j ≥ 1 1 a + a2+··· which we denote as x = [a0; a1, a2, ...]. Rational numbers correspond to finite con- tinued fractions and, furthermore, each rational number has two representations by continued fractions: p = [a0; a1, a2, ..., a ] = [a0; a1, a2, ..., a − 1, a ]. q n n n

10 106

105

104

1000

100

2 4 6 8 10

Figure 5: The size of the coefficients under Lm,p grows exponentially with the number of iterations. Here we plot the size on a logarithmic scale.

Using this notation, the kth-order convergent of x is the rational form of the first k + 1 terms of continued fraction representing x, that is,

pk = [a0; a1, a2, ..., ak]. qk

a Furthermore, b is a best approximation of x if c a x − > x − d b

≤ c 6 a a whenever 0 < d b and d = b . In other words, b is the simplified fraction which has the lowest error for any fraction with denominator no greater than b. The following are two properties of continued fractions [?]:

• Every best approximation of a real number x is a convergent.

• The value x of the convergent infinite continued fraction (for k ≥ 0) satisfies the following inequality: p 1 x − k < q q 2 k k

To slow the growth of the coefficients of the Landen transformation, we choose to approximate each coefficient by truncating its continued fraction sequence. Hence, we need to develop algorithms that will determine the optimal term at which to truncate a continued fraction sequence. In the future, we will look to these two properties of continued fractions to bound the error for our algorithms.

6 Choosing a Convergent

We need an algorithm that specifies where to truncate the continued fraction sequence of the Landen coefficients. After each iteration of Landen, we will approximate each coefficient by truncating its continued fraction sequence. The desired algorithm is

11 one that both reduces the size of the coefficients and does not accumulate significant error as we iterate Landen. Our original thought was that if k is large then we could truncate the continued fraction at the kth term without introducing much error. In testing, however, we noticed that the error was not similar among different continued fraction√ sequences. For example, if k=10, the percentage error for the continued fraction of 2 is 4.42096× 10−6% while it is 5.12705×10−11% for π. This algorithm is unpredictable so we chose to explore other ideas. The following two algorithms are based on the idea that if one term in a continued fraction sequence is extremely large, then the continued fraction can be truncated before that term without introducing much error. The large term will appear in the continued fraction sequence in the form 1 , where 0 < c < 1. This value will term+c be very small and will not contribute much to the value of the continued fraction itself. Our second algorithm implements these two ideas. It looks at each term in the sequence and truncates immediately before the first term whose value is greater than 100. Sometimes there may not be a term larger than 100. However, there could be a somewhat large term later in the sequence at which we could truncate the continued fraction. The third algorithm implements the norm defined as (position) × (value) as a means to handle such a sequence. It starts at a0 and terminates before the first term whose (position) ×(value) is greater than a specified threshold (we chose 1000). This allows us to truncate early if there is a large term and also allows us to truncate later in the sequence if there is a term of relatively large magnitude. As was previously stated, we need an algorithm that is accurate and reduces the size of the coefficients. Below are graphs of the size of the coefficients as a function of the number of iterations of our Landen (m = 3) algorithm for the rational function

7 + 2x2 + 4x3 + x7 R(x) = . 15707 + 18861x2 + 9432x4 + 2513x6 + 376x8 + 30x10 + x12 We measured the size of fractions and of rational functions as the maximum of the integer lengths (Fig. 6). The size for the first algorithm grows almost exponentially. This is undesirable as it will increase the runtime of our program. The second algo- rithm has increasing size, but does not appear to have exponential growth. It seems to have the least correlation between iteration and size. The third algorithm also appears to be growing exponentially. We calculated the percentage error after one iteration of the Landen algorithm. For the first algorithm it was 2.75 × 10−7%, for the second it was 4.62 × 10−11 %, and for the third it was 4.43 × 10−11%. Evidently, the second and third algorithms have similar error which is much less than that of the first algorithm. We concluded that our second algorithm was superior. Not only is it the best compromise between size and accuracy, but it also seems to have the most control over the growth of the coefficients when iterating the Landen transformation. In future sections, we will refer to this algorithm as CFReducem, where m is the specified threshold.

12 106 1 ´ 104 5000 105

1000 104 500

1000

100 100 (a) 2 4 6 8 10 (b) 2 4 6 8 10 12 5 ´ 104

1 ´ 104 1 ´ 104 5000 5000

1000 1000 500 500

100 100

(c) 2 4 6 8 10 12 (d) 2 4 6 8 10 12 Figure 6: Plots of size of coefficients against the number of iterations of the Landen transformation on a logarithmic scale(a) without truncation, (b) with the first trun- cation algorithm, (c) with the second truncation algorithm, and (d) with the third truncation algorithm for R given above.

7 Even Landen Transformations

A known property of the Landen transformations considered here is that they map even functions to even functions. Knowing a priori that Re is even, one can simplify the expression for L(Re). The computation of this simplified expression Le is faster for a specific even Re than is the computation of L. We sought to take advantage of these facts to streamline our integration technique. Our first attempt uses the simplified Le. Given a rational function R, we decompose it into its even and odd parts and then eliminate the odd part:

∞ ∞ R(x) + R(−x) R(x) − R(−x) R(x)dx = + dx Z−∞ Z−∞  2 2  ∞ R(x) + R(−x) = dx. Z−∞ 2

B(x) However, this doubles the order of the function. For R(x) = A(x) , the even part is

1 B(x)A(−x) + B(−x)A(x) R (x) = (R(x) + R(−x)) = . e 2 2A(x)A(−x)

13 Note that the degree of the denominator is doubled and the degree of the numerator is at most doubled. It turns out that because the degree of the denominator is doubled, applying Le is slower than applying L. For several randomly generated rational functions with denominator of degree p, the time required to apply L(R), L(Re), and Le(Re) was averaged for each p (Fig. 7). Le(Re) is a significant improvement to L(Re)

Time to Run Iterations Time HsL

40

30

20

10

Order of Denominator 4 6 8 10 12 14

Figure 7: Averaged over 100 random integrable rational functions, Le(Re) (dotted) ran faster than L(Re) (dashed), but L(R) (solid) was overall the fastest. with respect to computational time. Unfortunately, the original transformation L(R) is still the fastest algorithm. Clearly, the fact that Le(Re) eliminates the odd part of R is being trumped by the fact that the denominator of Re has twice the degree than that of R. We still do not want to disregard the fact that the Landen transformation maps even functions to even functions. Our next approach truncates coefficients of odd powers once they are “small enough.” Since the coefficients will eventually converge to binomial numbers, the coefficients of odd powers will converge to zero. Instead of iterating until these coefficients become zero, we prematurely estimate them as zero. In practice, once the coefficients of odd powers are within a certain distance of zero, we approximate them as zero. In this way, we are approximating nearly even functions as even ones. We found that this transformation runs faster than the regular transformation L(R) and converges at the same rate.

14 4

p 6

8

10

-16

-18

% log10H errL -20

-22

2 3 4 5 m

Figure 8: The base-ten log of the average percent error when computing the integral of 50 random integrable rational functions using ML for each m, p. The error is mostly uncorrelated with p and decreases exponentially with m.

8 A Combined Approach

Under L, coefficients converge to integers-, specifically binomials. A rational that is nearly an integer n has a continued fraction sequence of either the form [n; M, ...] or [n − 1; 1, M, ...] with M very large. We can consider two cases for choosing a continued fraction convergent to approximate the rational: either we truncate the continued fraction before M in which case our approximation is simply a nearest integer approximation, or we truncate after M in which case our approximation is useless because the number remains complicated. Our investigation from the previous section led us to notice that at a certain point, approximating the coefficients by the integers to which they converge is helpful. So, when one component is nearly aligned with the terminal point, we align it exactly because there is nothing better we can do to simplify it. We couple the ideas from the last section with our continued fraction reduction method to create a new operator. We arrived at an operator ML, a composition on Lm,p, which we iterate until the integral approximations are unchanged within a prescribed precision.

ML = Kp,ǫ ◦ Lm,p

The operator Kp,ǫ operates on rational functions R of order (p − 2, p) to produce

15 4 5 Time to complete HsL 2 4 0 4 m 6 3

p 8 2 10

Figure 9: The time in seconds to terminate the ML iterations for different (p, m), averaged over 50 random integrable functions for each (p, m). Although time grows exponentially in both p and m, it is much faster than the original transformation.

2 2−1 s (1+x )p/ R. It compares R to R = (1+x2)p/2 . For each coefficient ci (i = 0, ..., 2p) of R, if | − s| s eci ci < ǫ, then ci = ci ; otherwise, ci = CFReducem(ci). To illustrate the operator, we return to our example with L10,4 and R(x) = x2+x+1 e e x4−2x3+3x2−2x+2 . We have that 2930046x2 − 1199x + 2929328 L10 4(R) = , 2441288x4 + 3116x3 + 4882813x2 + 3116x + 2441525 1465023x2 − 1199x 2929328 × 1464664 2929328 + 1 = 2441288x4 3116x3 4882813x2 3116x . 2441525 2441525 + 2441525 + 2441525 + 2441525 + 1

K −4 ◦ L Apply 4,10 10,4 to obtain

444812x2 − 1199x 2929328 444703 2929328 + 1 K4 10−4 ◦ L10 4 × ( , , )(R) = 4 1289x3 2 1289x . 2441525 x + 1009989 + 2x + 1009989 + 1 We observe that every coefficient of an even power in the denominator is close to its terminal binomial number and the other coefficients have been reduced to less com- plicated rational numbers. The change here is less dramatic than when we compare n 3 Ln K −4 ◦ L L 10,4 and ( 4,10 10,4) with n at least 3, in which case the coefficients of 10,4(R) have some 700 to 1200 digits each. In our analysis, we applied the technique to a large number of randomized rational functions integrable over the real line for different p and m. We found that our tech- nique delivers good approximations with an incredibly shorter running time. First, the base-ten log of the average percent error when using ML decreases exponentially with m (Fig. 8). We next consider the average time in seconds to terminate the ML

16 5

0

4 6 6 5

8 4

p 10 3 m

12 2

Figure 10: Base-ten log of average time in seconds to terminate ML (lower) and L (higher). Averaged over 50 random integrable functions for each (p, m) with p ≤ 10, m ≤ 4 and extrapolated over a larger domain in this depiction. iterations for different (p, m) (Fig. 9). The average time grows exponentially with increasing p and m. In terms of algorithmic complexity in p and m, our method is no better than simply iterating L. However, the time to complete the process is still under five seconds for p ≤ 10 and m ≤ 5, compared with the original technique which terminates in over twenty-four hours in the same environment.1 The rate of expo- nential growth is also smaller (Fig. 10). Extrapolations from averaged data imply that the average time to terminate L for p = 12, m = 6 is about 32 years while it is measured to be under 10 seconds for ML.

9 New Directions

We have made progress toward applying the Landen-transformation technique to non-rational functions and have successfully realized the technique as a feasible com- putational tool. We are interested in formally assessing the bounds on the error of ML = Kp,ǫ ◦ Lm,p. We need to see the extent to which our technique is accurate and be able to classify degenerate cases for which ML fails, if these exist. We also

1These times correspond to high-level prototype implementations in Mathematica and therefore are relevant only in comparison to one another and in consideration of overall growth.

17 hope that our work toward applying the technique to non-rational functions will cul- minate in describing a subclass of non-rational functions for which we can use Pad´e approximants to apply ML.

10 Acknowledgments

This work has been supported by the National Science Foundation under grant 0754872, the National Security Agency under grant H98230-08-1-0063, the Gauss Research Foundation in Puerto Rico, and the Mathematical Sciences Research Insti- tute. We would like to extend a special thank-you to Ivelisse Rubio Canabal, Victor Moll, and Erin Beyerstedt for their guidance and patience, as well as Luis Medina, Candice Price, and the other students at the MSRI Undergraduate Program for their feedback and support.

11 Appendix: Description of Supplemented Code

11.1 Transform Generator SylvesterMatrix[poly1,poly2,var] Input: poly1 and poly2, two polynomials var, the variable of the polynomials Computes the Sylvester Matrix of the polynomials poly1 and poly2. This function was written by Michael Trott. Example: poly1 = x2 + 4, poly2 = 3x3 + x + 7, var = x SylvesterMatrix[poly1,poly2,var] =

1 0 4 0 0 0 1 0 4 0 0 0 1 0 4   3 0 1 7 0   0 3 0 1 7  

Res[poly1,poly2,var] Input: poly1 and poly2, two polynomials var, the variable of the polynomials Computes the resultant (the determinant of the Sylvester matrix) of polynomials poly1 and poly2. We use this function because it is faster than the one built into the Mathematica . Example: poly1 = x2 + 4, poly2 = 3x3 + x + 7, var = x Res[poly1,poly2,var] = 533

18 GetLandenTransform[p,m] Input: p, the degree of the denominator of the rational integrand m, the order of convergence

Computes a function that represents the Landen transformation Lm,p. The returned function has two arguments: the first is a list of length p-1 which represents the coefficients of the numerator (beginning with the constant term) and the second is a list of length p+1 which represents the coefficients of the denominator (beginning with the constant term). The function returns a pair of similar lists corresponding to the transformed rational function. Example: g=GetLandenTransform[4,2]; { } { } 24+8x+24x2 g[ 1,0,2 , 1,0,2,2,1 ] = 12+16x+32x2+16x3+16x4

11.2 Various Utilities ListToPoly[l,x] Input: l, a list of coefficients x, variable of the desired polynomial Output: A polynomial in x with coefficients l Converts a list of coefficients into a polynomial. Example: ListToPoly[{2,4,6},x] = 2 + 4x + 6x2

ListToRat[l,x] Input: l, a pair of lists of coefficients x, variable of a desired rational function Output: A rational in x with coefficients l Converts a pair of lists of coefficients into a rational function. {{ } { }} 1+2x+3x2 Example: ListToRat[ 1,2,3 , 4,5,6 , x] = 4+5x+6x2

SizeOfExpression[e] Input: e, an expression Output: The size of the expression Evaluates the size of an expression as the total number of operands. b e+f Example: SizeOfExpression[(a + c+d ) ] = 6

FractionLength[x] Input: x, a fraction Output: The size of the fraction Evaluates the length of a fraction as the max of integer lengths among its numerator and denominator. Example: FractionLength[12/245] = 3

19 RationalLength[l] Input: l, a pair of lists of the coefficients of a rational Output: The size of the rational Evaluates the length of a rational function as the total of fraction lengths of its coef- ficients. Example: RationalLength[{{1,1/32},{2}}] = 4

11.3 Randomizing Utilities RandomRat[n] Input: n, a control over the value of the fraction Output: A random rational number Gives a random irreducible rational number such that its numerator and denominator are between n and n/10 (n must be divisible by 10). 3 Example: RandomRat[20] = 5

RandomRat[n,m] Input: n, a control over the numerator, and m, a control over the denominator Output: A random rational number Gives a random irreducible rational number such that its numerator is between n and n/10 and its denominator is between m and m/10 (n and m must be divisible by 10). 12 Example: RandomRat[20,30] = 29

RandomNearlyInt[i,t:3] Input: i, an integer, and t, the desired nearness to the integer Output: A random rational number close to i Gives a random irreducible rational number that is i plus or minus a random irre- ducible rational of size around 10−t. 922628 Example: RandomNearlyInt[4,3] = 230501

RandomNearlyBinomial[n] Input: n, an integer Output: A random rational that is close to a binomial number Gives a random nearly integer rational as above that is near a random binomial num- ber given n. 192593 Example: RandomNearlyBinomial[5] = 192904

BadRatFuncDenominatorQ[l]

20 Input: l, a list representing a polynomial Output: True or false Returns the Boolean true if the polynomial described by the coefficient list l contains a real root. Example: The function x3 + x2 + x +1 has a real root at x = 0. BadRatFuncDenominatorQ[{1,1,1,1}] = True The function x2 +1 has no real roots. BadRatFuncDenominatorQ[{1,0,1}] = False

RandomGoodRatFuncDenominator[ln] Input: ln, a list of “sizes” Output: A list of coefficients of a polynomial Returns a list of coefficients of a polynomial with random rational coefficients with “size” prescribed by the list of “sizes” given by the list ln that has no real roots. { 2 4} { 7 34 1252 } Example: RandomGoodRatFuncDenominator[ 10, 10 , 10 ] = 2 , 39 , 4857

RandomGoodRatFuncNumerator[ln] Input: ln, a list of “sizes” Output: A list of coefficients of a polynomial Returns a list of coefficients of a polynomial with random rational coefficients with “size” prescribed by the list of “sizes” given by the list ln. { 3 2 2} { 625 7 99 62 } Example: RandomGoodRatFuncNumerator[ 10 , 10, 10 , 10 ] = 933 , 6 , 29 , 53

RandomGoodRatFunc[p,n] Input: p, the degree of the denominator n, the “size” of the random rational coefficients Output: An irreducible rational function Returns an irreducible rational fraction integrable on the real line and with random rational coefficients. Each coefficient has “size” n. 40 + 31 + 41 2 39 55 x 75 x Example: RandomGoodRatFunc[4,100] = 65 + 25 + 85 2+ 77 3+ 100 4 34 29 x 72 x 94 x 27 x RandomRat2[n,k:50] Input: n, the “size” of the fractional part k, the integer part Output: A random irreducible irrational number Returns a random irreducible irrational number with fractional part denominator and n numerator between n and 10 and with integer part between -k and k. If k is not in- putted then the function assumes a default value of k=50. 6959 3474 Example: RandomRat2[10000,5] = 3485 = 1 + 3485 109352 3732 RandomRat2[10000] = 5381 = 20 + 5281

21 RandomGoodRatFuncDenominator2[ln] Input: ln, a list of “sizes” Output: A list of coefficients of a polynomial Returns the list of coefficients of a polynomial with random rational coefficients with considerable integer part with ”size” prescribed by the list ln. The polynomial re- turned has no real roots. { 2 4} { 393 173 66037 } Example: RandomGoodRatFuncDenominator2[ 10, 10 , 10 ] = 8 , 43 , 7094

RandomGoodRatFuncNumerator2[ln] Input: ln, a list of “sizes” Output: A list of coefficients of a polynomial Returns a polynomial with random rational coefficients with considerable integer part with ”size” prescribed by ln. { 3 2 2} { 12009 − 391 213 − 981 } Example: RandomGoodRatFuncNumerator2[ 10 , 10, 10 , 10 ] = 313 , 9 , 28 , 40

RandomGoodRatFunc2[p,n] Input: p, the degree of the denominator n, the “size” of the random rational coefficients Output: An irreducible rational function Returns an irreducible rational function with no real poles and with random rational coefficients with considerable integer part. Each coefficient has “size” n. 730 − 1651 + 1944 2 13 82 x 85 x Example: RandomGoodRatFunc2[4,100] = − 1229 + 325 − 737 2− 231 3− 975 4 88 18 x 21 x 74 x 53 x MakeRandomNearlyBadRat[p,n] Input: p, the degree of the denominator n, the “size” of the random rational coefficients Output: An irreducible rational function Returns an irreducible rational function with nearly real poles (that is, the poles have very small imaginary part) and with random rational coefficients with considerable integer part. Each coefficient has “size” n. − 975 32 Example: MakeRandomNearlyBadRat[2,100] = 24681767808817363172702218473612829912429 + 56515124610 + 2 3063604021085610882110881075664326921156 9955495447 x x 28257562305 ± 91 ≈ ± × −8 which has poles at x = 9955495447 5559726622 i 2.8 1.6 10 .

MakeNearlySimplePoly[d,t] Input: d, the degree of the denominator t, a parameter Output: A random polynomial 2 d Returns a random polynomial that is nearly (1 + x ) 2 with nearness t. 9563243616820 135 9053672786507 2− Example: MakeNearlySimplePoly[4,10]= 9563243616107 + 4520107739176 x+ 4526836393479 x 603 3 1883743808117 ≈ × −11 2 − × −11 3 4 9342773469319 x + 1883743808519 1.00 + 2.98 10 x + 2.00x 6.45 10 x + 1.00x .

22 MakeNearlySimpleRat[p,t,L:1] Input: p, the degree of the denominator n, the “size” of the random rational coefficients L, a factor to distribute among the numerator Output: A random rational function p −1 (1+x2) 2 Returns a random rational function that is nearly L × p with nearness t. If L (1+x2) 2 is not inputted then it is set to the default L=1. Example: MakeNearlySimpleRat[4,10,2] 4334896210021 − 1632 10233163873652 2 2167448105356 2803521002305 x + 5116581937023 x = 1686773738159 533 5271021441403 3 − 497 8434440222871 1686773739150 + 1509242911556 x + 2635510720579 x 9676395471497 + 8434440222535 1.00 − 2.91 × 10−10x + 2.00x2 ≈ 2 × 1.00 + 3.53 × 10−10 + 2.00x2 − 5.14 × 10−11x3 + 1.00x4 11.4 Continued Fraction Reducers CFReduce1[x] Input: x, a rational number Output: The value of the truncated continued fraction sequence of x Reduces the number x by taking the prec-th convergent of its continued fraction se- quence. 74814 Example: prec = 8; CFReduce1[.992847]= 75353

CFReduce2[x] Input: x, a rational number Output: The value of the truncated continued fraction sequence of x Reduces the number x by taking the k-th convergent where k is the smallest s.t. ak ≥ thresh2 and k ≥ cutlbnd 93691 Example: cutlbnd = 5; thresh2 = 100; CFReduce2[.992847]= 94366

CFReduce3[x] Input: x, a rational number Output: The value of the truncated continued fraction sequence of x Reduces the number x by taking the k-th convergent where k is the smallest s.t. k × ak ≥ thresh3 and k ≥ cutlbnd 449578 Example: thresh2 = 1000, CFReduce3[.992847]= 452817

11.5 L and K ◦ L Iterators Landen[l,g]

23 Input: g, a Landen transform l, a triplet of two coefficient lists and a factor Output: The result of applying Landen transformation g on the rationa function described by l Applies the Landen transform g to a rational l, where l is a triplet of two coefficient lists and one common factor.

ExtractLanden[p,m] Input: p, the degree of the denominator of a rational function m, the desired order of convergence Output: The (p, m) Landen transform Gets the Landen transform of order (p, m) from the table of pre-generated transforms stored in transforms.dump. Example: ExtractLanden[6,6] results in the same transformation as GetLandenTransform[6,6]

MakeBinRatList[p] Input: p, the degree of the denominator (1+x2)p/2−1 Output: The coefficient lists of the binomial expansion of (1+x2)p/2 (1+x2)p/2−1 Makes a triplet of two coefficient lists that represent (1+x2)p/2 and one Null to tag this special triplet Example: MakeBinRatList[4]] = {{1, 0, 2, 0, 1}, {1, 0, 3, 0, 3, 0, 1}, Null}

KSCGFactorRatList[l,lmatch,prec,reducer] Input: l, a triplet of two coefficients lists and a factor, describing a rational function lmatch, the “simple” coefficients, usually the output of MakeBinRatList prec, the distance from the “simple” coefficients within which to approximate as “simple” reducer, a rational number reducing operator, usually CFReduce2 Output: A triplet of two coefficient lists and a factor, the result of reducing l Divides out constant coefficients and appends to a common factor. Compares l to lmatch coefficient by coefficient. Coefficient-wise, reduces (using given reducer) if the coefficient is not within prec of the corresponding coefficient in lmath, and match lmath if it is within prec of the coefficient.

RegularFactorRatList[l] Input: l, a triplet of two coefficients lists and a factor, describing a rational function Output: A triplet of the coefficients of the normalized function and the common factor Divides out constant coefficients and appends to common factor

KSCG[l0,g,intnearness,precision]

24 Input: l0, the coefficient-factor triplet g, a Landen transform intnearness, the nearness at which to approximate a coefficient as its terminal binomial precision, the desired precision to which to compute the final approximation. Output: A complete log of running the technique. For each iteration, a pair of the time to run and the coefficient-factor triplet. Applies the Landen transform repeatedly on a triplet, reducing as above after each iteration, until value is unchanged within the specified precision. Stores time to com- pute each iteration.

Regular[l0,g,precision,timeout:450] Input: l0, the coefficient-factor triplet g, a Landen transform precision, the desired precision timeout, maximum run-time Output: A complete log of running the technique. For each iteration, a pair of the time to run and the coefficient-factor triplet. Applies the Landen transform repeatedly to a coefficient-factor triplet until value is unchanged within precision or more than timeout seconds have passed. Timeout is printed to screen as well as tagged in the returned list as a Null at the end of the list.

References

[1] D. Manna & V. Moll, An iterative method for numerical integration of rational functions, Contemporary Mathematics, AMS, 2008.

[2] J. Borwein & P. Borwein, PI and the AGM: A Study in and Computational Complexity, , Wiley-Interscience, 1998.

[3] D. Manna & V. Moll, Landen Survey, Probability, Geometry and Integrable Systems, MSRI Publications, 2007:55.

[4] A. YA. Khinchin, Continued Fractions, The University of Chicago Press, 1964.

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The tables below present the frequency distribution of responses, by individual question. 1. Presently, I am confident I can understand: Mean 1=Not 2=A little 3=Somewhat 4=Highly 5=Extremely Item NA NR Total (St. confident confident confident confident confident Dev.), n 3.71 1.1. What research in 0(0%) 1(6%) 6(35%) 7(41%) 3(18%) 0(0%) 0(0%) 17 (0.85), mathematics is about 17 1.2. What are the advantages 4.24 of an undergraduate research 0(0%) 0(0%) 3(18%) 7(41%) 7(41%) 0(0%) 0(0%) 17 (0.75), experience 17 1.3. What are the job 2.71 opportunities for 2(12%) 5(29%) 7(41%) 2(12%) 1(6%) 0(0%) 0(0%) 17 (1.05), mathematics majors 17 1.4. What are the fellowship 2.75 3(18%) 4(24%) 5(29%) 2(12%) 2(12%) 0(0%) 1(6%) 17 and graduate study (1.29),

14 opportunities in mathematics 16 NR=No Response; NA=Not Applicable; Standard deviation calcuated assuming a sample (denominator is # responses -1) 2. Presently, I confident I can: Mean 1=Not 2=A little 3=Somewhat 4=Highly 5=Extremely Item NA NR Total (St. confident confident confident confident confident Dev.), n 2.76 2.1. Write a technical article 0(0%) 9(53%) 5(29%) 1(6%) 2(12%) 0(0%) 0(0%) 17 (1.03), 17 2.53 2.2. Design a scientific poster 3(18%) 6(35%) 5(29%) 2(12%) 1(6%) 0(0%) 0(0%) 17 (1.12), 17 3.35 2.3. Give oral presentations 1(6%) 2(12%) 6(35%) 6(35%) 2(12%) 0(0%) 0(0%) 17 (1.06), 17 3.47 2.4. Find data or articles in 0(0%) 1(6%) 8(47%) 7(41%) 1(6%) 0(0%) 0(0%) 17 (0.72), journals or elsewhere 17 2.5. Use a software like 3.06 Mathematica or Maple to 1(6%) 6(35%) 4(24%) 3(18%) 3(18%) 0(0%) 0(0%) 17 (1.25), program or solve some 17 mathematics problems NR=No Response; NA=Not Applicable; Standard deviation calcuated assuming a sample (denominator is # responses -1) 3. Presently, I am interested in… Mean 1=Not at all 2=A little 3=Somewhat 4=Highly 5=Extremely Item NA NR Total (St. interested interested interested interested interested Dev.), n

15 3.1. Discussing about 3.29 mathematics with friends or 2(12%) 2(12%) 5(29%) 5(29%) 3(18%) 0(0%) 0(0%) 17 (1.26), family 17 3.2. Reading articles about 3.35 mathematics in magazines, 1(6%) 3(18%) 5(29%) 5(29%) 3(18%) 0(0%) 0(0%) 17 (1.17), journals or on the internet 17 4.18 3.3. Taking additional 1(6%) 0(0%) 0(0%) 10(59%) 6(35%) 0(0%) 0(0%) 17 (0.95), courses in mathematics 17 3.4. Exploring career 4 (0.94), opportunities in 0(0%) 1(6%) 4(24%) 6(35%) 6(35%) 0(0%) 0(0%) 17 17 mathematics 3.5. Attending graduate 4.06 0(0%) 1(6%) 3(18%) 7(41%) 6(35%) 0(0%) 0(0%) 17 school in mathematics (0.9), 17 2.82 3.6. Attending graduate 4(24%) 3(18%) 5(29%) 2(12%) 3(18%) 0(0%) 0(0%) 17 (1.42), school in other area 17 3.7. Having another 4.25 undergraduate research 0(0%) 1(6%) 2(12%) 5(29%) 8(47%) 1(6%) 0(0%) 17 (0.93), experience 16 NR=No Response; NA=Not Applicable; Standard deviation calcuated assuming a sample (denominator is # responses -1) 4. What is your gender? Response # Respondents a. Male 9 b. Female 8 Total 17 5. What is your age group?

16 Response # Respondents a. 18 or younger 2 b. 19-21 12 c. 22-30 3 Total 17 6. What is your ethnic designation? Response # Respondents a. White/Caucasian 2 b. Black/African American 3 c. Hispanic or Latino/not White 8 e. Asian or Pacific Islander 2 f. Other 2 Total 17 7. What is your current GPA in a system that assumes a 4.00 as an A (highest score possible)? Response # Respondents a. 4.00-3.60 13 b.3.01-3.59 3 c. 2.51-3.00 1 Total 17 8. Have you worked in an undergraduate research project in mathematics at your home institution? Response # Respondents 1=No 11 (65%) 2=Yes 6 (35%) Not Applicable 0 (0%)

17 No Response 0 (0%) Total 17 9. Have you attended a summer research program before? Response # Respondents 1=No 11 (65%) 2=Yes 6 (35%) Not Applicable 0 (0%) No Response 0 (0%) Total 17

18 Instructions: Glance at the entire evaluation before you start filling it out. Please take sufficient time to fill it out. If you run out of space in a Comments section, use the last page. There are a couple of questions that should be filled out after your presentation; these will only take a few minutes. Your input is greatly appreciated. Thank you.

Pre-Research Seminar

Did you find the material, techniques and applications that you learned in the pre-research seminar interesting? Yes, very interesting. Yes, interesting. Somewhat interesting. No. Comments: 3 8 6

Was the seminar time used effectively? Yes, very effectively. Yes, effectively. Somewhat effectively. No. Comments: 3 11 3

The computational laboratory was designed to intertwine and supplement the mathematics being learned in the seminar. How successful was the implementation of the computational laboratory in achieving this goal? Very successful. Successful. Somewhat successful. Not successful. Comments: 9 6 1 1

On a scale of 1-5 please rate the usefulness of each of the following in helping you to learn the material presented in the pre-research seminar. X = does not apply, 1=not useful, 5 = very useful.

X 1 2 3 4 5 A 1 6 6 4Lectures B 3 6 3 5 Homework assignments C 3 4 7 1 2 Textbook(s) D 3 1 4 6 3 Notes written by research mentor E 1 2 5 9 Interaction and collaboration with seminar mates F 41One-on-one or group sessions with research leader, postdoc or 3 0 TAs

Comments:

19

What other activities helped or would have helped you to learn the mathematics needed for your research? Formal instruction, Mathematica, readings prior to the program, lectures, more basic computer introduction, having more background, review of basic topics

The pace at which new material was presented was: Too fast. Just right. Too slow. Comments:

4 12 1

The amount of homework during the first week was: About the correct amount to help you learn the material. Too much. Too little. Comments:

12 4 1

8. The help, support, feedback and encouragement from the professor, postdoc, TAs and other staff was: The correct amount. Just below the correct amount. Not enough. None. Comments:

12 4 1

9. Please comment on any other pre-research aspect of MSRI-UP. (e.g., if you could change anything about the pre-research phase, what would you change?) less homework, more formal instruction, more example problems, more time to study, more explanation for HW, more group rotations, more feedback on HW, more time to analyze student personalities before choosing gp

Research Projects

1. Your research project was: Clearly-defined. Not clearly-defined. Not defined at all. Comments: 11 5 1

20

2. The mathematical level of your research project was: Way too challenging. Challenging. Not very challenging Too easy. Comments: 3 14

3. The guidance and support on your project that you received from the research mentor and associates was: The correct amount. Just below the correct amount. Not enough. None. Comments: 12 3 2

4. Comment on the adequacy (or lack thereof) of the computing facilities for carrying out the work on your project. Need computers at night, not enough Mathematica licenses, faster computers

5. Comment on the adequacy (or lack thereof) of reference material available for carrying out the work on your project. Wonderful library, not enough references, cannot checkout books, great

6. Did you like working on a group research project? Yes, very much. Yes. A little. No. Comments:

10 6 1

7. The guidance on preparing your presentation and technical report was: The correct amount. Just below the correct amount. Not enough. None. Comments:

15 2

8. How satisfied with the results of your research project are you?

21 Very satisfied. Satisfied. Not satisfied. Comments:

9 7 1

9. How satisfied are you with the quality of your A. Presentation? Very satisfied. Satisfied. Not satisfied. Comments:: 14 3

B. Technical report? Very satisfied. Satisfied. Not satisfied. Comments: 8 6 2

10. What did you like most about your research project? Using Mathematica, excitement of new ideas, subject, discovering things, cool graphics, topic, working in gp on new problems, the group

11. What did you dislike most about your research project? Not enough guidance, working in a gp, writing the results, not enough time, needing more math background, vagueness of the problem

12. Please comment on any other aspect of the research part of MSRI-UP. (e.g., if you could change anything about your research experience, what would you change?) dorms and food were not good, more references, A/C

13. Only answer this question if you have participated in other undergraduate research projects or summer programs. How does your research experience during MSRI-UP compare with your other research experiences? MSRI-UP did better job creating collaborative environment, this was too short, this was a bad experience

22

Workshops, Colloquia, Other Academics not Evaluated Above

1. Do you feel that the colloquia were successful giving you a glimpse of other areas of mathematics? Yes, very much so. Yes, somewhat. Not really. Not at all. 1 9 5 2

2. Do you feel that the colloquia allowed you to meet researchers and faculty at universities with graduate programs? Yes, very much so. Yes, somewhat. Not really. Not at all.

5 7 5 3. Which was your favorite colloquium? Least favorite?

Favorite: Least Favorite: Williams: 1, Beck: 4, Orellana: 2, Wilson: 5, Arsuaga: 4, Wilson: 1, Williams: 1

4. On a scale of 0-4 please rate the usefulness of each of the following workshops. 0 = not useful, 4 = very useful.

0 1 2 3 4 Workshop 4 5 4 4 Graduate and fellowship workshop by Colette Patt. 1 4 Workshop on using LaTeX. 3 4 8 5 Workshop on preparing and giving a mathematics oral presentation. 4 7 6 Workshop on preparing and giving a mathematics poster presentation. Comments:

5. Is there a workshop, discussion or panel topic that you would have liked? Please describe.

23 Measuring Some MSRI-UP Objectives

1. Prior to MSRI-UP, had you worked on an undergraduate research project in mathematics? Yes. No. Comments: 6 11

2. After MSRI-UP, do you want work on an undergraduate research project in mathematics? Yes. No. Comments: 15 2

3. Presently, I am not a little somewhat highly extremely confident I can understand what research in mathematics is about. Comments: 3 12 2

4. Presently, I am not a little somewhat highly extremely confident I can understand what are the advantages of an undergraduate research experience. Comments:

2 9 6

5. Presently, I am not a little somewhat highly extremely confident I can understand what are the job opportunities for mathematics majors. Comments:

2 7 6 2

6. Presently, I am not a little somewhat highly extremely confident I can understand what are the fellowships and graduate study opportunities in mathematics. Comments: 1 4 10 2

24

7. Presently, I am not a little somewhat highly extremely confident I can write a technical article. Comments: 8 7 2

8. Presently, I am not a little somewhat highly extremely confident I can design a scientific poster Comments: 2 6 8 1

9. Presently, I am not a little somewhat highly extremely confident I can give an oral presentation. Comments: 1 9 7

10. Presently, I am not a little somewhat highly extremely confident I can find data or articles in journals or elsewhere Comments:

6 6 5

11. Presently, I am not a little somewhat highly extremely confident I can use a software like Mathematica or Maple to program or solve some mathematics problems. Comments: 1 3 7 6

12. Presently, I am not at all a little somewhat highly extremely interested in discussing about mathematics with friends or family.

25 Comments: 2 1 3 7 4

13. Presently, I am not at all a little somewhat highly extremely interested in reading articles about mathematics in magazines, journals or the internet. Comments:

1 5 7 4

14. Presently, I am not at all a little somewhat highly extremely interested in taking additional courses in mathematics. Comments:

2 6 9

15. Presently, I am not at all a little somewhat highly extremely interested in attending graduate school in mathematics. Comments:

1 4 4 8

16. Presently, I am not at all a little somewhat highly extremely interested in having another undergraduate research experience. Comments:

1 2 1 5 8

V. Non-Academic Aspects of MSRI-UP and General Questions

1. On a scale of 0-4 please rate how happy were you with each of the following. 0 = not happy at all, 4 = very happy.

26 0 1 2 3 4 A 2 5 5 4 Living arrangements. B 4 5 4 3 Eating arrangements. C 1 8 8 Saturday outings. D 1 1 1 3 Number of students in the program. 2 E 5 5 7 Transportation arrangements. F 1 3 4 Overall design, organization and administration of the program. 0

Comments (Use back of page if necessary):

Too many students, stay the last Saturday, organization was perfect

2. Which was your favorite outing? Why? San Francisco: 2, Kayaking: 12, Muir Woods: 1

3. Which was your least favorite outing? Why? Monterrey: 8, Muir Woods: 4

4. What are the things that you particularly liked about the program? Project, administration, tea time

5. What are the things that you particularly disliked about the program? People that do not applu should not be allowed to participate, dynamics with people, working late, few free time

6. Do you think that MSRI-UP has changed your outlook on your academic future? If so, how?

7. Please use the back to add any additional comments that you think are important or relevant to any aspect of MSRI-UP.

27

8. What is your gender? female male.

8 9

9. What is your age group? 18 or younger 19 - 21 22 - 30 more than 30

1 13 3 10. What is your ethnic designation? White/Caucasian Black/African American Hispanic/Latino Asian or Pacific Islander Other 2 3 8 2 1 prefer not to answer: 1

11. Has anyone in your family attended graduate school? yes no.

11 6 12. Have you attended mathematics or science meetings in the past? yes no.

10 7 13. Have you presented at a national meetings in the past? yes no.

4 13

28 CRITICAL ISSUES IN EDUCATION: THE TEACHING AND LEARNING OF ALGEBRA

MAY 14–16, 2008

For over two decades, the teaching and learning of algebra has been a focus of mathematics education at the precollege level. This workshop examined issues in algebra education at two critical points in the continuum from elementary school to undergraduate studies: at the transitions from arithmetic to algebra and from high school to university. In addition, the workshop involved participants in discussions about various ways to to structure an algebra curriculum across the entire K–12 curriculum. Algebra is an essential prerequisite in almost all collegiate mathematics courses, a fact that has caused many mathematicians to become intensely interested in school algebra. Education research has devoted a huge amount of energy to the study of algebra learning (and more recently, algebra teaching). And, save, perhaps, for deductive geometry, teachers know that algebra is one of the toughest subjects to teach, especially to underprepared students with poor backgrounds in arithmetic. This made MSRI an ideal host for the workshop, because it has established itself as a convener for all three subgroups of the mathematics community. The workshop was organized around three overarching questions. Each question was launched by a panel of scholars who are working in the relevant area. The panel was followed by several concurrent sessions, led by teachers or other practitioners, designed to give specifics examples to address the question. The concurrent sessions were followed by presentations that discussed or reacted to the issues brought up by the opening panel and the parallel sessions. Presentations, abstracts, and videos of the presentations are on the MSRI website at http://www.msri.org/calendar/workshops/WorkshopInfo/454/show workshop The organizing questions were: Question 1: What are some organizing principles around which one can cre- ate a coherent pre-college algebra program? Several influences in the latter half of the 20th century have caused educators to rethink the purpose of school algebra and its traditional centrality in the secondary curriculum. These in- fluences include the ubiquity of low-cost graphing (and now com- puter algebra) calculators, the “algebra for all” movement, the increasingly common introduction of algebra in middle or even el- ementary school, and curricular trends that emphasize modeling, applications, data analysis, and other organizers. Presentations by curriculum developers, researchers, and teachers articulated and illustrated underlying frameworks for developing algebra pro- grams. Question 2: What is known about effective ways for students to make the transition form arithmetic to algebra? 1 2 MAY 14–16, 2008

Several organizing committee members felt that this was the key question—that the successful transition form arithmetic to alge- bra is the linchpin for success in higher mathematics. Presenta- tions addresses several sub-quesitons: • What does research say about this transition? • What kinds of arithmetic experiences help preview and build the need for formal algebra? • In what ways does high school and undergraduate mathe- matics depend on fundamental ideas developed in the tran- sition from arithmetic to algebra? • What are some effective pedagogical approaches that help students develop a robust understanding of algebra? Question 3: What algebraic understandings are essential for success in be- ginning collegiate mathematics? Although the arithmetic→algebra transition is central for further study of algebra, there are other important pieces that need to be in place for students to succeed in college courses. Many of the presentations dealt with the need for certain kinds of techni- cal fluency: “symbol sense” and “mindful manipulation,” among others. Presentations also dealt with questions like these: • What kinds of problems should high school graduates be able to solve? • What algebraic habits of mind should students develop in high school? • What are the implications of current and emerging tech- nologies on these questions? Question 3 was also addressed by a special panel that discussed the partic- ular algebraic needs of prospective teachers.

The audience for the workshop included mathematicians, mathematics educators, classroom teachers, and education researchers who are concerned with improving the teaching and learning of algebra across the grades. Sessions featured direct experience with several curricular approaches to algebra, as well as reports from researchers, educators, and members of national committees that are charged with finding ways to increase student achievement in algebra. The hands-down highlight of the meeting was an impromptu plenary by Dan Chazan, who showed some of his Themat videos—animated cartoons of algebra classrooms, in which the stylized look of the characters allows viewers to concentrate on the mathematics and classroom dialogue presented by the authors. Participants also enjoyed the presentations by Deborah Ball and Bill McCallum on, respectively, the report by the National Math Panel and the forthcoming NCTM High School document. Positive reviews by participants mentioned the balanced agenda, the chance to hear from national experts, and the opportunity to discuss algebra teaching and learning with such a diverse clientele. Participants also felt that the days were too packed, that underlying tensions between mathematicians and educators surfaced too often, and that they wanted to hear presentations from people who were not so invested in particular agendas. CRITICAL ISSUES IN EDUCATION: THE TEACHING AND LEARNING OF ALGEBRA 3

Dissemination. Videos of all the presentations are on the MSRI website. There were note takers at each session, and these notes will be made available to the organizers. The notes will also be used to support the writing of a book in the MSRI series that will have chapters corresponding to our three questions. The following have agreed to author the chapters: Question 1: Hyman Bass Question 2: Mark Saul Question 3: Bill McCallum and Dan Chazan

Critical Issues in Education: Teaching and Learning Algebra Mathematical Sciences Research Institute, Berkeley, California May 14 – May 16, 2008

Wednesday, May 14, 2008

Charter Bus 1:45 PM Depart Double Tree Hotel Scheduled 2:00 PM Depart Hearst Mining Circle Departures UC Berkeley “Hill” shuttle departs UC Berkeley’s Hearst Mining Circle every half hour beginning at 7:40AM 2:00 – 2:30PM Coffee, tea in the Atrium Session 1.1 Plenary Session in the Simons Auditorium

2:30– 3:00PM Robert Bryant, MSRI Director Welcome, Overview, and Purpose of Deborah Ball, University of Michigan Workshop/Framing Questions

Question 1 What are some organizing principles around which one can create a coherent pre-college algebra program?

3:00-5:30PM Al Cuoco, Center for Mathematics Education Diane Resek, San Francisco State University Tom Sallee, University of California, Davis Panel discussion Zalman Usiskin, University of Chicago (See Abstracts) James Fey, University of Maryland

5:30-6:30PM Reception and light buffet dinner in the Atrium Session 1.2 Plenary Session in the Simons Auditorium

6:45-7:15PM Deborah Ball, University of Michigan The National Mathematics Advisory Panel report: Summing Up and Taking Stock (See Abstract)

7:15-7:45PM William McCallum, University of Arizona Report on the NCTM Lenses on High School Mathematics report (See Abstract)

7:45-8:45PM Hyman Bass, University of Michigan Discussants on the presentation Roger Howe, Yale University

Charter Bus Scheduled 9:00 PM Hearst Mining Circle, Double Tree Hotel Departures

Page 1 of 11

Thursday, May 15, 2008

Charter Bus 7:00AM Depart Double Tree Hotel Scheduled 7:15AM Depart Hearst Mining Circle Departures UC Berkeley “Hill” shuttle departs UC Berkeley’s Hearst Mining Circle every half hour beginning at 7:40 AM

7:30-8:00AM Coffee, tea in the Atrium Session 1.3 Plenary Session in the Atrium

8:00-8:15AM Al Cuoco, Center for Mathematics Education Overview of the day.

8:15-9:15AM Parallel Sessions: Question 1

1.3a Stephanie Ragucci, Core Plus/Andover High Simons Auditorium School (See Abstract) Annette Roskam, CME/Rice Lake High School

Baker Board Room 1.3b Carol Cho, CPM/Alhambra High School (See Abstract) Sybilla Beckmann, University of Georgia

1.3c Matt Bremer, IMP/Berkeley High School Commons Room Pat Thompson, Arizona State University (See Abstract)

9:15-9:45AM Coffee, Tea, Danish, etc. in the Atrium Session 1.4 Plenary Session in the Simons Auditorium

9:45-11:15AM Roger Howe, Yale University William McCallum, University of Arizona Discussants on Question 1. Betty Phillips, Michigan State University

11:15-1:00PM Lunch in the Atrium

Question 2 What is known about effective ways for students to make the transition from arithmetic to algebra?

Session 2.1 Plenary Session in the Simons Auditorium

1:00-3:00PM David Carraher, TERC Jo Ann Lobato, San Diego State University Panel discussion Alan Schoenfeld, University of California, (See Abstract) Berkeley Uri Treisman, University of Texas

3:00-3:30PM Coffee, Tea, Danish, etc. in the Atrium 3:30-4:30PM Parallel Sessions: Question 2

Page 2 of 11

2.2a Ted Courant, Bentley School Simons Auditorium Paul Goldenberg, CME (See Abstract)

2.2b Virginia Bastable, Mount Holyoke College Baker Board Room Susan Jo Russell, TERC (See Abstract) Deborah Schifter, Education Development Center

2.2c Betty Phillips, Michigan State University Commons Room Mark Saul, Bronxville Schools (Ret.)

Session 2.3 Plenary Session in the Simons Auditorium

4:30-6:30PM Hung-Hsi Wu, University of California, Berkeley Herb Clemens, chair, Ohio State University Question 2: The transition from arithmetic to algebra: Robert Moses, The Algebra Project further perspectives Paul Sally, University of Chicago (See Abstracts)

Charter Bus Scheduled 6:45PM Depart for Hearst Mining Circle and Double Tree Hotel. Departures

Friday, May 16, 2008

Charter Bus 7:00AM Depart Double Tree Hotel Scheduled 7:15AM Depart Hearst Mining Circle Departure UC Berkeley “Hill” shuttle departs UC Berkeley’s Hearst Mining Circle every half hour beginning at 7:40 AM

7:30-8:00AM Coffee, tea in the Atrium Session 2.4 Plenary Session in the Simons Auditorium

8:00-8:15AM James Fey, University of Maryland Overview of the Day.

8:15-9:45AM Hyman Bass, University of Michigan Megan Franke, University of California, Los Discussants on Question 2. Angeles Ed Silver, University of Michigan

9:45-10:15AM Coffee, tea in the Atrium

Question 3 What Algebraic understandings are essential for success in beginning collegiate mathematics?

Session 3.1 Plenary Session in the Simons Auditorium

10:15-11:45AM William McCallum, IME/University of Arizona Panel discussion. See Abstracts. Tom Roby, University of Connecticut Deborah Hughs-Hallett, IME/University of Arizona

Page 3 of 11

11:45-12:45PM Lunch in the Atrium 12:45-1:45PM Parallel Sessions: Question 3

3.1a William McCallum, University of Arizona Simons Auditorium Glenn Stevens, Boston University (See Abstract)

3.1b Dan Chazan, University of Maryland Baker Board Room James Fey, University of Maryland (See Abstract)

Session 3.2 Plenary Session in the Simons Auditorium

1:45-3:15PM Herb Clemens, Ohio State University Mark Saul, Bronxville Schools, ret. Discussants on Question 3. Ed Silver, University of Michigan

3:15-3:45PM Coffee, tea in the Atrium Session 3.3 Plenary Session in the Simons Auditorium

3:45-5:15PM Dan Chazan, University of Maryland Al Cuoco, Center for Mathematics Education Panel: Preparing teachers to teach algebra Hung-Hsi Wu, University of California, Berkeley (See Abstract)

5:15-5:45 Deborah Ball, University of Michigan Closing Session: Connections among the questions

5:45-6:45PM Reception in the Atrium

Charter Bus Scheduled 7:00PM Depart for Hearst Mining Circle and Double Tree Hotel. Departures

Currently Available Videos

• Al Cuoco, James Fey, Diane Resek, Tom Sallee, Zalman Usiskin , What are some organizing principles around which one can create a coherent pre-college algebra program? May 14,2008, 03:00 PM to 05:30 PM

• Al Cuoco, James Fey, Diane Resek, Tom Sallee, Zalman Usiskin , What are some organizing principles around which one can create a coherent pre-college algebra program? May 14,2008, 03:00 PM to 05:30 PM

• Al Cuoco, James Fey, Diane Resek, Tom Sallee, Zalman Usiskin , What are some organizing principles around which one can create a coherent pre-college algebra program? May 14,2008, 03:00 PM to 05:30 PM

• Al Cuoco, James Fey, Diane Resek, Tom Sallee, Zalman Usiskin , What are some organizing principles around which one can create a coherent pre-college algebra program? May 14,2008, 03:00 PM to 05:30 PM

Page 4 of 11

• Al Cuoco, James Fey, Diane Resek, Tom Sallee, Zalman Usiskin , What are some organizing principles around which one can create a coherent pre-college algebra program? May 14,2008, 03:00 PM to 05:30 PM

• Deborah Ball , The National Mathematics Advisory Panel Report: Summing Up and Taking Stock May 14,2008, 06:45 PM to 07:15 PM

• William McCallum , Report on the NCTM Lenses on High School Mathematics report May 14,2008, 07:15 PM to 12:00 AM

• Hyman Bass, Roger Howe , Discussants on the presentation May 14,2008, 07:45 PM to 08:45 PM

• Hyman Bass, Roger Howe , Discussants on the presentation May 14,2008, 07:45 PM to 08:45 PM

• Stephanie Ragucci, Annette Roskam , Problem Solving Using CME & Core-Plus May 15,2008, 08:15 AM to 09:15 AM

• Stephanie Ragucci, Annette Roskam , Problem Solving Using CME & Core-Plus May 15,2008, 08:15 AM to 09:15 AM

• Carol Cho , 1.3b Parallel Sessions: Question 1 May 15,2008, 08:15 AM to 09:15 AM

• Sybilla Beckmann , Solving algebra story problems with simple “strip diagrams,” solving them with algebra, and connecting the two approaches. May 15,2008, 08:15 AM to 09:15 AM

• Matt Bremer , 1.3c Parallel Sessions: Question 1 May 15,2008, 08:15 AM to 09:15 AM

• Pat Thompson , 1.3c Parallel Sessions: Question 1 May 15,2008, 08:15 AM to 09:15 AM

• Roger Howe, William McCallum, Betty Phillips , Discussants on the presentation May 15,2008, 09:45 AM to 11:15 AM

• Roger Howe, William McCallum, Betty Phillips , Discussants on the presentation May 15,2008, 09:45 AM to 11:15 AM

• Roger Howe, William McCallum, Betty Phillips , Discussants on the presentation May 15,2008, 09:45 AM to 11:15 AM

• David Carraher, Jo Ann Lobato, Alan Schoenfeld, Uri Treisman , What is known about effective ways for students to make the transition from arithmetic to algebra? May 15,2008, 01:00 PM to 03:00 PM

• David Carraher, Jo Ann Lobato, Alan Schoenfeld, Uri Treisman , What is known about effective ways for students to make the transition from arithmetic to algebra? May 15,2008, 01:00 PM to 03:00 PM

Page 5 of 11 • David Carraher, Jo Ann Lobato, Alan Schoenfeld, Uri Treisman , What is known about effective ways for students to make the transition from arithmetic to algebra? May 15,2008, 01:00 PM to 03:00 PM

• David Carraher, Jo Ann Lobato, Alan Schoenfeld, Uri Treisman , What is known about effective ways for students to make the transition from arithmetic to algebra? May 15,2008, 01:00 PM to 03:00 PM

• Ted Courant, Paul Goldenberg , Does 8th grade algebra prepare students for Geometry and high school mathematics? May 15,2008, 03:30 PM to 04:30 PM

• Ted Courant, Paul Goldenberg , How the ideas and language of algebra K-5 set the stage for algebra 8-12 May 15,2008, 03:30 PM to 04:30 PM

• Virginia Bastable, Susan Jo Russell, Deborah Schifter , Strengthening K-5 Arithmetic/Preparing for Algebra May 15,2008, 03:30 PM to 04:30 PM

• Virginia Bastable, Susan Jo Russell, Deborah Schifter , Strengthening K-5 Arithmetic/Preparing for Algebra May 15,2008, 03:30 PM to 04:30 PM

• Virginia Bastable, Susan Jo Russell, Deborah Schifter , Strengthening K-5 Arithmetic/Preparing for Algebra May 15,2008, 03:30 PM to 04:30 PM

• Betty Phillips, Mark Saul , 2.2c Parallel Sessions: Question 2 May 15,2008, 03:30 PM to 04:30 PM

• Betty Phillips, Mark Saul , 2.2c Parallel Sessions: Question 2 May 15,2008, 03:30 PM to 04:30 PM

• Herb Clemens, Robert Moses, Mary Jo Tavormina, Hung-Hsi Wu , The transition from arithmetic to algebra: further perspectives May 15,2008, 04:30 PM to 06:30 PM

• Herb Clemens, Robert Moses, Mary Jo Tavormina, Hung-Hsi Wu , The transition from arithmetic to algebra: further perspectives May 15,2008, 04:30 PM to 06:30 PM

• Herb Clemens, Robert Moses, Mary Jo Tavormina, Hung-Hsi Wu , The transition from arithmetic to algebra: further perspectives May 15,2008, 04:30 PM to 06:30 PM

• Herb Clemens, Robert Moses, Mary Jo Tavormina, Hung-Hsi Wu , The transition from arithmetic to algebra: further perspectives May 15,2008, 04:30 PM to 06:30 PM

• Hyman Bass, James Fey, Ed Silver , Discussants on the presentation May 16,2008, 08:15 AM to 09:45 AM

• Hyman Bass, James Fey, Ed Silver , Discussants on the presentation May 16,2008, 08:15 AM to 09:45 AM

• Hyman Bass, James Fey, Ed Silver , Discussants on the presentation May 16,2008, 08:15 AM to 09:45 AM

Page 6 of 11 • Deborah Hughes Hallett, William McCallum, Tom Roby , What Algebraic understandings are essential for success in beginning collegiate mathematics? May 16,2008, 10:15 AM to 11:45 AM

• Deborah Hughes Hallett, William McCallum, Tom Roby , What Algebraic understandings are essential for success in beginning collegiate mathematics? May 16,2008, 10:15 AM to 11:45 AM

• Deborah Hughes Hallett, William McCallum, Tom Roby , What Algebraic understandings are essential for success in beginning collegiate mathematics? May 16,2008, 10:15 AM to 11:45 AM

• William McCallum, Glenn Stevens , Question 3 Talk May 16,2008, 12:45 PM to 01:45 PM

• William McCallum, Glenn Stevens , Mining the early mathematics curriculum May 16,2008, 12:45 PM to 01:45 PM

• Dan Chazan, James Fey , What algebraic understandings do we wish future teachers might gain in college? May 16,2008, 12:45 PM to 01:45 PM

• Dan Chazan, James Fey , What algebraic understandings do we wish future teachers might gain in college? May 16,2008, 12:45 PM to 01:45 PM

• Herb Clemens, Mark Saul , Discussants on the presentation May 16,2008, 01:45 PM to 03:15 PM

• Herb Clemens, Mark Saul , Discussants on the presentation May 16,2008, 01:45 PM to 03:15 PM

• Dan Chazan, Al Cuoco, Hung-Hsi Wu , Preparing teachers to teach algebra May 16,2008, 03:45 PM to 05:15 PM

• Dan Chazan, Al Cuoco, Hung-Hsi Wu , Preparing teachers to teach algebra May 16,2008, 03:45 PM to 05:15 PM

• Dan Chazan, Al Cuoco, Hung-Hsi Wu , Preparing teachers to teach algebra May 16,2008, 03:45 PM to 05:15 PM

• Deborah Ball , Connections among the questions May 16,2008, 05:15 PM to 05:45 PM

Participant List

Name Role Institution Adiredja, Aditya P Notetaker UC Berkeley Agarwal, Mahesh Kumar Participant McMaster University Alexeev, Natalia Academic Sponsor University of Georgia Alibegovic, Emina Academic Sponsor University of Utah Anderson, Renee Wilkerson Participant Grant High School Anhalt, Mary Jo Participant Bakersfield College, Delano Ball, Deborah Loewenberg Organizer University of Michigan Bartlo, Joanna Rachel Academic Sponsor Portland State University Bass, Hyman Organizer/Speaker University of Michigan

Page 7 of 11 Bastable, Virginia Speaker Mount Holyoke College Beck, Matthias Academic Sponsor San Francisco State University Becker, Joanne Rossi Participant San Jose State University Beckmann, Sybilla Speaker University of Georgia Beissinger, Janet Simpson Participant University of Illinois at Chicago Berger, Lisa Academic Sponsor Stony Brook University Bernbaum Wilmot, Diana Joy Notetaker University of California, Berkeley Bertolone-Smith, Claudia Marie Participant Douglas County School District Blachman, Nancy Participant N/A Bloom, Irene Academic sponsor Arizona State University, downtown Phoenix Campus Bogley, William Academic Sponsor Oregon State University Boley, Heidi A. Participant Oakland Unified School District Bonesteel, Patty Academic Sponsor Wayne State University Boyce, Cate Participant Portland Public Schools Bravewoman, Mary Teresa Participant City College of San Francisco Bremer, Matt Speaker Berkeley High School Brennan, Brendan P Participant University of Hawaii Bressoud, David Speaker Macalester College Bryant, Robert Leamon Speaker MSRI Butler, Steve Participant Tamalpais Union High School District Cabana, Carlos Participant San Lorenzo HS Calahan, Heather Participant UCLA Mathematics Calvert, Margaret Participant Portland Public Schools Carraher, David Speaker Technical Education Research Centers Carroll, Cathy Participant WestEd Carroll, Maria Dreux Participant Alameda County Office of Education Castilla, Guillermo E Participant san jose city college Castro-Superfine, Alison Academic sponsor University of Illinois at Chicago Celestin, Lele Participant university of Dschang Champney, Danielle Notetaker UC Berkeley Chazan, Dan Speaker University of Maryland Childs, Kimberly McRae Participant Stephen F. Austin State University Cho, Carol Speaker Alhambra High School Clemens, Herb Organizer/Speaker Ohio State University Cohen-Corwin, Amy Academic Sponsor Rutgers University Colen, Yong Suk Participant Indiana University of PA (IUP) Collins, Anne M. Participant Lesley University Cooper, Sandy Academic Sponsor WSU Cossey, Ruth Participant Mills College Courant, Ted Speaker Bentley School Cuoco, Al Organizer/Speaker Center for Mathematics Education Currell, Marian Participant Civic Center Secondary School Currie, Gina Participant Washington State University D'Arcy, Jeanne Participant San Francisco Unified School District Davis, Frank E. Participant TERC Davis, Kathleen M. Participant Fresno Unified School District DeCarli, Elizabeth Participant Key Curriculum Press Diaz, Marco Antonio Participant UCLA Disston, Jacob l Participant willard middle school, berkeley unified school district Dougherty, Barbara Jo Participant University of Mississippi Douglas, Lewis Philip Participant Lawrence Hall of Science

Page 8 of 11 Dworkin, Lise Participant San Francisco Unified School District Earnest, Darrell Participant University of California, Berkeley Ferreira, Pamela Participant Walnut Grove Elementary, PUSD Fey, James Organizer/Speaker University of Maryland Fi, Cos Academic Sponsor The University of Iowa Fillingim, Jennifer Participant N/A Foster, David Participant Noyce Foundation Francisco, Evelyn Participant Oakland Technical High School Franke, Megan Organizer/Speaker University of California Fraser, Sherry Participant Interactive Mathematics Program, Sonoma State University Freeouf, Barbara C Participant Washington State Univ. Vancouver Gagnon, George Participant College of Engineering Galindo, Enrique Academic Sponsor Indiana University Garn, Harvey Participant Center for Math. Excellence and Equity Geltmeyer, Gerd-Henning John Participant San Ramon Valley USD Getz, Amy Participant Fort Lewis College Gilbert, Michael Participant University of Hawaii Glines, Kathleen Participant Maret School Goldenberg, Paul Speaker Center for Mathematics Education Gomez, Emiliano Participant UCB - University of California, Berkeley Gonsalves, Philip Participant Alameda County Office of Education Gottlieb, Dewey Participant Hawaii DOE Grebe, Phillip A Participant N/A Hackenberg, Amy J Participant Indiana University Haldar, Lina Chopra Participant University of California, Berkeley (Biotech) Harel, Guershon Academic Sponsor UCSD Harmon, Jendayi A. Math For America Math for America Hasan, Angela Laila Participant UCLA GSE&IS, Center X Helminck, Aloysius Gerardus Academic sponsor NC State University Henderson, David Allen Participant Cornell University Henderson, David W. Academic Sponsor Cornell University Hernandez, Andrea Carolina Math for America Math for America Hitomi, Stanley Participant San Ramon Valley Unified School District Holmes, Lisa Participant Mills College Howe, Roger Speaker Yale University Hughes Hallett, Deborah Speaker University of Arizona Hung, Marcus Participant San Francisco Unified School District James, Julie Participant Universtiy of Mississippi Jay, Melissa Rae Participant Tomlin Middle School Johnson, George C. Participant University of California California Teacher Advisory Council, National Teacher Advisory Jones, Juliana Eileen Participant Council Judson, Thomas Academic sponsor Harvard University Kach, Asher Moshe Participant University of Connecticut, Storrs Kappes, Mary Lee Participant Portland Public Schools Keeley, Judith Participant RI Department of Education Kendall, Sue Participant San Ramon Valley Unified School District Kennedy, Nadia Stoyanova Participant Stony Brook University Kessel, Cathy Participant self employed Kim, Myong-Hi Participant SUNY Old Westbury King, James Richard Academic sponsor University of Washington

Page 9 of 11 Kinzer, Cathy Jeanne Participant New Mexico State University Kohler, Brynja Academic Sponsor Utah State University Kopp, Jaine Participant UC Berkeley/LHS/CeMEE Koshlap, Marilyn Margaret Participant Laney College Kravin, Drew Participant Alameda County Office of Education Lam, Sandra C Participant San Francisco Unified School District Lambie, Catie Participant Kern County Superintendent of Schools Langbort, Carol R. Participant San Francisco State University Lehman, Rachel Cohen Academic Sponsor University of California, Irvine Lempp, Steffen Academic Sponsor University of Wisconsin, Madison Lenane, Susan Participant Maret School Levin, Mariana Notetaker UC Berkeley Lewis, Jim Participant University of Nebraska, Lincoln Lieberman, Joanne Participant CSU Monterey Bay Lienert, Carl Participant Fort Lewis College Lobato, Jo Ann Speaker San Diego State University Lomneth, Marcia Participant UC Berkeley, Lawrence Hall of Science Lovric, Miroslav Academic Sponsor McMaster University Lyon, Ann Participant Thurgood Marshall Academic HIgh School Madden, James J. Academic Sponsor LSU Marsh, William Academic sponsor N/A Martin, Danny Bernard Participant University of Illinois, Chicago Martinez, Adelita Participant Stanislaus County Office of Education Martinez, Samuel Cheito Participant Rowell Elementary School Matthews, Nancy L Academic Sponsor University of Oklahoma Mayfield-Ingram, Karen Participant Center for Equity & Excellence in Mathematics(CeMEE McCallum, William Gordon Speaker University of Arizona McGinn, Dan Participant University of Wisconsin Megginson, Robert Eugene Academic Sponsor University of Michigan, LSA Dean's Office melero, selia Participant OUSD Mezzio, Kelly Participant California High School millman, richard Academic Sponsor university of kentucky Mitchell, Jean M Participant CSU Monterey Bay Moses, Robert Speaker The Algebra Project Inc. Muller, Gretchen Participant Marin Teaching Network Murray, Margaret A.M. Participant ACT, Inc. Niazi, Ghazala Participant California High School Nkwanta, Asamoah Participant Morgan State University Noah, Josephine Participant Key Curriculum Press Odish, Faris academic sponsor Arizona State University, Tempe Campus Olkin, Julia Participant Cal State University East Bay Olson, Melfried Participant University of Hawaii Orr, John Academic Sponsor University of Nebraska Ovchinnikov, Sergei Participant San Francisco State University Pace, Deborah Ann Participant Stephen F. Austin State University Paterson, Anne F. Participant Santa Clara Unified School District Pederson, Stephanie Ann Participant Ida Crown Jewish Academy Pence, Barbara May Johnson Participant San Jose State University Peterson, Sara Participant City College of San Francisco Phillips, Betty Speaker Michigan State University Piontkowski, Dennis Participant City College of San Francisco

Page 10 of 11 Ragucci, Stephanie Speaker Andover High School Resek, Diane Speaker San Francisco State University Riehle, Patrick Participant Pearl City High School Roby, Tom Speaker University of Connecticut Roddick, Cheryl d Participant san jose state u. Roskam, Annette M Speaker CME Rothschild, Bruce Participant UCLA Russell, Susan Jo Speaker TERC Saba, Farrokh Participant UNLV University of Nevada, Las Vegas Sallee, Tom Speaker University of California, Davis Sally, Paul Speaker University of Chicago Saul, Mark E. Speaker National Science Foundation Schaar, Richard Academic Sponsor Texas Instruments, Inc. Schaefer, Diane Participant RI Department of Education Schifter, Deborah Speaker EDC Schmitt, Frederick Participant College of Marin Schoenfeld, Alan H. Organizer/Speaker UCB - University of California, Berkeley Seashore, Kimberly Participant U.C. Berkeley/LHS/CeMEE Segert, Jan Academic Sponsor University of Missouri Shah, Niral Participant University of California, Berkeley Shannon, Ann Participant Consultant Shaughnessy, Meghan Participant University of California, Berkeley Sherman, Jody Participant University of California, Berkeley Silver, Ed Organizer/Speaker University of Michigan Sitabkhan, Yasmin Participant UC Berkeley Slovin, Hannah Participant University of Hawaii Smith, Marianne Participant MSRI Smith, Melanie Rose Participant Math for America Steimle, Alice Participant University of Mississippi Stevens, Glenn Speaker Boston University Stevens, Harriette S. Participant UCB - University of California, Berkeley Tavormina, Mary Jo Participant Chicago Public Schools Terrell, Maria Academic Sponsor Cornell University Thaler, Alvin I. Participant NSF Thompson, Pat Speaker Arizona State University Tillema, Erik Participant Indiana University Indianapoli Titi, Edriss Academic Sponsor University of California - Irvine Treisman, Uri Speaker University of Texas, Austin Usiskin, Zalman Speaker University of Chicago Venenciano, Linda Participant University of Hawaii Ward, Sean P Participant Emery Secondary School White, Tobin Academic Sponsor UC Davis Wiles, Benjamin Christopher Academic sponsor Kansas State University Wilson, W Stephen Academic sponsor Johns Hopkins University Wolfson, Risa Ann Participant Lawrence Hall of Science Wong, Justine Academic Sponsor N/A Wright, David G. Academic Sponsor Brigham Young University Wu, Hung-Hsi Speaker University of California Young, Diane Hussey Participant Austin Independent School District Young, Yolanda Participant Oakland Unified School District

Page 11 of 11

Mathematical Systems Biology of Cancer II

Scientific Goals

A collaboration between LBNL and MSRI to provide scientific community outreach into the mathematics community in the area of cancer systems biology was founded in 2005 and extended in the fall of 2007. The 2007 effort again focused on the scientific underpinnings and analytical approaches being used by the NCI’s Integrative Cancer Biology Program (ICBP). The ICBP, which funded portions of the 2007 conference, focuses on developing systems approaches to biological problems with an emphais on biological processes involved in cancer. The ultimate goal of the 2007 series was to introduce these concepts and analytical methods to mathematicians.

Organizational Structure

The workshop was held over three days, October 24th to the 26th, 2007. Each day focused on a particular topic, where day 1 was dedicated to tutorials on computational and statistical methods in cancer system biology. Day two was centered around scientific discussions of biology, technology, and mathematical approaches to new scientific problems. Day three was discussions of particular efforts at data analysis of cancer systems biology.

Scientific Developments

Sixteen speakers presented their work in a total of eighteen presentations. In the tutorials session, Elizabeth Purdom (UC Berkeley) presented her work “Introduction to Exon Arrays “ as both a tutorial and a research seminar on the application of robust methods for identifying alternative splicing. Sach Mukherjee (University of Warwick, UK) described possible approaches to Bayesian reconstructions of signaling systems in a talk titled “Monte Carlo Methods For Learning Biological Networks”. Will Chen (Harvard Medical School) described properties of dynamical systems and fitting their behavior to real data in “Modeling Signal Transduction and Other Reaction Networks with ODEs”. Lang Li (Indiana University) in his talk “The Applications of Differential Equation Based Nonlinear Regression in Pharmacokinetics and Metabolomics” also described the utility of ordinary differential equations in modeling signaling systems. Serban Nacu (Genentech) described approaches for incorporating network analysis into the understanding of gene expression data in a talk titled “Searching for differentially expressed pathways in microarray data”. Ben Raphael (Brown University) presented work “Analyzing Structural Rearrangements in Cancer Genomes” where he described approaches to reconstructing the topology of cancer genomes (also in a research seminar on day 3). Finally, Susan Holmes (Stanford University) presented “New in situ data for the analysis of interactions between cancer and the immune system in auxillary lymph nodes” where she use classifiers to understand disease outcome from imaging data.

On the invited speaker day, five presentations were provided. Suzanne Gaudet (Harvard Medical School) described a systematic analysis of signaling systems in “Analysis of the regulators of caspase activation by death receptor ligands”. Neil Hayes (UNC) described clinical endpoints and genomic associations in “Genomic Classification of Lung Cancer for Clinical Use”. Lingchong You (Duke) in his talk “A Bistable Rb/E2F Switch: A Model for Mammalian Cell Cycle Entry” described a model of the cell cycle. Gary Nolan (Stanford University) presented an eclectic talk of biology, technology, and computational modeling in “Inference engines for primary cell signaling using patient samples”. Finally, Khuloud Jaqaman (Scripps) discussed the application of modeling approaches to the yeast cell cycle in “Stochastic models of yeast kinetochore-microtubule interactions”.

During the data analysis presentations six speakers presented. Mark Robinson (WEHI) presented “A Comparison of Affymetrix Gene Expression Arrays” where he outlined the performance of three types of array platforms. Elizabeth Purdom (UC Berkeley) further described alternative splicing detection in her talk, “Detection of Alternative Splicing with Exon Arrays”. Dustin Potter (Ohio State) described challenges associated with epigenomics in “The Analysis of Differential Methylation Hybridization Data”. Keith Baggerly (MD Anderson) presented his account of forensic biostatistics and bioinformatics in his work, “Cell Lines, Microarrays, Drugs and Disease: Trying to Predict Response to Chemotherapy”. Ben Raphael (Brown) described the novel mathematical framework needed to reconstruct genomes in “Derivation and Consequences of Rearranged Cancer Genomes”. And finally, Adam Olshen (Memorial Sloan Kettering) presented expanded work on cancer genome copy number analysis in “Segmentation of Array CGH Data with Application to Allele Specific Copy Number, Clonality, and Copy Number Variation”

Mathematical Systems Biology of Cancer II October 24, 2007 to October 26, 2007

Schedule Wednesday October 24, 2007

8:45AM - 9:00AM Welcome by MSRI 8:45AM - 5:00PM TUTORIALS 9:00AM - 9:45AM Elizabeth Purdom Introduction to Exon Arrays 9:45AM - 10:30AM Sach Mukherjee Monte Carlo Methods For Learning Biological Networks 11:00AM - 12:00PM William Chen Issues in Modeling Signal Transduction and Other Reaction Networks with ODEs 12:00PM - 1:30AM Lunch 1:30PM - 2:15PM Li Lang The Applications of Differential Equation Based Nonlinear Regression in Pharmacokinetics and Metabolomics 2:15PM - 3:00PM Serban Nacu Searching for differentially expressed pathways in microarray data 3:00PM - 3:30PM Tea break 3:30PM - 4:15PM Ben Raphael Analyzing Structural Rearrangements in Cancer Genomes 4:15PM - 5:00PM Susan Holmes New in situ data for the analysis of interactions between cancer and the immune system in auxillary lymph nodes 5:00PM - 6:00PM Reception

Thursday October 25, 2007

9:00AM - 5:00PM INVITED SPEAKER PRESENTATIONS 9:00AM - 10:00AM Suzanne Gaudet Analysis of the regulators of caspase activation by death receptor ligands 10:00AM - 10:30AM Coffee break 10:30AM - 11:30AM Neil Hayes Genomic Classification of Lung Cancer for Clinical Use 11:30AM - 12:30PM Lingchong You A Bistable Rb/E2F Switch: A Model for Mammalian Cell Cycle Entry 12:30PM - 1:30PM Lunch 1:30PM - 2:30PM Garry Nolan Inference engines for primary cell signaling using patient samples 2:30PM - 3:30PM Khuloud Jaqaman Stochastic models of yeast kinetochore-microtubule interactions 3:30PM - 4:00PM Tea break 4:00PM - 5:00PM Alan Perelson Modeling viral infections **MSRI Biology Colloquium Lecture Series**

Friday October 26, 2007

9:00AM - 5:00PM DATA ANALYSIS PRESENTATIONS 9:00AM - 9:45AM Introduction to ICBP data 9:45AM - 10:30AM Mark Robinson A Comparison of Affymetrix Gene Expression Arrays 10:30AM - 11:00AM Coffee break 11:00AM - 11:45AM Elizabeth Purdom Detection of Alternative Splicing with Exon Arrays 12:00PM - 1:30PM Lunch 1:30PM - 2:15PM Dustin Potter The Analysis of Differential Methylation Hybridization Data 2:15PM - 3:00PM Keith Baggerly Cell Lines, Microarrays, Drugs and Disease: Trying to Predict Response to Chemotherapy 3:00PM - 3:30PM Tea break 3:30PM - 4:15PM Ben Raphael Derivation and Consequences of Rearranged Cancer Genomes 4:15PM - 5:00PM Adam Olshen Segmentation of Array CGH Data with Application to Allele Specific Copy Number, Clonality, and Copy Number Variation

No Videos Available

Participant List

Name Role Institution Ahn, Soyeon Participant UC Berkeley Albertson, Donna G. Participant University of California San Francisco Arora, Jatinder k Participant LBNL Atherton, Juli K Participant UC Berkeley Baggerly, Keith Speaker UT M.D. Anderson Cancer Center Bani Asadi, Narges Participant Stanford University Barbour, Jason D Participant UC San Francisco Bayandorian, Hovig Participant Lawrence Berkeley Laboratory Bengtsson, Henrik Participant UC Berkeley Bhamidi, Sreekalyani Shankar Participant University of California Bradford, Jessica Participant Virginia Tech Capobianco, Enrico Participant Center for Advanced Studies, Research and Development in Sardinia Chang, Hang Participant LBNL Chappey, Colombe Participant Monogram Biosciences Chen, Jun Participant University of pennsylvania Chen, Rong Participant Stanford University Chen, William Speaker Harvard University Cohen, Dov Participant Sandia National Labs Crivelli, Silvia Noemi Participant LBNL Denning, John Participant Kaiser Permanente DeWoskin, Daniel Participant San Francisco State University Durinck, Steffen Participant LBL Durrett, Rick Participant Cornell University Flaherty, Patrick Participant Stanford University Fodor, Imola Participant Lawrence Livermore National Laboratory Fridlyand, Jane Participant Genentech Gallahan, Daniel Participant NIH/National Cancer Institute Gaudet, Suzanne Speaker MIT Gentles, Andrew Participant Stanford University Gibb, Bill Participant LBNL Gray, Joe W. Participant Lawrence Berkeley National Laboratory Han, Ju Participant Lawrence Berkeley National Laboratory Hayes, Neil Speaker University of North Carolina Heile, Frank Participant N/A Holmes, Susan Speaker Stanford University Hower, Valerie Marie Participant Wake Forest University Health Sciences/Virginia Bioinformatics Institute Itani, sleiman Participant Massachusetts Institute of Technology Jaqaman, Khuloud Participant The Scripps Research Institute Korsan, Bob Participant Decisions, Decisions! Laborde, Jose Miguel Participant Florida Atlantic University Lang, Li Speaker N/A Lenburg, Marc Participant Boston University School of Medicine Li, Caixia Participant University of California, San Francisco Li, Jingyi Participant UC Berkeley Lipkin, Efrem Participant CoDesign Liu, Zack Participant University of Pennsylvania Liu, Zhihua Participant Florida Atlantic University Lok, Larry Participant Molecular Sciences Institute Meza, Juan C. Participant LBNL Morgan, Alex Participant Stanford University Mukherjee, Sach Speaker University of Warwick Nacu, Serban Speaker Stanford University Ngunkeng, Grace Participant Inst for Pure and Applied Mathematics Nolan, Garry Speaker Stanford University Obeyesekere, mandri Participant UT M. D. Anderson Cancer Center Olshen, Adam B. Speaker Memorial Sloan-Kettering Cancer Center Paquet, Agnes Participant Monogram Biosciences Paquette, Jesse Participant UCSF Cancer Center Parvin, Bahram Participant LBNL Perelson, Alan Speaker Los Alamos National Laboratory Pilram, Atta Participant Bio Math Sciences Pinar, Ali Participant Lawrence Berkeley National Laboratory Pinkel, Daniel Participant University of California San Francisco Plevritis, Sylvia Participant Stanford University Poland, Bill Participant Pharsight Corporation Potter, Dustin Speaker Virginia Tech Purdom, Elizabeth Speaker Stanford University Qiao, Yingfeng Participant UCSF Qiu, Feng Participant California State University, East Bay Qiu, Peng Participant Stanford University Rajagopal, Ram Participant N/A Raphael, Ben Speaker University of California, San Diego Ravikumar, Bala Participant Sonoma State University Ray, Amrita Participant LBNL Rejniak, Katarzyna Anna Participant University of Dundee Ritz, Anna Participant Brown University Robinson, Mark Speaker Walter and Eliza Hall Institute Rocha, Guilherme Veiga Participant UC Berkeley Roth, Lindsey Participant Office of Environmental Health Hazard Assessment Roy, Ritu Participant UCSF Sachs, Karen Participant Stanford University Sadanandam, Anguraj Participant Lawrence Berkeley National Lab Sahoo, Debashis Participant Stanford University Sakuma, Thais Harumi Participant Federal State University of Rio de Janeiro Salari, Keyan Participant Stanford University Scheler, Gabriele Participant Stanford University Sims, Gregory E Participant Lawrence Berkeley National Lab Sopchak, Lynne Participant UCSC Speed, Terence P. Participant UC Berkeley Spellman, Paul Participant Lawrence Berkeley National Laboratory Sun, Shuying Participant The Ohio State University Taub, Margaret Participant UC Berkeley Tikhonova, Irina Participant N/A Tokuyasu, Taku Participant UCSF Tong, Frances Participant UC, Berkeley Venkatasubrahmanyam, Shivkumar Participant Stanford University Wang, Nancy Participant UC Berkeley Wang, Wenyi Participant Stanford University Wu, G. Albert Participant LBNL Wu, Henry Participant N/A Xiao, Yuanyuan Participant UCSF Yeh, Ru-Fang Participant UCSF You, Lingchong Speaker Duke University Zhang, Gao Participant University of Pennsylvania Zhang, Michael Yu Participant DOE Joint Genome Institute Zhang, Nancy Ruonan Participant Stanford University Zhao, Qian Participant UCSF Zhao, Shoujun Participant University of California, San Francisco Zwart, Peter Participant Lawrence Berkeley national Laboratories

MSRI 25th Anniversary Celebration

January 27–30, 2008

Organized by Alejandro Adem, Isadore Singer, and Robert Bryant

The focus of this scientific workshop was on currently active areas of mathematics in which MSRI has played a significant role. There were 16 research-expository talks of extremely high quality (detailed below), a panel on mathematics education chaired by Deborah Ball, and a panel on the past, present, and future of MSRI.

In addition, there were a reception and a banquet, at which the Nobel Laureate Stephen Chu spoke on the various definitions of energy in physics and their mathematical significance.

The talks and panels were recorded and are available for viewing on MSRI’s web site. There were over 150 registered participants.

Talks:

Persi Diaconis: “Harnessing Chance; The Case of Random Packing”

The twentieth century was witness to the taming of chance.

Today we are harnessing chance; to approximate impossible high-dimensional integrals, solve intractable counting problems, understand the structure of the genome, and many further tasks from grand to mundane. Mathematical understanding of these new techniques lags way behind applications.

I will illustrate with the physics/chemistry problem of random packing of hard spheres in a box. As usual, I will begin by trying to answer ‘who cares.’ This was the first modern application (Metropolis, et.al.). For Sparse systems we can give useful results using micro-local and geometric analysis techniques (work with Gilles Lebeau). The general problem, particularly the most interesting dense case, is wide open.

Andrei Okounkov: “Counting curves in 3-folds”

This will be a leisurely discussion of some conjectures and theorems about the enumerative geometry of algebraic curves in 3-folds and related combinatorial structures.

Ravi Vakil: “Murphy's Law in algebraic geometry: Badly-behaved moduli spaces”

We consider the question: “How bad can the deformation space of an object be?” (Alternatively: “What singularities can appear on a moduli space?”) The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space can be arbitrarily bad.” We show this for a number of important moduli spaces.

More precisely, up to smooth parameters, every singularity that can be described by equations with integer coefficients appears on moduli spaces parameterizing: smooth projective surfaces (or higher-dimensional manifolds); smooth curves in projective space (the space of stable maps, or the Hilbert scheme); plane curves with nodes and cusps; stable sheaves; isolated threefold singularities; and more. The objects themselves are not pathological, and are in fact as nice as can be. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise.

I will begin by telling you what ‘moduli spaces’ and ‘deformation spaces’ are. The complex-minded listener can work in the holomorphic category; the arithmetic listener can think in mixed or positive characteristic. This talk is intended to be (mostly) comprehensible to a broad audience.

Daniel Freed: “Remarks on Chern-Simons theory”

The Chern-Simons invariant was introduced into differential geometry in the early 1970s. Its quantum embodiment at the end of the 1980s quickly became a poster child quantum field theory for mathematicians: not only does it place knot polynomials in a manifestly three-dimensional context, but it also reveals many algebraic and topological aspects of quantum field theory in general. The mathematics involved strays far from its differential-geometric origins. I will review some developments in this area and use Chern-Simons theory as a window into the broader math-physics interaction.

Bryna Kra: “Beyond Fourier Analysis”

Much recent work in ergodic theory has been motivated by interactions with combinatorics and with number theory. A striking example is Szemeredi's Theorem, which states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon after Szemeredi's proof, Furstenberg gave a new proof using ergodic theory. This opened new questions in ergodic theory, and developments in ergodic theory, in turn, have lead to breakthroughs in additive combinatorics. While Fourier analysis is useful for understanding some patterns (such as arithmetic progressions of length 3), it does not suffice for understanding general patterns. It turns out that algebraic constraints (nilsystems) play a key role in understanding the more complicated phenomena, both in additive combinatorics and in ergodic theory. I will give an overview of the role of nilsystems in the recent developments, explaining the beginnings of a theory of higher order Fourier analysis that is the main tool for addressing open problems in the area.

George Papanicolaou: “Sensor Imaging”

Imaging to a mathematician is a special class of inverse problems in analysis, differential equations and probability, which are typically ill-posed.

They have played an important role in the development of mathematical methods that turned out to have broader significance and applicability.

But in the sciences and in engineering, imaging can mean many different things, including the recovery of the approximate location and properties of an object from the echoes of waves received at various sensors placed in the environment of the object.

Seismic imaging, sonar, radar, diagnostic imaging with ultrasound, etc., are examples. There is now an emerging interdisciplinary science of sensor imaging that has an interesting mathematical theory. In this lecture, I will describe the basic elements of this theory. I will also compare the theoretical performance of some special ultrasonic imaging systems to the actual performance of similar systems in nature, the ones used by the dolphin and by the bat.

Not surprisingly, evolution has created bio-sonar systems that perform much better than current theories predict. What is missing in our theoretical understanding?

Peter Ozsvath: “Heegaard Floer homology and knots”

Heegaard Floer homology is an invariant for low-dimensional manifolds defined using Heegaard diagrams and holomorphic disks, constructed in joint work with Zoltán Szabó. These constructions can be specialized to give an invariant for knots in the three-sphere, knot Floer homology, which has the structure of a bigraded whose graded Euler characteristic is the Alexander polynomial. Unlike the Alexander polynomial, however, knot Floer homology contains precise geometric information about the knot: It encodes the knot genus.

I will discuss applications of this theory, as well as several recent advances in explicitly calculating these invariants in elementary terms. I will describe joint work with collaborators including Ciprian Manolescu, Sucharit Sarkar, András Stipsicz, Zoltán Szabó, and Dylan Thurston.

Gunnar Carlsson: “Topology and Data”

The science and engineering disciplines are producing enormous volumes of data from many different experimental sources. The data comes in many forms, and developing methods for usefully analyzing it is of great importance. In this talk, we will discuss some methods arising out of topology for extracting qualitative information from these data sets. We will discuss persistent homology, topological methods for providing ‘roadmaps’ to the data, and homotopy theoretic methods for analyzing the stability of the road methods.

Kenneth Ribet: “Class groups and Galois representations”

When I arrived in Berkeley, the New Institute was a topic of active conversation and intense planning, but it was not yet open for business. Soon after I began working in this town, the MSRI opened its doors at the west edge of the campus, and we all watched the construction as it progressed on the hill. For this anniversary celebration, I will revisit the subject of my first mathematical lecture on Berkeley, which concerned the construction of unramified extensions of cyclotomic fields through modular forms and Galois representations. The technique that I exposed in that lecture is currently being applied in new contexts (Skinner–Urban, Dasgupta,…). Meanwhile, the theorem that I announced at the time was proved later on (c. 1988) in a completely different manner by techniques of Thaine, Kolyvagin, and Rubin.

Richard Melrose: “Configuration and Moduli Spaces”

(No abstract is available.)

Charles Fefferman: “Whitney's extension problem and interpolation of functions”

Fix positive integers m,n. Let E be a subset of R^n, and let f be a given real-valued function on E. How can we tell whether f extends to a C^m function F on the whole R^n? If F exists, how small can we make its C^m norm? What can we say about the derivatives of F at a given point? Can we take F to depend linearly on f? What if we demand only that F agree approximately with f on E? If E is finite, can we compute a nearly optimal F from f? How many operations does it take? What if we are allowed to discard a few points from E?

Many of the results presented are joint work with Bo'az Klartag.

Inez Fung: “Mathematics of Climate Change”

Climate models synthesize available observations, and forecast (and hindcast) climate change. This talk reviews the history and mathematics of climate modeling and present new mathematical challenges for predicting climate of the 21st century.

Vaughan Jones: “MSRI and interactions between operator algebras, physics and low dimensional topology”

In the academic year 1984-1985 MSRI ran two simultaneous year-long programs, one in operator algebras and the other in low dimensional topology. The interactions that occurred were at the beginning of a lively and enduring area of mathematics centered around what is now known as TQFT (topological quantum field theory).

MSRI meetings have continued to inspire and unify work in this and neighboring areas. I will attempt to give a historical survey of this and other closely related areas.

H. Blaine Lawson: “Dirichlet Duality and the Nonlinear Dirichlet Problem”

(No abstract is available.)

Paul Seidel: “Pair-of-pants decompositions”

I will explain some of the facts about pair-of-pants decompositions as they occur in low- dimensional topology; higher-dimensional generalizations suggested by tropical geometry (Mikhalkin); and some hints of how this might be useful in symplectic geometry. This subject is still young but hopefully we'll find out more in 2009/10, when all of these fields will be represented at MSRI.

Michael Hopkins: “Topological Field Theories”

The notion of a topological field theory has proved to be an extraordinarily useful scheme for organizing and known invariants of manifolds and for predicting the structure of new ones. In this talk I'll explain the notion of a topological field theory, and the way in which trying to classify them forces topologists to re-examine some of the oldest assumptions in topology. I'll also explain recent joint work with giving a complete classification of topological field theories in (very) low dimensions.

Panels:

Education Panel: “MSRI and Mathematics Education”

This panel, chaired by Deborah Ball, featured the panelists Deborah Ball, Herbert Clemens, Ricardo Cortez, Hugo Rossi, and Hung-Hsi Wu.

History Panel: “MSRI: and Mathematics Education”

This panel, chaired by Robert Bryant, featured the panelists Alejandro Adem, , Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Isadore Singer, and Alvin Thaler

Schedule

Sunday January 27, 2008

8:30AM - 9:30AM Breakfast and Refreshments Harnessing chance; the case of random 9:30AM - 10:30AM Persi Diaconis packing 10:30AM - 11:00AM Coffee Break 11:00AM - 12:00PM Andrei Okounkov Counting curves in 3-folds 12:00PM - 2:00PM Lunch Murphy's Law in algebraic geometry: Badly- 2:00PM - 3:00PM Ravi Vakil behaved moduli spaces 3:00PM - 3:30PM Afternoon Tea 3:30PM - 4:30PM Daniel Freed Remarks on Chern-Simons theory 4:30PM - 5:30PM Reception

Monday January 28, 2008

8:30AM - 9:30AM Breakfast and Refreshments 9:30AM - 10:30AM Bryna Kra Beyond Fourier Analysis 10:30AM - 11:00AM Coffee Break 11:00AM - 12:00PM George Papanicolaou Sensor Imaging 12:00PM - 1:30PM Lunch 1:30PM - 2:30PM Peter Ozsvath Heegaard Floer homology and knots 2:30PM - 3:30PM Topology and Data 3:30PM - 4:00PM Afternoon Tea Education Panel: Deborah Ball 4:00PM - 6:00PM Herbert Clemens: "Coalitions between mathematicians and educators seeking change: Shared purpose, shared means" Ricardo Cortez Hugo Rossi: "Math Circles at MSRI" Hung-Hsi Wu

Tuesday January 29, 2008

8:30AM - 9:30AM Breakfast and Refreshments 9:30AM - 10:30AM Kenneth Ribet TBD 11:00AM - 12:00PM Karen Uhlenbeck Gauge Theory: Old and New 12:00PM - 1:30PM Lunch Whitney's extension problem and 1:30PM - 2:30PM Charles Fefferman interpolation of functions 2:30PM - 3:30PM Inez Fung Mathematics of Climate Change Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Panel Discussion MSRI: Past, Present, and 4:00PM - 6:00PM Eisenbud, Calvin Moore, Future Isadore Singer, Alvin Thaler 6:00PM - 9:00PM Banquet: Special Guest Speaker Steven Chu

Wednesday January 30, 2008

8:30AM - 9:30AM Breakfast and Refreshments MSRI and interactions between operator 9:30AM - 10:30AM Vaughan Jones algebras, physics and low dimensional topology 10:30AM - 11:00AM Coffee Break Dirichlet Duality and the Nonlinear Dirichlet 11:00AM - 12:00PM H. Blaine Lawson Problem 12:00PM - 1:30PM Lunch 1:30PM - 2:30PM Pair-of-pants decompositions 2:30PM - 3:30PM Michael Hopkins TBD 3:30PM - 4:00PM Afternoon Tea

Currently Available Videos

• Persi Diaconis , HARNESSING CHANCE; THE CASE OF RANDOM PACKING January 27,2008, 08:30 AM to 09:30 AM

• Andrei Okounkov , Counting curves in 3-folds January 27,2008, 11:00 AM to 12:00 PM

• Ravi Vakil , Murphy's Law in algebraic geometry: Badly-behaved moduli spaces January 27,2008, 02:00 PM to 03:00 PM

• Daniel Freed , Remarks on Chern-Simons theory January 27,2008, 03:30 PM to 04:30 PM

• Bryna Kra , Beyond Fourier Analysis January 28,2008, 09:30 AM to 10:30 AM

• George Papanicolaou , Sensor Imaging January 28,2008, 11:00 AM to 12:00 PM

• Peter Ozsváth , Heegaard Floer homology and knots January 28,2008, 01:30 PM to 02:30 PM

• Gunnar Carlsson , Topology and Data January 28,2008, 02:30 PM to 03:30 PM

• Deborah Ball, Herb Clemens, Ricardo Cortez, Hugo Rossi, Hung-Hsi Wu , Education Panel January 28,2008, 04:00 PM to 06:00 PM

• Deborah Ball, Herb Clemens, Ricardo Cortez, Hugo Rossi, Hung-Hsi Wu , Education Panel January 28,2008, 04:00 PM to 06:00 PM

• Deborah Ball, Herb Clemens, Ricardo Cortez, Hugo Rossi, Hung-Hsi Wu , Education Panel January 28,2008, 04:00 PM to 06:00 PM

• Deborah Ball, Herb Clemens, Ricardo Cortez, Hugo Rossi, Hung-Hsi Wu , Education Panel January 28,2008, 04:00 PM to 06:00 PM

• Deborah Ball, Herb Clemens, Ricardo Cortez, Hugo Rossi, Hung-Hsi Wu , Education Panel January 28,2008, 04:00 PM to 06:00 PM

• Kenneth Ribet , Class groups and Galois representations January 29,2008, 09:30 AM to 10:30 AM

• Richard Melrose , Configuration and Moduli Spaces January 29,2008, 11:00 AM to 12:00 PM

• Charles Fefferman , Whitney's extension problem and interpolation of functions January 29,2008, 01:30 PM to 02:30 PM

• Inez Fung , Mathematics of Climate Change January 29,2008, 02:30 PM to 03:30 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM • Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Alejandro Adem, Lenore Blum, Robert Bryant, Tony Chan, David Eisenbud, Calvin Moore, Alvin Thaler , Panel Discussion MSRI: Past, Present, and Future January 29,2008, 04:00 PM to 06:00 PM

• Vaughan Jones , MSRI and interactions between operator algebras, physics and low dimensional topology January 30,2008, 09:30 AM to 10:30 AM

• H. Blaine Lawson , Dirichlet Duality and the Nonlinear Dirichlet Problem January 30,2008, 11:00 AM to 12:00 PM

• Paul Seidel , Pair-of-pants decompositions January 30,2008, 01:30 PM to 02:30 PM

• Michael Hopkins , Topological Field Theories January 30,2008, 02:30 PM to 03:30 PM

Participant List

Name Role Institution Abadie, Beatriz CAS Rep University of the Republic Abrams, Aaron David CAS Rep Emory University Adams, Allan Participant MIT Adem, Alejandro Organizer PIMS - Pacific Institute for the Mathematical Sciences Alperin, Roger C. CAS Rep San Jose State University Assaf, Sami H Participant MIT Axler, Sheldon Participant San Francisco State University Ball, Deborah Loewenberg Speaker University of Michigan Bass, Hyman Governance University of Michigan Beck, Matthias Participant San Francisco State University Beheshti Zavareh, Roya CAS Rep Washington University Benkart, Georgia M. Participant University of Wisconsin Bi, Shuchau Participant University of California Bleiler, Steven A. CAS Rep Portland State University Blum, Lenore Speaker Carnegie Mellon University Bourguignon, Jean-Pierre Participant IHÉS Bracho, Javier Participant Instituto de Matematicas, UNAM Bryant, Robert Leamon Organizer MSRI Buhler, Joe P. Participant CCR Burnside, Beth Participant UCB - University of California, Berkeley Carlson, Jon F. Participant University of Georgia Carlsson, Gunnar Speaker Stanford University Chan, Tony F. Speaker Mathematics and Physical Sciences (MPS) Chen, Haojie Participant University of Minnesota Chen, Sophie Participant UC Berkeley Chu, Steven Speaker UC, Berkeley Ciubotaru, Dan CAS Rep University of Utah Clemens, Herb Speaker Ohio State University Cochran, George CAS Rep Louisiana State University Cortez, Ricardo Speaker Tulane University Dadok, Jiri CAS Rep Indiana University Dasgupta, Samit CAS Rep Harvard University Daugherty, Zajj Participant University of Wisconsin di Francesco, Philippe Participant Service de Physique Theorique Diaconis, Persi Speaker Stanford University Diaz, Rafael Participant Universidad Politecnica de las Fuerzas Armadas Edwards, David CAS Rep University of Georgia Edwards, Robert Participant UCLA Eisenbud, David Speaker University of California Elkies, Noam CAS Rep Harvard University Engheta, Bahman CAS Rep University of California, Riverside Evans, Craig Participant UC, Berkeley Fefferman, Charles Speaker Columbia University Feldman, Jacob Participant UCB - University of California, Berkeley Feshbach, Mark F. Participant National Science Foundation Freed, Daniel S. Speaker University of Texas, Austin Friedlander, Susan CAS Rep Northwestern University Fung, Inez Speaker UC, Berkeley Galatius, Soren Participant Stanford University Goodman, Frederick Participant University of Iowa Goroff, Daniel L. Participant Harvey Mudd College Grant, David CAS Rep University of Colorado Hales, Alfred W. Participant Institute for Defense Analyses (CCR-LJ) Harbourne, Brian CAS Rep Univ Nebraska-Lincoln Hartke, Stephen CAS Rep University of Nebraska-Lincoln Helminck, Aloysius Gerardus CAS Rep NC State University Hemmer, David J. Participant State University at Buffalo, SUNY Hernandez, David Participant N/A Holmes, Susan Participant Stanford University Hopkins, Michael J. Speaker Harvard University Jones, Christopher R. T. Participant University of North Carolina, Chapel Hill Jones, Vaughan Speaker UCB - University of California, Berkeley Juteau, Daniel Pierre Participant Jussieu University Kalka, Morris CAS Rep Tulane University Kania-Bartoszynska, Joanna Participant National Science Foundation Kaper, Hans G. Participant NSF - National Science Foundation Kedem, Rinat Participant University of Illinois, Urbana-Champaign Kim, Hee Jung CAS Rep Louisiana State University Kirby, Robion C. Participant UCB - University of California, Berkeley Klawe, Maria M. Participant Harvey Mudd College Koev, Plamen CAS Rep North Carolina State University Kovchegov, Yevgeniy CAS Rep Oregon State University Kra, Bryna Speaker Northwestern University Kra, Irwin Participant Math for America Kujawa, Jonathan R Participant University of Oklahoma Lam, T. Y. Participant University of California Lawson, H. Blaine Speaker SUNY, Stony Brook Lawson, Kris Participant East Bay Express Lin, En-Bing CAS Rep Central Michigan University Lind, Douglas CAS Rep University of Washington Lockhart, Deborah Frank Participant NSF - National Science Foundation Lotay, Jason Participant University College Oxford Centro de Investigacion y Estudios Avanzados - Lupercio, Ernesto Participant CINVESTAV Makarov, Nikolai CAS Rep California Institute of Technology March, Peter Participant National Science Foundation Maroti, Attila Participant USC Mbirika, Abukuse (Aba) Participant University of Iowa Megginson, Robert Eugene Participant University of Michigan, LSA Dean's Office Melrose, Richard Burt Participant MIT - Massachusetts Institute of Technology Meza, Juan C. Participant LBNL Milewski, Paul CAS Rep University of Wisconsin Miller, C. Keith Participant UC, Berkeley Moci, Luca Participant Roma Tre Moore, Calvin C. Speaker UCB - University of California, Berkeley Moreno, Santiago CAS Rep University of British Columbia Nie, Jiawang CAS Rep UCSD Noll, Jennifer CAS Rep Portland State University Okounkov, Andrei Speaker Princeton University Orellana, Rosa Participant Dartmouth College Orrison, Michael Participant Harvey Mudd College Ovchinnikov, Sergei Participant San Francisco State University Ozsvath, Peter Steven Speaker Columbia University Panaretos, Victor Participant Ecole Polytechnique Fédérale de Lausanne Papanicolaou, George C. Speaker Stanford University Parlett, Beresford N. Participant UC, Berkeley Pereira, Mariana CAS Rep Facultad de Ciencias, Universidad de la Republica Politis, Dimitris CAS Rep University of California, San Diego Poonen, Bjorn Participant UCB - University of California, Berkeley Previato, Emma CAS Rep Boston University Prokhorenkov, Igor Participant TCU Ram, Arun Participant University of Melbourne Rehmeyer, Julie Participant N/A Ribet, Kenneth A. Speaker UC Berkeley Roberts, Paul Calvin Participant University of Utah Rossi, Hugo Speaker University of Utah Roth, Ilan Participant UC Berkeley Rufus, Anneli Participant East Bay Express Sadosky, Cora Participant Howard University Saint-Donat, Bernard Participant N/A Schmitt, Frederick Participant College of Marin Seidel, Paul Alfred Speaker MIT Simon, Horst D. CAS Rep LBNL Singer, Isadore M. Organizer MIT - Massachusetts Institute of Technology Singh, Anurag Kumar CAS Rep University of Utah Srinivasan, Bhama Participant University of Illinois, Chicago Strain, John Participant UCB - University of California, Berkeley Straus, Sandor Participant Firedoll Foundation Sun, Jingzhou Participant Johns Hopkins University Sun, Shenghao Participant UC Berkeley Swisher, Holly CAS Rep Oregon State University Symonds, Peter Participant University of Manchester Teleman, Constantin Participant University of California Thaler, Alvin I. Speaker NSF Thiem, Nat Participant University of Colorado Thomas, Jean C. Participant UC Berkeley Torres, Hector Participant N/A Treinen, Ray CAS Rep Kansas State University Uhlenbeck, Karen K. Speaker University of Texas, Austin Vakil, Ravi Speaker Stanford University Velasco, Mauricio Fernando Participant UC Berkeley Voiculescu, Dan Virgil Participant UCB - University of California, Berkeley Vojta, Paul Alan Participant UCB - University of California, Berkeley Vu, Vincent Quang Participant UCB - University of California, Berkeley Wagner, Michelle D. Participant National Security Agency Wahl, Bernt Participant N/A Weinstein, Alan Participant UCB - University of California, Berkeley Wood, Carol S. CAS Rep Dept of Math, Wesleyan University Wu, Hung-Hsi Speaker University of California Wunsch, Jared Participant Northwestern University Xu, Feng CAS Rep University of California, Riverside Yamamoto, Makoto Participant University of California Yan, Donghui Participant UCB - University of California, Berkeley Yetter, David N. CAS Rep Kansas State University Yianilos, Peter N. Participant N/A Zelevinsky, Andrei Participant Northeastern University Zheng, Bowei Participant UC Berkeley Zhu, Yi Participant SUNY Stony Brook

Exterior Differential Systems and the Method of Equivalence Held May 5-9, 2008 at MSRI Organized by Jeanne Clelland, William Shadwick (Chair), and George Wilkens

1 Exterior Differential Systems and the Method of Equiva- lence

February 2008 marks the 100th anniversary of the publication of Elie´ Cartan’s paper, “Les Sous-Groupes des Groupes Continus de Transformations,” and the introduction to the Method of Equivalence, Cartan’s tremendously powerful machinery for uncovering ge- ometry. Thanks in large part to the contributions of two subsequent masters of this mathe- matics, S.-S. Chern and Robby Gardner, Cartan’s Method of Equivalence and his theory of Exterior Differential Systems have found applications from pure mathematics to mechanics, engineering, and computer science. This workshop brought together leading practitioners in Exterior Differential Systems and the Method of Equivalence to explore recent developments including: the study of Rieman- nian submersions; holonomy; almost complex manifolds; representation theory and math- ematical physics; integrable systems and conservation laws; partial differential equations, including Backlund transformations and solitons; control theory, and a new central limit theorem. May 2008 also marks the 10th anniversary of Robby Gardners untimely death, and this workshop is held in his honor.

2 Organizational Structure

The workshop followed the standard MSRI 5-day workshop format. There were 19 hour-long talks (described in the next section), as well as an informal presentation by Ian Anderson on his DifferentialGeometry package, which is now included in the computer algebra system MAPLE. We had a wonderful workshop dinner at a local Chinese restaurant, where Pat Eberlein and Robert Bryant shared fond memories of Robby Gardner. There were 61 registered participants, and lively interactions abounded throughout the workshop. One graduate student was heard to comment, ”I feel like I’m attending a family reunion instead of a workshop,” and several participants expressed the opinion that this was the best conference that they had ever attended.

1 3 Scientific Program

Monday, May 5: 9:30 - 10:30: Robert Bryant, “Riemannian foliations in space forms” The PDE system that governs Riemannian foliations is examined using the methods of exterior differential systems and the moving frame. In various special cases and in low dimension, a complete classification is obtained. 11:00 - 12:00: Colleen Robles, “Rigidity of projective homogeneous varieties and Lie algebra cohomology” The problem of identifying homogeneous varieties from their local differential geometry dates back to Monge, and has been studied by Fubini, Griffiths and Harris, Hwang and Yamaguchi, and others. I will describe recent work with J.M. Landsberg that establishes a general recognition theorem. The key component is the resolution of exterior differential systems by Lie algebra cohomology. 2:00 - 3:00: Keizo Yamaguchi, “Contact geometry of second order” I want to summerize the recent developments of the subjects treated in the “Five Varibles Paper” of E.Cartan, which Robby loved to read. 4:00 - 5:00: Michael Eastwood, “Prolongation on contact manifolds” Prolongation is a classical technique used to explore the consequences of a system of PDE. In interesting examples, the PDE in question are closely tied to some geometric structure on the underlying manifold. The Killing equations or conformal Killing equations on a vector field provide classical examples in Riemannian geometry. In these cases, the general theory of prolongation (due to Goldschmidt, Spencer, and others) is perfectly suited to give the best possible constraints on the solution space. But sometimes it is desirable to modify the prolongation procedure itself in order better to take account of the underlying geometry. In CR geometry, for example, one should bear in mind and utilise the natural contact structure. The aim of this talk is briefly to present classical prolongation from the point of view of connections and then to explain how this theory should be modified on contact manifolds. The main ingredients are differential geometry and Lie theory. This is joint work with Rod Gover. Tuesday, May 6: 9:30 - 10:30: Ana Cascon, William Shadwick, “The geometry of probability distributions: a new central limit theorem” Omega functions are a recently discovered way of representing probability distributions with finite first moment. The geometry of omega functions and the equivalence problem for the affine group reveals remarkable structure in the space of such distributions. This includes a natural measure of dispersion about the mean, improvements on the Markov and Chebychev inequalities, a new affine invariant and a new central limit theorem. 11:00 - 12:00: Shankar Sastry, “Parking cars with N trailers and nonholonomy: reflections on the applications of BC3G”

2 1:30 - 2:30: Pawel Nurowski, “GL(2,R) geometry of ODEs” 2:30 - 3:30: Hubert Goldschmidt, “Infinitesimal isospectral deformations of symmetric spaces of compact type” Let (X, g) be an irreducible symmetric space of compact type. According to a result of Guillemin, the infinitesimal deformation corresponding to an isospectral deformation of the metric g belongs to the kernel of a certain Radon transform acting on the symmetric 2- forms on X. This is the motivation for defining the space I(X) of infinitesimal isospectral deformations of X as a subspace of the kernel of this Radon transform. If I(X) vanishes, an isospectral deformation of the metric g is trivial to first-order. We shall give an overview of our joint work with Jacques Gasqui concerning the space I(X): 1) A necessary condition for the vanishing of I(X) is that it be reduced, i.e., it is not the cover of another symmetric space. 2) If X is the reduced space of a Grassmannian, the space I(X) vanishes. This was known when X is a projective space which is not equal a sphere; using work of Duistermaat- Guillemin, this leads to spectral rigidity results for the projective space. 3) If X is the reduced space of the special Lagrangian Grassmannian SU(n)/SO(n), or of the SU(n), or of the symmetric space SU(2n)/Sp(n), with n ≥ 3, the space I(X) does not vanish and we give explicit constructions of non-trivial infinitesimal deformations. 4) We determine the space I(X) when X is the reduced space of the symmetric space SU(3)/SO(3) or of the unitary group SU(3). Wednesday, May 7: 9:30 - 10:30: Niky Kamran, “Wave equations and the Penrose process in Kerr geometry” I will present a rigorous proof of super-radiance, the wave analogue of the Penrose process, for the scalar wave equation in the Kerr metric. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau 11:00 - 12:00: Thomas Ivey, “B¨acklund transformations and Darboux integrability for non- linear wave equations” A B¨acklund transformation between two EDS is a common integrable extension; here, we consider transformations between hyperbolic Monge-Amp`ere(MA) systems defined by 6- dimensional double fibration over two 5-manifolds. We show that a hyperbolic MA system is linked by such a B¨acklund transformation to the standard wave equation if and only if the MA system is Darboux-integrable after one prolongation. This is joint work with Jeanne Clelland. Thursday, May 8: 9:30 - 10:30: Peter Vassiliou, “Geometric aspects of differential equations in the work of Robert B. Gardner” In his PhD thesis in 1965, Robert B. Gardner initiated the development of some tools and ideas for the geometric study of differential equations that have inspired subsequent generations of mathematicians. In this talk I will attempt to review a few strands of this aspect of Robert Gardners mathematical work and give a personal view of how these ideas have been further developed in more recent times.

3 11:00 - 12:00: Joseph Landsberg, “Lines and asymptotic lies of projective varieties” n n+1 Let X ⊂ CP be a hypersurface defined as the zero set of a degree d polynomial with d ≤ n. Such hypersurfaces have lines through each point x ∈ X. Let Cx ⊂ P(TxX) denote the set of tangent directions to lines on X passing through x. Jun-Muk Hwang asked how Cx varies as one varies x. The answer turns out to be interesting, with two natural exterior differential systems governing the motion. In addition to describing these EDS and some immediate consequences, I will also discuss applications to questions in computational complexity and algebraic geometry. This is joint work with C. Robles. 2:00 - 3:00: Christine Scharlach, “Symmetries for cubic forms and adapted frames” In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the cubic form C (resp. the difference tensor K). A hypersurface is said to admit a pointwise group symmetry if at every point h, S and K are preserved under the group action. Necessarily the possible groups must be subgroups of the isometry group and the symmetry condition helps to restrict the frame bundle. The study of submanifolds which admit a pointwise group symmetry was initi- 3 ated by Bryant; he studied 3-dimensional Lagrangian submanifolds of C . The concept was introduced in affine hypersurface theory by Vrancken. He gave a classification of 3- dimensional positive definite affine hyperspheres admitting pointwise isometries, which was extended by Lu and myself to hypersurfaces. In all cases the isometry group is SO(3), only in the latter the invariance of the shape operator is an extra condition. We will report on the classification of 3-dimensional indefinite affine hyperspheres (i. e. the isometry group is SO(1, 3)). Some of the classes are well known (constant curvature, homogeneous), but also we obtain many interesting classes which are warped products of 2-dimensional affine spheres with curves. 4:00 - 5:00: Phillip Griffiths, “Exterior differential systems in algebraic geometry” (Berkeley Math Dept. colloquium) Exterior differential systems (EDS) provide the basic tool for studying the geometry of differential equations. There is a canonical exterior differential system that governs the behavior of the Hodge structure in a family of algebraic varieties. We will review the basics of EDS’s and explain two central open questions. Then we will recall the definition of the canonical systems and give two applications to algebraic geometry. Friday, May 9: 9:30 - 10:30: Jeanne Clelland, “Sub-Finsler geometry in dimensions three and four” Motivated by examples from optimal control theory, we consider the notion of sub-Finsler geometry and show how it is a natural generalization of sub-Riemannian geometry. In dimensions three and four, we can compute a complete set of local invariants and derive geodesic equations, whose solutions represent optimal paths for the corresponding control theory problem. We will show examples and discuss some of the difficulties that arise in higher dimensions. This work is joint with Christopher Moseley and George Wilkens. 11:00 - 12:00: Ian Anderson, “Symmetry reduction and Darboux integrability” In this talk I shall describe some new applications of the theory of symmetry reduction to the study of Darboux integrable differential systems. I shall propose a nonlinear generalization

4 of the classical Laplace transform, address the problem of when the Cauchy problem for scalar, second order pde in the plane can be solved by quadratures and and outline a new proof of Moutard’s theorem. 1:00 - 2:00: Vladimir Matveev, “Integrals quadratic in velocities and solution of two prob- lems of Sophus Lie” In 1882 Sophus Lie asked to describe all 2-dimensional metrics admitting vector fields whose flows sends (unparameterized) geodesics to geodesics. I will explain the solution of this problem (part of the results are joint with R.L. Bryant and G. Manno). The results are published in arXiv:0705.3592, arXiv:0802.2344, and arXiv:0802.2346. The solution uses a description of metrics whose geodesic flows admit integrals quadratic in velocities; I will give an overview of this subject. Besides this theory, the solution consists of a trick and huge calculations. I will explain the trick and hope you will explain me what other classical problems one can hope to solve by using similar calculations. 2:30 - 3:30: Kris Jenssen, “Hyperbolic conservation laws with prescribed eigencurves” Motivated by recent examples of singular behavior in systems of hyperbolic conservation laws in one space dimension, we consider the following problem: Given n vector fields n R1, ..., Rn in R ; determine if there exists an n × n-system of hyperbolic conservation laws with a flux function whose Jacobian has the Ri as right eigenvectors. We formulate this as a problem for the eigenvalues of the system. For n ≥ 3 this typically gives an over- determined system of PDEs for the eigenvalues to which we apply Cartan-K¨ahler theory. Several concrete examples are also given. This is joint work with Irina Kogan (NCSU). 4:00 - 5:00: George Wilkens, “Second-order type-changing evolution equations with first- order intermediate equations” We present a partial classification for smooth, type-changing symplectic Monge-Ampere partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems. This is joint work with Jeanne Clelland and Marek Kossowski.

Videos of all talks are available at http://www.msri.org/calendar/workshops/WorkshopInfo/446/show workshop.

5

Exterior Differential Systems and the Method of Equivalence May 05, 2008 to May 09, 2008

Schedule Monday May 5, 2008 9:30AM - Robert Bryant Riemannian Foliations in Space Forms 10:30AM 10:30AM - Tea in the Lobby 11:00AM 11:00AM - Rigidity of projective homogeneous varieties and Lie Joseph Landsberg, Colleen Robles 12:00PM algebra cohomology 12:00PM - Lunch 2:00PM 2:00PM - Keizo Yamaguchi Contact Geometry of Second Order 3:00PM 3:00PM - Tea 4:00PM 4:00PM - Michael Eastwood Prolongation on Contact Manifolds 5:00PM Tuesday May 6, 2008 9:30AM - The Geometry of Probability Distributions: A New Central Ana Cascon, William Shadwick 10:30AM Limit Theorem 10:30AM - Tea 11:00AM 11:00AM - Shankar Sastry : Parking cars with N trailers and Nonholonomy: reflections on the 12:00PM applications of BC3G 12:00PM - Lunch 1:30PM 1:30PM - Pawel Nurowski GL(2,R) geometry of ODEs 2:30PM 2:30PM - Infinitesimal isospectral deformations of symmetric Hubert Goldschmidt 3:30PM spaces of compact type 3:00PM - Tea 3:30PM 4:00PM - MSRI Biology Colloquia: Talk by Dr. Garrett Odell in the Simons Auditorium at MSRI's Chern 5:00PM Hall 5:00PM - Reception 6:00PM Wednesday May 7, 2008

9:30AM - Wave equations and the Penrose process in Kerr geometry Niky Kamran 10:30AM 10:30AM - Tea 11:00AM 11:00AM - Backlund Transformations and Darboux Integrability for Thomas Ivey 12:00PM Nonlinear Wave Equations 12:00PM - Open Afternoon 6:30PM Banquet in memory of Robby Gardner @ The Great China 6:30PM - Robert Bryant, Patrick Eberlein, Restaurant, Berkeley 9:30PM William Shadwick Guests of honor: Harolyn and Kirsten Gardner. Thursday May 8, 2008 9:30AM - Geometric Aspects of differential equations in the work of Peter Vassiliou 10:30AM Robert B. Gardner 10:30AM - Tea 11:00AM 11:00AM - Integrals quadratic in velocities and solution of two Vladimir Matveev 12:00PM problems of Sophus Lie 12:00PM - Lunch 2:00PM 2:00PM - Christine Scharlach Symmetries for Cubic Forms and Adapted Frames 3:00PM 3:00PM - Tea 4:00PM 4:00PM - (Math Dept Colloquium) Exterior Differential Systems in Phillip Griffiths 5:00PM Algebraic Geometry Friday May 9, 2008 9:30AM - Jeanne Clelland Sub-Finsler geometry in dimensions three and four 10:30AM 10:30AM - Tea 11:00AM 11:00AM - Ian Anderson Symmetry reduction and Darboux Integrability 12:00PM 12:00PM - Lunch 2:00PM 2:00PM - Hyperbolic Conservation Laws with Prescribed Kris Jenssen 3:00PM Eigencurves 3:00PM - Tea 4:00PM 4:00PM - Second-order type-changing evolution equations with George Wilkens 5:00PM first-order intermediate equations

Currently Available Videos

• Robert Bryant , Riemannian Foliations in Space Forms May 5,2008, 09:30 AM to 10:30 AM

• Colleen Robles , Rigidity of projective homogeneous varieties and Lie algebra cohomology May 5,2008, 11:00 AM to 12:00 PM

• Keizo Yamaguchi , Contact Geometry of Second Order May 5,2008, 02:00 PM to 03:00 PM

• Michael Eastwood , Prolongation on Contact Manifolds May 5,2008, 04:00 PM to 05:00 PM

• Ana Cascon, William Shadwick , The Geometry of Probability Distributions: A New Central Limit Theorem May 6,2008, 09:30 AM to 10:30 AM

• Ana Cascon, William Shadwick , The Geometry of Probability Distributions: A New Central Limit Theorem May 6,2008, 09:30 AM to 10:30 AM

• Shankar Sastry , Parking cars with N trailers and Nonholonomy: reflections on the applications of BC3G May 6,2008, 11:00 AM to 12:00 PM

• Pawel Nurowski , GL(2,R) geometry of ODEs May 6,2008, 01:30 PM to 02:30 PM

• Hubert Goldschmidt , Infinitesimal isospectral deformations of symmetric spaces of compact type May 6,2008, 02:30 PM to 03:30 PM

• Niky Kamran , 09:30AM - 10:30AM Niky Kamran Wave equations and the Penrose process in Kerr geometry May 7,2008, 09:30 AM to 10:30 AM

• Thomas Ivey , Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations May 7,2008, 11:00 AM to 12:00 PM

• Peter Vassiliou , Geometric Aspects of differential equations in the work of Robert B. Gardner May 8,2008, 09:30 AM to 10:30 AM

• Joseph Landsberg , EDS and the Method of Equivalence Lines and Asymptotic Lines of Projective Varieties May 8,2008, 11:00 AM to 12:00 PM

• Christine Scharlach , Symmetries for Cubic Forms and Adapted Frames May 8,2008, 02:00 PM to 03:00 PM

• Jeanne Clelland , Sub-Finsler geometry in dimensions three and four May 9,2008, 09:30 AM to 10:30 AM

• Ian Anderson , Symmetry Reduction and Darboux Integrability May 9,2008, 11:00 AM to 12:00 PM

• Vladimir Matveev , Integrals quadratic in velocities and solution of two problems of Sophus Lie May 9,2008, 12:00 PM to 01:00 PM

• Kris Jenssen , Hyperbolic Conservation Laws with Prescribed Eigencurves May 9,2008, 02:00 PM to 03:00 PM

• George Wilkens , Second-order type-changing evolution equations with first-order intermediate equations May 9,2008, 04:00 PM to 05:00 PM

Participant List

Name Role Institution Anderson, Ian Participant N/A Bassirou, Diatta Participant Jackson State University Bryant, Robert Leamon Participant MSRI Cascon, Ana Speaker Omega Analysis Limited Chen, Wen-Haw Participant Tunghai University Clelland, Jeanne Organizer University of Colorado, Boulder Deutsch, Michael Participant Washington University Donaldson, Neil Participant UC Irvine Drager, Lance Participant Texas Tech University Eastwood, Michael Speaker University of Adelaide Eberlein, Patrick Berry Participant University of North Carolina, Chapel Hill Faran, James Participant SUNY at Buffalo Fox, Daniel Participant Duke University Gluck, Herman R. Participant University of Pennsylvania Goertsches, Oliver Participant UC Irvine Goldschmidt, Hubert Speaker Columbia University Griffiths, Phillip A. Speaker Institute for Advanced Study Han, Chong-kyu Participant Seoul National University Hengesbach, Conrad Participant Duke University Hubert, Evelyne Participant INRIA, Sophia Antipolis Ionel, Marianty Participant University of Toledo Ishikawa, Goo Participant Hokkaido University Ivey, Thomas Participant College of Charleston Jablonski, Michael Participant University of North Carolina Jensen, Gary R. Participant Washington University Jenssen, Kris Participant Penn State Kamran, Niky Participant McGill University Kogan, Irina A. Speaker North Carolina State University Korsan, Bob Participant Decisions, Decisions! kossowski, marek Participant University of North Carolina Kruglikov, Boris S. Participant University of Tromso Landsberg, Joseph M. Participant Texas A & M University Lee, Jeffrey Participant Texas Tech University Liu, Yang Participant University of Georgia Lotay, Jason Participant University College Oxford Mahdipour Sh., Ali Participant Iran University of Science and Technology Malkoun, Joseph Participant Stony Brook University Manno, Giovanni Participant Università del Salento Matveev, Vladimir S. Speaker University of Jena Mestdag, Tom Participant University of Michigan Mettler, Thomas Participant University of Fribourg Moseley, Christopher G. Participant Calvin College Nurowski, Pawel Institute of Theoretical Physics, University of Krzysztof Speaker Warsaw Odinette Renée, Abib Participant Université de Rouen Preston, Serge Participant Portland State University Ream, Robert Participant Utah State University Robles, Colleen Mary Speaker Texas A&M University Santoso, Jenny Participant University of Stuttgart Sastry, Shankar Participant N/A Scharlach, Christine Speaker Technische Universitaet Shadwick, William Francis Organizer Omega Analysis Limited Shibuya, Kazuhiro Participant Hokkaido university Smith, Abraham Participant Duke University Stackpole, Matthew Notetaker University of Colorado Strazzullo, Francesco Participant Utah State University Szereszewski, Adam Participant University of Warsaw Terng, Chuu-Lian Participant UC Irvine The, Dennis Participant McGill University Vassiliou, Peter John Speaker university of canberra Wafo Soh, Celestin Participant Jackson State University Wang, Erxiao Eric Participant National Univ. of Singapore Wilkens, George R Organizer University of Hawaii at Manoa Wolf, Joseph A. Participant UC Berkeley Wong-Kew, Richard Participant DRS Litigations Monitoring Project Xu, Feng Participant Duke University Yamaguchi, Keizo Speaker Hokkaido University

REPORT ON THE CONFERENCE “MODULAR FORMS AND ARITHMETIC”

FRANK CALEGARI, SAMIT DASGUPTA, BJORN POONEN, AND RICHARD TAYLOR

This is a report on the conference “Modular forms in arithmetic” held June 28 to July 2, 2008 at the University of California, Berkeley and the Mathematical Sciences Research Insti- tute. The conference consisted of 17 lectures by international experts in arithmetic geometry, with a focus on aspects connected with automorphic forms and Galois representations. It opened with a pair of colloquium-style lectures, which, as intended, attracted not only the conference participants but also a broader audience from around the San Francisco Bay Area. First, Barry Mazur spoke on Ribet’s construction of abelian extensions via the Galois representations attached to modular forms, and how its modern interpretation in terms of families of Galois representations connects to current research. Second, Kevin Buzzard gave a very general lecture on the role of modularity of elliptic curves in the proof of Fermat’s last theorem, and explained the larger picture to which this conjecturally belongs. These two lectures placed many of the later lectures in context. For instance, Mazur’s lecture set the stage for Samit Dasgupta’s lecture on the weak Gross-Stark conjecture and Jo¨elBella¨ıche’s lecture on extensions of p-adic Galois representations. Other lectures covered current research on the geometry of Shimura varieties (Mark Kisin

on the Fp-points, Elena Mantovan on integral models of compactifications), the insuffi- ciency of known cohomological obstructions for explaining non-existence of rational points on varieties (Bjorn Poonen), Rebolledo’s use of supersingular elliptic curves to prove some results towards uniform bounds for non-surjectivity of Galois representations associated to elliptic curves over Q (Lo¨ıcMerel), some partial results towards a function field Lang- Trotter conjecture (Nicholas Katz), the rigid p-adic geometry of modular curves (Robert Coleman), a conjectural strengthening of the Jacquet-Langlands correspondence that sees torsion (Frank Calegari), torsion points on abelian varieties (Matthew Baker), a new functor from representations to (φ, Γ)-modules (Marie-France Vign´eras), the interplay between ramification subgroups and finite subgroups of complex Lie groups arising in the lo- cal Langlands correspondence (), and speculations on Kolyvagin’s old ideas that give (perhaps overly optimisitic) hope for a proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q with analytic rank greater than 1 (William Stein). Two of the speakers, David Helm and Matthew Emerton discovered as a result of the 1 2 FRANK CALEGARI, SAMIT DASGUPTA, BJORN POONEN, AND RICHARD TAYLOR

conference that they were working on the same problem, extending the mod ` local Lang- lands correspondence to representations over an Artinian ring instead of just a field. Because the collaboration between Helm and Emerton is a striking concrete example of the success of the conference, we asked Helm to write a short but detailed summary of their work. He writes:

Both Matthew Emerton’s research and my own centered on the question of making the local Langlands correspondence for GL(2) work in families. That is, given a p-adic family of two-dimensional representations of the Galois group of a local field F , we sought a natural way of associating a p-adic family of

admissible representations of GL2(F ), in a way that induced the classical local Langlands correspondence on points of the two families. Neither of us managed to achieve this goal independently, but Emerton was able to show that given a family ρ of Galois representations, there was at most one family π of admissible representations satisfying a short list of properties that one would expect if π and ρ were related by local Langlands. He was not able to show that such a π always existed, however. Meanwhile, I had independently taken a more constructive approach, via deformation theory. My approach was able to associate a π to every ρ, but my construction was very ad hoc, and I was unable to find a convincing way of showing that the π I constructed was the “right” one. Emerton and I only became aware of each other’s work when the conference abstracts were posted; we soon found that our work dovetailed perfectly — his result uniquely characterised the families π I constructed, and my construction proved his conjecture that a π satisfying his properties always existed.

Finally, we note that the extended breaks between lectures gave ample opportunities for interaction between participants. Several junior participants commented to us afterwards that they appreciated the opportunity to meet and discuss mathematics with the leaders in the field. For instance, one student mentioned that, through talking to Barry Mazur during the breaks at the conference, she learned an idea that would let her circumvent a mathematical obstacle she was facing in the writing of her dissertation.

Modular Forms and Arithmetic June 28, 2008 to July 02, Schedule

Sat Jun 28 Sun Jun 29 Mon Jun 30 Tue Jul 1 Wed Jul 2 (UCB) (UCB) (MSRI) (MSRI) (MSRI)

8:30 - 9:00 AM Coffee and 8:30 - 9:00 Registration AM at UCB Coffee North Gate Hall

9:00 -

10:10 AM 9:00 - 10:00 9:00 - 10:00 Introduction AM AM and 9:00 - 10:00 9:00 - 10:00 Barry Mazur Frank Calegari AM AM Mathew

Emerton Construction Towards a Robert Coleman Marie-France of abelian Torsion Vigneras Level lowering extensions Jacquet- Wide Open for p-adic following Langlands Spaces modular Ken Ribet Correspondence forms for GL(2)

10:10 - 10:00 - 10:00 - 10:30 10:00 - 10:30 10:00 - 10:30 10:30 AM 10:30 AM AM AM AM Morning Morning Morning Break Morning Break Morning Break Break Break

10:30 - 10:30 - 11:30 10:30 - 11:30 10:30 - 11:30 11:30 AM AM 10:30 - 11:30 AM AM

AM Kevin David Helm Samit Dasgupta Elena Mantovan Buzzard Bjorn Poonen

On l-adic Ribet’s converse Integral models Ken Ribet families of Cohomological to Herbrand for toroidal and admissible obstructions to and the weak compactifications Fermat’s representations rational points Gross-Stark of Shimura Last of GL (Q ) conjecture varieties Theorem 2 p Free Afternoon 11:30 - 11:30 - 2:00 11:30 - 2:00 11:30 - 2:00 PM 2:00 PM PM PM Lunch Lunch Lunch Lunch 2:00 - 3:00 PM 2:00 - 3:00 2:00 - 3:00 PM PM Nicholas Katz William Stein Joel Bellaiche 2:00 - 3:00 PM

Lang- Kolyvagin’s Non trivial Matt Baker Trotter Approach to extensions of p- revisited, the Birch and adic Galois Torsion points on and lower Swinnerton- representations abelian varieties bounds for Dyer that are trivial Frobenius Conjecture at p traces 3:00 - 4:00 3:00 - 4:00 3:00 - 4:00 PM 3:00 - 4:00 PM PM PM Afternoon Afternoon Tea Afternoon Tea Afternoon Tea Tea

4:00 - 5:00 4:00 - 5:00 PM 4:00 - 5:00 PM PM Benedict Gross

Mark Kisin Loic Merel

Ramification Shimura theory and Modular varieties finite symbols for mod p subgroups of global fields Lie groups 7:00 PM

Banquet honoring Ken Ribet

on his 60th Birthday (UCB Faculty club)

Currently Available Videos

• Barry Mazur , Construction of abelian extensions following Ken Ribet June 28,2008, 09:10 AM to 10:10 AM

• Kevin Buzzard , Ken Ribet and Fermat's Last Theorem June 28,2008, 10:30 AM to 11:30 AM

• Nicholas Katz , Lang-Trotter revisited, and lower bounds for Frobenius traces June 28,2008, 02:00 PM to 03:00 PM

• Mark Kisin , Shimura varieties mod p June 28,2008, 04:00 PM to 05:00 PM

• Marie-France Vigneras , A functor from smooth OL-torsion representations to ( φ, Γ )-modules June 29,2008, 09:00 AM to 10:00 AM

• David Helm , On l-adic families of admissible representations of GL2(Qp) June 29,2008, 10:30 AM to 11:30 AM

• William Stein , Kolyvagin's Approach to the Birch and Swinnerton-Dyer Conjecture June 29,2008, 02:00 PM to 03:00 PM

• Benedict Gross , Ramification theory and finite subgroups of Lie groups June 29,2008, 04:00 PM to 05:00 PM

• Matthew Emerton , Level Lowering of p-adic Modular Forms June 30,2008, 09:00 AM to 10:00 AM

• Bjorn Poonen , Cohomological Obstructions to Rational Points. June 30,2008, 10:30 AM to 11:30 AM

• Frank Calegari , Towards a Torsion Jacquet_Langlands Correspondence for GL(2) July 1,2008, 09:00 AM to 10:00 AM

• Samit Dasgupta , Ribet's Converse to Herbrand and the Weak Gross-Stark Conjecture. July 1,2008, 10:30 AM to 11:30 AM

• Joel Bellaiche , Non Trival Extensions of p-adic Galois Representations that are Trival at p. July 1,2008, 02:00 PM to 03:00 PM

• Loic Merel , Modular Symbols for Global Fields July 1,2008, 04:00 PM to 05:00 PM

• Robert Coleman , Wide Open Spaces July 2,2008, 09:00 AM to 10:00 AM

• Elena Mantovan , Integral Models for Toroidal Compactications of Shimura Varieties. July 2,2008, 10:30 AM to 11:30 AM

• Matthew Baker , Torsion Points on Abelian Varieties July 2,2008, 02:00 PM to 03:00 PM

Participant List

Name Role Institution Achter, Jeffrey D. Participant Colorado State University Participant/ Adibhatla, Rajender CLAY University of Sheffield Agarwal, Mahesh Kumar Participant McMaster University Agashe, Amod Sadanand Participant Florida State University Agboola, Adebisi Participant University of California, Santa Barbara Allotta, Jeff Participant Northwestern University Arnold, Trevor Participant University of Washington Baker, Matthew Howard Speaker/ CLAY Georgia Tech University Bakhova, Maiia Jurevna Participant Louisiana State University Baran, Burcu Participant Universita' di Roma Bellaiche, Joel Speaker/ CLAY Columbia University Participant/ Brown, Jim CLAY Caltech Brumer, Armand Participant fordham university Buhler, Joe P. Participant CCR Participant/ Busuioc, Cecilia CLAY Boston University Buzzard, Kevin Speaker/ CLAY Imperial College, London Calegari, Frank Organizer/ CLAY Northwestern University Cardon, David A. Participant Brigham Young University Cheng, Yuan-You Participant N/A Choi, Suh Hyun Participant/CLAY Harvard University chung, min i Participant N/A Citro, Craig Louis Participant UCLA Coleman, Robert Speaker/ CLAY UCB Conrad, Brian David Participant stanford univ. Dasgupta, Samit Organizer/ CLAY Harvard University de Jong, Johan Speaker/ CLAY Columbia University Dembele, Lassina Participant University of Duisburg-Essen Dewar, Michael Patrick Participant/CLAY University of Illinois at Urbana-Champaign Dieulefait, Luis Victor Participant Harvard University El-Guindy, Ahmad Participant Texas A&M University Elkin, Arsen Participant Colorado State University Ellenberg, Jordan Participant University of Wisconsin, Madison Ellwood, David A. Organizer Boston University Emerton, Matthew James Speaker/ Clay Northwestern University Eriksson, Dennis Participant Tokyo University Participant/ Fité, Francesc CLAY Universitat Politècnica de Catalunya Freeman, David Participant University of California, Berkeley Fried, Michael D. Participant UC Irvine Participant/ Fuchs, Elena D CLAY Princeton University Garton, Derek William Participant/ University of Wisconsin—Madison CLAY Ghitza, Alexandru Edgar Participant Colby College Greicius, Aaron Participant UC, Berkeley Gross, Benedict H. Speaker/ CLAY Harvard University Gruendken, Linda Meike Participant University of Pennsylvania Participant/ Guitart, Xavier CLAY Universitat Politècnica de Catalunya Hagedorn, Tom Participant The College of Harby, John Participant N/A Hartshorne, Robert Participant UC Berkeley Helm, David Speaker/ CLAY University of Texas Ishikawa, Muriel Y. Participant N/A Jetchev, Dimitar Participant University of California, Berkeley Jones, Nathan C Participant CRM, Universite de Montreal Katz, Nicholas Speaker/ CLAY Princeton University Kedlaya, Kiran Sridhara Participant Massachusetts Institute of Technology Kharel, Savan Participant Indiana University Participant/ Kim, Byungchan CLAY University of Illinois at Urbana-Champaign Kisin, Mark Speaker/ CLAY University of Chicago Kramer, Ken Participant Queens College (CUNY) Lan, Kai-Wen Participant Harvard University Lang, William E. Participant Brigham Young University Lario, Joan-Carles Participant Universitat Politècnica de Catalunya Lenstra, Hendrik W. Speaker/ CLAY Universiteit Leiden Li, Wen-Ch'ing Winnie Participant Penn State University Ling, Jie Participant UW-Madison Participant/ Liu, Tong CLAY University of Pennsylvania Long Hoelscher, Jing Participant University of Arizona Long, Ling Participant Iowa State University Participant/ Luu, Martin CLAY Princeton University Lyo, Grace Participant MIT Lyons, Christopher Participant Caltech Mantilla, Guillermo Arturo Participant UW Madison Mantovan, Elena Speaker/ CLAY California Institute of Technology Mayer, Hartwig Participant Humboldt University of Berlin Mazur, Barry Speaker/ CLAY Harvard University McCallum, William Gordon Participant University of Arizona McMurdy, Ken Participant Ramapo College of New Jersey Institut de Mathématiques de Jussieu -- Université Denis Merel, Loic Speaker/ CLAY Diderot Mohamed, Moustafa Ibrahim Moustafa Participant University of Warwick Molina Blanco, Santiago Participant Universitat Politecnica de Catalunya Nicole, Marc-Hubert Participant Institut de mathematiques de Jussieu Niziol, Wieslawa Participant University of Utah Olsson, Martin Participant UC Berkeley Omar, Sami Participant University of Tunis Ono, Ken Participant University of Wisconsin Ozman, Ekin Participant University of Wisconsin-Madison Pakingan, Bryan Participant UC Berkeley Papaioannou, Athanasios Participant University of Chicago Park, Jae-Young Participant UC Berkeley Paulin, Alexander Participant Imperial College Poonen, Bjorn Organizer UCB - University of California, Berkeley Prasad, Dipendra Participant Tata Institute of Fundamental Research Ravgoza, Isabela Participant N/A Raygoza, Isabela Participant N/A Reeder, Mark Participant Boston College Ribet, Kenneth A. Participant UC Berkeley Robert, Francesc Creixell Participant Universitat Politecnica de Catalunya Satriano, Matt Participant Stanford University Schein, Michael M. Participant Hebrew University of Schoof, Rene' Participant Universita' di Roma Schuett, Matthias Participant Harvard University Sengun, Mehmet Haluk Participant Univ. of Wisconsin Seo, Soogil Participant Yonsei university Shahriyari, Leili Participant JHU Shin, Sug Woo Participant Harvard University Silverberg, Alice Participant University of California, Irvine Spencer, Mark Participant Springer Stein, William Arthur Speaker/ CLAY Univ of Washington Sun, Shenghao Participant UC Berkeley Takloo-Bighash, Ramin Participant UIC Tan, Fucheng Participant MIT Tate, John Participant U. Texas Austin Taylor, Richard Lawrence Organizer Harvard University Tornaría, Gonzalo Participant Facultad de Ciencias Trifkovic, Mak Participant Univ. of Victoria Tsaknias, Panagiotis Participant University of Sheffield Turkelli, Seyfi Participant University of Wisconsin Upton, Margaret Participant Texas A&M University Urban, Eric Speaker/ CLAY Columbia University Van Luijk, Ronald Martinus Participant Warwick University Van Order, Jeanine Marie Participant University of Cambridge Varilly, Anthony Participant UCB - University of California, Berkeley Vega, Maria Valentina Participant Texas A&M University Vigneras, Marie-France Speaker/ CLAY Université de Paris 7 (Diderot) Vincent, Christelle Participant University of Wisconsin - Madison Voight, John Michael Participant University of Vermont Walji, Nahid Participant California Institute of Technology Weigandt, James Emmanuel Participant Purdue University Weinert, Andreas Victor Participant University of Edinburgh Weinstein, Jared Participant UCLA Wong Kew, Rich Participant Postdoc Research Fellows Wood, Victoria Y. H. Participant UC Berkeley Yazdani, Soroosh Participant McMaster University Yoo, Hwajong Participant UC Berkeley Zhu, Hui June Participant State University of New York (SUNY) at Buffalo

Workshop Report: 2006 and 2007 Summer Graduate Workshops on Data Assimilation for the Carbon Cycle

Inez Fung University of California, Berkeley

Dates July 16-28, 2006 MSRI

July 8-13, 2007 National Center for Atmospheric Sciences (NCAR), Boulder, CO

Principal Organizers

Inez Fung, University of California, Berkeley Doug Nychka, NCAR Eugenia Kalnay, University of Maryland

Schedule of Talks

2006 Lectures: Monday, 17 July • Carbon Cycle, Inez Fung, University of California, Berkeley • Introduction to data assimilation and review of empirical methods (sections 5.1 and 5.2), Eugenia Kalnay, University of Maryland • The Likelihood, Bayesian Statistics and spatial data, Doug Nychka, NCAR Tuesday, 18 July • Atmospheric circulation and transport, Inez Fung, University of California, Berkeley • Introduction to least squares; multi-variate statistical data assimilation (Section 5.3 and part of Section 5.4), Eugenia Kalnay, University of Maryland • Bayes, the Kalman filter and variational methods, Doug Nychka, NCAR Wednesday, 19 July • Carbon source/sink: statement of the problem, Inez Fung, University of California, Berkeley • 3D-Var, OI and PSAS (parts of section 5.5, and sections of the thesis by Takemasa Miyoshi, • http://www.atmos.umd.edu/~ekalnay/MiyoshiThesis.pdf where he implemented 3D-Var and LEKF on the SPEEDY global primitive equations model), Eugenia Kalnay, University of Maryland • Engineering the Ensemble Kalman filter, Doug Nychka, NCAR Thursday, 20 July • Atmospheric, land and ocean observations, Scott Doney, Woods Hole Oceanographic institution • 4D-Var and Ensemble Kalman Filter (parts of section 5.6), Eugenia Kalnay, University of Maryland • Smoothers and solving inverse problems, Doug Nychka, NCAR Friday, 21 July • The Orbiting Carbon Observatory, Charles Miller, Califonia Institute of Technology • Comparison of 4D-Var, and different types of Ensemble Kalman Filter, Eugenia Kalnay, University of Maryland • Estimating parameters and using DART, Doug Nychka, NCAR Monday, 24 July • 4D Var for CO2 source/sinks - introduction to project, David Baker, NCAR • Ensemble Kalman Filter for CO2 source/sink estimation, David Baker, NCAR • Geostatistics - Principles of spatial analysis, Anna Michalak, University of Michigan Tuesday, 25 July

Page 1 of 6 • Inversion - history, computational requirements, Ian Enting, University of Melbourne and Anna Michalak, University of Michigan • Statistics of Inversions, Ian Enting, University of Melbourne • Statistics of analyzing inversion results, Anna Michalak, University of Michigan Wednesday, 26 July • Operational data assimilation of atmospheric trace gases from AIRS, IASI, CrIS, Chris Barnet, NOAA • Assimilating model parameters, David Schimel, NCAR • Automatic differentiation, Ian Enting, University of Melbourne Thursday, 27 July • Validation of satellite retrievals, Chris Barnet, NOAA • Carbon Observations in the Forbidding Ocean, Jim Bishop, Lawrence Berkeley National Laboratory

2007 Lectures: Monday, 9 July • The Carbon Cycle, Inez Fung, University of California, Berkeley • Data Assimilation and Forecasting the Weather, Eugenia Kalnay, University of Maryland • A Statistician's View of the Carbon Problem, Doug Nychka, NCAR • Synthesizing Information for the Environmental Sciences, James S. Clark, Duke University Tuesday, 10 July • Carbon Cycle: An Inverse Problem, Inez Fung, University of California, Berkeley • Distributions, Spatial Statistics and a Bayesian Perspective, Doug Nychka, NCAR • The Likelihood, the Prior and Bayes Theorem, Doug Nychka, NCAR • Ancient and Modern Ways to Evaluate the Posterior, Doug Nychka, NCAR • Data Assimilation for Tropospheric CO, Avelino Arellano, NCAR Wednesday, 11 July • Ensemble Kalman Filter in the Presence of Model Errors, Eugenia Kalnay, University of Maryland Thursday, 12 July • A Parametric and Process-oriented View of the Carbon System, David Schimel, NCAR • The TransCom3 Time-Dependent Global CO2 Flux Inversion ... and More, David Baker, NCAR • Aircraft CO2 Observations and Global Carbon Budgeting, Britton Stephens, NCAR Friday, 13 July • Simulation of Atmospheric CO2 Concentration with SPEEDY, Ji-Sun Kang, University of Maryland • The application of ensemble Kalman filter in adaptive observation and information content estimation studies, Junjie Liu, University of Maryland • Ground-based Observations of Total Column CO2, Gretchen Keppel Aleks, California Institute of Technology • Physical-Statistical Modeling, Rajib Paul, Ohio State University • A Funny Twist on Geostatistics, Ben Shaby, Cornell University • Ensemble Kalman Filter: The Movie, Doug Nychka, NCAR

Participant List

2006 Lecturers: David Baker, NCAR Chris Barnet, NOAA Jim Bishop, Lawrence Berkeley National Laboratory Scott Doney, Woods Hole Oceanographic Institution Ian Enting, University of Melbourne Inez Fung, University of California, Berkeley Eugenia Kalnay, University of Maryland

Page 2 of 6 Charles Miller, Califonia Institute of Technology Anna Michalak, University of Michigan Doug Nychka, NCAR David Schimel, NCAR

2006 Students: (8F, 19M. Ethnicity data was not collected) Alanood Alkhaled, University of Michigan Troy Bulter, Colorado State University Elliott Campbell, University of Iowa Haiyan Cheng, Virginia Tech Vani Cheruvu, ASP / NCAR James Ferguson, University of Victoria Humberto Godinez, Portland State Carl Hammarsten, Rice University Arta Jamshidi, Colorado State University Ji-Sun Kang, University of Maryland Charlie Koven, University of California, Berkeley Nir Krakauer, Caifornia Institute Of Technology Hiu Fung Roger Kwok, Hong Kong University of Science & Technology Quanlin Li, Duke University Junjie Liu, University of Maryland Chi On Andrew Lo, Hong Kong University of Science & Technology Erica McGrath-Spangler, Colorado State University Kazuyuki Miyazake, Japan Agency for Marine - Earth Science Kim Mueller, University of Michigan Kristjan Onu, University of Illinois, Urbana-Champaign Jun H. Park, University of Illinois, Urbana-Champaign Jeremy Praissman, University of Georgia Alexander Stine, University of California, Berkeley Abby Swann, University of California, Berkeley Jerry Tjiputra, University of Wisconsin Tadeusz Tworek, University of Illinois, Chicago Adam Wolf, Carnegie Institution

2007 Lecturers: Jeffrey Anderson, NCAR Avelino Arellano, NCAR David Baker, NCAR James Clark, Duke University Inez Fung, University of California, Berkeley Eugenia Kalnay, University of Maryland Douglas Nychka, NCAR David Schimel, NCAR Britton Stephens, NCAR

2007 Students: (13F, 17M. Ethnicity data was not collected) Gretchen Keppel Aleks, California Institute of Technology Alison Appling, Duke University Greg Barron, Gafford University of Arizona Jonathan Beezley, University of Colorado, Denver Jay Breidt, Colorado State University Troy Butler, Colorado State University, Ft. Collins Haiyan Cheng, Virginia Tech Dan Cooley, NCAR Sambingo Da Silva Cardoso, NCAR Ankur Desai, NCAR

Page 3 of 6 Sherri Heck, NCAR Taka Ito, Colorado State University, Ft. Collins Ji-Sun Kang, University of Maryland Yongku Kim, Ohio State University Nir Krakauer, University of California, Berkeley Ed Lee, NCAR Junjie Liu, University of Maryland Amy Nail, North Carolina State University Yumiko Nakatsuka, National Institute for Environmental Studies, Japan Yosuke Niwa, University of Tokyo Rajib Paul, Ohio State University Haifeng Qian, University of Maryland Ben Shaby, Cornell University Shagi-Di Shih, University of Wyoming Pierre C. Sibiry Traore, University of Florida Tamara Singleton, University of Maryland Elaine Spiller, SAMSI Zan Stine, University of California, Berkeley Francesca Terenzi, Columbia University Nedjeljka Zagar, NCAR

Short Summary of the Workshop Goals

These were the first two Summer Graduate Workshop on Carbon Data Assimilation, jointly sponsored by MSRI and NCAR. The workshops are designed to seed a new field of multi-disciplinary research – assimilation of asynchronous satellite observations of carbon dioxide concentrations into global atmospheric circulation models to produce synoptic maps of carbon dioxide from which surface exchange (source/sink functions) of carbon dioxide can be estimated. The workshops expose students and researchers in the geosciences, ecology and mathematics to multidisciplinary Earth science through an urgent science problem that demands a new mathematical approach. The urgency comes from the vast data stream anticipated from the satellite “Orbiting Carbon Observatory” scheduled for launch in December 2008, and from the societal need to manage carbon dioxide sources and sinks to slow down global warming.

Concrete outcomes of the workshop

The lecturers come from several disciplines, and so the workshop is a multi-disciplinary education for the lecturers as well as for the graduate students and postdocs. As a result of the 2006 workshop, a proposal to expand data assimilation for weather prediction to include the carbon cycle was submitted to, and has since been accepted by, the Department of Energy (PI: Kalnay and Fung). The research will be carried out using NERSC computers and will support several graduate students who participated in the Workshop.

The 2006 workshop was supplemented with computer exercises. NERSC kindly provided access for all participants. Exercises on data assimilation using Local Transform Ensemble Kalman Filter and 4D Var were provided by Drs. Kalnay and Baker, and Dr. Schimel provided exercises on MCMC. Participants were invited to adapt the exercises in their own research.

The 2007 workshop was supplemented with computational examples using the Data Assimilation Research Testbed (DART), developed by Dr. Jeff Anderson of NCAR. http://www.image.ucar.edu/DAReS/DART/ A common room at NCAR was set up with computers (2 participants per computer), and two programmer/analysts were on hand to assist all aspects of the software and hardware issues. Dr. Anderson guided the workshop participants, worked through the exercises and led the discussion of the results. DART software is publicly available, and so the participants gained not only exposure to new areas of research, but also tools they can adapt and apply to their own work.

Dissemination

Page 4 of 6 http://www.atmos.berkeley.edu/~inez/MSRI-NCAR_CarbonDA/ http://www.image.ucar.edu/Workshops/CDAS_2007/ http://www.msri.org/communications/vmath/semester/dr/2006070220070101/show_semester

Evaluations

Evaluation questionnaires were distributed to all students in both years. All respondents were highly satisfied with what they got out of attending the courses.

Page 5 of 6 Pictures

The Participants at MSRI, Berkeley, California, July 2006

The Participants at Boulder, Colorado, July 2007

Page 6 of 6 Report on the Graduate Workshop: A window into zeta and modular physics

The workshop which convened from June 16th-27th/2008 and involved some 31 students featured 5 speakers: Klaus Kirsten (co-organizer),Geoff Mason, Audrey Terras, Michael Tuite , and Floyd Williams(organizer)-Klaus and Michael with earned PhD's in physics. There were 2 lectures per day, with the exception of the last 2 days, which I comment on later. Lectures on zeta functions and modular forms with indications of their application to physics (Casimir energy, quantum mechanics, quantum chaos, black hole entropy in general relativity, Bose-Einstein condensates, etc.) were provided by Klaus, Audrey, and Floyd. There was some overlap on modular forms with the lectures of Geoff and Michael though their lectures focused mainly on an introduction to vertex operator algebras and their role in conformal field theory. Hard copies of notes (which also contained extra background material,and home work problems)and their posting on the web, together with lots of post lecture discussions helped to complement the effectiveness of the formal lectures.

In further detail, some of the basic topics covered were the following: The Riemann zeta function-it analytic continuation, functional equation, and special values, zeta regularization of infinity factorial and more generally of infinite dimensional determinants, Hurwitz, Epstein-Barnes zeta functions and their connections to eigen value problems ,Schrodinger's equations and corresponding spectral zeta functions, Casimir energy (=the special value of zeta at s= - 1/2),partition function for coupled harmonic oscillators and related Bose Einstein condensation for particles trapped by a harmonic oscillator potential, application of Barnes-Epstein zeta to gravitational corrections due to extra dimensions (Kaluza-Klein theory), Ihara, Selberg, and graph zeta functions and discussions regarding the Riemann hypothesis for them, trees, statistics, random matrices, modular forms (holomorphic Eisenstein series, q-expansions, remarks on Hecke L-functions)and a special lecture on the history of the development of the theory of modular forms,vertex algebras and vertex operators algebras (with much emphasis on the Heisenberg and Virasoro examples, and with emphasis on the "locality axiom "), connections with conformal field theory, partition functions, modular invariance, and rationality of vertex operator algebras.

The lectures of Geoff and Michael on vertex operator algebras required a bit more background for some of the students. In response to their needs, Geoff organized a clinic (which ran for some hours) which according to the students (who were most appreciative) made such a big difference.

In addition to the formal lectures, and Geoff's clinic,the workshop featured a Speaker's Seminar which provided the students some insight into current research of the speakers. Although this seminar was not mandatory for the students their attendance was probably in the 100% range, except for 1 day when there was competition with a World Cup game. The seminar topics included new contour methods for evaluating infinite dimensional determinants, applications of a deformation of the Patterson-Selberg zeta function to black hole entropy, and partition functions for higher genus conformal field theory.

The speakers made a point of engaging with the students after lectures, of being available to respond to questions, asking about their personal work and math/physics interests, and in general of attempting to gage their various reactions and needs regarding the lectures. As a result of conversations with Elizabeth Malmskog (a student from Colorado State),for example, I decided to have a very informal questions/answers session (in place of a formal lecture)the day before the end of the workshop. For over an hour the speakers, on the spot, simply answered a full range of questions ranging from "what is a Boson" to string theory and in particular "Calabi-Yau compactifications". We continued to be impressed with the advanced knowledge that some of the students had.

Also instead of a formal lecture, the workshop devoted the last day to providing students an opportunity to give a talk. Four students spoke: Jennie D'Ambroise, Paul Nelson, Savan Kharel, and Dr.Shabnam Beheshti. Savan, who has only just graduated from college, seems to be extremely gifted. He helped the speakers answer questions on string theory in the questions/answers session. The student talks dealt with Bose-Einstein condensates, higher dimensional Friedmann cosmology, a survey of zeta functions, string theory, and a soliton-black hole correspondence.

I realize that the backgrounds of the students were varied, even though some of them really were exceptional. As far as I could discern, the high level of their enthusiasm was maintained throughout. Many expressed heartfelt thanks, on the last day in particular, and felt grateful that such a workshop was offered.

The initial goal of the workshop was to provide for the students a bridge, or a window, into the vast arena of interactions between physics and number theory, with an emphasis on zeta functions and modular forms, and an introduction to modular and conformal field theory aspects of the theory of vertex operator algebras. Towards this end all of the proposed lectures, except one, were presented, with an extensive handout of notes, as has already been mentioned. The only planned topic that was not covered was further remarks on modular forms of negative weight, and specifically discussion of the Rademacher-Zuckerman exact formula and asympototics for their Fourier coefficients. This discussion was not given as we wanted to respond in favor of specific requests regarding a question/answer session and to offering some historic perspctives, as we have also previously mentioned. To this extent, and given the array of positive comments from the students(almost daily) ,it seems safe for us to conclude that our goal was largely accomplished. Of course we leave it to MSRI and to the students, their formal responses/comments/evaluations, to make a final definitive conclusion regarding success /or not/of the workshop.

Floyd L. Williams

A Window into Zeta and Modular Physics

June 16, 2008 to June 27, 2008 Schedule Tuesday, June 17 9:30 AM – 10:30 AM “Vertex Operators” by Geoff Mason 11:00 AM – 12:00 PM “Hurwitz, Barnes, and Epstein Zeta Functions” by Klaus Kirsten Wednesday, June 18 9:30 AM – 10:30 AM “Functional Education of Zeta, Special Values, and Infinity Factorial” by Floyd Williams 11:00 AM – 12:00 PM “Vertex Operator Algebras” by Geoff Mason Thursday, June 19 Speaker: Prof. Audrey Terras 9:30 AM – 10:30 AM “Riemann, Dedekind, Selberg, and Ihara Zetas” 11:00 AM – 12:00 PM “Ruelle Zeta and Prime Number Theorem for Graphs” 12:00 PM Gather in the Atrium to leave for the BBQ! Friday, June 20 9:30 AM – 10:30 AM “Application of Epstein-Barnes Zeta to Gravity in Extra Dimensions” by Floyd Williams 11:00 AM – 12:00 PM “Motivations, and the Casimir Energy as an Application of Zeta Function Techniques” by Klaus Kirsten Speaker’s Seminar: 1:30 PM – 2:30 PM Computation of Functional Determinants Via Contour Integration” by Klaus Kirsten Monday, June 23 9:30 AM – 10:30 AM “Partition sums and Zeta Functions” by Klaus Kirsten 11:00 AM – 12:00 PM “Edge and Path Zetas, Connection with random Matrices” by Audrey Terras Speaker Seminar 1:30 – 2:30 PM “Genus 2 partition functions” by Geoff Mason Tuesday, June 24 9:30 AM – 10:30 AM “Notes on my notes/unfinished business” by Floyd Williams 11:00 AM – 12:00 PM “Vertex operators 3” by Geoff Mason Speaker Seminar 1:30 – 2:30 PM “Quantum correction to black hole entropy via deformation of zeta” by Floyd Williams Wednesday, June 25 9:30 AM – 10:30 AM “Modular forms in vertex operator algebras I” by Michael Tuite 11:00 AM – 12:00 PM “Modular forms in vertex operator algebras II” by Michael Tuite Thursday, June 26 9:30 AM – 10:30 AM “Modular forms-Their history and profile in the scheme of mathematical things” by Floyd Williams 11:00 AM – 12:00 PM “Questions/answers session: What you wanted to know but were afraid to ask” by Floyd Williams Speaker Seminar 1:30 pm “Exceptional groups and the virasoro algebra” by Michael Tuite Friday, June 27 9:30 AM – 9:50 AM Savan Kharel “Arithmetic framework to string compactification and emergent space time” 9:55 AM – 10:15 AM Paul Nelson TBA 10:40 AM – 11:00 AM Jennie D’Ambroise “A brief overview of reformations of Einstein’s equations” 11:05 AM – 11:25 AM Dr. Shabnam Beheshti “How a soliton illuminates a black hole”

Currently Available Videos

• Floyd Williams, Riemann's zeta function June 16,2008, 09:30 AM to 10:30 AM

• Klaus Kirsten, Introduction to Hurewitz,Epstein,Barnes zeta June 16,2008, 11:00 AM to 12:00 PM

• Geoff Mason, Conformal field theory (CFT) for beginners June 17,2008, 09:30 AM to 10:30 AM

• Klaus Kirsten, A simple example of a Casimir enengy calculation(another example of zeta regularization) June 17,2008, 11:00 AM to 12:00 PM

• Floyd Williams, Modular forms of positive weight and their Fourier coefficients June 18,2008, 09:30 AM to 10:30 AM

• Geoff Mason, Conformal field theory (CFT) June 18,2008, 11:00 AM to 12:00 PM

• Audrey Terras, Introduction to Selberg,Ihara,Ruelle zeta June 19,2008, 09:30 AM to 10:30 AM

• Audrey Terras, zeta zeros/poles and distribution of primes June 19,2008, 11:00 AM to 12:00 PM

• Floyd Williams, Dedekind eta and its reciprocal with some string theory remarks June 20,2008, 09:30 AM to 10:30 AM

• Klaus Kirsten, Computation of functional determinants via contour integrals June 20,2008, 11:00 AM to 12:00 PM

• Klaus Kirsten, Polylog expansion of a partition function with some intuitive remarks on Bose-Einstein condensation June 23,2008, 09:30 AM to 10:30 AM

• Audrey Terras, Quantum chaos June 23,2008, 11:00 AM to 12:00 PM

• Floyd Williams, Modular forms of negative weight (more on the reciprocal of the Dedekind eta function,and a statement of the Rademacher-Zuckerman formula for Fourier coefficients) June 24,2008, 09:30 AM to 10:30 AM

• Geoff Mason, CFT June 24,2008, 11:00 AM to 12:00 PM

• Michael Tuite, A Window into Zeta an Modular Physics talk I June 25,2008, 09:30 AM to 10:30 AM

• Michael Tuite, A Window into Zeta an Modular Physics talk II June 25,2008, 11:00 AM to 12:00 PM

• Floyd Williams, Generalized Cardy formula for black hole entropy(applications of forms of non-positive weight) June 26,2008, 09:30 AM to 10:30 AM

• Jennie D'Ambroise, On Relating d-dimensional FRLW Cosmology to Bose- Einstein Condensates June 26,2008, 11:00 AM to 12:00 PM

• Paul Nelson, Zeta and L-functions June 27,2008, 09:30 AM to 10:15 AM

• Savan Kharel, Arithmetic of String Compactification and Emergent Spacetime. June 27,2008, 10:30 AM to 11:15 AM

• Shabnam Beheshti, How a soliton Illuminates a Blackhole June 27,2008, 11:15 AM to 12:00 PM

Participant List

Name Role Institution Ashley, Caleb Participant N/A Bakhova, Maiia Participant Louisiana State University Banerjee, Abhishek Participant Johns Hopkins University University of Massachusetts, Beheshti, Shabnam Participant Amherst Bi, Shuchau Participant University of California, Berkeley Boettner, Stefan Participant Tulane University Cohen, Sean Participant North Carolina State University Conway, Alex Participant Princeton University Crompton, Catherine Participant Emory University D'Ambroise, Jennie Participant N/A Farrington, Eleanor Participant Boston University Franze, Craig Participant N/A Gharahbeigi, Sara Participant Washington University, St. Louis Hurley, Donny Participant N/A Jensen, Erik Participant University of North Carolina Kharel, Savan Participant Indiana University Kim, Myoungil Participant Boston University Organizer/ Kirsten, Klaus Speaker Baylor University Kleinman, Aaron Participant University of California, Berkeley Kohl, Karen Participant Tulane University Krauel, Kayden Participant University of California, Santa Cruz Malikiosis, Romanos Participant University of California, Los Angeles Malmskog, Elizabeth Participant Colorado state University Marion, Samantha Participant University of Alberta Marks, Chris Participant University of California, Santa Cruz Mason, Geoff Speaker University of California, Santa Cruz Nelson, Paul Participant California Polytechnic Institute Nitz, Ted Participant N/A Pejic, Michael Participant N/A Powell, Kevin Participant N/A Quddus, Safdar Participant Washington University, St. Louis Roy, Michael Participant University of Colorado Shankar, Arul Participant Princeton University Steele, George Alexander Participant Boston University Sun, Jie Participant University of Alberta Terras, Audrey Speaker University of California, San Diego Tuite, Michael Speaker N/A Vinogradov, Ilya Participant Princeton University Walji, Nahid Participant California Polytechnic Institute Wechter, Matthew Participant University of Illinois, Chicago Whitcher, Ursula Anne Participant University of Washington Organizer/ University of Massachusetts, Williams, Floyd Speaker Amherst Wittenborn, Erika Participant University of Colorado