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ISSN 0002-9920 Notices of the American Mathematical Society 35 Monticello Place, Pawtucket, RI 02861 USA American Mathematical Society Distribution Center

of the American Mathematical Society April 2011 Volume 58, 4

Deformations of Bordered Surfaces and Convex page 530

Taking Math to Heart: Mathematical Challenges in Cardiac Electrophysiology page 542

A Brief but Historic Article of Siegel page 558

Remembering Paul Malliavin page 568

Volume 58, Number 4, Pages 521–648, April 2011

About the Cover: Collective behavior and individual rules (see page 567)

Trim: 8.25" x 10.75" 128 pages on 40 lb Velocity • Spine: 1/8" • Print Cover on 9pt Carolina “So r r y , t h a t ’s n o t c o r r e c t .” “Th a t ’s c o r r e c t .”

Two Online Homework Systems Went Head to Head. Only One Made the Grade.

What good is an online homework system if it can’t recognize right from wrong? Our sentiments exactly. Which is why we decided to compare WebAssign with the other leading homework system for math. The results were surprising. The other system failed to recognize correct answers to free response questions time and time again. That means students who were actually answering correctly were receiving failing grades. WebAssign, on the other hand, was designed to recognize and accept more iterations of a correct answer. In other words, WebAssign grades a lot more like a living, breathing professor and a lot less like, well, that other system. So, for those of you who thought that other system was the right answer for math, we respectfully say, “Sorry, that’s not correct.”

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We are surrounded by complex systems. Familiar examples include power grids, transportation systems, financial markets, the Internet, and structures underlying everything from the environment to the cells in our bodies. Mathematics and statistics can guide us in understanding these systems, enhancing their reliability, and improving their performance. Mathematical models can help uncover common principles that underlie the spontaneous organization, called emergent behavior, of flocks of birds, schools of fish, self-assembling materials, social networks, and other systems made up of interacting agents.

M ATHEMATICS AWARENESS MONTH April 2011 www.mathaware.org

SPONSORED BY THE JOINT POLICY BOARD FOR MATHEMATICS

American Mathematical Society American Statistical Association Mathematical Association of America Society for Industrial and Applied Mathematics

Image 1: Starlings 2, Tomas Jensen. From istock.com. Image 2: Electricity 1, Annemiek van der Kuil. Image 3: Hurricane Katrina, NASA. Image 4: A voltage-gated potassium channel (Kv1.2) visualized with the VMD software. Courtesy of the Theoretical and Computational Biophysics group, NIH Resource for Macromolecular Modeling and Bioinformatics, Beckman Institute, University of Illinois at Urbana-Champaign. Image 5: Bangkok Skytrain Sunset, David Iliff. Wikimedia Commons. GNU Free Documentation License. Background image: The Product Space. Image courtesy of Cesar Hidalgo, Center for International Development, Kennedy School of Government, . Notices of the American Mathematical Society April 2011 Communications

528 A Photographic Look at the Joint Mathematics 550 Meetings, New Orleans, 2011

550 3N Colored Points in a Plane Günter M. Ziegler

567 Collective Behavior and Individual Rules (About the Cover) 568 558

580 WHAT IS...a G2-? Spiro Karigiannis Features 588 Doceamus: Making Mathematics Work for The theme of Mathematics Awareness Month for 2011 is “Unrav- Minorities eling Complex Systems”. As part of this theme, the Notices this Manuel P. Berriozábal month includes an article about mathematics and cardiology. The issue also includes an article about small divisors in dynami- 590 and Poets Cai Tianxin cal systems (an idea also consonant with complex systems). The mathematical topics are rounded out with an unusual treatment 593 2011 Steele Prizes of Riemann surfaces. Finally, we offer recollections of noted Paul Malliavin. 597 2011 Conant Prize —Steven G. Krantz, Editor 599 2011 Morgan Prize 601 2011 Satter Prize 530 Deformations of Bordered Surfaces and 603 2011 Bôcher Prize Convex Polytopes

606 2011 Doob Prize Satyan L. Devadoss, Timothy Heath, and Wasin Vipismakul 608 2011 Eisenbud Prize 542 Taking Math to Heart: Mathematical 610 2011 in Challenges in Cardiac Electrophysiology Commentary John W. Cain

525 Opinion: Thriving with the 562 A Brief but Historic Article of Siegel NSF Rodrigo A. Pérez Sastry G. Pantula 568 Remembering Paul Malliavin 527 Letters to the Editor Daniel W. Stroock, Marc Yor, Jean-Pierre Kahane, 582 Nonsense on Stilts— Richard Gundy, Leonard Gross, Michèle Vergne A Book Review Reviewed by Olle Häggström

585 To and Beyond!—A Book Review Reviewed by Dan Rockmore Notices Departments of the American Mathematical Society About the Cover ...... 567

EDITOR: Steven G. Krantz Mathematics People ...... 612 ASSOCIATE EDITORS: Mok and Phong Receive 2009 Bergman Prize, Anantharaman Awarded Krishnaswami Alladi, David Bailey, , 2010 , Andrews Awarded 2010 Sacks Prize, AAAS Fellows Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert J. Daverman, Susan Friedlander, , for 2011, Cora Sadosky (1940–2010). Rafe Mazzeo, Harold Parks, Lisette de Pillis, , Mark Saul, Edward Spitznagel, John Swallow Mathematics Opportunities ...... 615 SENIOR WRITER and DEPUTY EDITOR: Schauder Medal Award Established; DMS Workforce Program in the Allyn Jackson Mathematical Sciences; Call for Proposals for 2012 NSF-CBMS Regional MANAGING EDITOR: Sandra Frost Conferences; NSF-CBMS Regional Conferences, 2011; AWM Gweneth CONTRIBUTING WRITER: Elaine Kehoe Humphreys Award; Project NExT: New Experiences in Teaching. CONTRIBUTING EDITOR: Randi D. Ruden For Your Information ...... 617 EDITORIAL ASSISTANT: David M. Collins PRODUCTION: Kyle Antonevich, Anna Hattoy, Mathematics Awareness Month—April 2011, Google Donation to Teresa Levy, Mary Medeiros, Stephen Moye, Erin Support IMO, Corrections. Murphy, Lori Nero, Karen Ouellette, Donna Salter, Deborah Smith, Peter Sykes, Patricia Zinni Reference and Book List ...... 618 ADVERTISING SALES: Anne Newcomb SUBSCRIPTION INFORMATION: Subscription prices Mathematics Calendar ...... 624 for Volume 58 (2011) are US$510 list; US$408 insti- tutional member; US$306 individual member; US$459 New Publications Offered by the AMS ...... 630 corporate member. (The subscription price for mem- bers is included in the annual dues.) A late charge of Classified Advertisements ...... 637 10% of the subscription price will be imposed upon orders received from nonmembers after January 1 of the subscription year. Add for postage: Surface Meetings and Conferences of the AMS ...... 639 delivery outside the and —US$27; in India—US$40; expedited delivery to destinations Meetings and Conferences Table of Contents ...... 647 in North America—US$35; elsewhere—US$120. Subscriptions and orders for AMS publications should be addressed to the American Mathematical Society, P.O. Box 845904, , MA 02284-5904 USA. All orders must be prepaid. ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be sent to [email protected] (classified ads) or notices-ads@ From the ams.org (display ads). SUBMISSIONS: Articles and letters may be sent to the editor by email at [email protected], by AMS Secretary fax at 314-935-6839, or by postal mail at Department of Mathematics, Washington University in St. Louis, Campus Box 1146, One Brookings Drive, St. Louis, MO Call for Nominations for 2012 Leroy P. Steele Prizes ...... 621 63130. Email is preferred. Correspondence with the managing editor may be sent to [email protected]. For more information, see the section “Reference and Call for Nominations for Position of AMS Secretary ...... 622 Book List”. NOTICES ON THE AMS WEBSITE: Supported by the Call for Nominations for AMS Award for Mathematics Programs AMS membership, most of this publication is freely That Make a Difference ...... 623 available electronically through the AMS website, the Society’s resource for delivering electronic prod- ucts and services. Use the URL http://www.ams. org/notices/ to access the Notices on the website. [Notices of the American Mathematical Society (ISSN 0002- 9920) is published monthly except bimonthly in June/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, RI 02940-6248 USA.] Publication here of the Society’s street address and the other information in brackets above is a technical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: [email protected]. © Copyright 2011 by the American Mathematical Society. I thank Randi D. Ruden for her splendid editorial work, and All rights reserved. for helping to assemble this issue. She is essential to everything Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines that I do. established to ensure permanence and durability. —Steven G. Krantz Opinions expressed in signed Notices articles are those of Editor the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Opinion

Thriving with the NSF In March 2010, an MPSAC working group on DES wrote in its report that, “Mathematics and statistics lie at the intersection It is an exciting time to be at the Division of Mathematical Sciences of all quantitative fields engaged in DES, through the power of of the National Science Foundation. It is a great place to work, their abstractions, and they swiftly convey breakthroughs in and a place where research and diversity THRIVE! In this article, one field into related ones.” The working group recommended I would like to focus on three items: my goals, new opportunities, obtaining significant funds to support DES research through and the budget. CAREER awards and workforce development in understand- My Goals: My first goal is to see that mathematical, statistical, ing and inference with massive and complex data, as well as and computational sciences THRIVE, not just simply survive, at to provide Research Experiences for Undergraduates supple- DMS. Second, I would like to diversify our workforce and broaden ments for DES training. See http://www.nsf.gov/dir/index. the participation at all levels of training. Finally, I would like DMS jsp?org=MPS and “A Report of the NSF Advisory Committee for to be the best place to work for a diverse group of energetic re- Cyberinfrastructure Task Force on Grand Challenges” to learn searchers and program assistants. more about CDS & E. This task force report recommends support THRIVE here is an acronym. I hope that: THematic (core) and for research programs in “advances in discretization methods, multidisciplinary Research is well funded; and that our research solvers, optimization, statistical methods for large data sets, and has a high Impact, is very innovative in solving major societal validation and uncertainty quantification” and training of “the issues, and is highly Visible. Finally, we want to Educate future next-generation of data-scientists who can work in a multidis- researchers, problem solvers, and critical thinkers. Our communi- ciplinary team of researchers in high performance computing, ties can help in making our excellent contributions visible to the mathematics, statistics, domain-specific sciences, etc.” In addition public and show how we are the backbone of innovation. Articles to the above, there are many NSF multidisciplinary activities that written for news media and magazines in other sciences, Math provide opportunities for our communities: BIO-MaPS, supporting Awareness Month, Math Moments, and Statistical Significance are research at the intersection of biological and mathematical and some examples of activities that help with the visibility of our physical sciences; math-bio initiatives with the National Institute professions. Also, we eagerly await the reports from a National of General Medical Sciences; initiatives with the Defense Threat Academy of Sciences study called Math Sciences 2025. Please visit Reduction Agency; CMG, supporting collaborations between http://www8.nationalacademies.org/cp/projectview. math and geosciences; SEES, focused on research in energy and aspx?key=49237 to provide feedback on this project. sustainability; FODAVA, concerned with data and visualization; and CIF21, Cyberinfrastructure for 21st Century Science and En- I hope to work with our communities to diversify our work- gineering. These are just a few examples. In order to surf the data force (students, postdocs, faculty, and leadership) and support tsunami and work in multidisciplinary teams, we need to train broadening participation at all levels. Hopefully, with your help, our students in core, computational, and communication skills. DMS panels and program officers represent diversity as well. Budget @ DMS: Typically, DMS invests 70% of its budget to Finally, I hope to be remembered as someone who was fair to all support disciplinary research, predominantly through individual our programs and valued the contributions of each person in all grants. Of this, about 10% is invested in multidisciplinary activi- divisions at NSF. I plan to build on the harmony that exists and ties. About 15% is invested in workforce-related activities, 10% make DMS an attractive place for a diverse group of folks who in math sciences institutes, and 5% in other activities. We are love to come to DMS to serve our professions. DMS (http://www. looking forward to the outcomes of the solicitations on research nsf.gov/div/index.jsp?div=DMS) has several positions open networks and for institutes. In spite of receiving a large number in various programs now. of high-quality proposals, the success rate for research funding Opportunities @ DMS: A recent working group of the NSF is below 30%, a of our limited budget. Thus, we are un- Mathematical and Physical Sciences Advisory Committee (MPSAC) able to fund many excellent proposals. On the other hand, we made a strong case for support for basic research and provided can’t fund proposals that are never submitted! It is encouraging several examples of how basic research from the past helped de- to see that our communities’ hard work in getting more of our velop many useful inventions (such as laser, GPS, cell phone, PET students to apply for NSF Graduate Research Fellowships is scans, … ). The working group report mentions, “Support for basic yielding positive results. We also encourage applications from a research is an essential part of the NSF mission” and “A successful diverse group of students from every institution, not just from innovation strategy requires significant investments across NSF a select few, to apply for Math Sciences Postdoctoral Research core programs.” DMS intends to continue its significant support Fellowships (MSPRF). for basic research in core areas of mathematics and statistics. We are under continuing resolution for the fiscal year 2011 In addition to the basic research, Data-Enabled Science (DES), budget until March 4, 2011, and we remain optimistic about both computing science, grand challenges in cyberinfrastructure, and the FY11 and FY12 budgets. Investment in research is a key to multidisciplinary research play a very important part in DMS innovation and economic competitiveness. It has no political activities. Prior to my arrival at NSF, our previous NSF director, boundaries. The statistical, mathematical, and computational Arden Bement, signed a memo from which I quote: “NSF should sciences have an impact on all other sciences, and other sciences create a program in Computational and Data-Enabled Science and in turn have an impact on our basic research. Engineering (CDS & E) … ”, and “CDS & E is now clearly recogniz- Thank you in advance for your support to achieve our com- able as a distinct intellectual and technological discipline lying mon goals. As you advance the frontiers of our disciplines, please at the intersection of applied mathematics, statistics, computer also take advantage of opportunities to solve our future societal science, core science and engineering disciplines. It is dedicated challenges in health, climate, energy, sustainability, and security, to the development and use of computational methods and data among many others. Keep up the great work! mining and management systems to enable scientific discovery Sastry G. Pantula and engineering innovation.” Massive and complex data are here Director, Division of Mathematical Sciences to stay and provide a diverse set of opportunities for both theo- National Science Foundation retical and applied areas of computational, mathematical, and [email protected] statistical sciences.

APRIL 2011 NOTICES OF THE AMS 525 Applied Mathematics Titles New!! from

Nonlinear Waves in Integrable and Turning 40? Nonintegrable Systems *IANKE9ANG Mathematicalhil Modeling and Computation 16 Partial Differential Equations: This book presents cutting-edge developments Analytical and Numerical Methods, in the theory and experiments of nonlinear Second Edition waves. Its coverage of analytical methods for Mark S. Gockenbach nonintegrable systems is the first of its kind. This undergraduate textbook introduces It also covers analytical methods for integrable students to partial differential equations with a equations, and comprehensively describes unique approach that emphasizes the modern efficient numerical methods for all major finite element method alongside the classical aspects of nonlinear wave computations. Turning 50? method of . sXXVI PAGESs3OFTCOVER sXX PAGESs(ARDCOVER )3".    s,IST0RICE )3".    s,IST0RICE 3)!--EMBER0RICEs#ODE-- 3)!--EMBER0RICEs#ODE/4 The Linear Sampling Method Nonlinear Programming: in Inverse Electromagnetic Concepts, Algorithms, and Scattering Applications to Chemical Processes &IORALBA#AKONI $AVID#OLTON ,ORENZ4"IEGLER AND0ETER-ONK MOS-SIAM Series on Optimization 10 CBMS-NSF Regional Conference Series This book addresses modern nonlinear in Applied Mathematics 80 programming (NLP) concepts and algorithms, The linear sampling method is the oldest and Turning 60? especially as they apply to challenging most developed of the qualitative methods applications in chemical process engineering. in inverse scattering theory. It is based on The author provides a firm grounding in solving a linear integral equation and then fundamental NLP properties and algorithms, using the equation’s solution as an indicator Celebrate your milestone and relates them to real-world problem function for the determination of the support by becoming an classes in process optimization. of the scattering object. sXVI PAGESs(ARDCOVER sX PAGESs3OFTCOVER )3".    s,IST0RICE )3".    s,IST0RICE AMS Life Member -/33)!--EMBER0RICEs#ODE-/ 3)!-#"-3-EMBER0RICEs#ODE#" Shapes and Geometries: Metrics, Generalized Concavity Analysis, Differential Calculus, and -ORDECAI!VRIEL 7ALTER%$IEWERT Optimization, Second Edition 3IEGFRIED3CHAIBLE AND)SRAEL:ANG -ICHEL#$ELFOURAND*EAN 0AUL:OLÏSIO Classics in Applied Mathematics 63 Advances in Design and Control 22 Originally published in 1988, this enduring This new edition provides a self-contained text remains the most comprehensive book presentation of the mathematical foundations, on generalized convexity and concavity. The Rates are US$2520, US$1680, constructions, and tools necessary for studying authors present generalized concave functions or US$840, depending on age. problems where the modeling, optimization, in a unified framework, exploring them Members by reciprocity (outside or control variable is the shape or the primarily from the domains of optimization structure of a geometric object. and economics. the U. S.) pay US$1260, US$840, sXXIV PAGESs(ARDCOVER sXVI PAGESs3OFTCOVER or US$420. )3".    s,IST0RICE )3".    s,IST0RICE 3)!--EMBER0RICEs#ODE$# 3)!--EMBER0RICEs#ODE#, Call your Member and Customer Services representative at To ORDER 0LEASEMENTIONKEYCODEh"./vWHENYOUORDER 800-321-4267 or email Order online: www.siam.org/catalog s5SEYOURCREDITCARD!-%8 -ASTER#ARD OR [email protected] 6)3! #ALL3)!-#USTOMER3ERVICEAT    WORLDWIDEORTOLLFREEAT  3)!- IN53!AND#ANADA&AX    % MAIL[email protected]s3ENDCHECKORMONEY ORDERTO3)!- $EPT"+./ -ARKET3TREET TH&LOOR 0HILADELPHIA 0!  www.ams.org SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS  Letters to the Editor

The Epistemological Surprise knowledge is true, and we know so. “In 1947, following World War II, the Examination Paradox Classically, the latter comes with the Treaty of gave this area [a region In Volume 57, Kritchman and Raz dis- former by the nature of the mod- in Transylvania] to Romania, so that cussed the Surprise Examination Par- els used. However, in 2000 Carlson is where it is found on today’s maps.” adox: wherein, a teacher announces introduced more general models, Since this statement could give the K(K(φ) → φ) impression that that region of Tran- a surprise exam next week. The stu- where can fail while K(φ) → φ sylvania had never before belonged dents reason a Friday exam would holds: we might not know to Romania, here is a synopsis of the be unsurprising. Similarly the exam knowledge is true, even if it is. By K(K(φ) → φ) history of Transylvania in the twenti- cannot be Thursday, Wednesday, etc., no longer demanding , eth century. The union of Transylva- so cannot ever occur. Formalizations with our new definition of surprise, nia with Romania was proclaimed at of the announcement fall into the the paradox vanishes (provably, using Alba Iulia (1918) by the elected rep- logical and epistemological schools Carlson’s models). The relaxed de- mands are analogous to Gödel’s In- resentatives of the ethnic Romanian (as Chow called them in 1998). In majority in Transylvania, following the logical school, pre-Kritchman, we completeness, strengthening a bridge between and epistemology. -Hungary’s defeat in World say facts are surprising if they are War I. The Allies confirmed the union unprovable, given the announcement —Sam Alexander in the Treaty of Trianon (1920). The and observed nonoccurrences. By region of Northern Transylvania was Ohio State University imitating the above informal reason- granted to Hungary by the Second Vi- [email protected] ing, the formalized announcement is enna (Diktat) Award (1940) under the contradictory. (Received December 15, 2010) arbitration of Nazi and Fas- Kritchman observes, since this an- cist Italy. The Treaty of Paris (1947) nouncement formalization is contra- Program for the Joint Meetings returned the region to Romania. dictory, the resulting formalization When my January 2011 issue of the of “surprise” fails: contradictions Notices arrived today, I was quite —Marius Stefan prove anything; being unsurprised by surprised by the extraordinary heft [email protected] things provable from contradiction, of the volume. This surprise turned nothing surprises us (Halpern and to disappointment when I discovered (Received January 4, 2011) Moses observed the same in 1986 but that the unusual weight was due to a reacted differently than Kritchman). A Mathematician’s Answer to printing of the program for the an- the Historical Question It is better (Kritchman suggests) to nual Joint Meetings, taking up over I agree with Marius Stefan in that my say the students are surprised by an 100 pages and nearly half of the comment on Transylvania is “irrel- exam if they cannot prove it before- issue. In this day and age of environ- evant to the article’s topic”. My only hand or if they can prove another mental conservation and tight bud- aim in including it was to clarify the exam occurred earlier, when it actu- gets, was it really necessary to print reason why the town of Marosva- ally didn’t (whence they could prove the entire program into thousands sarhely (which played a significant anything). Modifying the formaliza- of copies of the Notices, especially role in the life of János Bolyai) can tion, the paradox disappears, leading when this information is all available to a new proof of Gödel’s Incomplete- be found on the map of Romania online? At the least, I’d think the Soci- today— for the benefit of those read- ness . ety could get by with mailing separate I shall outline how Kritchman’s ers who haven’t followed the compli- programs only to those attending the cated history of Transylvania. above insight translates to the epis- meeting. Having just paid my Society temological school. Introduce a new The analysis of this complex his- dues for the coming year, I hope they torical issue is pertinent neither to connective K, where K(φ) denotes won’t be put to such needless print- this article nor to the journal itself. “the students know (φ) the night be- ing and mailing expenses in 2011. Moreover, Marius Stefan does not fore the exam”. Tentatively, call (φ) contradict my statement, he merely surprising if ¬K(φ). The teacher’s —Greg Friedman expands on it. announcement formalizes easily as a Department of Mathematics Finally, I trust that the really set of statements involving K; these Texas Christian University significant historical misconcep- statements are shown, imitating the [email protected] tion that my article‚ “Real face of informal paradox, to contradict cer- János Bolyai” is trying to address tain basic axioms of knowledge. (Received December 24, 2010) has not escaped his attention. Kritchman leads us to instead call (φ) surprising if (¬K(φ)) ∨ K(1 = 0). Transylvanian History T. Dénes To remove the paradox, though, we The article “Real face of János Bolyai” [email protected] must also account for Gödel. Most by T. Dénes, which appeared in the classic axiomatizations of knowledge January 2011 issue of the Notices, (Received February 2, 2011) include K(φ) → φ and K(K(φ) → φ): contains the following statement:

APRIL 2011 NOTICES OF THE AMS 527 528 Photographs by E. David Luria, Diane Boumenot (AMS), and Annette Emerson (AMS). N OTICES

OF

THE AMS V OLUME 58,N UMBER 4 See photo key on page 614

APRIL 2011 NOTICES OF THE AMS 529 Deformations of Bordered Surfaces and Convex Polytopes Satyan L. Devadoss, Timothy Heath, and Wasin Vipismakul

he moduli space of Riemann surfaces of extension of the work by Liu [16] on the moduli genus g with n marked particles is influ- of J-holomorphic curves and open Gromov-Witten ential in many areasof mathematics and invariants. theoretical physics, ranging from quan- A third field of intersection comes from the tum cohomology to number theory to world of operads and algebraic structures. For Tfluid dynamics. This space has a natural extension instance, Kaufmann, Livernet, and Penner provide by considering surfaces with boundary, appear- several topological and homological operads based ing, for instance, alongside open-closed string on arcs on bordered surfaces [14]. Indeed, the theory and holomorphic curves with Lagrangian natural convex polytopes appearing in our setting submanifolds. One goal of this article is to pro- fit comfortably in the framework of higher category vide an accessible understanding of these moduli theory and the study of A∞ and L∞ structures seen spaces by constructing a stratification based on the from generalizations of associahedra, cyclohedra, pair-of-pants decomposition and the well-known and multiplihedra [8]. phenomenon of bubbling. Along the way, we will An overview of the paper is as follows: We begin classify all such spaces that can be realized as with a review of the definitions of interest, followed convex polytopes, relating them to the famous by construction of the moduli spaces, providing associahedron. details of several low-dimensional examples and There are several (overlapping) fields that touch their stratification. The polytopal spaces are then upon these ideas. Recent work spearheaded by classified, along with a description of the as- Fomin, Shapiro, and Thurston [11] has established sociahedron and polytopes. A new a combinatorial world of cluster algebras related to is introduced based on the moduli space bordered surfaces with marked points, introduc- of the annulus. Finally, the combinatorial and al- ing notions of triangulated surfaces and tagged gebraic properties of this polytope are explored arc complexes. Another perspective comes from and related to the multiplihedron. symplectic geometry, where an analytic approach has been taken by Fukaya and others [12] to con- Definitions struct moduli spaces of bordered surfaces. Indeed, A smooth connected oriented bordered Riemann this article can be viewed, at a high level, as an surface S of type (g, h) has genus g ≥ 0 with h ≥ 0 Satyan L. Devadoss is associate professor of mathemat- disjoint ordered circles B1,...,Bh for its boundary. ics at Williams College. His email address is satyan. We assume the surface is compact whose bound- [email protected]. ary is equipped with the holomorphic structure Timothy Heath is a graduate student at Columbia Univer- induced by a holomorphic atlas on the surface. sity. His email address is timheath@math. Specifically, the boundary circles will always be columbia.edu. given the orientation induced by the complex Wasin Vipismakul is a graduate student at the Univer- structure. The surface has a marking set M of sity of Texas. His email address is wvipismakul@math. type (n, m) if there are n labeled marked points in utexas.edu. the interior of S (called punctures) and mi labeled

530 Notices of the AMS Volume 58, Number 4 marked points on the boundary component Bi , Let S be a surface without boundary. A decom- where m = m1, . . . , mh. Throughout the paper, position of (S, M) into pairs of pants is a collection we define m := m1 + + mh. of disjoint pairs of pants on S such that the union of their closures covers the entire surface and the Definition. The set (S, M) fulfilling the above re- pairwise intersection of their closures is either quirements is called a marked bordered Riemann empty or a union of marked points and closed surface. We say (S, M) is stable if its automorphism geodesic curves on S. Indeed, all the marked points group is finite. of M appear as boundary components of pairs of pants in any decomposition of S. A disjoint set (a) (b) of curves decomposing the surface into pairs of pants will be realized by a unique set of disjoint geodesics, since there is only one geodesic in each homotopy class. We now extend this decomposition to include marked surfaces with boundary. Consider the complex double (SC, σ ) of a marked bordered Riemann surface (S, M). If P is a decomposition of SC into pairs of pants, then σ (P) is another Figure 1. (a) An example of a marked bordered decomposition into pairs of pants. The following Riemann surface (b) along with its complex is a generalization of the work of Seppälä: double. Lemma 1 (16, Section 4). There exists a decompo- sition of SC into pairs of pants P such that σ (P) = Figure 1(a) shows an example of (S, M) where S is P and the decomposing curves are simple closed of type (1, 3) and M is of type (3, 1, 2, 0). Indeed, geodesics of SC. any stable markedborderedRiemannsurface has a Figure 2(a) shows examples of some of the geo- unique hyperbolic metric such that it is compatible desic arcs from a decomposition of SC, where part with the complex structure, where all the boundary (b) shows the corresponding decomposition for S. circles are geodesics, all punctures are cusps, and Notice that there are three types of decomposing all boundary marked points are half cusps. We geodesics γ. assume all our spaces (S, M) are stable throughout this paper. (1) Involution σ fixes all points on γ: The geodesic must be a boundary curve of S, The complex double SC of a bordered Riemann surface S is the oriented double cover of S without such as the curve labeled x in Figure 2(b). 1 (2) Involution σ fixes no points on γ: The boundary. It is formed by gluing S and its mirror geodesic must be a closed curve on S, such image along their boundaries; see [2] for a detailed as the curve labeled y in Figure 2(b). construction. For example, the disk is the surface (3) Involution σ fixes two points on γ: The of type (0, 1) whose complex double is a sphere, geodesic must be an arc on S, with its whereas the annulus is the surface of type (0, 2) endpoints on the boundary of the surface, whose complex double is a torus. Figure 1(b) shows such as the curve labeled z in Figure 2(b). the complex double of (S, M) from part (a). In the case when S has no boundary, the double SC is simply the trivial disconnected double-cover of S. (a) y (b) The pair (SC, σ ) is called a symmetric Riemann surface, where σ : SC → SC is the antiholomorphic involution. The symmetric Riemann surface with a marking set M of type (n, m) has an involu- x x tion σ together with 2n distinct interior points z z {p1, . . . , pn, q1, . . . , qn} such that σ (pi ) = qi , along with m boundary points {b1, . . . , bm} such that y y σ (bi ) = bi. Definition. A pair of pants is a sphere from which Figure 2. Examples of some geodesic arcs three points or disjoint closed disks have been re- from a pair of pants decomposition of (a) the moved. A pair of pants can be equipped with a complex double and (b) its marked bordered unique hyperbolic structure compatible with the Riemann surface. complex structure such that the boundary curves are geodesics. We assign a weight to each type of decompos- 1The complex double and the Schottky double of a ing geodesic in a pair of pants decomposition, surface coincide, since the surface is orientable. corresponding to the number of Fenchel-Nielsen

April 2011 Notices of the AMS 531 coordinates needed to describe the geodesic. The The Moduli Space geodesic of type (2) above has weight two be- Given the definition of a marked bordered Riemann cause it needs two Fenchel-Nielsen coordinates to surface, we are now in position to study its moduli describe it (length and twisting angle), whereas space. For a bordered Riemann surface S of type geodesics of types (1) and (3) have weight one (g, h) with marking set M of type (n, m), we

(needing only their length coordinates). denote M(g,h)(n,m) as its compactified moduli space. Lemma 2 (1, Chapter 2). Every pair of pants de- Analytic methods can be used for the construction composition of a marked bordered Riemann sur- of this moduli space that follow from several face (S, M) of type (g, h) with marking set (n, m) important, foundational cases. The topology of has a total weight of the moduli space Mg,n of algebraic curves of genus g with n marked points was provided by (1) 6g + 3h − 6 + 2n + m Abikoff [1]. Later, Seppälä gave a topology for from the weighted decomposing curves. the moduli space of real algebraic curves [18]. Based on the discussion above, we can reformu- Finally, Liu modified this for marked bordered late the decomposing curves on marked bordered Riemann surfaces; the reader is encouraged to Riemann surfaces in a combinatorial setting: consult [16, Section 4] for a detailed treatment of the construction of M(g,h)(n,m). Definition. An arc is a curve on S such that its endpoints are on the boundary of S, it does not Theorem 3 (16, Section 4). The moduli space intersect M nor itself, and it cannot be deformed M(g,h)(n,m) of marked bordered Riemann surfaces arbitrarily close to a point on S or in M. An arc is equipped with a (Fenchel-Nielson) topology that corresponds to a geodesic decomposing curve of is Hausdorff. The space is compact and orientable type (3) above. A loop is an arc whose endpoints with real dimension 6g + 3h − 6 + 2n + m. are identified. There are two types of loops: A The dimension of this space should be familiar: 1-loop can be deformed to a boundary circle It is the total weight of the decomposing curves S of having no marked points, associated with of the surface given in Equation (1). Indeed, the a decomposing curve of type (1) above. Those stratification of this space is given by collections belonging to curves of type (2) are called 2-loops. of compatible arcs and loops (corresponding to We consider isotopy classes of arcs and loops. decomposing geodesics) on (S, M) where a collec- Two arcs or loops are compatible if there are tion of geodesics of total weight k corresponds to a curves in their respective isotopy classes that do (real) codimension k stratum of the moduli space. not intersect. The weighting of geodesics based on The compactification of this space is obtained by their Fenchel-Nielsen coordinates now extends to the contraction of the arcs and loops—the degen- weights assigned to arcs and loops: Every arc and eration of a decomposing geodesic γ as its length 1-loop has weight one, and a 2-loop has weight collapses to γ = 0. two. There are four possible results obtained from Figure 3 provides examples of marked bordered a contraction; each one of them could be viewed Riemann surfaces (all of genus 0). Parts (a)–(c) show naturally in the complex double setting. The first examples of compatible arcs and loops. Part (d) is the contraction of an arc with endpoints on shows examples of arcs and loops that are not the same boundary, as shown by an example in allowed. Here, either they are intersecting the Figure 4. The arc of part (a) becomes a double point marked point set M or they are trivial arcs and loops, which can be deformed arbitrarily close to a (a) (b) point on S or in M. We bring up this distinction to denote the combinatorial difference between our situation and the world of arc complexes, recently highlighted by Fomin, Shapiro, and Thurston [11].

Figure 4. Contraction of an arc with endpoints on the same boundary.

on the boundary in (b), being shared by the two (a) (b) (c) (d) surfaces.2 Similarly, one can have a contraction of Figure 3. Parts (a)–(c) show compatible arcs an arc with endpoints on two distinct boundary and loops, whereas (d) shows arcs and loops components as pictured in Figure 5. In both these that are not allowed. 2We abuse terminology slightly by sometimes referring to these double points as marked points.

532 Notices of the AMS Volume 58, Number 4 cases, we are allowed to normalize this degen- this comes from the weighting of the geodesics. eration by pulling off (and doubling) the nodal Since the loop drawn on the surface representing singularity, as in Figure 5(c). the central vertex has weight two, the codimension of this strata increases by two. (a) (b) (c)

Figure 5. Contraction of an arc with endpoints on distinct boundary components.

The other two possibilities of contraction lie with loops, as displayed in Figure 6. Part (a) shows the contraction of a 1-loop collapsing into a puncture, whereas part (b) gives the contraction Figure 8. The space M(0,,1)(2,,1) is a topological of a 2-loop resulting in the classical notion of disk. bubbling.

(a) (b) Example. Thus far, we have considered moduli spaces of surfaces with one boundary component. Figure 9 shows an example with two boundary components, the space M(0,2)(0,1,1) . Like Figure 8, the space is a topological disk, though with a different stratification of boundary pieces. It too is not a CW-complex, again due to a loop of weight two. Figure 6. Contraction of (a) the 1-loop and (b) the 2-loop.

Example. Consider the moduli space M(0,1)(2,0) of a disk with two punctures. By Theorem 3, the dimension of this space is one. There are only two boundary strata, one with an arc and another with a loop, showing this space to be an interval. Figure 7 displays this example. The left endpoint is the moduli space of a thrice-punctured sphere, whereas the right endpoint is the product of the moduli spaces of punctured disks with a marked point on the boundary.

Figure 9. The space M(0,,2)(0,,1,,1) is a topological disk.

Example. The moduli space M(0,2)(1,0) is the pen-

Figure 7. The space M(0,,1)(2,,0) is an interval. tagon, as pictured in Figure 10. As we will see be- low, this can be reinterpreted as the 2D associahedron. Example. The moduli space M can be seen (0,1)(2,1) Example. By Theorem 3, the moduli space as a topological disk with five boundary strata. M is three-dimensional. Figure 11 shows It has three vertices and two edges, given in Fig- (0,3)(0,0) the labeling of this space, seen as a ure 8, and is two-dimensional due to Theorem 3. with three quadrilaterals and six . Simi- Note that the central vertex in this disk shows that lar to above, this can viewed as 3D associahedron. this space is not a CW-complex. The reason for

April 2011 Notices of the AMS 533 vertices of Kn are enumerated by the Catalan Cn−1, and its construction as a polytope was given independently by Haiman (unpublished) and Lee [15]. Proposition 4. Four associahedra appear as mod- uli spaces of marked bordered surfaces:

(1) M(0,0)(3,0) is the 0D associahedron K2. (2) M(0,1)(2,0) is the 1D associahedron K3. (3) M(0,2)(1,0) is the 2D associahedron K4. (4) M(0,3)(0,0) is the 3D associahedron K5.

Proof. The space M(0,0)(3,0) is simply a point corre- sponding to a pair of pants, thus trivially having a unique pair of pants decomposition. The remain- ing three cases follow from the examples depicted above in Figures 7, 10, and 11, respectively. Fig- ure 12 summarizes the four surfaces encountered. 

Figure 10. The space M(0,,2)(1,,0) is the 2D associahedron.

Figure 12. Four surfaces whose moduli spaces yield associahedra given in Proposition 4.

It is natural to ask which other marked bordered moduli spaces are convex polytopes, carrying a rich underlying combinatorial structure similar to the associahedra. In other words, we wish to classify those moduli spaces whose stratifications are identical to the face posets of polytopes. We start with the following result: Theorem 5. The following moduli spaces are poly- topal.

(1) M(0,1)(0,m) with m marked points on the boundary of a disk. Figure 11. The space M(0,,3)(0,,0) is the 3D (2) M(0,1)(1,m) with m marked points on the associahedron. boundary of a disk with a puncture. (3) M(0,2)(0,m,0) with m marked points on one boundary circle of an annulus. Convex Polytopes The first two cases will be proven below in From the previous examples, we see that some Propositions 7 and 8, where they are shown to be of these moduli spaces are convex polytopes, the associahedron and cyclohedron, respectively, whereas others fail to be even CW-complexes. As and their surfaces are depicted in Figure 13(a) we see below, those with polytopal structures are and (b). We devote a later section to proving the all examples of the associahedron polytope. third case in Theorem 13, a new Definition. Let A(n) be the poset of all diagonal- called the halohedron, whose surface is given in izations of a convex n-sided polygon, ordered such part (c) of the figure. It turns out that these are that a ≺ a′ if a is obtained from a′ by adding new noncrossing diagonals. The associahedron Kn−1 is (a) (b) (c) a convex polytope of dimension n − 3 whose face poset is isomorphic to A(n).

The associahedron was originally defined by Figure 13. (a) Associahedra, (b) cyclohedra, (c) Stasheff for use in homotopy theory in connection halohedra. with associativity properties of H-spaces [19]. The

534 Notices of the AMS Volume 58, Number 4 the only polytopes appearing as moduli of marked Remark. Since each 2-loop has weight two, Lemma bordered surfaces. 2 and Theorem 3 show that any such loop will Remark. Parts (1) and (2) of Theorem 5 can be rein- increase the codimension of the corresponding terpreted as types A and B generalized associahe- strata by two. This automatically disqualifies the dra of Fomin and Zelevinsky [10]. Therefore, it is stratification from being that of a polytope. In- tempting to think that this new polytope in part (3) deed, the cases outlined in Theorem 6 are exactly could be the type D version. This is, however, not those that allow 2-loops. the case, as we see in the section Combinatorial We close this section with two well-known and Algebraic Structures. results.

Theorem 6. All other moduli spaces of stable bor- Proposition 7 (6, Section 2). The space M(0,1)(0,m) dered marked surfaces not mentioned in Proposi- of m marked points on the boundary of a disk is tion 4 and Theorem 5 are not polytopes. the associahedron Km−1. Proof. The overview of the proof is as follows: Sketch of Proof. Construct a dual between m We find certain moduli spaces that do not have marked points on the boundary of a disk and an polytopal structures and then show these moduli m-gon by identifying each marked point to an spaces appearing as lower dimensional strata to edge of the polygon, in . Then each arc the list above. Since polytopes must have poly- on the disk corresponds to a diagonal of the poly- topal faces, this will show that none of the spaces gon, and since arcs are compatible, the diagonals on the list are polytopal. are noncrossing. Figure 15 shows an example for Outside of the spaces in Proposition 4 and The- the 2D case.  orem 5, the cases of M(g,h)(n,m) that result in stable spaces that we must consider are the following: (1) when g > 0; (a) (b) (2) when h + n > 2 and h + n + m > 3; (3) when h = 2, and both m1 > 0 and m2 > 0. We begin with item (1), when g > 0. The simplest case to consider is that of a torus with one punc- ture. (A torus with no punctures is not stable, with infinite automorphism group, and so cannot be considered.) The moduli space M(1,0)(1,0) of a torus with one puncture is two-dimensional (from The- orem 3) and has the stratification of a 2-sphere, constructed from a 2-cell and a 0-cell. Thus, any other surface of nonzero genus, regardless of the Figure 15. The bijection between M(0,,1)(0,,5) and number of marked points or boundary circles, K4 . will always have M(1,0)(1,0) appearing in its bound- ary strata; Figure 14 shows such a case where this punctured torus appears in the boundary of A close kin to the associahedron is the cyclohe- M(2,3)(1,0). This immediately eliminates all g > 0 dron. This polytope originally manifested in the cases from being polytopal. For the remaining work of Bott and Taubes [4] with respect to knot cases, we need only focus on the g = 0 case. invariants and was later given its name by Stasheff. Definition. Let B(n) be the poset of all diagonal- izations of a convex 2n-sided centrally symmet- ric polygon, ordered such that a ≺ a′ if a is ob- tained from a′ by adding new noncrossing diag- onals. Here, a diagonal will either mean a pair of Figure 14. The appearance of M(1,,0)(1,,0) in the boundary of higher genus moduli. centrally symmetric diagonals or a diameter of the polygon. The cyclohedron Wn is a convex polytope of dimension n−1 whose face poset is isomorphic Now consider item (2), when h = 0 and n > 3. to B(n). The strata of the resulting moduli space M is 0,n Proposition 8 (7, Section 1). The space M well known, indexed by labeled trees, and is not a (0,1)(1,m) of m marked points on the boundary of a disk with polytope. For all other values, the space M (0,1)(2,1) a puncture is the cyclohedron W . appears as a boundary strata. As we have shown m in Figure 8, this space is not a polytope. Similarly, Sketch of Proof. Construct a dual to the m marked for item (3), the moduli space M(0,2)(0,1,1) appears points on the boundary of the disk to the symmet- in all strata, with Figure 9 showing the polytopal ric 2m-gon by identifying each marked point to a failure.  pair of antipodal edges of the polygon, in cyclic

April 2011 Notices of the AMS 535 order. Then each arc on the disk corresponds to a Definition. Let G be a connected graph. A tube is pair of symmetric diagonals of the polygon; in par- a set of nodes of G whose induced graph is a con- ticular, the arcs that partition the puncture from nected proper subgraph of G. Two tubes u1 and u2 the boundary points map to the diameters of the may interact on the graph as follows:

polygon. Since arcs are compatible, the diagonals (1) Tubes are nested if u1 ⊂ u2. are noncrossing. Figure 16 shows an example for (2) Tubes intersect if u1 ∩ u2 ≠ ∅ and u1 ⊂ u2  the 2D case. and u2 ⊂ u1. (3) Tubes are adjacent if u1 ∩u2 = ∅ and u1 ∪ u2 is a tube in G. (a) (b) Tubes are compatible if they do not intersect and are not adjacent. A tubing U of G is a set of tubes of G such that every pair of tubes in U is compatible. Theorem 9 (5, Section 3). For a graph G with n nodes, the graph associahedron KG is a simple convex polytope of dimension n−1 whose face poset is isomorphic to the set of tubings of G, ordered such that U ≺ U ′ if U is obtained from U ′ by adding tubes.

Figure 16. Cyclohedron W3 using brackets and Corollary 10. When G is a path with n nodes, KG polygons. becomes the associahedron Kn+1. Similarly, when G is a cycle with n nodes, KG is the cyclohedron Wn. Figure 17 shows the 2D examples of these cases (a) (b) of graph associahedra, having underlying graphs as paths and cycles, respectively, with three nodes. Compare with Figures 15 and 16. There exists a natural construction of graph associahedra from iterated truncations of the simplex: For a graph G with n nodes, let △G be the (n−1)-simplex in which each facet (codimension one face) corresponds to a particular node. Each proper subset of nodes of G corresponds to a unique face of △G, defined by the intersection Figure 17. Graph associahedra of the (a) path of the faces associated with those nodes, and the and (b) cycle with three nodes as underlying corresponds to the face that is the entire graphs. polytope △G. Theorem 11 (5, Section 2). For a graph G, trun- cating faces of △G that correspond to tubes in in- Remark. For associahedra, the moduli space of m creasing order of dimension results in the graph marked points on the boundary of a disk, any arc associahedron KG. must have at least two marked points on either We now create a new class of polytope that side of the partition in order to maintain stabil- mirror graph associahedra, except now we are ity. For the cyclohedra, however, the puncture in interested in truncations of cubes. We begin with the disk allows us to bypass this condition since the notion of design tubes. a punctured disk with one marked point on the boundary is stable. Definition. Let G be a connected graph. A round tube is a set of nodes of G whose induced graph is a Graph Associahedra and Truncations of connected (and not necessarily proper) subgraph Cubes of G.A square tube is a single node of G. Such We now turn our focus on the new polytope tubes are called design tubes of G. Two design arising from the moduli space of marked points tubes are compatible if on an annulus, as shown in Figure 13(c). We wish, (1) they are both round, they are not adjacent however, to place this polytope in a larger context and do not intersect. based on truncations of simplices and cubes. Let (2) otherwise, they are not nested. us begin with motivating definitions of graph A design tubing U of G is a collection of design associahedra; the reader is encouraged to see [5, tubes of G such that every pair of tubes in U is Section 1] for details. compatible.

536 Notices of the AMS Volume 58, Number 4 Figure 18 shows examples of design tubings. can readily be seen as △G. Since the graph cubea- Note that unlike ordinary tubes, round tubes do hedron is obtained by iterated truncation of the not have to be proper subgraphs of G. Based on faces labeled with round tubes, we see from Theo- design tubings, we construct a polytope, not from rem 11 that this converts △G into the graph asso- truncations of simplices but cubes: For a graph ciahedron KG. The 2n facets of CG coming from

G are in bijection with the design tubes of G cap- turing one vertex. The remaining facets of CG as well as its face poset structure follow immediately from Theorem 9.  Figure 18. Design tubings. We are now in position to justify the introduc- tion of graph cubeahedra in the context of moduli G with n nodes, we define G to be the n-cube spaces. where each pair of opposite facets corresponds to a particular node of G. Specifically, one facet Theorem 13. Let G be a cycle with m nodes. The in the pair represents that node as a round tube moduli space M(0,2)(0,m,0) of m marked points on and the other represents it as a square tube. Each one boundary circle of an annulus is the graph subset of nodes of G, chosen to be either round cubeahedron CG. We denote this special case as Hm or square, corresponds to a unique face of G, and call it the halohedron. defined by the intersection of the faces associated with those nodes. The empty set corresponds to Proof. There are three kinds of boundaries ap- the face that is the entire polytope G. pearing in the moduli space. The loop capturing the unmarked circle of the annulus is in bijection Definition. For a graph G, truncate faces of G with the round tube of the entire graph, as shown that correspond to round tubes in increasing order in Figure 20(a). The arcs capturing k marked of dimension. The resulting convex polytope CG is the graph cubeahedron. Example. Figure 19 displays the construction of CG when G is a cycle with three nodes. The facets of the 3-cube are labeled with nodes of G, each pair of opposite facets being round or square. The first step is the truncation of the corner vertex labeled with the round tube being the entire graph. Then (a) (b) (c) (d) the three edges labeled by tubes are truncated. Figure 20. Bijection between M(0,,2)(0,,m,0) and Hm .

boundary points correspond to the round tube surrounding k − 1 nodes of the cycle, displayed in parts (b) and (c). Finally, arcs between the two boundary circles of the annulus are in bijection with associated square tubes of the graph, as in (d). The identification of these poset structures is then immediate.  Figure 19. Iterated truncations of the 3-cube based on a 3-cycle. Figure 21(a) shows the labeling of H2 based on arcs and loops on an annulus. By construction of the graph cubeahedron, this pentagon should Theorem 12. For a graph G with n nodes, the be viewed as a truncation of the square. The 3D graph cubeahedron CG is a simple convex poly- halohedron H3 is depicted on the right side of tope of dimension n whose face poset is isomorphic Figure 19. to the set of design tubings of G, ordered such that U ≺ U ′ if U is obtained from U ′ by adding tubes. Remark. It is natural to expect these Hm polytopes to appear in facets of other moduli spaces. Fig- Proof. The initial truncation of the vertex of G ure 22 shows why three pentagons of Figure 11 where all n round-tubed facets intersect creates are actually halohedra in disguise: The arc is con- an (n − 1)-simplex. The labeling of this simplex is tracted to a marked point, which can be doubled the round tube corresponding to the entire graph and pulled open due to normalization. The result- G. The round-tube labeled k-faces of induce a G ing surface is the pentagon H2. labeling of the (k − 1)-faces of this simplex, which

April 2011 Notices of the AMS 537 (a) (b) the circular diagonal, whereas the diameter is not, since they intersect. On the other hand, two di- ameters are compatible since they are considered not to cross due to the central vertex. Figure 21(b)

shows the case of H2, now labeled using poly- gons, showing the compatibility of the different diagonals. We summarize this below. Definition. Let C(n) be the poset of all diagonal- izations of a convex 2n-sided centrally symmet- ric polygon with a central vertex, ordered such that a ≺ a′ if a is obtained from a′ by adding Figure 21. (a) The two-dimensional H2 and (b) new noncrossing diagonals. Here, a diagonal will its labeling with polygons. either mean a circle around the central vertex, a pair of centrally symmetric diagonals, or a diame-

ter of the polygon. The halohedron Hn is a convex polytope of dimension n whose face poset is iso- morphic to C(n). Let us now turn to the case of CG when G is a Figure 22. Certain pentagons of Figure 11 are path: actually the polygon H . 2 Proposition 14. If G is a path with n nodes, then CG is the associahedron Kn+2.

Combinatorial and Algebraic Structures Proof. Let G∗ be a path with n + 1 nodes. From We close by examining the graph cubeahedron Corollary 10, the associahedron Kn+2 is the graph CG in more detail, especially in the cases when associahedron KG∗. We therefore show a bijec- G is a path and a cycle. We have discussed in tion between design tubings of G (a path with n the section Convex Polytopes the poset structure nodes) and regular tubings of G∗ (a path with n + of associahedra Kn and cyclohedra Wn in terms 1 nodes), as in Figure 24. Each round tube of G of polygons. We now talk about the polygonal maps to its corresponding tube of G∗. Any square version of the halohedron Hn: Consider a convex tube around the k-th node of G maps to the tube 2n-sided centrally symmetric polygon with an ad- of G surrounding the {k + 1, k + 2, . . . , n, n + 1} ditional central vertex, as given in Figure 23(a). nodes. This mapping between tubes naturally ex- The three kinds of boundary strata which appear tends to tubings of G and G∗ since the round and in M(0,2)(0,m,0) can be interpreted as adding diago- square tubes of G cannot be nested. The bijection nals to the symmetric polygon. Part (b) shows that follows.  a “circle diagonal” can be drawn around the ver- tex, parts (c) and (d) show how a pair of centrally symmetric diagonals can be used, and a diameter can appear as in part (e).

(a) (b) (c) (d) (e) Figure 24. Bijection between design tubes and 2 2 2 2 2 regular tubes on paths. 1 3 1 3 1 3 1 3 1 3

3 1 3 1 3 1 3 1 3 1 We know from Theorem 11 that associahedra 2 2 2 2 2 can be obtained by truncations of the simplex. 1 2 1 2 1 2 1 2 1 2 But since CG is obtained by truncations of cubes, Proposition 14 ensures that associahedra can be obtained this way as well. Such an example is 3 3 3 3 3 depicted in Figure 25, where the 4D associahedron K appears as iterated truncations of the 4-cube. Figure 23. Polygonal labeling of Hn . 6

Proposition 15. The facets of Hn are

Due to the central vertex of the symmetric (1) one copy of Wn polygon, it is important to distinguish a pair of (2) n copies of Kn+1 and diameters as in (d) versus one diameter as in (e). (3) n copies of Km × Hn−m+1 for each m = Indeed, the pair of diagonals is compatible with 2, 3, . . . , n.

538 Notices of the AMS Volume 58, Number 4 is the cyclohedron Wn, whose face poset is iso- morphic to the ways to associate n objects on a circle. We establish such an algebraic structure for the halohedron Hn. We begin by looking at the algebraic structure behind another classically known polytope, the multiplihedron. The multiplihedra polytopes, denoted Jn, were first discovered by Stasheff in [20]. They play a similar role for maps of loop spaces as associa- hedra play for loop spaces. The multiplihedron Jn is a polytope of dimension n − 1 whose vertices correspond to ways of associating n objects and Figure 25. The iterated truncation of the applying an operation f . At a high level, the mul- 4-cube resulting in K6 . tiplihedron is based on maps f : A → B, where neither the range nor the domain is strictly as- sociative. The left side of Figure 27 shows the Proof. Recall that there are three kinds of codi- 2D multiplihedron J3 with its vertices labeled. mension one boundary strata appearing in the These polytopes have appeared in numerous ar- eas related to deformation and category theories. moduli space M(0,2)(0,n,0) . First, there is the unique loop around the unmarked boundary Notably, work by Forcey [9] finally proved Jn to be circle, which Proposition 8 reveals as the cyclohe- a polytope while giving it a geometric realization. dron Wn. Second, there exist arcs between distinct boundary circles, as shown in Figure 26(a). This f (a(bc)) f (a) f (bc) f (a(bc)) f (a) f (bc) figure establishes a bijection between this facet and CG, where G is a path with n − 1 nodes. By f ((ab)c) Theorem 14 above, such facets are associahedra f ((ab)c) Kn+1. There are n such associahedra since there are n such arcs. Finally, there exist arcs capturing m marked f (a)( f (b) f (c)) boundary points. Such arcs can be contracted and then normalized, as shown in Figure 4. This f (ab) f (c) ( f (a) f (b)) f (c) f (ab) f (c) f (a) f (b) f (c) leads to a product structure of moduli spaces Figure 27. The multiplihedron and the graph M(0,1)(0,m+1) and M(0,2)(0,n−m+1,0), where the +1 in both markings appears from the contracted arc. cubeahedron of a path. Proposition 7 and Theorem 13 show this facet to be K ×H . The enumeration follows because m n−m+1 Recently, Mau and Woodward [17] were able to there are n sets of m consecutive marked points view the multiplihedra as a compactification of the on the boundary circle.  moduli spaces of quilted disks, interpreted from the perspective of cohomological field theories. Here, a quilted disk is defined as a closed disk with a circle (“quilt”) tangent to a unique point in the boundary, along with certain properties. Halohedra and the more general graph cube- (a) (b) (c) (d) ahedra fit into this larger context. Figure 28 serves Figure 26. Bijection between certain facets of as our guide. Part (a) of the figure shows the halohedra and associahedra. moduli version of the halohedron. Two of its facets are seen as (b) the cyclohedron and (c) the associahedron. If the interior of the annulus is now From an algebraic perspective, the face poset of colored black, as in Figure 28(d), viewed not as a the associahedron Kn is isomorphic to the poset topological hole but as a “quilted circle”, we obtain of ways to associate n objects on an interval. In- deed, the associahedra characterize the structure (a) (b) (c) (d) (e) of weakly associative products. Classical exam- ples of weakly associative product structures are the An spaces, topological H-spaces with weakly associative multiplication of points. The notion Figure 28. The polytopes (a) halohedra, (b) of “weakness” should be understood as “up to cyclohedra, (c) associahedra, (d) homotopy”, where there is a path in the space cyclo-multiplihedra, (e) multiplihedra. from (ab)c to a(bc). The cyclic version of this

April 2011 Notices of the AMS 539 Strictly Associative G path G cycle general G Both A and B cube [3] cube cube Only A composihedra [9] cycle composihedra graph composihedra Only B associahedra halohedra graph cubeahedra Neither A nor B multiplihedra [20] cycle multiplihedra graph multiplihedra [8]

a cyclic version of the multiplihedron. When an arc Acknowledgments is drawn from the quilt to the boundary as in part We thank Chiu-Chu Melissa Liu for her enthu- (e), symbolizing (after contraction) a quilted circle siasm, kindness, and patience in explaining her tangent to the boundary, we obtain the quilted work, along with Ben Fehrman, Stefan Forcey, disk viewpoint of the multiplihedron of Mau and Chris Manon, Jim Stasheff, and Aditi Vashist for Woodward. helpful conversations. The first author also thanks The multiplihedron is based on maps f : A → B, MSRI for their hospitality, support, and stimulat- where neither the range nor the domain is strictly ing atmosphere in fall 2009 during the Tropical associative. From this perspective, there are sev- Geometry and Symplectic Topology programs. eral important quotients of the multiplihedron, as The authors were partially supported by NSF given by the table above. grant DMS-0353634 and the SMALL program at The case of associativity of both range and do- Williams College, where this work began. main is discussed by Boardman and Vogt [3], where the result is shown to be the n-dimensional cube. References The case of an associative domain is described 1. W. Abikoff, The Real Analytic Theory of Teichmüller by Forcey [9], where the new quotient of the Space, Springer-Verlag, 1980. multiplihedron is called the composihedron; these 2. N. Alling and N. Greenleaf, Foundations of the polytopes are the shapes of the axioms governing Theory of Klein Surfaces, Springer-Verlag, 1971. composition in higher enriched category theory. 3. J. Boardman and R. Vogt, Homotopy Invari- The classical case of a strictly associative range ant Algebraic Structures on Topological Spaces, (for n letters in a path) was originally described in Springer-Verlag, 1973. [20], where Stasheff shows that the multiplihedron 4. R. Bott and C. Taubes, On the self-linking of knots, Journal of Mathematical Physics 35 (1994), J becomes the associahedron K . The right side n n+1 5247–5287. of Figure 27 shows the underlying algebraic struc- 5. M. Carr and S. Devadoss, Coxeter complexes and ture of the 2D associahedron viewed as the graph graph-associahedra, Topology and its Applications cubeahedron of a path by Proposition 14. 153 (2006), 2155–2168. 6. S. Devadoss, Tessellations of moduli spaces and a b c d e f the mosaic operad, in Homotopy Invariant Algebraic Structures, Contemporary Mathematics 239 (1999), 91–114. f ((a(bcd))(ef)) f (ab) f (c) f ((de)f) f (a) f (bc) f (de) f (f) 7. , A space of cyclohedra, Discrete and Computational Geometry 29 (2003), 61–75. 8. S. Devadoss and S. Forcey, Marked tubes and Figure 29. Bijection between design tubes and the graph multiplihedron, Algebraic and Geometric associativity. Topology 8 (2008), 2081–2108. 9. S. Forcey, realizations of the multi- plihedra, Topology and its Applications 156 (2008), Figure 29 sketches a bijection between the as- 326–347. sociahedra (as design tubings of five nodes from 10. S. Fomin and A. Zelevinsky, Y -systems and gen- Figure 24) and associativity/function operations eralized associahedra, Annals of Mathematics 158 (2003), 977–1018. on six letters. Recall that the face poset of the cyclo- 11. S. Fomin, M. Shapiro, and D. Thurston, Cluster hedron Wn is isomorphic to the ways to associate n algebras and triangulated surfaces, Part I: Cluster objects cyclically. This is a natural generalization complexes, Acta Mathematica 201 (2008), 83–146. of Kn where its face poset is recognized as the 12. K. Fukaya, Y-G Oh, H. Ohta, and K. Ono, ways of associating n objects linearly. Instead, if Lagrangian Intersection Floer Theory, American Mathematical Society, 2009. we interpret Kn+1 in the context of Proposition 14, the same natural cyclic generalization gives us the 13. S. Katz and C. C. Liu, Enumerative geometry of sta- ble maps with Lagrangian boundary conditions and halohedron H . In a broader context, the general- n multiple covers of the disc, Advances in Theory of ization of the multiplihedron to an arbitrary graph Mathematical Physics 5 (2002), 1–49. is given in [8]. A geometric realization of these 14. R. Kaufmann, M. Livernet, and R. Penner, Arc op- objects as well as their combinatorial interplay is erads and arc algebras, Geometry and Topology 7 provided there as well. (2003), 511–568.

540 Notices of the AMS Volume 58, Number 4 MATHEMATICS AT THE N ATIONAL S ECURITY A GENCY 15. C. Lee, The associahedron and triangulations of the n-gon, European Journal of 10 (1989), 551–560. Make a calculated difference 16. M. Liu, Moduli of J-holomorphic curves with with what you know. Lagrangian boundary conditions and open Gromov- Witten invariants for an S1-equivariant pair, preprint, arxiv:math.0210257. 17. S. Mau and C. Woodward, Geometric realiza- tions of the multiplihedron and its complexification, Compositio Mathematica, to appear. 18. M. Seppälä, Moduli spaces of stable real algebraic curves, Annals Scientific Ecole Normal Superior 24 (1991), 519–544. 19. J. Stasheff, Homotopy associativity of H-spaces, Transactions of the AMS 108 (1963), 275–292. 20. , H-spaces from a Homotopy Point of View, Springer-Verlag, 1970.

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April 2011 Notices of the AMS 541 Taking Math to Heart: Mathematical Challenges in Cardiac Electrophysiology John W. Cain

n his website, University of Utah math- was sofar aheadof its time that it is mind-boggling ematics professor James Keener poses to think that it was constructed without the luxury the question, “Did you know that heart of modern computing. Soon after Hodgkin and attacks can give you mathematics!?” Huxley shared the 1963 Nobel Prize in Medicine Indeed, there are a host of important or Physiology for their efforts, the first of many researchO problems in cardiology that appear ideal adaptations of their model to cardiac tissue was for unified attack by mathematicians, clinicians, proposed. Such models are the subject of our first and biomedical engineers. What follows is a survey two challenge problems. of six ongoing Challenge Problems that (i) seem tractable and (ii) draw from a variety of mathe- Electrophysiology matical subdisciplines. We hope that this article Examining the electrical activity in a person’s will serve as a “call to arms” for mathematicians body can reveal a great deal of physiological in- so that we, as a community, can contribute to an formation. At some point in our lives, many of improved understanding of cardiac abnormalities. The emphasis of this article will be on cardiac us will undergo an electroencephalogram (EEG), a electrophysiology, because some of the most excit- recording of electrical activity in the brain, or an ing research problems in mathematical cardiology electrocardiogram (ECG), a recording of electrical involve electrical wave propagation in heart tis- activity in the heart. To understand where these sue. The quantitative study of electrophysiology tiny electrical currents originate, we must “zoom has a fascinating history, with its notable mile- in” to the molecular level. Bodily fluids such as stones touched by tragedy and triumph. Nearly blood contain dissolved salts and, consequently, a century has passed since the tragically prema- contain positively charged sodium, potassium, ture death of George Ralph Mines (1886–1914), and calcium ions. As these ions traverse cell a brilliant physiologist who apparently died from membranes, the resulting electrical currents elicit self-experimentation in his laboratory. Perhaps changes in the voltage v across the membrane. Mines would be comforted to know that his In the absence of electrical stimulation, v rests pioneering research continues to influence the at approximately -85 millivolts, the negative sign mathematical study of reentrant arrhythmia (see indicating that a cell’s interior contains less pos- Temporal Pattern Challenge below). Nearly half a itive charge than its exterior. Electrical stimuli century after Mines’s death, in a stunningly elegant can cause a resting cardiac cell to respond in a blend of mathematics and experimentation, British rather dramatic fashion. Namely, if a sufficiently physiologists Alan Hodgkin and Andrew Huxley strong stimulus current is applied to a suffi- introduced a model of electrical propagation in the ciently well-rested cell, then the cell experiences squid giant axon [11]. Their mathematical model an potential: v suddenly spikes and remains elevated for a prolonged interval (Figure 1). The John W. Cain is professor of mathematics at Virginia Commonwealth University. His email address is jwcain@ existence of threshold stimulus strengths pro- vcu.edu. vides a mechanism by which a cell can distinguish

542 Notices of the AMS Volume 58, Number 4 between “background noise” and real electrical systems of ordinary differential equations. Lower stimuli [16]. dimensional systems are easier to analyze, allow- ing us to characterize how certain physiological parameters affect the dynamics. For example, APDn DI n APDn+1 DI n+1 FitzHugh-Nagumo [7] reduction of the Hodgkin- (a) Huxley model can, under suitable rescaling, be written as B dv (b) ǫ = f (v, w) = Av(v − α)(1 − v) − w dt dw voltage = g(v, w) = v − βw, (c) dt where ǫ, A, α, and β are positive parameters, and 0 < α < 1. This two-variable system can be time analyzed using standard phase-plane techniques Figure 1. Action potentials in a paced cardiac and, in the case that ǫ ≪ 1, one may extract cell. Bold circles correspond to periodically asymptotic solutions via singular perturbation applied stimuli (period B). (a) Slow pacing techniques. Adapting this single-cell model to the (large B) yields a normal response in which tissue level, the resulting system of partial differ- every stimulus elicits an action potential. (b) ential equations has been well studied, revealing Faster pacing (smaller B) can cause alternans, (i) existence of traveling pulse solutions (solitary an abnormal alternation of APD. (c) Under action potentials); (ii) existence of periodic wave- extremely rapid pacing (very small B), every train solutions; (iii) stability of these solutions and second stimulus fails to produce an action how they evolve from initial data; (iv) asymptotic potential. estimates of action potential duration and velocity in terms of the parameters; (v) existence of spiral (2-D tissue) and scroll (3-D tissue) wave solutions; and (vi) asymptotic estimates of the rotation fre- Modeling the Action Potential quencies of spiral and scroll waves. We remark Mathematically modeling the cardiac action po- that spiral waves are important in the genesis of tential is an attractive research topic, in part certain arrhythmias (see the last two Challenge because such models tend to be rooted in the Problems below). For one of the earliest detailed Nobel Prize-winning work of Hodgkin and Huxley. mathematical analyses of the FitzHugh-Nagumo For a well-written, modern mathematical treat- system, see Keener [15]. ment of how that model was constructed, see the Although low-dimensional models allow us to text of Keener and Sneyd [16]. The key idea is to characterize how certain physiological parame- model the cardiac cell membrane as an electrical ters affect the dynamics, such models may lack circuit. The membrane acts both as a capacitor important details known about cardiac electro- because it supports a charge differential and as physiology, thereby limiting their clinical use. By a variable resistor because it can open and close contrast, higher-dimensional systems may suc- ion channels to regulate the inward and outward cessfully mimic many features of the action potential, but their resistance to mathematical flow of current. Letting Cm denote the capacitance of a cardiac cell membrane, then the capacitive analysis makes it difficult to understand how solutions depend upon parameters and initial current Cmdv/dt must balance the total ionic cur- rent Iionic. In other words, Cmdv/dt + Iionic = 0. In conditions. building a realistic model, the tricky part is to determine the specific form of Iionic. Herein lies Simulating Whole-Heart Dynamics a challenge for mathematicians, amounting to a Action potentials can propagate through cardiac balancing problem. tissue because the individual cells are electri- Modeling Challenge: Simultaneously keep the cally coupled. In the domain formed by heart tissue, transmembrane voltage has both spatial model minimally complicated so that it is amenable to , but make the model suf- and temporal dependence: v = v(x, y, z, t) where ficiently detailed that it can reproduce as much (x, y, z) ∈ . The usual way to model electri- cal wave propagation in cardiac tissue is via the clinically relevant data as possible. equation Reference [23] provides the address of a large Internet repository of ionic models. Although the sm ∂v (1) Cm + Iionic = ∇ (σ ∇v), models span a wide range of complexity, vir- ν ∂t tually all of them are based upon the original where sm/ν is the cell surface area per unit volume, Hodgkin-Huxley formalism and are presented as σ is a matrix of conductivities, Iionic is the total

April 2011 Notices of the AMS 543 440 450

330 350 stable fixed point

alternans 220 250 APD (ms) f(DI) (ms) 110 150 (a) (b) 0 0 0 100 200300 400 415 435 455 475 495 DI (ms) B (ms) Figure 2. (a) Example of a restitution function f . (b) The corresponding for the mapping (2). The period-doubling bifurcation leads to alternans, an abnormal period-2 alternation of APD.

ionic current that flows across the cell membrane, Restitution and the gradient ∇v is taken with respect to the Constructing an ionic model (1) of the action poten- spatial variables. Neumann boundary conditions tial requires careful description of Iionic. However, are enforced on the boundary ∂ . For a derivation to be clinically useful, a model should be able to of equation (1), see Chapter 11 of [16]. do more than just reproduce traces of v and/or Equation (1) presents a nice challenge for nu- the various transmembrane currents that affect v. merical analysts. Modeling groups around the One of the ultimate goals of cardiac modeling is world, including those led by Peter Hunter (Auck- to understand mechanisms for the onset of ar- land Bioengineering Institute, University of Auck- rhythmias that, by definition, are all about timing. land) and Rob MacLeod (Scientific Computing and The ability to accurately predict the duration and Imaging Institute, University of Utah), have made propagation speed of an action potential is an considerable progress in tackling the following. important benchmark for an ionic model. Simulation Challenge: Numerically solve (1) with Electrical restitution is a special feature of car- diac tissue that can be loosely defined as follows: (i) a physiologically realistic choice of Iionic; (ii) a domain that mimics the geometry of the whole the more well-rested the tissue is, the longer the heart; and (iii) enough computational efficiency to duration of each action potential, and the faster simulate many heartbeats, in order to better un- they propagate. More quantitatively, suppose that derstand how arrhythmias may suddenly develop. a cell is repeatedly stimulated (paced) with period B, eliciting a sequence of action potentials. Define Of course, operating within all of these con- APDn, the action potential duration (APD) of the straints is difficult. As a rule, the more physiologi- nth action potential, as the amount of time during cally detailed the model, the larger the number of which v remains elevated above some specified differential equations and parameters that govern threshold between the nth and (n + 1)st stimuli Iionic. The domain is quite complicated because (Figure 1). By restitution of APD, we mean the de- the heart has four distinct chambers (left and right pendence of APD on the pacing period B—typically atria and ventricles) and is connected to various APD decreases as B decreases. large veins and arteries (e.g., pulmonary veins and The amount of rest that the cell receives be- arteries, superior and inferior vena cava, and the tween consecutive action potentials is known as aorta). To further complicate matters, different the diastolic interval (DI). As illustrated in Fig- types of cardiac tissue (atrial, ventricular, Purkinje ure 1a, the DI preceding the nth action potential fiber) have different conduction properties, imply- is simply DIn−1 = B − APDn−1. Numerous authors ing that the conductivity tensor σ , as well as Iionic, have modeled restitution using a one-dimensional have spatial dependence. mapping For animations of action potential propagation = = − in a simulated heart, please see the websites [24] (2) APDn f (DIn−1) f (B APDn−1). and [25]. In particular, the latter website con- The graph of the APD restitution function f tains some lovely movies showing action-potential can be measured experimentally by recording the propagation around anatomical obstacles (e.g., steady-state APD values for a range of different dead tissue), as well as the formation of abnormal pacing periods B. Depending upon the shape of f , spiral waves. the mapping (2) may suffer from physiologically

544 Notices of the AMS Volume 58, Number 4 undesirable period-doubling bifurcations as the RR Interval pacing period B is varied (Figure 2). The resulting abnormal alternation of APD is called alternans, and its onset can be understood by straightforward analysis [9] of equation (2). Assuming that f has the qualitative appearance indicated in Figure 2a, then the mapping (2) has a unique fixed point satisfying APD∗ = f (B − APD∗). The fixed point is a stable ′ ∗ if |f (B − APD )| < 1, and this condition Figure 3. Schematic diagram of one lead of an implicitly determines the critical pacing interval electrocardiogram (ECG). The RR interval is the B at which the bifurcation to alternans occurs. In time between consecutive peaks. the literature, the conjecture that alternans would occur if the slope of f exceeds 1 was known as the restitution hypothesis. Given the complexity discuss three important problems involving car- of the heart, it should not be surprising that diac rhythm: (i) analysis of heart rate variability, (ii) the restitution hypothesis is false—alternans may predicting spontaneous initiation and termination occur if restitution functions have shallow slope or of arrhythmias, and (iii) techniques for controlling may fail to appear even when restitution functions arrhythmias. The first two of these have been past are steep. Over the past two decades, groups of themes of the annual Computers in Cardiology mathematicians, physicists, biomedical engineers, Challenge [26]. and physiologists have attempted to modify the restitution hypothesis, a collective effort that I will Heart Rate Variability refer to as the: A perfectly regular heart rhythm is actually a sign Alternans Challenge: Derive a mathematical crite- of potentially serious pathologies. The heart rate rion that accurately predicts the onset of alternans. is regulated by the autonomic nervous system (ANS), baroreceptors, and other factors. The ANS Although it may serve as a reasonable model uses the neurotransmitters norepinephrine and in certain dynamical regimes, equation (2) is too acetylcholine to speed up or slow down the heart, simplistic to capture all of the relevant behavior respectively, and tiny fluctuations in the levels of cardiac rhythm. First, in order to reproduce of these neurotransmitters induce some degree experimentally obtained restitution data, the one- of variability in the intervals between consecutive dimensional mapping (2) should be replaced with beats. The interbeat interval can be identified a higher dimensional mapping [19]. Second, a with the RR interval in an ECG (see Figure 3), cardiac cell needs a certain threshold amount and attempts to quantify heart rate variability of recovery time θ before it is able to produce (HRV) usually involve analyzing time series of RR another action potential, and any stimuli that are intervals. Mathematicians and statisticians can be applied before the cell recovers its excitability of assistance by rising to the following: are simply ignored (Figure 1c). Thus, in order to HRV-Time Series Challenge: Devise quantitative account for abnormally rapid rhythms (small B), methods for distinguishing between the RR time the mapping (2) should be replaced by APDn = series of normal subjects and those with cardiac f (kB−APDn−1), where k is the least positive integer pathologies. Can some pathologies be diagnosed for which kB − APDn > θ. solely by analysis of RR time series, and, if so, which We remark that, as action potentials propagate ones? through tissue, their propagation speed exhibits Analysis of RR time series was the theme of the same sort of dependence upon how well rested the 2002 Computers in Cardiology Challenge and the tissue is (i.e., speed depends upon DI). Letting continues to be one of the most active research ini- c(DI) denote the restitution function for action tiatives in mathematical cardiology. Before briefly potential speed, the graph of c is qualitatively surveying three past attempts to quantify HRV, let similar to that of f . us consider two natural questions related to this Challenge. First, could the variance of a sequence Rhythm u1, u2, . . . , un of RR intervals be a useful diagnos- Coordinated, rhythmic contraction of the cardiac tic? Although an extremely low variance is a sign muscle is vital for the heart to perform its primary of serious trouble, many patients with potentially function: pumping oxygenated blood throughout fatal cardiac abnormalities can exhibit perfectly the body. Improving our ability to diagnose and normal variance. For example, infants who suffer treat abnormal rhythms (arrhythmias) is critical in an aborted sudden infant death syndrome (SIDS) our fight against heart disease, the leading cause episode may have almost identical RR-interval vari- of death in the United States. In this section, we ance as normal infants [17]. To spot more subtle

April 2011 Notices of the AMS 545 pathologies, we need methods for quantifying the of the vector xi . (4) Let (m, r) denote the average “regularity” of a cardiac rhythm (see also Figure 4). of the of these probabilities. Repeat Φ the above steps with a larger window size m + 1 and calculate (m+1, r). (5) Approximate entropy 0.8 0.68 is defined as Φ . 0.7 0.63 ApEn(m, r) = lim[ (m + 1, r) − (m, r)]. n→∞ 0.6 0.58 Note that the limit is takenΦ as the numberΦ of data

0.5 0.53 points grows without bound.

RR interval (seconds) (a) (b) Although ApEn may be clinically useful in pre- 0.4 0.48 dicting aborted SIDS episodes, as explained below, 0 200 400 600 800 0 200 400 600 800 beat number, n beat number, n . detecting other pathologies may require a finer approach. Figure 4. Two signals with identical mean and Multiscale Entropy Analysis. Subsequent variance but much different approximate work, much of it by the same research group, entropies. (a) RR intervals u versus beat n points out that ApEn has several drawbacks. number n from a healthy patient. Data were Whereas ApEn performs well in distinguishing obtained from the MIT-BIH Normal Sinus Rhythm database at PhysioNet [26]. (b) A sine between normal and abnormally regular rhythms, wave with the same mean (0.594 s) and ApEn can be misleading when it comes to rec- variance (0.00224 s2 ) as the data in panel (a). ognizing abnormally irregular rhythms (which are likely to produce higher ApEn values). The ApEn statistic fails to account for the multiple Second, given the existing array of diagnostic tests scales involved in regulating cardiac rhythm—the that clinicians have at their disposal, what advan- physiological control mechanisms for heart rate tages might “automated” mathematical/statistical span a wide range of spatial scales (subcellular methods convey? Techniques such as multiscale to systemic) and temporal scales. These issues entropy (MSE) analysis (see below) could likely be are addressed in [4], which illustrates how a useful in diagnoses, risk stratification, or detecting multiscale entropy (MSE) method can be used drug toxicity [5]. Consider, for example, that the di- to analyze cardiac rhythm. The MSE method agnosis of congestive heart failure often involves a can successfully distinguish between normal battery of tests such as an echocardiogram, chest rhythms and two different routes to heart disease: X-ray, and ECG. In an encouraging finding [4], increased regularity due to heart failure versus MSE analysis of routine twenty-four-hour Holter increased due to arrhythmias such monitor recordings demonstrates that patients as atrial fibrillation. with congestive heart failure are statistically well Fast Fourier Transforms and Power Spectra. separated from normal subjects. Another way to seek regular patterns within se- Approximate Entropy, ApEn. Pincus and Gold- quences of data points is to take the fast Fourier berger [17] noted that, for the purposes of transform (FFT) and create a power spectral den- distinguishing between normal infants and those sity plot [1]. Any tall, narrow spikes in such plots with aborted SIDS episodes, a statistic known as correspond to dominant frequencies and are a sig- the approximate entropy (ApEn) appears to pro- nature of regularity. Power spectral density plots vide a useful diagnostic. In their data, ApEn is for normal infants and aborted-SIDS infants both roughly twice as large in a normal infant relative have been shown to exhibit broadband noise. How- to an infant with an aborted SIDS episode. The ever, the plots for normal infants tend to be more definition of ApEn = ApEn(m, r) incorporates the broadbanded, with power distributed over a wider conditional probability that data patterns that re- range of frequencies [17]. Although this does in- main close (i.e., within some tolerance r) over a dicate that normal infants have less regularity in window of m observations will also remain close if their heart rhythms, it remains to be seen whether the window size is increased to m +1. More specif- the FFT can give a clinically useful diagnostic. ically, ApEn is calculated as follows: (1) Given a Spontaneous Initiation and Termination of sequence u1, u2, . . . , un of RR intervals and a win- dow size m ∈ N, for each i = 1, 2, . . . , n − m + 1 Arrhythmias define vectors xi = (ui , ui+1, . . . , ui+m−1) consisting Anatomical obstacles such as regions of dead (non- of m consecutive data points. (2) For each pair conducting) cardiac tissue can interfere with the of vectors xi and xj , compute their distance (e.g., normal propagation of electrical signals. Normally, maximum difference between corresponding com- the heart relies upon its own native pacemaker ponents). (3) Given a tolerance r ∈ R+, for each cells to supply electrical stimuli that generate i = 1, 2, . . . , n − m + 1, compute the (estimated) propagating action potentials. Both the timing of probability that a vector will be within distance r the stimuli and the positioning of the pacemaker

546 Notices of the AMS Volume 58, Number 4 few modeling studies focus specifically on tempo- ral patterns of intermittent bursts of rapid activity. One interesting exception was provided by Bub et al. [2], who blended mathematics with experiment to investigate bursting dynamics of rotors. Their experiments with cultures of embryonic chick heart cells revealed intermittent bursts of rotor (a) (b) waves, each lasting on the order of half a minute Figure 5. Wavefront of an action potential and with consecutive bursts separated by approx- propagating around a circular, nonconducting imately forty seconds. Using a cellular automaton obstacle in homogeneous tissue. (a) Snapshots model of an idealized two-dimensional sheet of of a single wave front propagating left to cardiac cells, they qualitatively reproduced the right. The wavefront breaks at the obstacle but same bursting dynamics. reemerges on the other side. (b) A spiral wave Because cellular automata models are difficult propagates unidirectionally (counterclockwise to analyze, other mathematical studies tend to in- in this case) while anchored to the obstacle. corporate another type of discrete-time model. As a first step toward understanding whether tissue can support a sustained spiral wave, numerous authors (e.g., [13, 20, 21]) have used partial dif- relative to the obstacle are critical in determining ference equations (P Es) to model propagation whether an arrhythmia will result. For example, of action potentials in idealized, one-dimensional Figure 5a shows five snapshots of a solitary action circular domains, such as the one formed by the potential wavefront colliding with a circular, non- boundary of the obstacle in Figure 5b. For exam- conducting obstacle in a square sheet of otherwise ple, Ito and Glass [13] introduced the following homogeneous tissue. In this case, the wavefront discrete, restitution-based model of a reentrant breaks at the obstacle but reemerges (with a slight action potential in a ring composed of m cells, deflection) on the far side of the obstacle. The each of length L: deflection of this wavefront causes spatial het- (3) erogeneity in the amount of local recovery time that the cells experience. One specific pathology DI = −f (DI ) that can be induced by such heterogeneity is the i,n i,n−1 − formation of spiral waves (Figure 5b) that become i 1 L m L anchored to the obstacle [12]. The figure shows a + + , c(I D n,j ) c(I D n−1,j ) counterclockwise rotating spiral wave whose tip j=1 j=1 is pinned to the nonconducting obstacle. Spiral i = 1, 2, . . . , m. waves in cardiac tissue can be quite dangerous, Here, f (DI) and c(DI) are the restitution func- because the period of the rotation is often signifi- tions for action potential duration and speed, cantly faster than the period of the heart’s native respectively, and are defined only for DI > θ, pacemaker cells. As the rotating spiral arm col- the threshold amount of rest required to sus- lides with advancing wavefronts emanating from tain propagation. Although this P E incorporates the pacemaker, the higher frequency spiral arm less physiological detail than the one-dimensional may seize control of more territory, ultimately version of the PDE in equation (1), solution of taking over the pacing of the heart and resulting P E (3) is far less computationally intensive. This in tachycardia (faster-than-normal rhythm). Com- allows for simulation over long time scales, which petition between the heart’s pacemaker and these is important in studying intermittent reentrant abnormal reentrant spiral waves can lead to spo- tachycardia. radic episodes of tachycardia, which is the subject Equation (3) does not account for the competi- of the following. tion between the action potentials supplied by the heart’s native pacemaker and those that are recir- Temporal Pattern Challenge: Create a mathemat- culated around the obstacle. Recently, Sedaghat ical model that reproduces temporal patterns of and collaborators [20] modified (3) to allow for spontaneous initiation and termination of tachy- spontaneous transitions between normal pacing, cardia. in which the rhythm is driven by the heart’s na- This problem is a variant of the 2004 Computers tive pacemaker, and (abnormal) reentry, in which in Cardiology Challenge. the rhythm is driven by a recirculating action There is a significant ongoing effort devoted potential in a ring-shaped pathway. It remains to this Challenge and related problems, and, in to be seen whether such models can reproduce particular, it is known that there are many mecha- the erratic patterns of intermittent arrhythmias, nisms and tissue geometries that can support the patterns that involve time scales ranging from creation and destruction of spiral waves. However, seconds to months.

April 2011 Notices of the AMS 547 Controlling the Rhythm each iteration. If is replaced by + ǫ (xn − xn−1) Upon detecting an arrhythmia, the natural next where ǫ > 0 is a feedback gain parameter, then, step is to (try to) stop it. Virtually every hospital- depending upon the choice of ǫ, we can (i) stabi- themed television program includes scenes in lize previously unstable fixed points; (ii) prevent which a manual external defibrillator is used to period-doubling cascades from occurring; and (iii) violently shock a patient during cardiac arrest. control chaos. Although wonderful for dramatic effect, this sort Although the aforementioned feedback control of defibrillation has adverse physiological side algorithms succeed in small patches of tissue, effects. Certain patients with heart disease re- both experimental [3] and theoretical [6] studies ceive a more humane alternative: battery-operated suggest that applying such schemes locally (via an implantable cardioverter defibrillators (ICDs) that implantable electrode) cannot control the rhythm intervene when onset of an arrhythmia is detected. over large enough spatial domains to be useful Although newer ICDs are better at distinguishing in the whole heart. This issue lies at the core of between non-life-threatening tachycardia and life- the above Challenge problem—controlling whole- threatening fibrillation, when an ICD elects to heart dynamics is difficult! defibrillate, it can cause excruciating pain. Rather than having an ICD deliver a high-frequency train Discussion and Further Reading of strong stimuli, is it possible (via careful timing) This article is intended to serve as an open invi- to deliver a train of tiny stimuli that accomplish tation to the mathematics community to join the the same goal? Below is a brief discussion of two fight for an improved understanding of cardiac attempts to tackle our final challenge. electrophysiology and arrhythmias. The featured Control Challenge: Devise a robust feedback Challenges were chosen, in part, because each control algorithm that can suppress abnormal problem has aspects that will appeal to vari- rhythms in the whole heart. ous mathematical subdisciplines (e.g., numerical In both experiments and numerical simulations, analysis, discrete dynamical systems, topology, Isomura et al. [12] were able to terminate an differential equations, and mathematical statis- abnormal spiral wave by applying a brief, high- tics). The reader is urged to consider how his or frequency train of tiny electrical stimuli. This is her own areas of mathematical expertise might precisely what an ICD does during antitachycardia aid in these exciting research efforts. pacing, and this technique is often effective in The above presentation should be accompanied restoring a normal rhythm. Prior explanations by several disclaimers. First, much of this article of why antitachycardia pacing works tend to be represents a mathematician’s interpretation of heuristic as opposed to mathematical. It would various cardiac phenomena. For example, Figure 5 be of great physiological interest if mathematical is a mathematical idealization of the substrate for analysis could show either (i) why existing anti- a pinned spiral wave and is quite different from tachycardia pacing techniques often work or (ii) the anatomical circuits that a cardiologist would that there is a better way. associate with tachycardia. Readers interested in a Another abnormal rhythm that has been suc- more accurate portrayal of reality are encouraged cessfully controlled both experimentally and in to consult with cardiologists and electrophysiol- simulations is known as T-wave alternans. Mathe- ogists. Second, it must be emphasized that there matically, the onset of alternans can be understood is already a vast literature dedicated to these six by examining the bifurcation diagram in Figure 2b. Challenges. Rather than vainly attempting to com- In the one-dimensional mapping (2), a period- pile a comprehensive bibliography, the references doubling bifurcation may occur as the pacing are merely intended to provide a few leads for period B is decreased. The bifurcation causes APD those who might be interested in this fascinating alternans, a pattern in which APD values exhibit beat-to-beat alternation. field. Both experiments [10] and theory [14] indicate For more information on mathematical cardi- that simple feedback control can terminate alter- ology (or mathematical physiology in general), nans in small patches of cardiac tissue. The idea there are several books that serve as good starting is to make small perturbations to the period B in points. Keener and Sneyd’s text [16], Mathemati- order to prevent the period-doubling bifurcation cal Physiology, provides an excellent mathematical from occurring in the restitution mapping (2). As treatment of electrophysiology and heart rhythm. an illustration, consider the famous discrete logis- Some older books, such as Glass and Mackey [8], tic mapping xn+1 = xn(1 − xn), where x0 ∈ [0, 1] Plonsey and Barr [18], and Winfree [22], in- and 0 ≤ ≤ 4. As the parameter is increased, a clude well-written introductions to mathematical cascade of period-doubling bifurcations ultimately cardiology. Finally, most of the websites that leads to chaos when ≈ 3.5699. This cascade can are referenced in this article contain extensive be prevented if small adjustments are made to at bibliographies.

548 Notices of the AMS Volume 58, Number 4 Acknowledgments [17] S. M. Pincus and A. L. Goldberger, Physiological Support of the National Institutes of Health under time-series analysis: What does regularity quan- tify?, Am. J. Physiol. Heart Circ. Physiol. 266 (1994), grant T15 HL088517-02 is gratefully acknowl- H1643–H1656. edged. I would also like to thank Professors D. G. [18] R. Plonsey and R. C. Barr, Bioelectricity: A Quan- Schaeffer, H. Sedaghat, and W. J. Terrell for their titative Approach, 2nd ed., Kluwer, New York, valuable advice and feedback. 2000. [19] D. G. Schaeffer, J. W. Cain, D. J. Gauthier, S. S. Kalb, W. Krassowska, R. A. Oliver, E. G. Tolka- References cheva, and W. Ying, An ionically based mapping [1] S. Akselrod, D. Gordon, F. A. Ubel, D. C. Shan- model with memory for cardiac restitution, Bull. non, A. C. Berger, and R. J. Cohen, Power spectrum Math. Bio. 69 (2007), 459–482. analysis of heart rate fluctuation: A quantita- [20] H. Sedaghat, M. A. Wood, J. W. Cain, C. K. Cheng, tive probe of beat-to-beat cardiovascular control, C. M. Baumgarten, and D. M. Chan, Complex Science 213 (1981), 220–222. temporal patterns of spontaneous initiation and [2] G. Bub, L. Glass, N. G. Publicover, and A. Shrier, termination of reentry in a loop of cardiac tissue, Bursting calcium rotors in cultured cardiac myocyte J. Theor. Biol. 254 (2008), 14–26. monolayers, Proc. Natl. Acad. Sci. USA 95 (1998), [21] M. D. Stubna, R. H. Rand, and R. F. Gilmour, 10283–10287. Jr., Analysis of a nonlinear partial difference equa- [3] D. J. Christini, M. L. Riccio, C. A. Culianu, J. J. tion, and its application to cardiac dynamics, Fox, A. Karma, and R. F. Gilmour, Jr., Control of J. Difference Equations and Applications 8 (2002), electrical alternans in canine cardiac Purkinje fibers, 1147–1169. Phys. Rev. Lett. 96 (2006), 104101. [22] A.T. Winfree, When Time Breaks Down: The Three- [4] M. Costa, A. L. Goldberger, and C.-K. Peng, Mul- Dimensional Dynamics of Electrochemical Waves tiscale entropy analysis of biological signals, Phys. and Cardiac Arrhythmias, Rev. E 71 (2005), 021906. Press, Princeton, 1987. [5] M. Costa, C.-K. Peng, and A. L. Goldberger, Multi- [23] http:////models.cellml.org/cellml scale analysis of heart rate dynamics: Entropy and [24] http://www.scholarpedia.org/article/ time irreversibility measures, Cardiovasc. Eng. 8 Cardiac_arrhythmia (2008), 88–93. [25] http://www.math.sjtu.edu.cn/faculty/ [6] B. Echebarria and A. Karma, Instability and spa- wying/ tiotemporal dynamics of alternans in paced cardiac [26] http://www.physionet.org/ tissue, Phys. Rev. Lett. 88 (2002), 208101. [7] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J. 1 (1961), 445–466. [8] L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press, Princeton, NJ, 1988. [9] M. R. Guevara, G. Ward, A. Shrier, and L. Glass, Electrical alternans and period-doubling bifur- cations, IEEE Computers in Cardiology (1984), 167–170. [10] G. M. Hall and D. J. Gauthier, Experimental con- trol of cardiac muscle alternans, Phys. Rev. Lett. 88 (2002), 198102. [11] A. L. Hodgkin and A. F. Huxley, A quantitative de- scription of membrane current and its application to conduction and excitation in nerve, J. Physiol. London 117 (1952), 500–544. [12] A. Isomura, M. Hörning, K. Agladze, and K. Yoshikawa, Eliminating spiral waves pinned to an anatomical obstacle in cardiac myocytes by high-frequency stimuli, Phys. Rev. E 78 (2008), 066216. [13] H. Ito and L. Glass, Theory of reentrant excitation in a ring of cardiac tissue, Physica D 56 (1992), 84– 106. [14] P. N. Jordan and D. J. Christini, Adaptive diastolic interval control of cardiac action potential duration alternans, J. Cardiovasc. Electrophysiol. 15 (2004), 1177–1185. [15] J. P. Keener, Waves in excitable media, SIAM J. Appl. Math. 39 (1980), 528–548. [16] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998.

April 2011 Notices of the AMS 549 3N Colored Points in a Plane

Günter M. Ziegler

Why do I want to tell you about the colored Tverberg theorem? Well: • the setting sounds so harmless, so elementary, as if children were playing with a few colored points in the plane; • it comes with anecdotes, like the one about a Nor- wegian mathematician freezing in a hotel room; • the latest twist in the story, and the key to a solution, has not been in a very technical proof but rather in the right wording of the problem and of the final outcome; • an interesting mix of geometric, combinatorial, algebraic, and topological methods have been used; • after almost two decades, progress is still being made (and I am happy I could contribute to it); The claim would be that for any such point and finally configuration in the plane there is a partition into • there is a lot more to do: intriguing conjectures triples, such that the corresponding triangles have that again sound harmless, elementary, playful. a point in common:

A Conjecture by Birch “On 3N points in a plane” is the title of a short paper [6] by Bryan John Birch from 1959. Its main result is right on the first page: Birch’s Theorem 1: Any 3N points in the plane determine N triangles that have a point in common. In order to illustrate this, we may assume that the 3N points are in general position. For N = 4 we have 3N = 12 points, and the situation might look like the following diagram:

Günter M. Ziegler is professor of mathematics at Freie Universität , D-14195 Berlin, Germany. His email address is [email protected].

550 Notices of the AMS Volume 58, Number 4 Birch’s proof for this is remarkable, as it uses of the disk points inside. By Brouwer’s fixed point a topological fixed point theorem—years before theorem this vector field must have a zero—which Lovász in 1978 proved the Kneser conjecture using turns out to be a center point.  the Borsuk–Ulam theorem, which is commonly seen In the same little paper [6] Birch observes that as the starting point of topological combinatorics the bound “3N” is not really tight, as one can also [9]. Birch indeed derives his result from the center get a result about only 3N − 2 points: point theorem, which had been first provided by Birch’s Theorem 1*: Any 3N − 2 points in the Bernhard H. Neumann in 1945 [15]: plane can be partitioned into N subsets whose con- Center Point Theorem. For any 3N points in the vex hulls have a point in common. plane there is a point c, called a “center point”, such The subsets thus consist of one, two, or three that any half plane that contains c must necessarily points, and hence their convex hulls are triangles, also contain at least N out of the given 3N points. edges, or single points—and we require that these Birch criticized Neumann’s proof for this result: N convex sets have a point in common. The drawing “his proof, though elementary, is long, and does not illustrates this for N = 4: extend to higher dimensions”. He thus proceeded to give his own proof, which also provides an n-dimensional center point theorem. However, apparently Birch did not know that Richard Rado had achieved the result much earlier, published in 1946 [16]. Rado’s proof for this is elementary geometry (it uses Helly’s theorem—“if any n sets in a finite family of convex sets in Rn have a point in common, then all the sets have a point in common”); see, e.g., [12, Sect. 1.4].  Here is how Birch derived his Theorem 1 from the center point theorem: Label the 3N points “1, 2,...,N, 1, 2,...,N, 1, 2,...,N” in clockwise or- der around the center point c:

4 1 A solution could thus be given by one single 2 3 point that lies in N − 1 triangles, or there could be N − 2 triangles that all contain the intersection c point of two further edges: 2 3

3 1 4 2 1 4

Then the center point c lies in each triangle determined by three points with the same label i. Indeed, each half space with c on the boundary contains at least N contiguous points from the circular sequence, and thus at least one point of each label i. And here is the idea used by Birch to prove the center point theorem: If a point x is not a And it is also easy to see that fewer than 3N − 2 center point, then some, but not all, of the half points will in general not be divisible into N subsets spaces that have x on the boundary are “bad”, whose convex hulls intersect. (Not even the affine i.e., contain less than N of the 3N points. These hulls will intersect, for codimension reasons.) In half planes at x point you into directions to look that sense, Birch’s Theorem 1* is sharp. for improvement. By averaging, Birch uses that to Who is Bryan John Birch? Where does this define, on a disk that contains the 3N points, a problem come from? His paper contains only a continuous (!) vector field which on the boundary small hint at the end:

April 2011 Notices of the AMS 551 In conclusion, I would like to thank remember details of what one was Professor Eggleston for his enter- thinking about 54 years ago. taining lectures, which led me to I submitted a thesis in competi- perpetrate this note. tion for a Trinity Junior Research But of course I could try to ask him (he was born in Fellowship in September 1956; this 1931 and is an emeritus professor in Cambridge, thesis was a compendium of vari- England). My email was answered promptly: ous bits and pieces, including “3N points in a plane”. The judges of the competition came from all faculties, so one had to include a summary that laymen could read. I remem- ber that my summary included a picture of the application of the first non-Helly case (7 points in a plane) to the configuration formed by the seven brightest stars in the Pleiades. I don’t remember why the paper wasn’t submitted till 1959: I was working on several other things—I switched to additive an- alytic number theory, and then to elliptic curvery—and very probably 1959 was when I gave up trying to prove higher dimensional cases of Tverberg’s theorem; another pos- sibility is that I was jogged into writing up by the publication of Eggleston’s tract.

Mariana Cook 2008 Indeed, Birch became famous for his work on

© elliptic curves—in particular since it led to a one- Bryan John Birch million-dollar millennium problem that carries his name, the “Birch and Swinnerton-Dyer conjecture”. Another one of his conjectures did not carry I was an undergraduate at Trinity such a cash prize but turned out to be important (College, Cambridge) from 1951-4, and influential nevertheless: again in his paper “On and in my third year I took Part III 3N points in a plane”, on the same page as his of the Mathematical Tripos; then Theorem 1*, we find a conjectured n-dimensional as now, one attended 6 to 8 “start- version of it: ing graduate student level” courses Birch’s Conjecture: Any (n+1)N−n points in Rn and took a cross-section exam at can be partitioned into N subsets, whose convex the end of the year; nowadays of hulls have a point in common. course it is no longer taken by 3rd And when Birch now writes that probably 1959 year students. One of the courses I was when he gave up trying to prove “higher took, in the spring of 1954 I would dimensional cases of Tverberg’s theorem”, then guess, was a very pleasant course he reverses the historical order of things: he gave on “Convexity” by H. G. Eggleston; up trying to prove higher dimensional versions of this course formed the for Birch’s theorem, that is, cases of his own conjecture. his Cambridge Tract, published in Tverberg came later. 1958 according to my references. Eggleston’s course contained sev- Tverberg’s Theorem eral proofs of the isoperimetric Helge Tverberg, born in 1935, is a Norwegian inequality, as well as Helly’s theo- mathematician. In 1961 he attended a workshop on rem (of course) and I was involved at University College London, with more complicated plane geom- where on the sidelines there was also a course on etry when I started research in the convexity, presumably taught by Rogers [12, p. 16]: Geometry of Numbers under Ian I found this material fascinating, Cassels; so it wasn’t unnatural for and read upon it more back in me to be thinking about “3N Points Bergen. Helly’s Theorem was es- in a Plane”; but one tends not to pecially fascinating, and, in my

552 Notices of the AMS Volume 58, Number 4 reading, I came upon the following I recall that the weather was bitterly application. Let S be a set of 3N cold in Manchester. I awoke very points in the plane. Then, there is a early one morning shivering, as the point p, not necessarily in S, such electric heater in the hotel room that every half-plane containing p had gone off, and I did not have an contains at least N points from S. extra shilling to feed the meter. So, It struck me that this would follow instead of falling back to sleep, I simply if it were always possible to reviewed the problem once more, split S into N triplets so that the N and then the solution dawned on triangles so formed would have a me! common point p. For, a half-plane I explained it to Birch, and, after containing p would contain at least an agreeable day of mathematical 1 vertex from each triangle. conversation with him, returned Thus Tverberg ran into Birch’s problem in an to Norway to start writing up the attempt to prove Rado’s center point theorem— result. just the opposite direction from that taken by Birch disagrees on this: he remembers that Tverberg Birch. was not all that interested in explaining his solution, and rather more in seeing a bit of England on his last day. But it’s not our job to resolve this apparent contradiction here. In any case, in 1966 (submitted May 8, 1964) Tverberg’s paper “A generalization of Radon’s theorem” [20] appeared, which he refers to as “T66”. The Tverberg theorem is his most famous result, which he came back to again and again. Thus “A generalization of Radon’s theorem, II” appeared in 1981 with a new proof, and “On generalizations of Radon’s theorem and the ham sandwich theorem” (joint work with Siniša Vre´cica)in 1993, which contains a tantalizing conjecture, an extension of Tverberg’s theorem to transversals. The original Tverberg theorem now has several different proofs, including those by Tverberg, Roudneff, Sarkaria, and more recently by Zvagel’skii. An especially of the Mathematisches Forschungsinstitut elegant version is due to Karanbir Sarkaria [17], with further simplifications by Shmuel Onn [4] Oberwolfach. Archives [11] [12, Sect. 8.3]. Helge Tverberg, 1981. Tverberg’s theorem, as proved in the “T66” paper, happens to be exactly Birch’s conjecture. Nevertheless, we would phrase it differently today. In 1962 Tverberg attended the ICM in Stockholm, Thus from now on we use d to denote the dimension Sweden, and there, after a dinner with Bryan Birch (that is, d = 2 in Birch’s classical theorem). We use and Hallard Croft from England, before parting the letter r for the number of subsets we want to at some street corner, he told Croft about his partition into (which was previously denoted by N). problem about 3N points in the plane. Croft had to And we use the letter N to denote N := (d+1)(r −1) disappoint him, the result already being known and (and thus unfortunately it now means something having been published by Birch—but he suggested completely different than before). And instead of that Tverberg could try his luck on the higher- discussing N +1 points in the plane and the convex dimensional case, which Birch had been unable to hulls of subsets, we now consider an N-dimensional do. simplex ∆N (which has N + 1 = (d + 1)(r − 1) + 1 Then in 1963 Tverberg first solved the three- vertices) and an affine map that in particular d dimensional case by a complicated proof that positions the N + 1 vertices of ∆N in R . Thus consisted of seven separate cases and offered no Tverberg’s theorem gets its modern form: hope for an extension to higher dimensions. Birch’s Conjecture = Tverberg’s Theorem: Let One year later, in 1964, he then obtained a travel d ≥ 1, r ≥ 2, and N := (d + 1)(r − 1). For every d stipend to England, where he wanted to discuss the affine map f : ∆N → R there are r disjoint faces of problem with Birch (then in Manchester) and with ∆N whose images under f intersect. Richard Rado (at the ). Rado The following diagram illustrates this result for had also obtained partial results. What happened the small parameters d = 2 and r = 2, where we then, Tverberg describes as follows [21, p. 16/17]: get N = 3, and thus have to consider a map of

April 2011 Notices of the AMS 553 a tetrahedron (three-dimensional simplex) to the “… We Need a Colored Version …” plane. A team of three Hungarians, Imre Bárány, Zoltan Füredi, and László Lovász, in an influential com- putational geometry paper “On the number of halving planes” (conference proceedings version 1988, journal publication 1990 [3]) stumbled upon a situation with three disjoint sets A, B, C of points in the plane and observed: For this we need a colored version of Tverberg’s theorem. In their paper they needed only a very simple small special case: Let A, B, C be sets of t red, green, resp. blue points in the plane, then one can find r = 3 disjoint triples consisting of one point of each color such that the convex hulls of the triples have a point in common. They gave a proof for t = 7, asserted they also The Topological Tverberg Theorem had a proof for t = 4, but also noted that they had The modern version of the Tverberg theorem is no counterexample even for t = 3. not only more succinct (and a bit more abstract), The call for a colored version of Tverberg’s theorem was seen as a challenge and attacked but it also has the advantage of suggesting a immediately. The first answer, by Imre Bárány and generalization, known as the “topological Tverberg David Larman in 1991, treated the case of 3r points theorem”, which the Hungarian mathematicians in the plane, with three different colors: Imre Bárány and András Sz˝ucs,together with the “… given r red, r white, r green Russian Senya B. Shlosman, presented in 1981. points in the plane …”. Topological Tverberg Theorem: [5] Let d ≥ 1, r ≥ 2, and N := (d+1)(r −1). For every continuous d map f : ∆N → R there are r disjoint faces of ∆N whose images under f intersect.

(Question: Why these particular three colors? I The name “topological Tverberg theorem” hap- have recently asked David Larman; he didn’t know. pens to be pretty bad terminology, not only since Perhaps his coauthor managed to slip in the colors neither Tverberg nor the theorem is “topological” of the Hungarian flag?) but—much more seriously—since in the version Anyway, here is the answer suggested by Bárány I have just stated this is not a theorem, but only and Larman: Colored Tverberg Theorem: Let d ≥ 1 and a conjecture. Indeed, Bárány et al. claimed and r ≥ 2, and f : -→ Rd affine (order at least proved this only for the case where r ≥ 2 is a prime. ∆N continuous), where the N + 1 vertices of carry Only later was this extended to the case of prime ∆N d + 1 different colors, and every color class Ci has powers r, first by Murad Özaydin in an unpublished size |Ci | ≥ t for a sufficiently large t. Then ∆N has preprint from 1987. We refer to the wonderful r disjoint rainbow faces, whose images under f textbook Using the Borsuk–Ulam Theorem by Jiˇrí intersect. Matoušek [13] for details and references. In any Of course again here neither Tverberg nor case the conjecture remains open and a challenge the theorem is colored. And the statement just up to now for d ≥ 2 and nonprime powers r. proposed is not a useful theorem, as it does not

554 Notices of the AMS Volume 58, Number 4 specify the “sufficiently large t”. A rainbow face This result was a surprise to us (and even more refers to a d-dimensional face of the simplex whose to others, perhaps), since we arrived at it via d + 1 vertices carry the d + 1 different colors. In considerable detours and it needed a substantial the case d = 2 thus we have at least 3t points in change of perspective. Indeed, we arrived at a the plane, which carry the three different colors. new colored Tverberg theorem that uses more The claim is that then there are r rainbow triangles colors, requires different assumptions about the that have a point in common: color classes, and contains the classical Tverberg theorem as a special case—and turns out to be much easier to prove.

A New Colored Tverberg Theorem Here it is: New Colored Tverberg Theorem [7]: Let d ≥ 1, d r ≥ 2 prime, N := (d + 1)(r − 1) and f : ∆N -→ R affine (or at least continuous), where the N + 1 ver- tices of ∆N have at least d + 2 different colors, and each color class Ci has size |Ci | ≤ r − 1. The ∆N has r disjoint rainbow faces whose images under f intersect. Thus, for example, we consider the following situation in the plane (d = 2) for r = 5, in which the N + 1 = 3 · 4 + 1 = 13 points have four different colors and no color is used more than r − 1 = 4 times (see the following figure). For d = 2 Bárány and Larman proved this, and indeed they obtained a “sharp” colored version of Birch’s theorem: for d = 2 it suffices to require that t ≥ r. For d = 1 the analogous colored version of Tverberg’s theorem is a nice exercise. For d > 2 Bárány and Larman presented it as a conjecture. This was answered by a breakthrough paper by Rade Živaljevi´cand Siniša Vre´cicafrom Belgrade, Yugoslavia, published in 1992 [24]. They introduced new concepts and methods to topological combi- natorics (in particular, “chessboard complexes”) and thus could show that the colored Tverberg, in the version just given, holds for t ≥ 2r − 1, if r is a prime, and thus also for t ≥ 4r − 3 and all r ≥ 2, due to Bertrand’s postulate that there is always a prime between n and 2n [1, Chap. 2]. Živaljevi´cand Vre´cica’sbreakthrough got a lot of attention. In particular, JiˇríMatoušek in Prague was so excited that he gave a course that eventually In comparison to the original colored Tverberg led to the textbook [13] mentioned before, which theorem the number of colors has changed (not develops all the mathematics from scratch that d + 1 any more, but at least d + 2), the sizes of is needed to arrive eventually, in the last section the color classes have changed (not “large enough” of the book, at the Živaljevi´c–Vre´cicaproof of the any more, and at least r, but instead less than r), colored Tverberg theorem. and the definition of a rainbow face has changed Nevertheless, the gap between t = r and t ≥ (they do not have to carry all the colors any more, 4r − 3 remained: the colored Tverberg theorem and indeed they can’t, but now they are defined as of Živaljevi´cand Vre´cicais not sharp. And also faces on which no vertex color appears more than it is not a generalization or strengthening of the once). One of the solutions looks like the figure at classical, color-free Tverberg theorem. the top of the next page. Try to find one yourself, A sharp version was obtained only very recently, before you turn the page! again by a team of three. Pavle V. M. Blagojevi´c Our theorem admits the special case that all from Belgrade, my Berlin Ph.D. student Benjamin color classes have size 1; thus all the vertices Matschke, and I were ready in October 2009 to of the simplex ∆N have distinct colors, and thus present a proof that t = r suffices when r + 1 is a all faces have the rainbow property, and thus we prime (and thus t = r + o(r) is good enough for obtain the original topological Tverberg theorem large r). by Bárány–Shlosman–Sz˝uczas a special case.

April 2011 Notices of the AMS 555 the topological space on the left-hand side is a simplicial complex that encodes the sets of r points on r disjoint rainbow faces of ∆N ; the right-hand side encodes the r-tuples of points (not all equal) they would map to. The classical “there is no such map” result of this type is the Borsuk–Ulam theorem, which says that there is no continuous map n n−1 F : S -→Z2 S that would be compatible with the antipodal action of the group Z2. For our more complicated setup we have even provided three different proofs. The easiest one may be found in [8]; it uses the mapping degree. One interesting aspect of this proof is that it asks us to count the number of Tverberg partitions for a In a second special case we could have d + 1 certain point configuration, where we get the result r − color classes of size 1, and one final color class (r − 1)!d . We eventually conclude that (r − 1)!d would consist of a single additional vertex with a must be divisible by r, which is not true if r is a separate color. This special case turns out to be prime. (See also [23] for a sketch of this proof, and important since we can derive the sharp classical [14] for an elementary “topology-free” version.) Tverberg theorem from it (for primes r), but also The second proof (which we found first) is since this special case indeed also implies the technically more demanding; it uses equivariant theorem in full generality. Both these reductions obstruction theory, which one can learn from are elementary geometric (and the ideas needed Tammo tom Dieck’s book on transformation groups have been used before in similar contexts). [19, Sect. II.3] and then apply to our concrete Thus we must prove the new colored Tverberg situation. Here one has to act with care, as the theorem for the case of color classes |C0| = |C1| = action of the symmetric group is not free. This · · · = |Cd | = r − 1 and |Cd+1| = 1. For this we use proof is not only more difficult than the first one, the by-now classical configuration space/test map but it also yields more: we get that the configuration scheme, which can be learned from the textbook space/test map proof scheme works, even if we use [13]—which recently on German eBay appeared in the full symmetric group, if and only if (r − 1)!d is the category “Books > Children’s & Youth Literature not divisible by r, that is, if r is a prime and in the > Fun & Games > Knowledge for Children”. So this uninteresting case r = 4, d = 1. In all other cases, is certainly a book you and your family cannot do the equivariant map F in question does exist, and without: we cannot conclude anything. The third proof, also from [8], is the most complicated one: it computes the Fadell–Husseini index, an ideal of the cohomology ring of the group that we have acting. However, it also yields even more: We get the full theorem directly, without previous reduction to the special case of color classes of sizes d − 1 resp. 1, and thus it can be extended to a proof of the transversal generalization of the new colored Tverberg theorem.

“Proofs should be communicated only by consenting adults in private” — Victor Klee (U. Washington)

According to this scheme we have to show Questions, Problems, Challenges that a certain equivariant map does not exist. 1. As mentioned above, for the classical Tverberg More precisely, we want to show that there is no theorem we have “elementary” linear algebra proofs continuous map that would work for all r ≥ 2. Is there a similar proof also for the affine case of the new colored F : ( )∗(d+1) ∗ [r] -→ SN−1 ∆r−1,r G Tverberg theorem? that is compatible with the action of a finite group 2. The Tverberg , whether colored or G (here a cyclic or a symmetric group). Here not, promise to us the existence of a specified type

556 Notices of the AMS Volume 58, Number 4 of partition of a point configuration. How would [3] I. Bárány, Z. Füredi, and L. Lovász, On the number one find one? Is it easy to find such a partition, of halving planes, Combinatorica 10 (1990), 175– can one compute one in polynomial time? This is 183. [4] I. Bárány and S. Onn, Carathéodory’s theorem, not clear at all—not even for the probably much colourful and applicable, Intuitive Geometry (Bu- simpler colored Carathéodory theorem of Bárány dapest, 1995), Bolyai Soc. Math. Studies 6, , [2] [12, Sect. 8.2]. 1997, János Bolyai Math. Soc., 11–21. 3. Whoever plays around with instances of the [5] I. Bárány, S. B. Shlosman, and A. Sz˝ucs, On a Tverberg theorem will notice that typically there is topological generalization of a theorem of Tverberg, not only one Tverberg partition, but indeed many J. London Math. Soc. (2) 23 (1981), 158–164. [6] B. J. Birch, On 3N points in a plane, Math. Proc. of them. For the configuration of (d + 1)(r − 1) + 1 Cambridge Phil. Soc. 55 (1959), 289–293. d points in R that is suggested by our third figure, [7] P. V. M. Blagojevi´c,B. Matschke, and G. M. Ziegler, there are exactly (r − 1)!d distinct partitions. Optimal bounds for the colored Tverberg problem, Gerard Sierksma from Groningen, , preprint, October 2009, 10 pages; revised Novem- has conjectured that there are always (that is, for ber 2009, 11 pages; http://arXiv.org/abs/0910. all point configurations) at least (r − 1)!d distinct 4987. [8] , Optimal bounds for a colorful Tverberg– Tverberg partitions. Indeed, he has offered one Vre´cicatype problem, preprint, November 2009, Adv. whole Gouda cheese for a proof of his conjecture, Math. (2011), doi:10.1016/j.aim.2011.01.009, which is why it is known as “Sierksma’s Dutch to appear; http://arXiv.org/abs/0911.2692. Cheese Problem”. This problem is open even for [9] M. de Longueville, 25 years proof of the Kneser the case of d = 2. Lower bounds on the number of conjecture, EMS-Newsletter No. 53 (2004), 16– solutions have been obtained by Aleksandar Vuˇci´c 19, http://www.emis.de/newsletter/current/ current9.pdf. and Rade Živaljevi´c[22] in the case of primes and [10] S. Hell, On the number of Tverberg partitions in by Stephan Hell [10] for the case of prime powers. the prime power case, European J. Combinatorics 28 It may well be that good lower bounds are easier (2007), 347–355. to prove for the Tverberg theorem with colors and [11] G. Kalai, Sarkaria’s proof of Tverberg’s the- that those could eventually be put together to yield orem, Two blog entries, November 2008, tight bounds for the case without colors. http://gilkalai.wordpress.com/2008/11/24/ sarkarias-proof-of-tverbergs-theorem-1/. 4. The case when r is not a prime power [12] J. Matoušek, Lectures on Discrete Geometry, Grad- continues to be the greatest challenge. For d = 2 uate Texts in Math. 212, Springer, New York, Torsten Schöneborn and I have recast this into a 2002. graph drawing problem [18]. Thus, for the smallest [13] , Using the Borsuk–Ulam Theorem. Lectures on case of r = 6, we would ask whether every drawing Topological Methods in Combinatorics and Geometry, K Universitext, Springer, Heidelberg, 2003. of the complete graph 16 in the plane either has [14] J. Matoušek, M. Tancer, and U. Wagner, A a vertex that is surrounded by five triangles (with geometric proof of the colored Tverberg winding number not equal to zero) or whether theorem, preprint, August 2010, 11 pages, some crossing of two disjoint edges is surrounded http://arXiv.org/abs/1008.5275. by four triangles. Counterexample, anyone? [15] B. H. Neumann, On an invariant of plane regions and mass distributions, J. London Math. Soc. 20 (1945), 226-237. “I was of course flabbergasted by the variety [16] R. Rado, A theorem on general , J. London of generalisations that have blossomed Math. Soc. 21 (1946), 291–300. in that particular garden!” [17] K. Sarkaria, Tverberg’s theorem via number fields, — Bryan Birch (2010) Israel J. Math. 79 (1992), 317–320. [18] T. Schöneborn and G. M. Ziegler, The topolog- ical Tverberg problem and winding numbers, J. Acknowledgments Combinatorial Theory, Ser. A 112 (2005), 82–104. I am grateful to my coauthors, Pavle Blagojevi´c [19] T. tom Dieck, Transformation Groups, Studies in and Benjamin Matschke, for the wonderful long- Mathematics, vol. 8, Walter de Gruyter, Berlin, 1987. [20] H. Tverberg, A generalization of Radon’s theorem, distance collaboration and to Lufthansa for the J. London Math. Soc. 41 (1966), 123–128. business-class upgrades on the way to ICM in [21] , A combinatorial mathematician in Norway: Hyderabad (when I wrote the original German some personal reflections, Discrete Math. 241 (2001), version of this paper for Mitteilungen der DMV pp. 11–22. 18 (2010), pp. 164–170) and on the way back to [22] A. Vuˇci´c and R. T. Živaljevi´c, Note on a conjecture of Sierksma, Discrete Comput. Geom. 9 (1993), 339– Germany (when I did the translation to English). 349. [23] R. T. Živaljevi´c and S. Vre´cica, Chessboard References complexes indomitable, preprint, November 2009, [1] M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 11 pages, http://arxiv.org/abs/0911.3512v1. Springer, Heidelberg, Berlin, 4th ed., 2009. [24] , The colored Tverberg’s problem and com- [2] I. Bárány, A generalization of Carathéodory’s plexes of injective functions, J. Combin. Theory, Ser. theorem, Discrete Math. 40 (1982), 141–152. A 61 (1992), 309–318.

April 2011 Notices of the AMS 557 A Brief but Historic Article of Siegel Rodrigo A. Pérez

À Adrien Douady et les hiboux

he two papers published by Carl L. Siegel together, even though the only prerequisite is to in 1942 were printed on ten consecutive be able to compute the radius of convergence of a pages in the October issue of the Annals power series using the root test. of Mathematics. In one of these papers The initial aim of this article was simply to Siegel gave the first positive solution to give an easy-to-read account of the original proof, Ta small denominator problem, and by doing so but soon I found myself tracking the ideas that he showed that there was hope for a successful must have gone into play as Siegel found his attack on one of the most important problems arguments and prepared them for publication. of the previous sixty years. It was a remarkable As a result, the proof I first produced is now achievement that has earned [15] acclaim as substantially simplified and annotated. Motivated “one of the landmark papers of the undergraduates with a semester of analysis under 1 twentieth century.” their belts should be able to follow the entire To justify this high opinion, we need to under- argument. Graduate students writing a paper for stand Siegel’s proof and its historical background. the first time may find it interesting to pursue In this article I will explain a comparative reading of this material and [15], ◦ What small denominators are and why they which is widely available through jstor. As a note are important. of warning for them, I kept Siegel’s notation for the ◦ The linearization problem and its status most part but made some changes (particularly in 1942. regarding subindices) that simplify the exposition ◦ Siegel’s original proof, including the cor- and keep compatibility between sections. rection of a minor gap. The following two sections give some histori- ◦ Some of the major mathematical develop- cal background on small denominators. The next ments in the wake of [15]. section explains the problem, and the subsequent section describes diophantine conditions. The the- How to Read This Article orem and its proof span the remaining sections. Siegel was a master of concise writing. In only six Note that the two lemmas are numbered as in pages he included a motivation for the result and [15]; nevertheless, Lemma 1 is proved at the end presented an intricate yet self-contained proof. because it is only incidental to the main argument. One drawback of his exposition is that it takes The remarks at the end give a minimal account considerable ingenuity to see how all the pieces fit of events after the publication of [15]. The reader interested only in the historical aspects of this Rodrigo A. Pérez is assistant professor in the Department story can safely skip the section on diophantine of Mathematical Sciences at Indiana University–Purdue University Indianapolis. His email address is rperez@ conditions and the last five sections. math.iupui.edu. Work supported by NSF grant DMS-0701557. What Is a Small Denominator? 1The quote is from [3, p. 482]. The present title is derived Before we can answer this question, consider a 2 from similar praise in [18, p. 6]. harmonic oscillator x¨ + ω1x = 0, whose general

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ω http://www-history.mcs.st-and.ac.uk/ x Siegel L. C. . 3 2 x(t) ω → ± hs rqec s1.2hus(one hours 12.42 is frequency whose ω = ω x ¨ 1 = 0 2 ω A u vnwe hti o h ae it case, the not is that when even but , + 2 . = approaches or eddb ag ae to waves large by needed hours 4 A cos 1 ω scoeto close is hsls ucini unbounded is function last This . cos cos 1 2 (ω x (ω (ω + rnia ua semidiurnal lunar principal 1 cos t ω 1 1 + t) t 1 2 resonance ω + (ω ω ϕ) x( x(t) − − ω ± 1 ϕ) 2 fw d periodic a add we If . 0 cos ω 1 ω 2 − ) ≠ e sfcso the on focus us let , t) 2 2 1 = = ersnsaperi- a represents h uteton quotient the , ω cos (ω ω = x( ˙ 1 − 1 2 ω 0 , 2 t (ω 0 − , sin t) 2 2 hc sso is which ) ω sarational a is ω (ω , 2 = 1 t) 2 2 1 .Then 0. t) . i the n oie fteAMS the of Notices 76 h rtepsto ftemtosof methods the of exposition famous first and a The 1784 in between 1786. Laplace published by memoir explained three-part finally was centuries. It eighteenth and seventeenth better the the of parts during astronomers troubled and tions trajectories. undisturbed is their 48 Jupiter from as which a far during as though years, displaced goes 918 about effect of cumulative cycle The along orbits. advance largest slowly the three that is around regions up which spaced build equally other, to by each tends conjunction, exerted on near planets interval. perturbation time two the this the consequence, of thirds a two occur As and conjunctions third one other at Two again. alignment v risadfrStr ocvrtesame the cover to Saturn for 8 and 8 cover orbits to 21760.362 five Jupiter takes it for sun), days the and Saturn between time the (21518.440 sun orbits days). to two complete the identical to Saturn around by nearly taken is orbits days) five (21662.945 complete to Jupiter orbits their ω in are daily and sun) cover the they around angle average in- Saturn observed the and as Jupiter earliest the known of is inequality The to phenomenon this relative orbits. of meant stance the is of term” period “long or- planetary Here, are in bits. irregularities denominators long-term small to linked mechanics, celestial In Mechanics Celestial motivation work. initial Siegel’s the for was system, In- Poincaré, solar the eluded sun. in which stability the of mutual around question the the planets me- deed, is two hamiltonian of setting of perturbation prototypical theory The perturbative chanics. the in may that resonances coefficients. near big multiple yield series of the face of the convergence on which establish to problem, simply(!) denominator is small the describe tion a that say We denominators. the aini fe xrse yapwrsre whose terms series have power pertur- a coefficients by the expressed present, often is are bation terms nonlinear If other. mo- frequencies periodic of distinct tions two when occurs resonance ◦ 1 5 erresonance near hsdsrpnywsntcdi culobserva- actual in noticed was discrepancy This fe ojnto cus(.. uie aligns Jupiter (i.e., occurs conjunction a After (the Saturn and Jupiter of motions mean The commonly most found are denominators Small oespitctdpeoeo hnsimple than phenomenon sophisticated more A /ω ′ 40 ω mω 2 ′′ 2 ss ls o5 to close so is = 1 necs ftoobt,tu reaching thus orbits, two of excess in + 120 nω . 4548 2 suuulysal o ecan we Now small. unusually is hnvrtelna combina- linear the whenever ′′ ω e a,rsetvl.Since respectively. day, per . 1 5 ω ′ and mω 1 / n auna a s21 as far as Saturn and ,tetm eddby needed time the 2, = 1 ω ◦ 299 + 5 2 ω ′ . nω neatwt each with interact 40 1 . 1283 ′′ and 2 necs of excess in ( ,n m, ω ′′ 2 e day per display ∈ great Z in ) ′ 559 The Linearization Problem Small denominators appear in many other settings in which irrational frequencies resonate. Siegel focused on a model problem in which no physi- cal considerations obscure the small-denominator issue. ∞ r Let f (z) = r=1 ar z be a nonlinear complex analytic function with a fixed point at 0. The value f ′(0) is called the multiplier of 0 and will be denoted λ. Here, λ is assumed different from 0. Figure 1. If the ratio of periods was exactly The linearization problem asks if there is a ∞ k 2/5, Saturn would cover two thirds of its function ϕ(z) = k=1 ckz satisfying in the time that Jupiter covers five thirds of its = ◦ own. (1) ϕ(λz) f ϕ (z). Note that if such a map exists, multiplication by a constant c before applying ϕ simply rescales the perturbation theory is also due to Laplace and ap- domain, so z ֏ ϕ(cz) is also a linearizing map. ′ peared in the first two volumes of his Mécanique By setting c = 1/ϕ (0), the coefficient c1 can be céleste, published in 1799. By the second half of assumed to be 1. the nineteenth century, perturbation theory had The Kœnigs-Poincaré theorem [12, p. 77] guar- been developed to a very high degree. In 1860–67, antees a solution to the linearization equation (1) C.-E. Delaunay published two 900-page volumes whenever |λ| ≠ 1. If λn = 1, an easy computation [6] in which he computed the orbit of the moon shows that f is linearizable if and only if f n = id. under the perturbative influence of the sun. The This leaves λ = e2πiθ with irrational θ as the three resulting series for the moon’s latitude, lon- most interesting case, and this condition will be gitude, and parallax include all terms up to order 7 assumed from now on. Geometrically, (1) says that and span 121 pages altogether. f is conjugate to an around the Chapter III of Delaunay’s opus contains the fixed point. The maximal domain of linearization first analytic description of small denominators is known today as a Siegel disk. [6, pg. 87]. The hamiltonian of the perturbed mo- tion of the moon is a function R of the mean motions ωM , ωS of the moon and the sun (n and n′ in [6]), and two other astronomical quantities. When the periodic component of R is written ex- plicitly, the solutions to Hamilton’s equations feature trigonometric series with linear com- binations of ωM and ωS in the denominators. Delaunay pointed out that because of this, higher order terms can be larger than first-order terms, making a truncated approximation useless. Although small denominators show up in other contexts in celestial mechanics, the underlying setting is always a series of the form ei(mω)t a , m m ω m∈(Zn)∗ Figure 2. The Julia set of f (z) = z2 + c, where c where ω is a vector of frequencies. If ω has many was chosen so that f has√ a Siegel disk with near resonances, the coefficients may grow too 3 rotation number θ = ( 2 − 1) ≈ 0.259921 ... large too often, threatening the convergence of The disk is the large highlighted region in the the series. H. Poincaré was the first to recognize center. and address this difficulty. In an often quoted frag- ment of the Méthodes Nouvelles [14, §148–149], he admitted that his methods did not guarantee the convergence of these series, but he granted the (re- In terms of the power series of f and ϕ, mote) possibility that some particular convergent equation (1) has the form r cases may exist. Poincaré was remarkably pre- ∞ ∞ ∞ scient in guessing both the existence of solutions k ℓ ck λz = ar  cℓz  ; and the difficulty of the proofs. k=1 r=1 ℓ=1  

560 Notices of the AMS Volume 58, Number 4 or, singling out the first term on the right (note bound on − log |λn −1| when λn −1 becomes small. n −1 ν that a1 = λ), Taking exponentials gives |λ − 1| ≤ n for n r ∞ ∞ ∞ larger than some M. This can be changed to a k k ℓ condition for every n by letting K be the larger (2) ck λ − λ z = ar  cℓz  . ν n −1 k=2 r=2 ℓ=1 of 1 and max n |λ − 1| . Then, setting   n≤M 2 Since c1 is taken to be 1, equation (2) gives a ν = ν + log2 K gives concrete recursive description of the sequence | n − |−1 ≤ ν ν { } k − k (6) λ 1 Kn < (2n) . ck . It states that ck λ λ z is the sum of all zk-monomials present in the right-hand Recall that λ = e2πiθ , so |λn −1| = 2| sin(πnθ)|. r ∞ ℓ side. Now, the expression ar ℓ=1 cℓz pro- More precisely [12, p. 129], if m is the nearest n duces zk-monomials exactly when 2 ≤ r ≤ k. These integer to nθ, i.e., if |nθ −m| < 1/2, then |λ −1| = ℓ1 ℓr 2 sin(π|nθ − m|), and since the graph of sin(πx) monomials have the form ar (cℓ1 z )...(cℓr z ), where the powers of z add up to k, so for k ≥ 2, lies between the lines y = 2x and y = πx when (3) 0 ≤ x ≤ 1/2, k 1 4|nθ − m| ≤ |λn − 1| < 2π|nθ − m|. c =  a c ... c  . k k − r ℓ1 ℓr λ λ = r 2 ℓ1+...+ℓr =k  Thus, (6) is equivalent to Equation (3) seems to solve the linearization m Q (7) θ − > , problem as it defines explicitly the coefficients of ν+1 n n a power series solution to (1). However, depending which is the defining property of a diophantine on the value of λ, the absolute values number θ of order ν + 1. Denote the set of all θ 1 1 satisfying (7) by D(ν + 1). It turns out that the (4) ε := = k |λk+1 − λ| |λk − 1| lowest ν such that D(ν + 1) is nonempty is ν = 1, (note the index discrepancy) and that D(2) has measure 0. On the other hand, D(ν) has full measure on [0, 1]. This implies can get very large very often. This threatens the ν>2 that λ satisfies (5) with probability one. convergence of the power series for ϕ, and, indeed, it is possible that the series solution is only formal. The Theorem We will call εk an sd-term, leaving the reader at liberty to decide whether sd stands for “Small With the above notation, Siegel’s result can be Denominator” or “Siegel Disk”. stated in its full strength as follows: √ Poincaré’s opinion on the scarcity of lineariz- 5ν+1 Theorem. Given ν ≥ 1, let bν = (3 − 8/) 2 . If able maps was prevalent for half a century. In 1917 θ ∈ D(ν + 1), then all normalized f of multiplier G. Pfeiffer [13] constructed the first nonlineariz- λ = e2πiθ are linearizable, and the linearizing map able example, and in 1928 H. Cremer [5] found a ϕ has radius of convergence at least bν . dense Gδ set of angles for which no rational func- tion is linearizable. In this atmosphere, Siegel’s In other words, all normalized f with multiplier paper came as a surprise, as he found a large λ in a subset of full measure of the circle have family of angles θ (satisfying condition (5)) for a guaranteed radius of linearization. The bound which linearization is possible. To state his result depends only on the diophantineness order of λ we need the following definition and a discussion and is given explicitly. of diophantine conditions. Definition. The power series of f is convergent, The Majorant Method r−1 To prove that the radius of convergence of ϕ is so there is a smallest a > 0 satisfying |ar | ≤ a for all r. If f is replaced by the conjugate function positive, Siegel used Cauchy’s majorant method af (z/a), the multiplier remains intact, but we can to estimate the exponential rate of growth of the { } assume |ar | ≤ 1. Such f is said to be normalized. sequence ck . This requires several steps. First, note from (3) that the absolute values |ck| are Diophantine Conditions bounded by the real sequence {ck} defined by c = 1 and Siegel required that λ satisfy 1 (5) log |λn − 1| = O(log n) as n → ∞. k (8) ck = εk−1  cℓ1 ... cℓr  This says that there is a constant ν > 0 such that = r 2 ℓ1+...+ℓr =k  for sufficiently large n, (recall the normalization assumption |ar | ≤ 1). log |λn − 1| ≤ ν log n. Since log |λn − 1| is at most log 2 (for λm accu- 2See the comment following the proof of Lemma 2 for a mulating at −1), condition (5) makes sense as a justification of this choice of logarithmic base.

April 2011 Notices of the AMS 561 ℓ The structure of these numbers is somehow their generating function y(x) = τℓx satisfies obscured by their recursive definition. Following the functional equation (8), the first coefficients are ∞ y = x + y r c2 = ε1 [c1c1] = ε1 , = r 2 = + + = + c3 ε2 [c1c2 c2c1] [c1c1c1] 2ε2ε1 ε2 , essentially by the same line of reasoning that (9) produces (3) from (2). Since the above is just y2 c4 = ε3 [c1c3 + c2c2 + c3c1] + [c1c1c2 + c1c2c1 = + y x 1−y , it follows that + c c c ] + [c c c c ] √ 2 1 1 1 1 1 1 1 + x − 1 − 6x + x2 = + + + + y(x) = , 4ε3ε2ε1 2ε3ε2 ε3ε1ε1 3ε3ε1 ε3. 4

Thus, ck is the sum of many expressions, each so the radius of convergence of y is the absolute of which is the product of several sd-terms (not value of the smallest root of 1 − 6x + x2; i.e., √ necessarily different). It is possible to describe (3− 8 ). In particular,√ the sequence√ {τk} grows as explicitly which products appear, but that is not a power of (3 − 8)−1 = (3 + 8 ). More accurately, important here and will be omitted. it is known [11] that it has the asymptotic behavior Let τ be the number of products in the ex- √ k k W 3 + 8 pansion of ck, and δk the maximum of their (12) τ ∼ , k 3/2 values. From (9), it is clear for instance that k 1 √ τ4 = 4 + 2 + 1 + 3 + 1 = 11, but the precise col- where W = 18π − 4 / = 0.069478 ... lection of sd-terms whose product realizes the 4 maximum δk will depend on λ. Note, however, The Subtle Estimate that (setting aside the factor εk−1 of ck) the largest product of sd-terms in the expansion of c appears The bulk of Siegel’s proof is concerned with show- k in some product c ... c and therefore has to ing that the sequence {δk} defined by recursion ℓ1 ℓr be the product of the largest products in each of (10) has an exponential bound whenever the sd- terms satisfy (6). Using the notation in (4), the cℓ1 ,..., cℓr . In other words, δk is given by δ1 = 1 and diophantine condition reads ν (13) εk ≤ (2k) . (10) δk = εk−1 max {δℓ1 ...δℓr } (k ≥ 2). ℓ1+...+ℓr =k 2≤r≤k This will be called the basic estimate. Since each This recursive definition allows for a much more δk is a product of O(k) sd-terms, (13) is far from giving an efficient bound on the growth of the efficient computation of δk. sequence. Siegel’s insight, and one of the reasons Cauchy’s majorant method is based on the his result was so influential, was the realization obvious fact that that once an sd-term is large, it takes several steps before another sd-term can have comparable size. (11) |ck| ≤ ck ≤ δkτk, This is made precise in the following argument. Since and on the observation that the values τk are given q p−q p q by τ1 = 1 and the recursion λ (λ − 1) = (λ − 1) − (λ − 1), k and |λq| = 1, we get via the triangle inequality, τ =  τ ... τ  k ℓ1 ℓr p−q p q = |λ − 1| ≤ |λ − 1| + |λ − 1|. r 2 ℓ1+...+ℓr =k  (compare with (8)). In sd-notation the above reads The theorem will follow from (11) and the −1 −1 −1 −1 εp−q ≤ εp + εq ≤ 2 min{εp, εq} . exponential bounds (12) on {τk} and (23) on {δk}. It is useful to keep in mind that the definition of Then, applying the basic estimate (13) to εp−q,

the numbers τk is related to the structure of the ν+1 ν (14) min{εp, εq} ≤ 2 (p − q) . linearization equation (1), while the definition of This is much better than the trivial min{ε , ε } ≤ δk reflects the effect of the angle θ on the sd-terms p q min{(2p)ν ,(2q)ν } and will be called the subtle εk. estimate. Siegel must have been pleased with the simplicity of this core idea, because he actually τk Grows Exponentially allowed himself a small boasting note at this point The Schröder numbers {τk} are well known in combinatorics (see [16, sequence A001003], [17], [15, p. 610]: and references therein). The initial values are 1, 1, “This simple remark is the main 3, 11, 45, 197, 903, 4279, 20793, 103049, ... , and argument of the whole proof.”

562 Notices of the AMS Volume 58, Number 4 −1 A Bound on a Product of SD-Terms The case δ1 ≤ AC suggests setting A = C and How does an estimate on the least of two sd-terms solving yield an upper estimate on a product of sd-terms? Ck−1 The following proof of Lemma 2 reformulates the (16) δ ≤ k B inductive argument in [15] to answer this question. k To simplify notation, let N = 22ν+1 (a constant that for all k. A solution to (16) can always be upgraded depends on the diophantineness value). to one that satisfies Lemma 2. + ≥ (a) (b) Given r 1 indices k0 > . . . > kr 1, B > 0 and C ≥ 2B the following holds: r r by a suitable increase in C. The extra conditions r+1 ν ν (15) εkp < N k0 kp−1 − kp . (a) and (b) have the advantage that p=0 p=1 B Cj1+j2−2 1 1 Cj1+j2−1 Note that it is the indices, rather than the sd- δ δ ≤ = C−1 + j1 j2 B B B terms themselves, that are arranged by size in j1 j2 j1 j2 (j1 + j2) (strictly) descending order. (a) Cj1+j2−1 (b) Cj1+j2−1 ≤ C−12B ≤ , (j + j )B (j + j )B Proof. The proof is by induction. The basic esti- 1 2 1 2 mate (13) covers the case r = 0. If r = 1, the basic and more generally, and subtle estimates give CJ−1 ν ν ν ν+1 ν ε ε ≤ (2 max{k , k }) (2 |k − k | ) (17) δj1 ... δjt ≤ , k0 k1 0 1 0 1 JB 2 ν ν < N k0 |k0 − k1| . whenever j1 + ... + jt = J. Now consider the case of r + 1 ≥ 3 sd-terms, Now suppose we are in possession of numbers and let εkj be the smallest one. By the induction hy- B and C that satisfy (a), (b), and (16) for all k ≥ 1 pothesis, the remaining sd-terms satisfy (15) with smaller than k0. Then inequality (19) below can be the index kj missing. If j = 0, then εkj = εk0 is reached by the following argument: bounded by In the decomposition (10) of k0 the sum of ν+1 ν ν indices of all deltas is equal to k . In particular, 2 (k0 − k1) < N(k0 − k1) 0 there can be at most one index larger than k /2. If and (15) holds. A similar argument applies when 0 this is the case, write δ = ε δ where j = r. k0 k0−1 k1 1 k1 > k0/2, and consider the decomposition (10) of When 0 < j < r, the inductive bound on εk0 ... ν k1. There may still be an index larger than k0. If εkj−1 εkj+1 ...εkr contains the factor (kj−1−kj+1) . so, write δk = εk −1 δk 2, and continue this Let a, b be such that {ka, kb} = {kj−1, kj+1} and 1 1 2 | − | ≥ | − | − ≤ process until the decomposition of some δkr has ka kj kj kb . It follows that (kj−1 kj+1) + ν 1 ν no delta with index larger than k0. This produces 2|ka − kj |, while (14) gives εkj ≤ 2 |kj − kb| . r the following tower Then the product p=0 εkp of all r + 1 sd-terms is bounded by δk0 = εk0−1 δk1 1 ,

Nr k (k − k )ν... (k − k )ν ε ...(k − k )ν δ = ε − δ , 0 0 1 j−1 j+1 kj r−1 r k1 k1 1 k2 2 r ν ν ν ν+1 ν ν ≤ N k0 (k0 − k1) ...2 |ka − kj | 2 |kj − kb| ...(kr−1 − kr ) = r+1 − ν − ν − ν − ν . N k0 (k0 k1) ... (kj−1 kj ) (kj kj+1) ... (kr−1 kr ) . (18) . It is worth noting that in the factorization δkr−1 = εkr−1−1 δkr r , N = 2ν 2ν+1 used in the last equality, both δ = ε δ ... δ , powers of 2 come from different sources. The kr kr −1 ℓ1 ℓs ν factor 2 is due to the fact that the interval where k0 > k1 > . . . > kr > k0/2. Note that each [k , k ] is shorter than twice the longer of j−1 j+1 p lumps together many unnamed deltas. Their ν+1 [k − , k ] and [k , k + ]. The factor 2 on the j 1 j j j 1 indices add up to kp−1 − kp, and are all at most k0. other hand comes from (6), where logarithmic The indices ℓq add up to kr and are also at most base 2 was chosen simply so that N has a clean k0. These conditions on indices will be necessary expression. in order to apply Lemma 1 in (22). Let us collapse the tower by repeated substitu- δk Grows Exponentially tion to get The stage is set to find an exponential bound on δk. r r It may be impossible to reconstruct what Siegel did δ = ε (δ ... δ ). to discover a proof, but here is a plausible scenario. k0 kp−1 p ℓ1 ℓs = = k B p 0 p 1 Start by writing δk ≤ AC /k with the intention of exploiting the recursive decomposition (10) to Then, applying Lemma 2 to the product of sd- find values A, B, C that make the inequality true. terms, inequality (17) to each p, and inequality April 2011 Notices of the AMS 563 (16) to each δℓq gives for all k0 ≥ 1, and the theorem is proved. (19) r The Auxiliary Inequality r+1 ν ν Although the proof is correct, the middle inequality δk0 ≤ N k0 kp−1 − kp  p=1 in line (8) of [15] does not hold when k is odd and   k r s t = . The following proof of Lemma 1 simplifies C(kp−1−kp)−1 Cℓq −1 2     B B Siegel’s exposition and avoids this minor lapse by = (kp−1 − kp) = ℓq p 1  q 1  considering instead inequality (29), which calls = Nr+1 C−r−s for a separate treatment of the cases t = 2 and = r s k 2, 3, 4. ν − ν−B −B k0 k0 (kp−1 kp) ℓq  C . Observation 1. The cubic polynomial P(x) = (R − = =  p 1 q 1  x)(x − S)2 with R > S has derivative P ′(x) = (2R + Recall that this inequality is contingent on S − 3x)(x − S), so S is a critical point. The second finding B, C that satisfy (a), (b), and (16) with derivative is P ′′(x) = 2R+4S−6x, which, evaluated 1 ≤ k < k0. The goal is to discover a smart choice at x = S, is 2R − 2S > 0. Thus S is the only local of B and C so that the right-hand side of (19) is minimum of P. It follows that if the interval I = Ck0−1 also bounded by B . Accordingly, the next step [a, b] lies to the right of S, then k0 −B is to find a way to extract a factor k0 from the min{P(x)} = min{P(a), P(b)}. middle parenthesis. More precisely, we look for an x∈I auxiliary inequality of the form Lemma 1. Let three integers k ≥ 2, r ≥ 0, and s ≥ 2

r s be given. If the integers x1, . . . , xr and y1, . . . , ys be- ν−B −B −ν−B k r s (20)  (kp−1 − kp) ℓ  ≤ k , + = q 0 long to 1,..., 2 and satisfy p=1 xp q=1 yq p=1 q=1 s   Ξ k with q=1 yq > k/2, then where may depend on r and s, but not on k0. r s 3 Since the sum of factors (kp−1 − kp) + ℓq is k0, 2 k Ξ (24) xp y ≥ . a simple heuristic for an inequality like (20) to be q r+s−1 p=1 q=1 2 possible is that the sums of exponents on the left and right sides balance out. In the present case Proof. Let t = r + s ≥ 2. Since 2t − 2 ≤ 2t−1, it this means (ν − B) − B = −ν − B, or suffices to prove (c) B = 2ν. k 3 (25) x y 2 ≥ . p q This is compatible with condition (a). As it turns 2t − 2 out, the heuristic works, and Siegel found the Some cases are immediate. If t = 2, then r = 0 and inequality (24) of Lemma 1 (see the following s = 2, so y1 = y2 = k/2 and (25) holds. Also, (25) section). Together with (c) and the index conditions holds trivially when k ≤ 2t − 2; this is the case for mentioned after (18), inequality (24) yields the k = 2, 3, 4 when t ≥ 3. It remains to consider what following version of (20): happens when t ≥ 3, k ≥ 5, and k > 2t − 2, or (21) equivalently, r s −3ν − − k0 − ν 2ν ≤ k  (kp−1 kp) ℓq  + − , ≤ ≤ r s 1 (26) 3 t 2 . = = 2 p 1 q 1  which means that (19) is bounded by The smallest product xp is realized when r −1 factors are equal to one, and the remaining factor −3ν k is what is left of xp. Thus, xp has the lower r+1 −r−s ν 0 k0 (22) N C k0 C r+s−1 bound xp − (r −1). Analogously, the product 2 yq can be estimated from below by yq − (s − k0−1 2ν+1 r+1 3ν r+s−1 −r−s+1 C 1). However, a sharper bound is available when = 2 2 C 2ν k0 k yq − (s − 1) > , for in that case the least r+s−1 2 22ν+1 23ν Ck0−1 product is realized by s − 2 factors equal to one, ≤ 2ν . k C k0 a factor equal to , and a factor equal to the 2 k Ck0−1 The last expression is smaller than when rest (so that no yq is larger than 2 ). In short, k2ν 0 (27) C ≥ 25ν+1. This last condition is compatible with (b) 5ν+1 and (c); and so, substituting B = 2ν and C = 2 xp ≥ xp − r + 1 , in (19) and (22) yields a proof that k yq − s + 1 if yq − s + 1≤ , ≥ 2 5ν+1 k0−1 yq  k k k 2  y − s − + 2 if y − s + 1≥ . (23) δ ≤ q 2 2 q 2 k0 2ν k0 

564 Notices of the AMS Volume 58, Number 4 k k The sum yq can take values between + 1 larger than the latter when t = 3 and when t = , 2 2 and (k − r). The analysis that follows breaks into the same inequality is valid in the full range of t, two cases. so continuing from (29), k k Case 1: If + 1 ≤ yq ≤ + s − 1 , then 2 2 2 2 k k − 1 k k − 1 (27) gives − t + 2 ≥ = (28) 2 2 2 2t − 2 2 x y 2 ≥ k − y − r + 1 y − s + 1 . k − 1 2 k 3 p q q q − ≥ (t 1) Let R = k − r + 1 and S = s − 1, so (28) reads k 2t − 2 3 3 x y 2 ≥ P y . Now, R > S so Observa- 4 2 k k p q q 2 > .  2 tion 1 applies. The product xp yq is bounded 5 2t − 2 2t − 2 from below by the minimum of Pin the range of yq . Since the range is included in the (larger) Conclusion interval I = k − r + 1 , k + s − 1 , and The influence of [15] was due in part to the 2 2 elementary nature of the majorant method, which k 2 − r + 1 > S by (26), the product xp yq 2 I hope to have conveyed; but its major impact is bounded by the least of was conceptual. Although the small-denominator 2 problem he solved was simpler than those found P k − r + 1 = k k − t + 2 2 2 2 in celestial mechanics, Siegel’s proof showed that and the convergence issue could be handled. His main 2 observation was that it is possible to quantify P k + s − 1 = k − t + 2 k . 2 2 2 how frequently small denominators of comparable But size can appear. This was a fruitful idea that he 2 extended to similar problems in several variables. k k − t + 2 ≤ 2 2 Moreover, the notion that number theory was k k − t + 2 k − t + 2 ≤ relevant to dynamical systems (via diophantine 2 2 2 approximations) served as a catalyst for much k + k − k − + 2 1 2 1 2 t 2 ensuing research. In time, the study of rotation domains became a subject of its own, with major because t ≥ 3. The last line is smaller than 2 contributions by A. Brjuno, T. Cherry, M. Her- k − t + 2 k , so for y in this range, 2 2 q man, J.-C. Yoccoz, R. Pérez-Marco, M. Shishikura, 2 C. McMullen, X. Buff, and A. Chéritat. In this area, x y 2 ≥ k k − t + 2 . p q 2 2 much of the effort was directed to geometric con- k + − ≤ ≤ − siderations; for instance, the behavior of critical Case 2: If 2 s 1 yq (k r), then (27) gives points near the boundary of a Siegel disk or the existence of Siegel disks with smooth boundary. 2 2 2 k k xp yq ≥ k − yq −r +1 yq −s− +2 . Alas, the approach in [15] did not apply in the 2 2 setting of hamiltonian dynamics where techniques = − + = + k − were most urgently sought. In 1954, at the Interna- In this case, let R k r 1 and S s 2 2, tional Congress of Mathematicians in Amsterdam, so (26) implies R > S. Now the range of yq is A. Kolmogorov announced a theorem (inspired in the interval I = k + s − 1 , (k − r) , which 2 in part by Siegel) that would change the face of clearly lies to the right of S. Observation 1 bounds dynamical systems in the form of KAM theory: 2 xp yq from below by the least of In an integrable hamiltonian system the phase 2 k k k space is foliated by invariant tori, each with an P + s − 1 = − t + 2 d 2 2 2 associated rotation vector ω ∈ R . The solutions and within a given torus are conjugate to the linear 2 2 − = k − + k translation p ֏ p + ωt. When the entries of ω are P(k r) 2 t 2 2 . rationally independent, the solutions are dense in Obviously the former is smaller, so x y 2 ≥ p q the torus and are said to be quasi-periodic. 2 k − + k 2 t 2 2 ; but at the end of Case 1 this Prior to 1954 it was expected that a small per- 2 turbation of an integrable system would destroy was shown to be larger than k k − t + 2 , so 2 2 this structure, so that most trajectories would (29) break away from their original tori and wander 2 k k − 1 2 around . That is, the assumption was x y 2 ≥ k k −t +2 ≥ −t +2 p q 2 2 2 2 that the perturbed system should be ergodic. for all valid values of yq. Kolmogorov showed that the structure of the Now, (k − 1)/2 − t +2 is a linear function of t, perturbed solutions is much more interesting than while (k−1)/(2t −2) is convex. Since the former is that. While many solutions do wiggle about phase

April 2011 Notices of the AMS 565 space, diophantine3 rotation vectors still give rise [2] Xavier Buff and Arnaud Chéritat, Ensembles de to (deformed) invariant tori where solutions are Julia quadratiques de mesure de Lebesgue stricte- quasi-periodic. Thus the regions of chaotic and ment positive, C. R. Math. Acad. Sci. Paris 341(11) regular behavior are inextricably blended together, (2005), 669–674. [3] Xavier Buff, Christian Henriksen, and John H. and each has positive measure. In other words, Hubbard, Farey curves, Experiment. Math. 10(4) the ergodic hypothesis has to be discarded. (2001), 481–486. Notice how this result takes us from study- [4] Luigi Chierchia and Corrado Falcolini, Com- ing the convergence of individual series to a pensations in small divisor problems, Comm. Math. global study of the space of solutions. The the- Phys. 175(1) (1996), 135–160. ory does not verify convergence for individual [5] Hubert Cremer, Zum Zentrumproblem, Math. Ann. 98 initial conditions, but rather guarantees a positive (1) (1928), 151–163. [6] Charles-E. Delaunay, 1867, Théorie du Mouve- probability of convergence, while making clear the ment de la Lune, 2 Vols. in Mem. Acad. Sci. 28 and 29 role of the diophantine condition. In a sense, the (Mallet-Bachelier, Paris, 1860, and Gauthier-Villars, small-denominator problem has been bypassed. Paris, 1867). After Kolmogorov’s announcement, techniques [7] Christopher Garrett, Tidal resonance in the Bay like the majorant method were abandoned (even by of Fundy and Gulf of Maine, Nature 238(5365) Siegel) in favor of global analysis in the KAM spirit. (1972), 441–443. [8] Boris Hasselblatt and Anatole Katok, The de- One of the few people to revisit Siegel’s method velopment of dynamics in the 20th century and was A. Brjuno [1]. He improved the original proof the contribution of Jürgen Moser, and described (what Yoccoz [18] would later prove Dynam. Systems 22(5) (2002), 1343–1364. is) the largest class of angles θ for which every [9] Michael-R. Herman, Recent results and some analytic function f with fixed point 0 of multiplier open questions on Siegel’s linearization theorem e2πiθ is linearizable. References to a few other of germs of complex analytic diffeomorphisms of Cn applications of the majorant method can be found near a fixed point, VIIIth International Congress on Mathematical Physics (Marseille, 1986), World Sci. in [4]. Publishing, Singapore, 1987, pp. 138–184. Many questions remain open. Are there [10] John H. Hubbard, The KAM theorem, Kolmogorov’s bounded Siegel disks whose boundary is not Heritage in Mathematics, Springer, Berlin, 2007, pp. a Jordan curve? What is the structure around 215–238. Cremer points, where linearization is impossible? [11] Donald E. Knuth, The Art of Computer Program- In 2005 X. Buff and A. Chéritat completed a ming, Addison-Wesley Publishing Co., Reading, project, started by A. Douady in the 1990s, Mass.-London-Amsterdam, 2nd edition, 1975, Vol- ume 1: Fundamental algorithms, Addison-Wesley to construct polynomial Julia sets of positive Series in Computer Science and Information measure. The strategy is to approximate a Cremer Processing. polynomial (whose Julia set has no interior) by [12] , Dynamics in One Complex Variable, a sequence of linearizable polynomials while volume 160 of Annals of Mathematics Stud- delicately controlling the reduction in area of ies, Princeton University Press, Princeton, NJ, 3rd the corresponding Siegel disks. Their results [2] edition, 2006. promise new life for an interesting subject. [13] G. A. Pfeiffer, On the conformal mapping of curvi- linear angles. The functional equation ϕ[f (x)] = a1ϕ(x), Trans. Amer. Math. Soc. 18(2) (1917), Acknowledgments 185–198. I am indebted to M. Aspenberg for the long [14] Henri Poincaré, Les Méthodes Nouvelles de la conversations that led to this paper, to R. Roeder Mécanique Céleste. Tome II. Méthodes de MM. for many illuminating suggestions, to J. Smillie for Newcomb, Gyldén, Lindstedt et Bohlin, Dover Publications, New York, NY, 1957. editorial advice, and to the many people that have [15] Carl L. Siegel, Iteration of analytic functions, Ann. read or listened to portions of this material. of Math. 43(2) (1942), 607–612. Note: Figure 2 was created with - [16] Neil J. A. Sloane, (2006), The On- Stream, a research-oriented program used Line Encyclopedia of Integer Sequences, to explore dynamical systems. Available at http://www.research.att.com/~njas/ sequences/. http://code.google.com/p/fractalstream. [17] Richard P. Stanley, Hipparchus, Plutarch, Schröder, and Hough, Amer. Math. Monthly 104(4) References (1997), 344–350. [1] Alexander D. Brjuno, Analytic form of differential [18] Jean-C. Yoccoz, Petits diviseurs en dimension equations, I, II, Trudy Moskov. Mat. Obšˇc. 25 (1971), 1, Société Mathématique de , Paris, 1995, 119–262; ibid. 26 (1972), 199–239. Astérisque No. 231 (1995).

3 κ Zd Compare the condition |ωv| ≥ |v|τ (for all v ∈ \{0}) with (7).

566 Notices of the AMS Volume 58, Number 4 About the Cover

want to infer from field observations the real Collective Behavior and mechanisms governing collective motion. Computers have advanced far enough over Individual Rules the past thirty years to produce stunning simulations, but many of these have been based on imagined rather than biologically This April the theme for “Math Awareness Month” is verified schemes. Recent advances in digital complexity. It is a huge theme, comprising a wide range of imaging have now made it possible to obtain diverse topics, and not all of it directly connected to math- high-quality dynamic data on large groups. ematics. But one of its most interesting components is These data are crucial to construct biologi- the emergence of highly structured behavior of large cally realistic models, as opposed to strictly assemblies governed by relatively simple rules of local hypothetical ones. interaction, and this shows great potential as a point for mathematical attack. Early studies involved only small The work we did in obtaining trajectories of groups of animals, but as computers advanced rapidly in hundreds of ducks within flocks allowed us to power these were followed by striking if largely specula- come to concrete conclusions about the nature tive computer simulations. However, in recent years tech- of their interactions. We found that repulsion nology, for example increasing sophistication of digital forces were an order of magnitude larger than photography, has brought about a resurgence of work on attraction and alignment forces, and that there was also a strong interaction with neighbors real populations. Among the most impressive studies are directly in front. those of bird flocks—pigeons, starlings, and ducks—but there have also been others concerned with schools of fish More information on the project can be found in the article and large groups of insects such as locusts. by Lukeman, Li, and Edelstein-Keshet in the July 13, 2010, The images on the cover were supplied by Ryan Luke- issue of the Proceedings of the National Academy of Sci- man (now on the faculty of St. Francis Xavier University in ence. Nova Scotia), and are from his Ph.D. thesis at the University Here is a brief list of suggestions of other things to of British Columbia. Lukeman took thousands of photo- look at: graphs of surf scoters moving around in salt water near Boids. One of the most famous computer simulations docks in downtown Vancouver. The background shows a was Craig Reynolds’ boids. The webpage http://www. segment of his photographs. The inset gives some idea of red3d.com/cwr/boids/ is comprehensive. how the photographs were processed. Lukeman tells us, Starlings. The images on the Web are fascinating. The main page for the 2007 STARFLAG meeting is a good place The figures illustrate four steps in processing to start: http://angel.elte.hu/starling/meeting. our photographs. (1) The top figure is a typi- html. cal image from a series of hundreds of images Much of the best work is by Andrea Cavagna and taken at 3 fps to capture the motion of surf his group. Here are two popular articles about it: scoters. (2) I then pass the image as a three- http://lansingwbu.blogspot.com/2010/01/ layered matrix into Matlab, and use image how-do-thousands-of-starlings-flock.html. processing to extract individual birds from http://www.americanscientist.org/issues/ the image. Image processing involves several num2/2011/1/flights-of-fancy/1. different kinds of analysis, followed by manual Pigeons. You can get a good idea of work on pigeons work to ensure that every individual has been from: http://www.wired.com/wiredscience/2010/04/ marked. (3) The center of mass of each duck is pigeon-flock-pecking-order/. computed, then overlaid on the original image Fish. Some of the most beautiful images on the Web: for visual checking. (4) By linking positions http://pinewooddesign.co.uk/2008/06/25/ in successive frames using particle tracking thousands-of-golden-rays-glide-silently- software, trajectories are constructed for each through-the-ocean-photos/. individual in the frame. I also correct the image Insects. There are two interesting groups at Princeton: distortion using simple trigonometry. The cor- http://icouzin.princeton.edu/; http://paw. rected positions and velocities are plotted at princeton.edu/issues/2010/05/12/pages/9635/ one instant in time in the last frame. index.xml. Other biological collectives. Even bacteria do it. Generally, useful data on collective animal http://tglab.princeton.edu/. motion is very difficult to obtain in the field. Computer simulations have been valuable, —Bill Casselman Graphics Editor but the real challenge is to determine which ([email protected]) of these models are biologically relevant. We

APRIL 2011 NOTICES OF THE AMS 567 Remembering Paul Malliavin

Daniel W. Stroock and Malliavin contin- ued to think about Marc Yor complex variable theory and ex- panded his inter- Paul George Malliavin ests to include On June 3, 2010, Paul Malliavin died at the Ameri- analytic functions can Hospital in Paris. At the time of his death, he of more than one was four months short of his eighty-fifth birthday. variable. This line Malliavin was a major mathematical figure of research culmi- throughout his career. He studied under Szolem nated in his joint Mandelbrojt, who had returned to France after paper, with his wife World War II from the United States, where he Marie-Paule Malli- had been on the faculty of what, at the time, was avin. As Gundy the Rice Institute. Both Malliavin and Jean-Pierre explains in his Kahane received their degrees under Mandelbrojt essay here, this in 1954, and Yitzhak Katznelson received his from Paul Malliavin, paper represents Mandelbrojt a couple of years later. Thus, in less surrounded by books, a departure from than three years, Mandelbrojt produced three stu- circa 2000. classical, purely an- dents who would go on to become major figures alytic thinking about analytic functions and in mid-twentieth-century . potential theory. Instead, the ideas in the Malliavin Malliavin’s own singular contribution to har- and Malliavin paper can be seen as descendants of monic analysis is described here by Kahane. Like stochastic analytic techniques with which Joseph many other definitive solutions to mathematical Doob had given a novel derivation of the Fatou problems, Malliavin’s solution to the spectral syn- theorem for analytic functions on the disk. thesis problem killed the field, with the ironic It would appear that Malliavin’s excursion into consequence that few young mathematicians even probability theory made a lasting impression on know the statement of the problem, much less the him. Ever since Laplace, France has had a proud name of the person responsible for its solution. tradition in probability theory. At the turn of the Not one to rest on his laurels, Malliavin soon twentieth century, Emile Borel and Henri Lebesgue turned his attention in new directions. His early were laying the foundations on which Andrey work won him an invitation to visit Arne Beurl- Kolmogorov would build the axiom system which

Photographs, unless otherwiseing noted, courtesy of Thérèse Malliavin. at the Institute for Advanced Study, where, has become the generally accepted one for the as Kahane explains here, during a second visit, mathematical analysis of random phenomena. At Malliavin and Beurling completely solved two fun- the same time, Henri Poincaré’s student Louis damental problems in classical complex variable Bachelier was constructing the model which has theory. After completing his project with Beurling, recently provided employment for many young mathematicians in the financial industry. Daniel W. Stroock is professor of mathematics at the Mas- That tradition was continued in France by Paul sachusetts Institute of Technology. His email address is Lévy, whose uncanny understanding of stochastic [email protected]. processes became, once it was explained to the Marc Yor is professor of mathematics at the Université rest of us by Kyoshi Itô, the basis for much of Pierre et Marie Curie. His email address is deaproba@ the work that probabilists have done ever since. proba.jussieu.fr. Further, under the masterful tutelage of Jacques

568 Notices of the AMS Volume 58, Number 4 Neveu and, on the more Not content to have been its inventor, Malliavin analytic side, Gustave played a leading role in the application of the Choquet, postwar France Malliavin calculus. Over the past twenty-five years, was producing a new co- he and his collaborators produced a large body of hort of mathematicians work in which his calculus played a central role. whose primary interest The essay here by Leonard Gross gives a glimpse was probability theory. into one of the programs in which Malliavin was Until recently, perhaps involved at the time of his death. Missing here the most influential of are accounts of the many other projects in which these was Choquet’s stu- Malliavin was engaged. For example, together with dent Paul-André Meyer, his son-in-law Anton Thalmaier, Malliavin wrote a who realized that, in their book in 2000 in which various applications of his haste, Lévy, Doob, and calculus to mathematical finance are proposed. others had treated sev- Finally, we have included here an homage to The young boy eral notions too casually, Malliavin composed by Michele Vergne. Her essay Malliavin, circa a situation that Meyer, portrays Malliavin the man, not just Malliavin 1940. first by himself and later the mathematician. Most people, even those who with his student Claude Dellacherie, remedied. have made profound contributions, are unable In spite of the existence of many active French to sustain their vigor and eventually enter their research groups in probability theory, Malliavin dotage. Malliavin never did. Indeed, his student charted his own course. He came to the subject and longtime collaborator Hélène Airault was at as an analyst with wide-ranging interests, and he his bedside as he was dying, and she reports brought to the subject a vision which only someone that he was discussing mathematics from behind with his encyclopedic knowledge of mathematics the oxygen mask covering his face. He was a could provide. Free from prejudice about what remarkable individual, and, at least for those of us topics and methods are or are not “probabilistic”, who were privileged to know him, the world will Malliavin trained his formidable technical exper- be a less interesting place now that he is gone. tise on aspects of the field that had not been fully considered. His initial project was to under- stand Brownian motion on a Riemannian manifold Jean-Pierre Kahane from a differential geometric standpoint. Building on a key observation made by David Elworthy Malliavin and Fourier Analysis and James Eells, Malliavin understood that the It is worth reading Malliavin’s articles again and Brownian motion on a manifold could be real- again, and it would be useful to have them collected ized by “rolling” a Euclidean Brownian motion in and published together. He worked in several the tangent space onto the manifold. In order to branches of mathematics, but all his mathemati- overcome the technical difficulties posed by the cal endeavors have in common a truly exceptional nondifferentiability of Brownian paths, he lifted vision which dominates the raw technical ac- everything to the bundle of orthonormal frames, complishment: he insisted on understanding the where he could apply well-established techniques problem “from above” before he would delve into from Itô stochastic differential calculus. As a the jungle of details involved in its solution. Here result, he gave an elegant construction of the I will restrict myself to a description of Malliavin’s Brownian paths on the manifold, one in which most important contribution to Fourier analysis. they came equipped with an intrinsic notion of Although it is only one of Malliavin’s many achieve- parallel transport. Malliavin’s ideas were quickly ments, it exemplifies the vision that he brought to absorbed and exploited by Jean-Michel Bismut, all his work. who used them in his proof of the Atiyah–Singer Rather than presenting the events in chrono- index theorem. logical order, I will start with Malliavin’s Compte- Having thoroughly assimilated Itô calculus, Rendus note of April 13, 1959,1 the one that won Malliavin began to realize that Itô’s stochastic him instant recognition. Afterward, I will look back differential equations could be viewed as a pre- on his 1954 thesis and forward to his collaboration scription for defining nonlinear transformations with Beurling in the 1960s. of Wiener space, transformations that, although they are defined only up to a set of Wiener measure Jean-Pierre Kahane is professor emeritus at the Univer- 0, are nonetheless “smooth” and, as such, are sus- sité de Paris-Sud. His email address is jean-pierre. ceptible to analysis. This realization was the origin [email protected]. of what Malliavin called the stochastic calculus of This is a translation (slightly adapted for the Notices) of variations and what one of the present authors an article published in La Gazette des mathematicians, dubbed the “Malliavin calculus”, a cursory résumé no. 126, October 2010. of whose initial formulation is given below. 1“On the impossibility of spectral synthesis on the line”.

April 2011 Notices of the AMS 569 There is much more that I might have, and maximal ideals containing it. Alternatively, if is maybe ought to have, included, but perhaps the dual group of G and A( ) is Wiener’s algebra, this brief selection will hasten the day when whose elements are the Fourier transformsΓ of his collected works are made available. elements of L1(G), then theΓ synthesis problem is the same as that of determining whether every The Spectral Synthesis Problem closed ideal of A( ) is the ideal of functions in A( ) In harmonic analysis, synthesis refers to the re- that vanish on some closed subset of . If instead construction of a sound, signal, function, or some of A( ) one looksΓ at space of continuous functionsΓ other quantity from its harmonics. For example, on that, if is not compact, vanish atΓ infinity, the periodic functions can be reconstructed via Fourier analogousΓ question has a positive answer. In fact, series. More generally, such reconstruction is pos- in thatΓ setting,Γ the problem reduces to showing sible for functions that are almost periodic in the that if f ∈ C( ) vanishes on a closed set E and is sense of H. Bohr, functions that are quasi-periodic a Radon measure on that is supported on E, then in the senseof Paleyand Wiener, and those that are , f = f dΓ = 0. For A( ), the problem can be mean-periodic in the sense expressed in an analogousΓ way, only the space of of . In all Radon measures has to beΓ replaced by the space these cases, one associates of “pseudo-measures”. That is, one wants to know with an element f of a speci- whether if f ∈ A( ) vanishes on a closed E ⊆ fied function space the closed and if T is a pseudo-measure that is supported on subspace τ(f ) generated by E, then it is necessarilyΓ true that T , f = 0. Γ the harmonics of f , and the In the case in which G is a Euclidean space, harmonics of f are the genera- the space of pseudo-measures can be identified tors of the simplest subspaces as the space of tempered Schwartz distributions contained in τ(f ). (With the ex- with bounded . In his 1948 3 ception of the mean-periodic counterexample for R , Schwartz took E to be the 2 case, in the examples cited unit sphere S and T to be the derivative in some these subspaces are necessarily direction (say, for definiteness, the radial direction) 2 one-dimensional.) of the surface measure σ for S . Because, as |u| → Harcourt Studios. In terms of these subspaces, ∞, σˆ (u) = O 1 , one knows that Tˆ (u) = O(1) Malliavin as a young |u| the synthesis problem is that of and therefore that T is a pseudo-measure. Thus man. determining whether the har- Schwartz’s counterexample reduces to the trivial monics contained in τ(f ) generate τ(f ). This is task of finding a test function that vanishes on S2 a very general question, but suppose that one and has nonvanishing derivative in the direction restricts one’s attention to spaces of bounded in which σ was differentiated to get T . functions. For example, consider the space L∞(R) What is the choice of E, T , and f when E is the with the weak topology it has as the dual of L1(R). line R or circle T? Malliavin’s idea was to start with (If one uses the strong topology, one recovers f instead of E and to choose f so that the formal ′ Bohr’s almost-periodic functions.) The same ques- composition δ0 ◦ f of the derivative of Dirac delta ∞ tion can be asked about L (G) with the weak function δ0 with f can be interpreted as a pseudo- topology when G is a locally compact Abelian measure whose support is the zero set of f . This group such as Rd , Z, or Zd . When G is compact, idea is beautiful. Wholly aside from the technical synthesis always holds, but Laurent Schwartz had challenge posed by its successful implementation, shown [22] in 1948 that it fails when G = R3 or just the realization that it might work is a tour de Rd for any d ≥ 3. Prior to Malliavin, the answer force. remained unknown in other cases. In particular, it Malliavin’s idea applied equally well to R and was unknown for the crucial cases in which G is to Z, and its extension to general, noncompact G’s R or Z, and finding the answer was a challenge to (i.e., G’s for which is not discrete) is relatively every analyst of the day. easy. Furthermore, in what is a nice example of In his note [15], Malliavin solved the problem the way in which probabilityΓ theory can simplify when G = R, andhe gavethe generalsolution inhis otherwise complicated analytic constructions, the Annales de l’IHES article [16]. Namely, he proved use of random trigonometric series can greatly there that synthesis fails for L∞(G) whenever G simplify his construction of f (cf. [4] and [10]). is a locally compact Abelian group which is not Nowadays there exist many other proofs of compact. Malliavin’s theorem. Varopoulos used his theory The problem has many equivalent forms. By of tensor algebras to derive the general result duality, it can be seen as a question about the from the case, handled by Schwartz, when G = R3. structure of the closed ideals in the convolution Returning to the idea of producing E before f , algebra L1(G), in terms of which the question is Körner produced a strange set E that is meager in whether such an ideal is the intersection of the the sense that C(E) = A(E) ≡ {ϕ ↾ E : ϕ ∈ A( )}

Γ 570 Notices of the AMS Volume 58, Number 4 (such a set E is call a “Helson set”) and yet is (2) the computation of the “totality radius” of sufficiently robust that it carries a pseudo-measure a given sequence . That is, find the upper bound √ whose Fourier transform tends to 0 at infinity of those a ≥ 0 such that the set e x−1 λ : λ ∈ (such a set is said to be generates L2 (−a,Λ a) . a “set of multiplicity”). Λ Although their results did not appear until Körner’s construction is 1967, the authors knew them as early as 1961, and complicated, but it had these results remain jewels in function theory. been known for a long The solution to the first problem makes use of time (cf. [11] and [7]) the logarithmic integral that the existence of a ∞ dx Helson set of multiplicity log |f (x)| 2 would show that spectral −∞ 1 + x synthesis fails. and is explained in detail by Paul Koosis in his Malliavin’s theorem has monographs [12] and [13]. Their answer is that been the subject of many an entire function is of the sort in (1) if and only lectures, commentaries, Identity photo, if its logarithmic integral converges. The most and scholarly articles (cf. circa 1975. important part of their answer is the statement that [18], [21], and [7]). How- if f is an entire function whose logarithmic integral ever, it marked the end of the era in which converges, then there is an entire function g of attention was focused on Wiener’s algebra A( ), arbitrarily small exponential type that is bounded which was considered to be an essential object for on R and for which f g is also of exponential analysis. Nonetheless, contrary to what one mightΓ type and bounded on R. Combining this statement have supposed, his theorem did not mark the end with Fourier analysis, one obtains the description of commutative harmonic analysis, only the end of of those hyperfunctions that can be regularized a particular period. Today the subject is alive and by convolution functions having arbitrarily small well, having been rejuvenated by the introduction support. of new directions in which to go. In order to solve the second problem, Beurling and Malliavin introduced a new notion of density, The Thesis and the Theorem of Beurling– which they called the “effective density”, for a Malliavin sequence . One way to define this notion involves In 1959 Malliavin was a professor in the faculty the notion of a “BM-regular” sequence: a sequence of sciences at the university in Caen. He already ′ for whichΛ there exists a D( ′) ∈ [0, ∞), the had a solid reputation in complex analysis. Like “density” of ′, such that me, he had been a student of Szolem Mandelbrojt. Λ Λ ′ dr Szolem used to tell stories and to ask questions. n(r)Λ − D( )r < ∞, 1 + r 2 He had done a joint piece of work with Norbert ′ ′ ′ Wiener, and he posed to Malliavin the following where n(r) ≡ card {λΛ ∈ : λ ≤ r} is the ′ question, which had its origins in that joint work: counting function for . The effective density What can be said about the set of real zeroes of a sequence is the infimumΛ over BM-regular ′ ⊆ of D( ′). MimickingΛ analogous ideas in of a holomorphic√ function f in the right half-space {z = x + −1 y ∈ C : x > 0} which satisfies an measure theory,Λ one can associate with each of inequality of the form |f (z)| ≤ M(x) for some M : theseΛ Λ densitiesΛ notions of interior and exterior (0, ∞) → (0, ∞)? density, in which case Beurling and Malliavin’s In his doctorat d’etat thesis, Malliavin gave a effective density becomes the exterior density complete and definitive answer to this question. associated with BM-regular sequences. Beurling Simultaneously, his thesis contains several beau- and Malliavin’s results are easy to describe, but tiful new results in functional as well as complex the methods by which they are proved are very analysis. In addition, it won Malliavin an invitation elaborate and require intricate refinements of in 1954–1955 to the Princeton Institute for Ad- ideas from potential theory (cf. [1] and [2]). vanced Study, and it was there that he met Arne Beurling. But it was when Malliavin returned to the References I.A.S. in 1960–1961 that he and Beurling launched [1] A. Beurling and P. Malliavin, On Fourier trans- their extremely fruitful collaboration. During that forms of measures with compact support, Acta year, they were able to completely solve two hard Math. 107 (1962), 291–309. [2] , On the closure of characters and the zeros and intimately related problems: of entire functions, Acta Math. 118 (1967), 79–93. (1) the characterization of those entire functions [3] J. Bourgain, Sidon sets and Riesz products, Ann. that can written as the quotient of two entire Inst. Fourier 35, No. 1 (1985), 137–148. functions, both of which are of exponential type [4] J.-P. Kahane, Sur un théorème de Paul Malliavin, C. and are bounded on the real line. R. Acad. Sci. A–248 (1959), 2943–2944.

April 2011 Notices of the AMS 571 [5] , Travaux de Beurling et Malliavin, Séminaire study of singular integrals. Most of this work Bourbaki, Décembre 1961, exposé 225. was done by mathematicians working in the tradi- [6] , Une nouvelle réciproque du théorème tion known as the Calderón-Zygmund school. The de Wiener–Lévy, C. R. Acad. Sci. A–264 (1967), major players during this period were Zygmund 104–106. himself [4] and his students, Alberto Calderón, [7] Jean–Pierre Kahane and R. Salem, Ensembles par- Elias Stein, , and later, Stein’s student faits et séries trigonométriques, Hermann, Paris, 1963. Charles Fefferman. Although Paul Malliavin did [8] Y. Katznelson, Sur les fonctions opérant sur not participate in this group, he and his wife did l’algèbres des séries de Fourier absolument conver- make a significant contribution, described below, gentes, C. R. Acad. Sci. A–247 (1958), 404–406. to work of their American colleagues [9]. [9] Y. Katznelson and P. Malliavin, Vérification Here is a simplified description of the origin statistique de la conjecture de la dichotomie sur of the Malliavin theorem: Let u(x, y) be a har- une classe d’algèbre de restriction, C. R. Acad. Sci. monic function ( u = 0), defined in the unit A–262 (1966), 490–492. disc x2 + y 2 < 1. We can always find another R. Kaufman [10] , Gap series and an example to harmonic functionu(x,˜ y) such that ∂ u = ∂ u˜, Malliavin’s theorem, Pacific J. Math. 28 (1969), x y ∂ u˜ = ∂ u (the Cauchy-Riemann equations). If the 117–119. x y → [11] T. Körner, A pseudofunction on a Helson set, map (x, y) (u(x, y), u(x,˜ y)) is one-to-one, its 2 2 Astérisque 5 (1973), 3–224 and 231–239. Jacobian J(x, y) = |∇(u(x, y))| = |∇(u(x,˜ y))| , [12] P. Koosis, The Logarithmic Integral, vols. I and II, by the Cauchy-Riemann equations. Thus, if we Cambridge University Press, 1988 and 1992. wish to calculate the square root of the area of [13] , Leçons sur le Théorème de Beurling et the image of a set in the disc, we must com- Malliavin, Publications CRM, Montréal, 1996. pute an integral. The following functional of u [14] P. Malliavin, Thèse, Acta Math. 93 (1954), was introduced by LusinΓ and called the “area inte- 179–255. gral”: A(u)( ) := { |∇u(x, y)|2χ (x, y)d(x, y)}1/2, [15] , Sur l’impossibilité de la synthèse spectrale where the set = (θ) is a Stoltz domain, a cone sur la droite, C. R. Acad. Sci. A–248, 2155–2157. Γ with its axis of symmetry a radius from 0 to a [16] , Impossibilité de la synthèse spectrale sur les Γ groupes abéliens non compacts, Publ. Math. I.H.E.S. point θ on theΓ boundaryΓ of the disc. A remark- (1959), 61–68. able fact is the following: The set of boundary [17] , Sur l’impossibilité de la synthèse spec- points θ : A(u)(θ) < ∞ coincides with the set of trale dans une algèbre de fonctions presque– boundary points where N(u)(θ) := sup(|u(x, y)| : périodiques, C R. Acad. Sci. A–248 (1959), 1756– (x, y) ∈ (θ)) < ∞ (up to a set of measure zero); 1758; voir aussi Calcul symbolique et sous–algèbres moreover, the harmonic function u(x, y) has a limit 1 de L (G), Bull. Soc. Math., Stockholm 1962, along allΓ paths converging to θ within the cone 368–378. (θ). Calderón [3] and Stein [9] extended this [18] , Ensembles de résolution spectrale, Proc. local equivalence to harmonic functions u(x, y) Intern. Congr. Math. Stockholm, 1962, 368–378. where x ∈ Rn, y > 0. Somewhat later Stein and [19] P. Malliavin and M.–P. Malliavin, Caractérisation Γ arithmétique d’une classe d’ensembles de Helson, Weiss [10] proved a version of the F. and M. Riesz C. R. Acad. Sci. A–264 (1967), 192–193. theorem for harmonic functions defined on the n [20] G. Pisier, Arithmetical characterisation of Sidon generalized half-plane (R , y > 0). In so doing, 1 n+1 sets, Bull. Amer. Math. Soc. 8 (1983), 87–89. they defined an H space for functions on R+ . [21] W. Rudin, Fourier Analysis on Groups, Wiley, 1962. However, it wasn’t until 1970, ten years later, that [22] L. Schwartz, Sur une propriété de synthèse spec- the functionals A(u), N(u) were shown to char- trale dans les groupes non compacts, C. R. Acad. Sci. acterize the classical H1 space [2]. The extension A–227 (1948), 424–426. of this theorem in [2] to the Stein-Weiss H1-space [23] N. Varopoulos, Sur un théorème de M. Paul of several variables is but one of a number of Malliavin, C. R. Acad. Sci. 263 (19667), 834–836. important results in a groundbreaking paper by Fefferman and Stein [5]. Given these developments, other contexts and Richard Gundy conjectures spring to mind. A natural context, suggested by Fefferman and Stein, is the Cartesian The Contribution of Paul and Marie-Paule product of two unit discs, D1 × D2 (or two half- Malliavin to the Study of Boundary Values planes). In either case, the appropriate Laplacian of Harmonic Functions on the Bidisc is 12 := ( 1)( 2). The functionals A(u), N(u) are In the period 1950–1980, much progress was made defined on the product boundary, say ∂D1 × ∂D2. understanding the boundary behavior of harmonic In this context, the problem is totally different: for functions of several variables in the context of the the classical case, the radius variable 0 < r < 1 is essentially a totally ordered dilation parameter. Richard Gundy is professor of statistics and mathematics In the bidisc, the pair of radii (r1, r2) are still at Rutgers University. His email address is gundy@rci. dilations, but they form a set of parameters that rutgers.edu. is only partially ordered. This produces a major

572 Notices of the AMS Volume 58, Number 4 011 ball the btcefrtefloigrao:I h disc the In reason: following the for obstacle ucinl eeeuvln.I hswy e of set a way, this In equivalent. were functionals enn h alai siae,w eeal to by able and the were result, that we same estimates, show the Malliavin obtained the [7] refining I and where the Stein set obtained the on [1] result: Brossard converse Jean Subsequently, that proof the contains χ Malliavin A Paul and put To mildly! courage. it indefatigable estimates and required subse- ingenuity the with the through plow in to produced managed blizzard are Undaunted, the that computation. quent sees terms one However, error when words. of few comes easy a payback is in the this which summarize Now to to applicable. domain enough a is bidisc, theorem entire function Green’s a the obtains on one doing, defined so In boundary. entire the the an to by region sawtooth bidisc the of character- function the istic extend collaboration: Malliavin the the is Enter product. setting Cartesian in two-parameter arises that the region sawtooth the Unfortunately, on defined to applicable functions easily only is theorem Green’s of bidisc, of one function distribution of the function say distribution functionals, these the of timation eta aia function maximal gential every the cones taking N(u)(θ) by all defined of region union sawtooth ap- a theorem, to Green’s plied is technique workhorse the April Photo by Xiangdong Li. nCiawt i ieMrePue spring Marie-Paule, wife 2 his with China In Γ 2 (θ (u)(θ 010. h ratruhpprb Marie-Paule by paper breakthrough The 1 )χ (θ 2 Γ 1 B (θ 1 θ , ≤ θ , n + 2 2 λ) )d(x 2 1 ) ) approximation { = re’ hoe rvdsa es- an provides theorem Green’s . : hr h orsodn nontan- corresponding the where = L 1 N(u) p X y , nrs,0 “norms”, A(u) 1 = ( Γ )d(x |∇ (θ) m(θ ( atsa rdc domains product Cartesian sfiieams everywhere almost finite is ,ρ) θ, u 1 2 sfiie Independently, finite. is | where y , 2 : N(u)(θ N(u) ()>λ) > A(u) 2 |∇ + 0 : hti mohu to up smooth is that e idea key ) < p < sfiiefralmost for finite is not ≤ ’qieMalliavin l’équipe A(u)(θ) oee,i the in However, . u ρ 2 1 | θ , ≤ eesrl a necessarily 2 rvddby provided ∞ ntrsof terms in , |∇ + 2 1 ) fthese of , θ sfinite. is ≤ ∈ u 12 λ D S n | ( 2 (or } or oie fteAMS the of Notices ) ) . , References upstairs really I the what in remember could locked said. I “It’s wish I was I zee all bedroom.” of about flustered, think know Completely could you term?” do Malliavin!” fourth-order eez “What Thees pause) pause)... (another Planet (long on somewhere “Allo from Earth. call when phone day, a unremarkable received an I Agatha of New close in of the from home at at spring dinner Jersey, for scene the down sitting in was a I probably 1978, evening, me) One (to Christie: following The recalls significance. vignette mathematical a than degree of geneous alai ae otisarte mysteri- rather a term |∇ error contains ous paper Malliavin developments. In these [6].) and from theses 1978 followed several of articles quoted, Lectures just Flour papers to St. addition the in may of found refinements, details be with the contribution these of Malliavin of exposition the survey an including comprehensive results, (A created. was called be can that results 10. 1. 8. 7. 2. 9. 6. 5. 4. 3. u ncnlso,Iwn ormr htthe that remark to want I conclusion, In oiusl úlitgaedar s finie, est d’aire l’intégrale Math. oú là moniques Brossard J. ui-adrnpu e ocin biharmoniques, Math. Sc. fonctions Bull les pour Lusin-Calderón Malliavin M.-P. 1029. disc, Gundy F. R. 1978. Berlin, Springer-Verlag, Math., in VIII Flour Saint class Math. Amer. the Trans. of characterization function maximal Gundy F. R. Burkholder, D. .M Stein M. E. II, variables several Stein M. E. L’espace indices: Gundy F. R. J., variables, N. Fefferman Princeton, C. Press, Univ. 1950. complex Princeton several 145–165, of in functions variables, of values Calderón boundary P. A. Zygmund, Calderón P. A. oi ucin fsvrlvralsI, variables several 103 of functions monic 2 | 2 )χ 16) 25–62. (1960), rc al cd c.USA Sci. Acad. Natl. Proc. Γ 2 (θ e série 1 caMath. Acta )χ rn.Ae.Mt.Soc. Math. Amer. Trans. nteter fhroi ucin of functions harmonic of theory the On , ngltspu atnae ne deux et un à martingales pour Inégalités , and opreetdsfntosbihar- fonctions des Comportement , and Γ otiuin oFuirAnalysis Fourier to Contributions 2 (θ 103 and p 5–3,Srne etr Notes Lecture Springer 252–334, pp. , e natermo acniwc and Marcinkiewicz of theorem a On , four 2 série .M Stein M. E. T ) H .Weiss G. and and 17) 77–95. (1979), } .M Stein M. E. 4 caMath. Acta p 157 oal eas ti homo- is it because notable , (u)(θ , o e hstr a more has term this me, For . 129 101 cl ’t ePoaiié de Probabilités de d’Été École .Malliavin P. .Zygmund A. 17) 137–153. (1971), H 17) 25–62. (1972), 1 17) 357–384. (1977), p nteter fhar- of theory the On , θ , , and , hoyfrtebidisc the for theory H 76 , 2 p 106 H ) hoyfrtepoly- the for theory 17) o ,1026– 3, no. (1979), p : .Silverstein M. sae fseveral of -spaces { = 68 16) 137–174. (1961), nérlsde Intégrales , oeo the on Note , 15) 55–61. (1950), caMath. Acta ( |∇ ul Sc. Bull. u 1 pp. , | H 2 A , p , 573 Daniel W. Stroock results in a measure Wh that is absolutely con- tinuous with respect to W and has a remarkably Malliavin the Probabilist simple Radon–Nikodym Rh, all of whose powers Like Norbert Wiener, Paul Malliavin came to proba- are integrable. Using Cameron and Martin’s result, bility theory from harmonic analysis, and, like one can show that differentiation Dh of a function on Wiener space in the direction of an h ∈ H Wiener, his analytic origins were apparent in ⊤ everything he did there. admits an adjoint Dh , and the existence of this adjoint allows one to make Sobolev-type exten- Under the influence of Paul Lévy, most post- ⊤ war (i.e., Word War II) probabilists have studied sions of Dh and Dh as closed, densely defined 2 W R stochastic processes as a collection of random operators on L ( ; ). In addition, one can show { ≥ } paths. For them, the measure determining the that if hk : k 1 is an orthonormal basis in H, distribution of those paths is an éminence grise then the operator, known to probabilists as the that is best left in the shadows. This perspec- Ornstein–Uhlenbeck operator, tive gained prominence because of the successes ∞ N = D⊤ D , it had in the hands of such masters as K. Itô hk hk and J. L. Doob, and no doubt it is responsible k=1 for some of the most stunning achievements of is self-adjoint and is independent of the partic- probabilists during the last sixty years. However, ular choice of orthonormal basis. Furthermore, this was not Wiener’s perspective, and it was not Wiener’s spaces of Malliavin’s either. Instead, for them, the principal homogeneous chaos object is the measure. Thus, according to Wiener, are the eigenspaces Brownian motion is a certain Gaussian measure for N . In fact, N ϕ = W , now called Wiener measure, on the space, nϕ for ϕ ∈ Z(n), Wiener space, of continuous paths, and, insofar a fact that accounts as possible, he analyzed and exploited it in the for people doing same ways that Gauss’s and related measures had Euclidean quantum been in finite-dimensional settings. For example, field theory calling it is a familiar fact that the Hermite polynomials N the number oper- are a natural, orthogonal basis for the standard ator. All this should Gauss measure on RN , and Wiener showed that come as no sur- there is an analogous orthogonal basis for his prise to anyone who measure on pathspace. More precisely, just as in has dealt with Gauss- RN , it is best to group together all the Hermite ian measures and polynomials of a fixed degree n and to consider Hermite polynomials the subspace spanned by them, so Wiener looked Induction in 1979 into in finite dimensions, the French Académie at the spaces Z(n) that are obtained by closing in where Cameron and des Sciences. L2(W; R) the linear span of the nth order Hermite Martin’s result is a polynomials on Wiener space. His motivation for trivial change of vari- looking at these subspaces was that he wanted to ables and the operator N is just the ground state interpret Z(n) as the subspace of L2(W; R) consist- representation of the Hermite operator (a.k.a., the ing of functions that have homogeneous nth order harmonic oscillator). However, there are techni- randomness, and, with his usual flair for words, cal difficulties that have to be overcome before he dubbed them the subspaces of homogeneous one can transfer the finite-dimensional results to chaos.1 Put another way, Wiener was attempting a infinite dimensions. spectral decomposition of L2(W; R) in which the The preceding discussion provides a context in spectral parameter is randomness, as opposed to which to describe one of Malliavin’s most impor- something more conventional, such as frequency. tant contributions to probability theory. What he In a related example, two of Wiener’s disciples, realized is that the Ornstein–Uhlenbeck operator R. H. Cameron and W. T. Martin, discovered that N can be used as the starting point for a robust Wiener measure is as translation invariant as any integration-by-parts formula for Wiener measure. measure in infinite dimensions has a right to He was far from the first to attempt integration be. Namely, they showed that if H is the Hilbert by parts for functions on Wiener space. Indeed, subspace of Wiener space whose elements h are Cameron and his student M. Donsker had been do- absolutely continuous and have square integrable ing it for years, and integration by parts in Wiener derivative, then translation of W by an h ∈ H space had been a basic tool of Euclidean quan- tum field theorists. However, earlier versions had 1It should be pointed out that Wiener’s own treatment of involved functions that are classically (i.e., in the this subject was somewhat awkward and that it was Itô sense of Fréchet) differentiable, whereas Malliavin who put it on a firm mathematical foundation. wanted to apply it to functions that are not even

574 Notices of the AMS Volume 58, Number 4 classically continuous. Specifically, with the goal where the symmetric, bilinear operation F,G has of proving elliptic regularity results, he wanted the crucial properties that it is nonnegative and to do integration by parts when the functions are satisfies solutions to Itô stochastic differential equations. If ′ 1 R R one thinks of an ordinary differential equation as (2) ϕ ◦ F,G = ϕ ◦ FF,G for ϕ ∈ Cb ( ; ). giving the prescription for turning a straight line Starting from (2), he argued that, because into the integral curve of a vector field, then Itô’s stochastic differential equations can be thought ϕ′ ◦FF,F = N Fϕ◦F −FN (ϕ◦F)−ϕ◦FN F, of as the analogous prescription for converting Brownian (i.e., Wiener) paths into the paths of a then if F,F > 0, one can write more general diffusion. In particular, if X( , x, w) N Fϕ ◦ F FN (ϕ ◦ F) ϕ ◦ F is the diffusion path starting at x corresponding ϕ′ ◦ F = − − ; to Wiener path w and ,FF ,FF ,FF and therefore, because N is self-adjoint, u(t, x) = f X(t, x, w) W (dw), ϕ′ ◦ F dW = then u will solve the diffusion equation ∂t u Lu with initial data f , where L is the associated dif- 1 F = ϕ ◦ F FN − N fusion operator, the (possibly degenerate) elliptic ,FF ,FF operator that appears in Kolmogorov’s backwards N F − dW , equation. ,FF Solutions to Itô’s equations have much to recommend them, but from a classical analytic which, by taking advantage of (1) and (2), can be perspective they are dreadful when viewed as rewritten as functions of w. Indeed, they are defined only up (3) to a set of W -measure 0, and so they are not 2N F F, F,F ϕ′◦F dW = − ϕ◦F + dW . amenable to any classical notion of differentia- ,FF ,FF tion. Undaunted by this formidable technicality, Malliavin realized that w ⇝ X(t, x, w) nonetheless The virtue of (3) is that it allows one to draw ought to be differentiable in the sense of Sobolev. conclusions about the distribution of F. Indeed, if After all, in infinite dimensions, there is no Sobolev is the distribution of F, then (3) says that, in the embedding theorem to prevent there from being language of Schwartz distributions, ∂ = −ψ, functions that are classically discontinuous and where ψ : R → R is the function such that ψ ◦ F yet infinitely differentiable in the sense of Sobolev. is the conditional expectation value EW σ (F) (n) To wit, any element of Z will be smooth in the of sense of Sobolev, but few elements of even Z(1) 2N F F, F,F Ψ ≡ + . will be classically continuous. With this in mind, ,FF ,FF Malliavin constructed a Schwartz-type space of p 2 Elementary analysisΨ shows that if ψ ∈ L (; R) functions on Wiener space. for some p > 1, then admits a density f that Rather than use powers of the operators D h is uniformly Hölder continuous with a Hölder to measure regularity, he used powers of N , exponent depending only on p. Furthermore, because the spectral and stability properties of N p ≤ p make it easier to understand questions about its since ψ L (;R) L (W ;R), one can estimate p domain than the corresponding questions about ψ L (;R) entirely in terms of Wiener integrals. Of course, thereΨ are at least two technical the domain of Dh. Because everything had to be based on properties of N , his integration- details that have to be confronted in order to by-parts formula had to be an application of the carry out Malliavin’s program. For one thing, one self-adjointness of N , and for this purpose he took has to check that F is in the domain of the advantage of the Leibniz rule satisfied by second- operations that one wants to perform on it. When order elliptic operators. That is, for R-valued F F is the solution to an Itô equation in which the and G on Wiener space, coefficients are smooth, Malliavin’s description of his Schwartz space in terms of N makes (1) N (FG) = FN G + GN F + F,G, this a difficult problem. The Japanese school, especially S. Watanabe and his student H. Sugita, 2 This had been done before by P. Kree, but Kree’s con- vastly simplified matters by describing the same struction had, at least for the applications that Malliavin Schwartz space in terms of the operators D . A key had in mind, a fatal flaw: the space he built was not an h algebra. In the absence of a Sobolev embedding theorem, ingredient in their approach was provided by P. A. the only way to achieve an algebra is to abandon L2 and Meyer, who showed that E. M. Stein’s Littlewood– deal with all Lp spaces simultaneously. Paley theory for symmetric semigroups can be

April 2011 Notices of the AMS 575 used to prove that for any orthonormal basis applications. All this is still in the early stages of {hk : k ≥ 1} and any p ∈ (1, ∞), development, but there can be no doubt that the 1 program that Malliavin initiated is one of the great ∞ 2 1 2 contributions to modern probability theory. N 2 F p R ∼ |D F| , L (W ; )  hk  k=1 p L (W ;R)   Leonard Gross where ∼ means here that one side is dominated by a constant times the other. Among the many areas of mathematics to which The second detail is a more interesting one. Paul Malliavin contributed in the past twenty years, In fact, it is less a detail than the heart of the the general problem of constructing an interesting whole program: namely, in deriving (3), it was theory of infinite-dimensional , on which assumed that F,F is strictly positive, and in there is a useful measure for doing harmonic anal- applying (3) it is necessary to have sufficient ysis, was the focus of a large part of his work. control on its positivity to estimate the Lp(W ; R)- Such a manifold must be able to support a notion norm of . Gaining such control can be a very of differentiation that relates well to the measure. challenging problem. In applications to solutions For example, one should be able to carry out Ψ to Itô’s equa- integration by parts at the very least. Preferably, tions, F,F can there should also be some kind of natural second- be recognized order, infinite-dimensional “elliptic” differential as a path- operator , which plays the role of a Laplacian. wise measure After Cameron and Martin’s work in the 1940s of ellipticity. and 1950s, showing that “advanced calculus” on In particular, Wiener space, using Wiener measure, could be when the dif- developed in an interesting way, it became reason- fusion operator able to seek interesting examples of not necessarily L is uniformly linear infinite-dimensional manifolds on which to elliptic, it is develop some analog of finite-dimensional har- relatively easy monic analysis. Early work in this direction began to check that in the 1960s (H. H. Kuo, D. Elworthy, J. Eells). Malli- the correspond- In conversation, circa 2000. avin initially sought examples in the form of spaces ing F,F will of continuous functions from an interval or circle have reciprocal moments of all orders. How- into a finite-dimensional Riemannian manifold M. ever, Malliavin was not ready to settle for a The measure induced by Brownian motion on M rederivation of classical elliptic regularity results. (or by pinned Brownian motion) would provide, in He wanted to show that his method could also be this case, a natural measure of interest on these applied to derive regularity results of the sort that spaces of continuous paths into M. The required Hörmander had proved for subelliptic operators. integration by parts theorems were established in Although Malliavin pointed the way, it required the late 1980s and early 1990s by J. M. Bismut, considerable effort by several authors to achieve B. Driver, E. Hsu, and Malliavin and his wife. his goal, and it must be admitted that in the end But in the last ten years Malliavin aimed at their effort was not rewarded by the discovery of producing a natural Brownian motion on the dif- many facts that more traditional analytic methods feomorphism group of the circle, thereby replacing had not already revealed. the finite-dimensional manifold M by an infinite- Expanding on the remark at the end of the dimensional one. As is often the case, construction preceding paragraph, one should recognize why of a Brownian motion in a manifold is more or it is that Malliavin’s ideas do not give an effi- less equivalent to the construction of a heat semi- cient way of looking at questions such as elliptic group et acting on functions over the manifold. regularity. Indeed, his approach takes an inher- In case the manifold is the diffeomorphism group ently finite-dimensional problem, lifts it to an of the circle, there is a natural infinite-dimensional infinite-dimensional setting, performs the analy- Laplacian. It is determined by regarding the tan- sis in infinite dimensions, and then projects that gent space at the identity to be vector fields on analysis back down to finite dimensions. This is a 1 S of Sobolev class H3/2. The index 3/2 arises little too much like going to a neighbor’s house by for several seemingly very different reasons. It is way of the moon: it works, but it is not efficient. the natural Sobolev class for action on the ar- Thus it was not until Malliavin’s ideas were applied gument of loops into a compact Lie group. It is to intrinsically infinite-dimensional problems that they came into their own. There is now a ma- Leonard Gross is professor of mathematics at Cor- jor industry, populated (somewhat worryingly) in nell University. His email address is gross@math. part by financial engineers, who are making such cornell.edu.

576 Notices of the AMS Volume 58, Number 4 also the correct Sobolev index for capturing the by the space of shapes, the procedure of mapping Weil-Petersson metric on this Lie algebra. a given homeomorphism φ to the pair f , h, the so- It is well understood, at least for linear spaces, called “welding" procedure, would give a Brownian that if the symbol of an operator is the norm motion on the space of shapes if one could extend on some infinite-dimensional H, the welding procedure to those homeomorphisms then the associated heat semigroup et cannot of S1 that are needed to support a Brownian motion act reasonably on functions over H but will act well on functions over a suitable enlargement as above, namely a large subset of the Hölderian of H. Otherwise said, the associated Brownian homeomorphisms. This is accomplished in the motion will have continuous sample paths into the paper [6], which welds deep stochastic analysis larger space but will jump out of H immediately. techniques with analytic function theory. The analog for nonlinear manifolds, such as the diffeomorphism group, is much more difficult. References Malliavin and his coauthors showed, over the 1. H. Airault and P. Malliavin, Regularized Brown- last ten years, that, for the natural Laplacian ian motion on the Siegel disk of infinite dimension, (whose symbol is the Weil-Petersson metric), it Ukraïn. Mat. Zh. 52 (2000), no. 9, 1158–1165. suffices to enlarge the group of H3/2 homeomor- MR1816929 (2002b:60098) phisms of S1 to the group of Hölder continuous 2. , Unitarizing probability measures for rep- homeomorphisms. This enlargement, along with resentations of Virasoro algebra, J. Math. Pures the obvious weakening of the topology, supports Appl. (9), 80 (2001), no. 6, 627–667. MR 1842293 the associated Brownian motion with continuous (2003a:58057) sample paths. 3. , Quasi-invariance of Brownian measures on the group of circle homeomorphisms and The evolution of this work can be traced through infinite-dimensional Riemannian geometry, J. Funct. the papers [10], [1], [9], [7], [4], [2], [5], [11], [3], [12], Anal. 241 (2006), no. 1, 99–142. MR 2264248 [13], [14], [8], and [6], many of which were written (2008b:60119) jointly with one or more of Malliavin’s coworkers, 4. Hélène Airault, Paul Malliavin, and Anton Hélène Airault, Ana-Bela Cruzeiro, Jiagang Ren, Thalmaier, Support of Virasoro unitarizing mea- and Anton Thalmaier. The program was completed sures, C. R. Math. Acad. Sci. Paris 335 (2002), no. only very recently, in the paper [6]. 7, 621–626. MR 1941305 (2004c:58070) The titles of some of these papers seem to have 5. , Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equa- more to do with analytic function theory than tions by a continuity method along stochastic flows, diffeomorphisms of the circle. It happens that a Math. Pures Appl. (9) 83 (2004), no. 8, 955–1018. Brownian motion on the diffeomorphism group MR 2082490 (2005i:58044) 1 of S induces, at least informally, a Brownian 6. , Brownian measures on Jordan-Virasoro motion on the space of closed Jordan curves in curves associated to the Weil-Petersson metric, the plane, as well as on certain spaces of univalent J. Funct. Anal. 259 (2010), no. 12, 3037–3079. holomorphic functions on the unit disc. At an MR 2727640 informal level the link is easy to understand: 7. Hélène Airault and Jiagang Ren, Modulus of con- Suppose that J is a Jordan curve and that f is a tinuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle”, J. Funct. holomorphic map from the open unit disk D onto Anal. 196 (2002), no. 2, 395–426. MR 1943096 the interior of the region bounded by J. It is known (2003i:58066) that f extends to a continuous function fˆ on the 8. Ana-Bela Cruzeiro and Paul Malliavin, Stochas- closure of D and that fˆ is a bijection of S1 onto J. tic calculus of variations on complex line bundle Moreover, there is a holomorphic function h from and construction of unitarizing measures for the the exterior of the unit disk onto the unbounded Poincaré disk, J. Funct. Anal. 256 (2009), no. 2, component of the complement of J. This also 385–408. MR 2476947 (2010b:31011) 9. Shizan Fang, Canonical Brownian motion on the dif- extends to the closure of the exterior of the unit ˆ feomorphism group of the circle, J. Funct. Anal. 196 disk, producing another continuous bijection h (2002), no. 1, 162–179. MR 1941996 (2003k:58060) 1 from S onto J. Insisting that limz→∞ h(z)/z > 0 10. Paul Malliavin, The canonic diffusion above the makes the choice of h unique. On the other hand, diffeomorphism group of the circle, C. R. Acad. f fails to be unique, but only to the extent that Sci. Paris Sér. I Math. 329 (1999), no. 4, 325–329. its argument can be changed by a D-preserving MR 1713340 (2000e:60129) Möbius transformation. For any such function f the 11. , Heat measures and unitarizing measures for map (fˆ )−1 ◦ hˆ : S1 → S1 is a homeomorphism φ of Berezinian representations on the space of univalent functions in the unit disk, Perspectives in analysis, S1. What homeomorphisms can arise in this way? It ∞ 253–268, Math. Phys. Stud. 27, Springer, Berlin, 2005. is known that if φ is C , then there exists a Jordan MR 2215728 (2007b:58019) curve and corresponding functions f and h whose 12. , Itô atlas, its application to mathematical fi- ratio, as above, is φ. Ignoring the nonuniqueness nance and to exponentiation of infinite dimensional of f , which only requires replacing Jordan curves Lie algebras, Stochastic analysis and applications,

April 2011 Notices of the AMS 577 501–514, Abel Symp. 2, Springer, Berlin, 2007. few crumpled sheets of paper out of his pocket. MR 2397802 (2009c:60140) “Madame, I would like to have your opinion of my 13. , Invariant or quasi-invariant probability speech,” and, in a low voice, he starts to whisper: measures for infinite dimensional groups. I. Non- “Already a hundred years ago, Elie Cartan... .” of Euler hydrodynamic, Jpn. J. Math. 3 Then comes his turn to speak, and in a loud voice, (2008), no. 1, 1–17. MR 2390181 (2009b:60169) he declaims: “Already a hundred years ago... .” 14. , Invariant or quasi-invariant probability mea- sures for infinite dimensional groups: II. Unitarizing Bravo, clap, clap, and his candidate is elected. measures or Berezinian measures, Jpn. J. Math. 3 The Poisson Summation Formula: Malliavin (2008), no. 1, 19–47. MR 2390182 (2009b:60170) and I are ecstatic about the Poisson summation formula. No doubt, he knows all its finest and deepest aspects. I don’t, but I nonetheless think it Michèle Vergne is the most beautiful formula in mathematics. Two Things That Malliavin Loves, Mathemat- Malliavin and I ics and Influencing People’s Destiny: These are Pisa, June 2010:IaminPisa,Ilearnviaemailabout not unrelated. Speaking about a colleague, he often the death of Paul Malliavin. If anyone seemed to lauds the beauty of his work: “Demailly’s annula- me to be immortal among immortals, it was he. tion theorem is extraordinary; Madame, consider I think of him introducing me to the Académie that it does not require pointwise estimates, but and guiding me on a visit to the Institut Library only in the mean... .” Villani’s work overjoys him. and the Bibliothèque Mazarine. We could see the A Reception at the Malliavins’ Home: I am in- Seine through the windows. He told me about vited to a reception at the Malliavins’ home. I go Cardinal Richelieu, to whom I think he attributed with my daughter Marianne. She was eight years the sentence: “I shall regret the beauty of this old at the time. We enter a paved courtyard, we place when in the other world”. go up some stairs, we ring a bell, and we enter I.H.P., May 1968: I am twenty-five-years old, an apartment, immersed in semi-darkness. Over- long hair, in blue jeans. My heroine is Louise flowing bookshelves cover the walls; the seats are Michel. I can well see myself sent to hard labor for antique, I fear that the furniture would disintegrate my ideas. into dust were the curtains to be drawn. Malliavin Slogans, demonstra- talks to my daughter, he finds an old, illustrated tions: the system must edition of The Children of Captain Grant for her. be changed. The dusty She sits on a window sill and reads passionately, “Institut Henri Poincaré” while the other guests, mainly mathematicians must be destroyed. from all over the world, tell each other about General meetings, decla- their lives. Marie-Paule becomes nervous: the pe- rations. When Malliavin tits fours must be eaten, the Bertillon sorbets must intervenes with his soft be tasted... . voice, I shout: “Malliavin Temptations: Malliavin phones me, he wants is a bourgeois, Malli- to propose me as a candidate for the Académie. I avin is a fascist”. If the object: “My father and mother are dead, it is too red guard were gathering late to please them, it would give me no personal their battalions, I would pleasure.” Malliavin responds: “Madame, we are be with them and would not Académicians for our pleasure, but to serve send Malliavin out to our country”. He then invites me to come to the serve the people. Malli- Académie and leads me to the salle des séances.

Photo by D. Stroock. avin continues to smile In the dim light, the white marble heads observe Visit to Versailles in 2008. serenely. Ever since then, the scene. That night, I have a dream: there is a when I meet him on the streets of Paris, he tips his lit niche, and inside the niche, a bottle of whiskey hat and addresses me with a ceremonious: “Chère sitting on a pedestal (I had just seen again Rio Madame”. Bravo). I realize that, more than anything else, At the Theater, January 2009: Malliavin is I wish to have a draught of whiskey and that I presenting a candidate for membership in the would also like to enter the Académie. I phone Académie des Sciences. It won’t be an easy win. I Malliavin: “Yes, I agree to be nominated.” Anyway, sit beside him at the green, oval desk. He draws a I am totally incapable of saying no to Malliavin. No: I am elected. My sister Gilberte does not Michèle Vergne is professor of mathematics at the Insti- survive for my election, and my sister Martine is tut de Mathématiques de Jussieu. Her email address is about to die. Now other plans are afoot behind the [email protected]. scene: a representative of France for the execu- This is a translation (slightly adapted for the Notices) of tive committee of the International Mathematical an article published in La Gazette des mathematicians, Union is needed. Malliavin and Jacques-Louis Lions no. 126, October 2010. contact me, they have decided it should be me.

578 Notices of the AMS Volume 58, Number 4 Malliavin calls me daily. “No, I do not want to, I cannot.” After each phone call, a wave of anxiety suffocates me. I feel like a fraud. Honors: I become accustomed to taking plea- sure in honors. Today is the séance solennelle (solemn convocation) at the Académie of Sciences. Going down the stairs between the raised sabers of the republican guards seems natural. Malliavin is wearing his ceremonial outfit. Befit- tingly, the Académie’s paleontologist has a sword with a dinosaur pommel. Malliavin is happy to be among his peers. He knows them all. He pushes a chair forward for Pierre Lelong, who is nearly ninety. He listens politely to Denisse, he teases Der- court, he says a kind word to me. He jokes: “Here, we we all love researchers studying longevity and who search for a happiness pill to give the elderly”; and, all the while, he is predicting the election of Beaulieu [who specializes in geriatrics] as the new president of the Académie des Sciences. Maneuvers: Malliavin has a plan for X: he sends stacks of mail, phones, counts his cards. He scrutinizes the weaknesses of the opposition’s plans for Y . If the maneuver fails, it’s a triumph for Z, and Y is chosen. Malliavin gives me sibylline advice. I interpret it as follows: as soon as someone is chosen for our section, we should all forget whatever bad things we once thought about him. Should we do likewise with the dead? It might be wise, since we will have to spend eternity by their sides. Haar Measures and Malliavin Measures: Is there some sort of Haar measure for loop groups? Which are the “natural” groups that admit uni- tary representations? Locally compact groups, thanks to Haar measure, but the unitary group also has a unitary representation. We may also construct ergodic measures for some natural groups, such as the infinite permutation group. These are questions that interest Paul Malliavin and Marie-Paule. I naïvely believe that mathematical ideas have no genitors and come forth out of cauliflowers. But, no, Haar measure did not exist before Haar, Malliavin calculus did not exist before Malliavin any more than Itô’s integral existed before Itô. Malliavin has a more human opinion: he is almost in tears when he learns that Itô received the Gauss Prize. Itô dies shortly thereafter, and the world without Itô seems less beautiful to Malliavin. In the same way, for me the world without Malliavin is not quite the same. I miss him.

April 2011 Notices of the AMS 579 ?WHATIS... a G2-Manifold? Spiro Karigiannis

AG2-manifold is a Riemannian manifold whose 2000 Alexei Kovalev found a different construction holonomy group is contained in the exceptional Lie of compact manifolds with G2 holonomy that pro- group G2. In addition to explaining this definition duced several hundred more nonexplicit examples. and describing some of the basic properties of These two are the only known compact construc- G2-manifolds, we will discuss their similarities and tions to date. An excellent survey of G2 geometry differences from Kähler manifolds in general and and some of the compact examples is [3]. from Calabi-Yau manifolds in particular. In terms of Riemannian holonomy, the aspect of The holonomy group of a Riemannian manifold the group G2 that is important is not that it is one is a compact Lie group that in some sense pro- of the five exceptional Lie groups, but rather that vides a global measure of the local curvature of it is the automorphism group of the octonions O, the manifold. If we assume certain nice conditions an eight-dimensional nonassociative real division on the manifold and its metric, then, of the five algebra. The octonions come equipped with a pos- exceptional Lie groups, only G2 can arise as such itive definite inner product, and the span of the a holonomy group. Berger’s classification in the identity element 1 is called the real octonions while 1950s could not rule them out, but it was generally its is called the imaginary believed that such metrics could not exist. Then in octonions Im(O) ≅ R7. This is entirely analogous 1987 Robert Bryant successfully demonstrated the to the quaternions H, except that the nonassocia- existence of local examples. Two years later, Bryant tivity introduces some complications. This analogy and Simon Salamon found the first complete, non- allows us to define a cross product on R7 as follows. compact examples of such metrics, on total spaces Let u, v ∈ R7 ≅ Im(O), and define u × v = Im(uv), of certain vector bundles, using symmetry methods. where uv denotes the octonion product. (In fact Since then physicists have found many examples of the real part of uv is equal to −u, v, just as it is noncompact holonomy G2 metrics with symmetry. for quaternions, where , denotes the Euclidean Finally, in 1994 caused great surprise inner product.) This cross product satisfies the by proving the existence of hundreds of compact following relations: examples. His proof is nonconstructive, using hard analysis involving the existence and uniqueness of u × v = −v × u, solutions of a nonlinear elliptic equation, much as u × v, u = 0, Yau’s solution of the Calabi conjecture gives a non- ||u × v||2 = ||u ∧ v||2, constructive proof of the existence and uniqueness of Calabi-Yau metrics (holonomy SU(m) metrics) on exactly like the cross product on R3 ≅ Im(H). Kähler manifolds satisfying certain conditions. In However, there is a difference. Unlike the cross product in R3, the following expression is not zero: Spiro Karigiannis is assistant professor of mathemat- ics at the University of Waterloo. His email address is u × (v × w) + u, vw − u, wv [email protected]. but is instead a measure of the failure of the An earlier version of this article appeared as an answer associativity: (uv)w − u(vw) ≠ 0. We note that on to a question on MathOverflow and can be found here: R7 http://mathoverflow.net/questions/49357/g-2- we can define a 3-form (a totally skew-symmetric and-geometry. The author thanks Pete L. Clark for trilinear form) using the cross product as follows: suggesting that it might be suitable for a “WHAT IS...?” ϕ(u, v, w) = u × v, w. For reasons we will not article for the Notices of the AMS, and also thanks the address here, this form is called the associative referee for useful suggestions. 3-form.

580 Notices of the AMS Volume 58, Number 4 A seven-dimensional smooth manifold M is said in the G2 case, the metric and the cross product are to admit a G2-structure if there is a reduction of the both determined nonlinearly from ϕ. However, the structure group of its frame bundle from GL(7, R) analogy is not perfect, because one can show that to the group G2, viewed naturally as a subgroup of when ∇ϕ = 0, the Ricci curvature of gϕ necessar- SO(7). This implies that a G2-structure determines ily vanishes. So G2-manifolds are always Ricci-flat! a Riemannian metric and an orientation. In fact, (This is one reason that physicists are interested on a manifold with G2-structure, there exists a in such manifolds—they play a role as “compactifi- “nondegenerate” 3-form ϕ for which, at any point cations” in eleven-dimensional M-theory analogous p on M, there exist local coordinates near p such to the role of Calabi-Yau 3-folds in ten-dimensional that, at p, the form ϕ is exactly the associative string theory. See [1] for a survey of the role of 7 3-form on R discussed above. Moreover, there is G2-manifolds in physics.) Thus in some sense G2- a way to canonically determine both a metric and manifolds are more like Ricci-flat Kähler manifolds, an orientation in a highly nonlinear way from the which are the Calabi-Yau manifolds. 3-form ϕ. Then one can define a cross product × In fact, if we allow the holonomy to be a by using the metric to “raise an index” on ϕ. In proper subgroup of G2, there are many exam- summary, a manifold (M, ϕ) with G2-structure ples of G2-manifolds. For example, the flat torus comes equipped with a metric, cross product, T 7, or the product manifolds T 3 × CY2 or S1 × CY3, 3-form, and orientation that satisfy where CYn is a Calabi-Yau n-fold, have holonomy groups properly contained in G . These are in some ϕ(u, v, w) = u × v, w. 2 sense “trivial” examples because they reduce to This is exactly analogous to the data of an almost lower-dimensional constructions. Manifolds with Hermitian manifold, which comes with a metric, an full holonomy G2 are also called irreducible G2- almost complex structure J, a 2-form ω, and an manifolds, and it is precisely these manifolds orientation that satisfy that Bryant, Bryant–Salamon, Joyce, and Kovalev constructed. ω(u, v) = Ju, v. We are still lacking a “Calabi-Yau type” theorem Essentially, a manifold admits a G2-structure if we that would give necessary and sufficient condi- can identify each of its tangent spaces with the tions for a compact seven-manifold that admits O R7 imaginary octonions Im( ) ≅ in a smoothly G2-structures to admit a G2-structure that is par- varying way, just as an almost Hermitian manifold allel (∇ϕ = 0). Indeed, we don’t even know what is one in which we can identify each of its tangent the conjecture should be. Some topological ob- spaces with Cm (together with its Euclidean inner structions are known, but we are far from knowing product) in a smoothly varying way. Manifolds sufficient conditions. In fact, rather than comparing with G2-structure also admit distinguished classes this problem to the Calabi conjecture, we should of calibrated submanifolds similar to the pseudo- instead compare it to a different problem that holomorphic curves of almost Hermitian manifolds. it resembles more closely, namely, the following. See [2] for more about calibrated submanifolds. Suppose M2n is a compact, smooth, 2n-dimensional For a manifold to admit a G2-structure, necessary manifold that admits almost complex structures. and sufficient conditions are that it be orientable What are necessary and sufficient conditions for and spin, equivalent to the vanishing of its first two M to admit Kähler metrics? We certainly know Stiefel-Whitney classes. Hence there are lots of such many necessary topological conditions, but we are seven-manifolds, just as there are lots of almost nowhere near knowing sufficient conditions. Hermitian manifolds. But the story does not end What makes the Calabi conjecture tractable (al- there. though certainly difficult) is the fact that we already Let (M, ϕ) be a manifold with G2-structure. Since start with a Kähler manifold (holonomy U(m) met- it determines a Riemannian metric gϕ, there is an ric) and try to reduce the holonomy by only one induced Levi-Civita covariant derivative ∇, and one dimension, to SU(m). Then the ∂∂¯-lemma in Kähler can ask if ∇ϕ = 0. If this is the case, (M, ϕ) geometry allows us to reduce the Calabi conjecture is called a G2-manifold, and one can show that to an (albeit fully nonlinear) elliptic PDE for a sin- the Riemannian holonomy of gϕ is contained in gle scalar function. Any analogous “conjecture” in the group G2 ⊂ SO(7). Finding such “parallel” G2- either the Kähler or the G2 cases would have to structures is very hard, because one must solve involve a system of PDEs, which are much more a fully nonlinear partial differential equation for difficult to deal with. the unknown 3-form ϕ. The G2-manifolds are in some ways analogous to Kähler manifolds, which Further Reading are exactly those almost Hermitian manifolds that [1] S. Gukov, M-theory on manifolds with exceptional satisfy ∇ω = 0. Kähler manifolds are much easier holonomy, Fortschr. Phys. 51 (2003), 719–731. to find, partly because the metric g and the almost [2] R. Harvey and H. B. Lawson, Calibrated geometries, complex structure J on an almost Hermitian mani- Acta Math. 148 (1982), 47–157. fold are essentially independent (they just have to [3] D. Joyce, Compact Manifolds with Special Holonomy, satisfy a mild condition of compatibility), whereas Oxford University Press, 2000.

April 2011 Notices of the AMS 581 Book Review Nonsense on Stilts Reviewed by Olle Häggström

Nonsense on Stilts: How to Tell Science from makes it unreasonable to apply identical criteria Bunk to all sciences, but there is more. Pigliucci stresses, quite rightly, the need for a general philosophy of Massimo Pigliucci science to recognize that some sciences, notably Press, May 2010 history and parts of evolutionary biology, aim to US$20.00, 336 pages map sequences of events in the past rather than ISBN-13: 978-02266-678-67 to formulate general laws applicable today and in the future. The task that biologist-turned-philosopher of sci- Nonsense on Stilts is reader-friendly and enter- ence Massimo Pigliucci sets himself in his book taining. Part of the reason for this is Pigliucci’s Nonsense on Stilts: How to Tell Science from Bunk preference for concrete examples as opposed to is very ambitious. With a broad audience of inter- losing the reader in abstract theory. Many authors ested laymen in mind, he aspires, as the subtitle would probably have begun with a chapter or two indicates, to show how to tell science from bunk— on the history of science; Pigliucci instead post- the so-called demarcation problem. And he is pones this deep into the book until the reader, not content with just a theoretical discussion (al- motivated by examples, realizes the need for a though the book offers some of that, too) but historical perspective. Up to that point he has wants to equip the layman with the intellectual discussed not only sciences such as sociobiology, tools to tell one from the other in, for instance, string theory, and the SETI search for extraterres- media reports on scientific issues. trial intelligence, which have all been the target A straightforward “if and only if” criterion for of controversy of one kind or another, but also what is science is too much to hope for, no matter areas that fall squarely in the pseudoscience cate- how much some fans of Karl Popper may think gory, in which his two favorite examples are those otherwise. In his first chapter, Pigliucci discusses that also attract the largest amount of attention what he calls “hard” versus “soft” sciences and in public debate: intelligent design and climate demonstrates the implausibility of formulating change denialism. In a later chapter he treats the simple criteria that work across all fields. Hard postmodern current in the sociology of science sciences, in his view, are those whose objects un- and related areas, whose hollowness was so mas- der study are either simple (relatively speaking) terfully exposed by physicist Alan Sokal in his or admit reductionist analysis via division into 1996 practical joke that has become known as the simple constituents; typical examples are physics Sokal Hoax.1 and chemistry. At the soft science end of the A common theme (albeit with various twists) in spectrum, we find, e.g., sociology, in which the Pigliucci’s discussions about sociobiology, string fundamental constituents of the systems under theory, and SETI is the need to relax, at least study are human beings—incomparably more com- in a short time perspective, the strict Popperian plicated than the atoms or elementary particles orthodoxy about falsifiability. In the first two of physics and chemistry. This difference alone cases I am happy to embrace his position, but as

Olle Häggström is professor of mathematics at Chalmers 1For Sokal’s own reflections on the incident, see A. Sokal, University of Technology in Gothenburg, Sweden. His Beyond the Hoax: Science, Philosophy and Culture, email address is [email protected]. Oxford University Press, 2008.

582 Notices of the AMS Volume 58, Number 4 regards the third I disagree. Neutrally described, warming. On the other hand, it is unrealistic that the purpose of SETI is to shed light on which of the man-on-the-street (or even the typical mathe- the two hypotheses matics professor) should be acquainted with the E = {extraterrestrial intelligence exists} entire chain of scientific arguments behind the conclusions, all the way down to the quantum and physics of the absorption spectrum of a CO2 mol- Ec = {extraterrestrial intelligence does not exist} ecule. So in practice there’s no way around the problem of judging whom to trust. On the other is actually true. To plug the project into falsifica- hand we certainly do not want to stop thinking for tionist formalism, we need to break the symmetry ourselves. It’s a tricky balance. and take either E or Ec to be the null hypothesis, The broad range of the contents of Nonsense on i.e., the hypothesis we should then try to falsify. If, like Pigliucci, we take E to be the null hypoth- Stilts is impressive but also tends to make the book esis, then we would be hard pressed to think of somewhat patchy (a patchiness that is inherited by a way (at least with current technology) that it the present review). Another consequence of the might be falsified. But if instead we take Ec as the broad scope is that the author needs to have done null hypothesis, then SETI will fit beautifully as a a vast amount of homework. As far as I can tell, textbook example of falsificationism in practice. Pigliucci has mostly succeeded in this—with the Pigliucci’s language is quite relaxed. Every now occasional exception. Since I share his know-it-all and then, he falls into sarcasm when describing disposition, I cannot resist listing some of the work that needs to be categorized as either weak cases in which he apparently doesn’t quite know science or even pseudoscience. Here’s a typical what he is talking about or where his arguments example: in a discussion (page 159) about the fail to convince. (If nothing else, I hope in this way (lack of) credibility of the conservative think-tank to instill the reader with the aforementioned cozy American Enterprise Institute on issues of climate feeling.) change, he asks, after mentioning some compro- • In his chapter on hard versus soft sciences, 2 mising facts, whether he “needs to say more to Pigliucci takes a classical paper by Platt as a push your baloney detector all the way up to red starting point for a discussion about why the alert”. Such passages add to the entertainment hard science of physics has shown greater and value of the book, but also have the potential more manifest advances than the comparatively downside of not putting the reader in the opti- softer ecology. He rejects, somewhat indignantly, mal state of sharpened senses for digestion of the idea that physicists might be more gifted than new thoughts and subtle distinctions. Rather, it ecologists (page 9). Shortly afterward, he considers tends to produce a cozy feeling: think how clever as perfectly plausible the idea that part of the we are, the author and I, especially compared to explanation may lie in the fact that physics enjoys these simpletons! A related quality of the book is higher prestige than ecology among American high Pigliucci’s straightforward, sometimes blunt, way school nerds, giving it a comparative advantage in of expressing his opinion on controversial issues. the recruitment of the best young talents. Viewed Again, this makes for enjoyable and interesting separately, each of these judgments by Pigliucci reading, although it is not always trivial to distin- makes some sense, but his failure to note how guish matters of fact from the author’s personal they contradict each other is a strange lapse. opinions. • In a discussion about statistical hypothesis How, then, does Pigliucci succeed in his ambi- testing, Pigliucci writes (page 79) that “one way to tion to solve the demarcation problem? He does understand what a p-value says is to think of it shed some useful light on it, but the problem as the probability (given certain assumptions) that of how a nonexpert should go about distinguish- the observed data are due to chance, as opposed ing science from bunk seems to be simply too to being the results of a nonrandom phenome- difficult to admit a clear-cut answer. When Pigli- non.” Here he commits the well-known fallacy of ucci summarizes his advice toward the end of the transposed conditional, i.e., he confuses the the book, it is striking how much emphasis he probability of the data given the null hypothesis puts on judging the credibility of the purported with the probability of the null hypothesis given scientist or the messenger, as opposed to judging the data.3 the quality of the arguments themselves. In other words, he advocates a large element of appeal to 2J. Platt, Strong Inference, Science 146 (1964), Oct. 16, authority, which may seem unsatisfactory. This, 347–353, http://ecoplexity.org/files/Platt. however, is probably unavoidable, especially in pdf. complex issues such as climate change. On one 3See, for instance, Cohen, J., The Earth is round (p. 5), hand, it is obviously important that citizens have American Psychologist 49 (1994), no. 12, 997–1003, an idea about where science stands concerning the http://www.ics.uci.edu/~sternh/courses/210/ link between greenhouse gas emissions and global cohen94_pval.pdf.

April 2011 Notices of the AMS 583 • On the topic of global warming, Pigliucci irreproachable—or so it may seem. After having discusses natural climate variations in the past explained the correspondence theory (pages 236– (page 136). Among climate change denialists, these 237), Pigliucci declares it untenable. What, then, are sometimes viewed as a decisive argument is the alternative? I do not doubt for one second against the idea of an ongoing anthropogenic that Pigliucci is a better philosopher of science global warming. Pigliucci notes, correctly, that the than I am, but precisely for this reason he should existence of natural climate variation is well known have been able to do better than merely declaring and uncontroversial in climate science. But when that “in philosophical circles, the correspondence he writes that “Greenland—which today is largely theory of truth has been largely superseded by covered by ice—was given that name because it more sophisticated epistemological positions”. was a lush land during the so-called Medieval I am in fact a bit confused by Pigliucci’s ar- Warming Period”, he is mistaken in his implication gument for declaring the correspondence theory that the Greenland ice cover was absent during the bankrupt. He holds that since science will never MWP (which, according to most definitions, took provide definite answers about what the world is place between AD 950 and 1250). The (relatively) like, we can never be sure about the truth that hospitable climate that is sometimes referred to the correspondence theory speaks of. Well, yes, concerns parts of Greenland’s coastal areas—its but if we agree that we can never know for sure inland was, like today, covered by ice. whether it is raining (for instance, we might be • Again on the topic of global warming, Pigliucci hallucinating), why is it so much worse that we explains the greenhouse effect by first outlining can never know whether it is true that it is rain- how a greenhouse works and then stating that the ing? But apparently it is, to the extent that the greenhouse effect of atmospheric carbon dioxide correspondence theory must be abandoned. is “the exact same phenomenon at the scale of the Furthermore, since Pigliucci neglects telling whole planet” (page 136). But here he is confused us what the “more sophisticated epistemological about how an actual greenhouse works. Its ceiling positions” are, the reader is left wondering how serves primarily to prevent not outgoing radiation these may come to grips with the problem that but convection. The term “greenhouse effect” is makes the correspondence theory so unacceptable. thus a bit of a misnomer (and Pigliucci is not the I would think (perhaps naively) that a theory of first to be confused by it), but it is so established truth that allows us to conclude that it is true that that we should simply get used to it, just like how it is raining, without knowing that it is actually we accept without complaints the term “sunrise” raining, violates the very notion of truth. These despite the phenomenon being caused by the are questions that call for answers, but Pigliucci Earth’s movement, not the sun’s. doesn’t even offer a hint. • Before Galileo, it was generally held that grav- Apart from these lapses and a few more, I ity works in such a way that heavy objects fall find Nonsense on Stilts to be a fairly convincing faster than lighter ones. Galileo realized the un- book. I would expect the average reader of the tenability of this view using the following thought Notices to be about equally convinced (and I do experiment. Imagine two rocks, one heavier than think that even the purest of pure mathematicians the other. Now join the two rocks by a rope, and who rarely or never interacts professionally with drop them. The pair of rocks will now (i) fall faster any applied researchers still has a lot to gain than the light rock would have done on its own, from occasionally paying some attention to the because the latter will be pulled downward, via philosophy of science). On the other hand, the book the rope, by the heavy rock. Similarly, the pair does have a marked tendency toward preaching to will (ii) fall slower than the heavy rock on its own, the choir. It is doubtful whether it would do much because the heavy rock will be pulled upwards, via to convince a reader who doesn’t already share the the rope, by the light rock. On the other hand, the author’s proscience stance on issues such as global pair is obviously heavier than the heavy rock on its warming or the evolution-creationism struggle. own, so by the old theory it will (iii) fall faster than the heavy rock on its own. Conclusions (ii) and (iii) contradict each other, so the old theory must be wrong. So far according to Galileo. Pigliucci tries to recount this argument (page 220), but fails to mention (iii), and claims incorrectly that (i) and (ii) contradict each other. • One issue that is sometimes discussed in the philosophy of science is what we mean by truth. According to the so-called correspondence theory of truth, a statement P is true if and only if P. For instance, the statement “it is raining” is true if and only if it is raining. Simple but

584 Notices of the AMS Volume 58, Number 4 Book Review To Complexity and Beyond! Review of Complexity: A Guided Tour

Reviewed by Dan Rockmore

Complexity: A Guided Tour time scales but also Melanie Mitchell across scales as we Oxford University Press, 2009 attempt to relate the US$29.95, 368 pages structures (and un- ISBN-13: 978-0195124415 derstandings) from one scale to the If the interests of the funding agencies provide any next, only to find sort of measuring stick for the coming of age of that they do not a discipline, then it would seem that the study of scale themselves, “complex systems” has recently become must-see a phenomenon science. You would be hard pressed to find in any often referred to as collection of recent requests for proposals an or- “emergence”. The ganization that is not looking to explicitly leverage standard straw man the framework of complex systems, to engender a of comparison here deeper understanding of its basic interests, be they is particle physics. health, defense, communications, ecology, econom- There, presumably, ics, markets, society, and even culture. the basic behavior That these phenomena all might admit a uni- of matter is well explained by molecular under- fied methodological approach suggests that they standings, which in turn is supported by atomic share some basic characteristics. One of the first studies, and so on; one turtle happily perched people to see these disciplinary trees as a forest upon the next. In comparison, understandings of was the polymath Herbert Simon (1916–2001). neuron-to-neuron activity via the Hodgkin-Huxley Simon worked at the highest levels in the economic, equations does not (yet?) lead us step-by-step (or social, and computer sciences. He was a founder of turtle-by-turtle) to any understanding of mind; a artificial intelligence, winner of the simple formulation of the interactions between one and of the Nobel Prize in Economics, and a leading buyer and one seller does not allow us to predict figure in modern psychology. the dynamics of the New York Stock Exchange. In his eloquent “The Organization of Complex Nevertheless, in each of these (and other) instances, Systems” [1] (the paper that named the field), Simon structure does exist at the macro and micro levels, points out that what these and other phenomena as well as many levels in between. How? Why? In have in common is a “Darwinian nature”, that is to Simon’s view the explanation is partly that evolved say, an evolutionary character given by a partially systems are generally only “weakly decomposable” ordered hierarchical structure whose components and that a main goal of studying such a diversity of interact on widely varying time scales and in whose (complex) systems is the “formulation of laws that development we see selection, adaptation, muta- express the invariant relations between successive tion, hysteresis, and contingency. Perhaps most levels in the hierarchy” which ultimately require a notably, we find a good deal of “nonlinearity”, not “Mendelian” explanation. only in particular measurable qualities at specific It would be something of an overstatement to sug- Dan Rockmore is John C. Kemeny Parents Professor and gest that the single paper by Simon launched a thou- chair of mathematics and professor of computer science sand others, but in the time between then and now at Dartmouth College. His email address is rockmore@ a field of “complex systems” has indeed emerged, cs.dartmouth.edu. realizing the idea that the general mechanisms

APRIL 2011 NOTICES OF THE AMS 585 of evolution as expressed in a wide range of Mitchell remains a member of SFI’s (geographically phenomena could be profitably addressed in distributed) external faculty (as am I), and we are a rigorous and coherent way, as an inherently also colleagues on SFI’s Science Steering Commit- a-disciplinary (in the sense of classical disciplinary tee. She knows many of the big players in the field, categorizations) activity. both past and present, and the tone of the book is Mitch Waldrop’s Complexity: The Emerging a friendly and personal one. This gives the book Science at the Edge of Order and Chaos [2] was something of a Zelig-like feel to it, but appropriate the book that put complex systems on the map. to one who has been working in the field from a Waldrop tells his story through the tale of the birth time close to its recent origins. of the , arguably the preeminent The sheer range of phenomena that fall under research center of complex systems studies in the the complex systems heading, each with its own world, but in 1984 little more than a somewhat jargon and important details, might give pause to inchoate thought shared among several senior sci- even the most seasoned expositor. Mitchell’s well- entists at Los Alamos National Laboratory. All of written book (already a winner of the 2010 Phi Beta these researchers were unhappy with LANL’s trend Kappa Science Book Award and on the long list toward compartmentalized and specialized science for the Royal Society’s 2010 book prize) meets the and thus both implicitly and explicitly concerned challenge by entwining concepts and topical “case that the understandings of the micro were not get- studies” through a five-part structure and taking ting at the big questions and problems of the day. pains to connect the dots as often as is possible. Waldrop gives us a good sense of the emergence Of the former, we find expositions of what I would of the subject and the creation of SFI as something call the “old” complex systems (such as some of of a manifestation of a scientific and technologi- those mentioned above) as well as the new. Of the cal zeitgeist. We learn of an initial small meeting old, the treatments here are generally pithier and between economists and physicists (motivated by clearer—a nice example of a genetic algorithm that era’s financial crisis), Brian Arthur and the for search is perhaps the best I’ve ever read. The discovery of the paradoxical “increasing returns” chestnuts of complexity are augmented by useful and the El Farol problem, John Holland’s invention overviews of more recent work. I enjoyed the updat- of evolutionary computing and the early excitement ing of cellular automata by a good explanation of about algorithmic and computational approaches the “particle” approach to studying CAs. The new to learning. Stuart Kauffman’s invention of the science of networks is given significant room, mak- N-K network, the first model of a genetic circuit, ing up the theme of the penultimate section. In this also plays a starring role, as do game theory and chapter and others in the book, much is made—and Robert Axelrod’s “tournament” for playing iterated rightfully so—of the way in which massive data sets Prisoner’s Dilemma that revealed the optimality of all sorts of phenomena have truly been a game- of the “Tit-for-Tat” strategy (proposed by Anatol changer in complexity analysis. Network theory was Rapoport). We see how the new research freedom before we had in hand large real-world (and excitement) afforded by increased access to networks to explore. Scaling is addressed, both in computing, in part enabled by the beginnings of and of itself, as well as in the context of the recent personal computing, catalyzed the work, helping work of Geoffrey West, Jim Brown, and Brian En- to make possible early investigations of artificial quist (and others) on the mechanistic explanations life and cellular automata. Waldrop’s book is a true for allometric scaling in terms of space-filling and good popular science read, and it is fair to say resource delivery systems. In these discussions as that it helped to create much of the early inter- well as others, I appreciated the even-handedness est in complex systems (along with Roger Lewin’s with which Mitchell presented controversial work. nearly simultaneous Complexity: Life at the Edge She brings the scientific debate out for display, of Chaos [3]). and the reader gets the sense of being drawn into Some twenty years later, the time is ripe to the conversation and unconsciously or consciously pay another visit to this place and see what kinds spurred to engage with the results and the science. of modernizations have been effected. To this In places such as these I felt less like a receiver of end, our view of complex systems science gets a information (as I did in the lengthy discussions of long-awaited updating and deepening in Melanie genetics) than a participant in a conversation and Mitchell’s Complexity: A Guided Tour. Mitchell is a was all the more interested in it for that experience. complex systems insider, well known and respected The “conceptual” parts are a welcome addition in the complexity research community with particu- and, to my taste, perhaps the most interesting lar expertise in the study of artificial intelligence, aspect of the book. Information and information cellular automata, and evolutionary computing. processing, as well as computation, provide grand Mitchell was a student of Douglas Hofstadter at the themes in the book (and are clearly Mitchell’s strong , where she learned genetic suit), as does evolution. A discourse on the various algorithms from John Holland. She then moved definitions of “complexity” and the difficulties of to the Santa Fe Institute, where she was a faculty finding a good definition that seems to work for member for several years, then up the road to living phenomena was fun. The art of modeling and Los Alamos and back to SFI. She is now professor the power (and potential pitfalls) also receive their of computer science at Portland State University. due. The threading of these grand ideas through

586 NOTICES OF THE AMS VOLUME 58, NUMBER 4 the topical studies gives one a good sense of the truly changed the field remains fertile ground connectivity of the various phenomena and thus the for the development of more powerful statistical sense that there is a subject here to be studied. It methods to understand the “patterns of activity” also helps bring out the inherent interdisciplinar- in complex phenomena. To this end, for example, ity of the subject. A closing chapter on “The Past one can point to the new field of computational and Future of the Sciences of Complexity” gives topology, a natural (mutated!) descendant of al- a nice historical contextualization of all that has gebraic topology, itself born of the study of that preceded it and also considers some of the objec- ever-intriguing complex system, the solar system. tions raised by those who don’t see complex sys- Turing’s interest in the abstraction of computation tems as a subject. generally, in the inanimate and animate, gave birth In Mitchell’s own words, the book is intended to automata theory, but more recently, the lambda for “everyone”, scientist and interested general calculus has been proposed as a more natural reader alike, and in this she has presented herself framework for understanding the “logic” of the with a significant challenge. Given such a goal, a cell [10]. Perhaps the notion of information and its certain amount of unevenness is to be expected, processing as instantiated across living systems with the hills, plains, and valleys defined in the (e.g., cells, ecologies, societies, markets) requires eyes and minds of the beholders. For this particular a different kind of understanding or abstraction. beholder, mathematician by training, and one who Indeed, in her concluding and thought-provoking has become increasingly more applied than pure chapter on the past and future of complex sys- and developed an interest in complex systems, I tems studies, Mitchell suggests that a whole new couldn’t keep myself from wishing that even more mathematics might be necessary to perform the of the “new” might be found in this book. We par- sought-after syntheses of complex systems studies. ticularly note Mitchell’s clarity of exposition: that That said, these are in some sense quibbles, Prisoner’s Dilemma and Tit-for-Tat would quickly and there is assuredly something in this book for bring us to the exciting new work in algorithmic everyone. With its generally clear writing and fine [4] or evolutionary game theory [5] or policing and bibliography, for the uninitiated, Melanie Mitchell’s conflict [6] and more discussion of the extraordi- Complexity is a great way to take a first voyage to nary new possibilities for social science enabled by the complex system of complex systems. We learn the Internet [7]; that the story of Lorenz’s discovery of its history, visit some of the modern hotspots, of chaos might bring us to more recent work in and, even more, are given a sense of the connec- collective dynamics related to both flocking and tions between these different times and places and neuroscience [8] instead of artificial life and CAs are invited to speculate about how it will all look (indeed, there is a relatively heavy emphasis on in a few years. It left me, and will surely leave any aspects of “evolutionary computing”), which has open-minded traveler, eager to continue to learn had a multitude of exposition; that we might now about this relatively unexplored and fertile world. find some space for the very exciting new work Ultimately, that’s precisely the hallmark of an ex- cellent tour guide. being done on understanding a general notion of metabolism and its implications for new theories of References the origin of life [9]. It was also somewhat surpris- H. A. Simon ing to find almost no discussion of econophysics, [1] , The organization of complex sys- tems, in H. H. Pattee (ed.), Hierarchy Theory, but some might say that in this market, that is a G. Braziller, New York, 1973, pp. 3–27. selling point…. [2] M. Waldrop, Complexity: The Emerging Science at Another part of this longing to see the new in the Edge of Order and Chaos, Simon and Schuster, this well-written book is my own personal hope New York, 1992. that more of the mathematics community might be- [3] R. Lewin, Complexity, Life at the , Mac- come engaged in thinking about complex systems. Millan Publishing Co., New York, 1992. Historically, the subject—even before it was a sub- [4] N. Nisan, T. Roughgarden, E. Tardos, and V. V. ject—has been fertile ground for mathematicians, Vazirani, Algorithmic Game Theory, Cambridge and that is still true. Of course, deep connections University Press, 2007. already exist to the world of nonlinear dynamics, in [5] K. Sigmund, Evolutionary Games and Population Dy- both pure and applied contexts. To extend results namics, Cambridge University Press, 1998. from smooth dynamics to the strange topologies [6] J. C. Flack, M. C. Girvan, F. B. M. de Waal, and D. C. of real life networks—or at least large classes of Krakauer, Policing stabilizes construction of social them—is of great importance as people wrestle niches in primates, Nature 439 (2006), 426–429. David Lazer to understand dynamics on networks. Progress [7] et al., Perspective: Social science: Com- here also has the potential to add some significant putational social science, Science 323 (2009), 721–732. [8] I. D. Couzin, Collective minds, Nature 445 (2007), understanding to the agent-based models that are (7129) 715–715. often used to simulate complicated phenomena. In [9] J. Trefil, H. Morowitz, and E. Smith, The origin of this direction, rigorous results in cellular automata life, American Scientist 97 (2009), 206. are still of interest, while a great challenge is the [10] W. Fontana and L. W. Buss, What would be con- creation of models that produce the multiple time- served if “the tape were played twice”?, Proc. Natl. scales of dynamics recognized by Simon decades Acad. Sci. USA 91 (1994), 757–761. ago. The oft-mentioned data explosion that has

APRIL 2011 NOTICES OF THE AMS 587 DOCEAMUS doceamus . . . let us teach

Making Mathematics Work for Minorities: A Challenge for Our Profession; A Service to Our Nation

Manuel P. Berriozábal

During the 1980s and the 1990s, both the Ameri- On various occasions, I have vigorously dis- can Mathematical Society and the Mathematical agreed with some of my colleagues, minority and Association of America expressed their deep con- majority, about the efficacy of some so-called cern about the underrepresentation of American reforms. minorities, specifically American Indian, African At this point, I wish to cite several pieces of American, and Hispanics, in the mathematics pro- research that share my point of view about the fession and more generally in the STEM (Science, effective educational preparation of students at Technology, Engineering, and Mathematics) pro- the precollege level, particularly minority students fessions. These minority groups constitute about and/or students from low-income families. In 1996 30% of our population. To reach these groups, our Bonnie Grossen, an editor for the Journal of Effec- educational establishment has implemented teach- tive School Practices, wrote an overview entitled ing reforms at all levels: elementary, middle school, “The Story Behind Project Follow Through” [5] high school, and collegiate. Yet recent manpower for a series of research articles on Project Follow studies conducted by the Congressional Research Through. Beginning in 1967 and over the next ten Service and the National Science Foundation [1], years, twenty-two sponsors worked with over 180 [2], [3], [4] reveal that the underrepresentation is sites in an effort to break the cycle of poverty still there. In short, these reforms offer us little to by improving education. Project Follow Through brag about. sought to evaluate the efforts by both economi- In my opinion, a major problem is that teaching cally and academically impoverished schools to methodology and content matter that in previ- achieve a level of education comparable with ous decades successfully prepared our majority mainstream America during that time. This study population as future citizens and STEM profes- was conducted through 1995. The study found sionals were considered by some members of our that minority students benefited from explicitly educational establishment as inappropriate for organized and intensive academics in a teacher- our minority population if we wanted to produce led classroom, as exemplified by the approach some type of parity. called Direct Instruction. In 2001 Clifford Adelman prepared a U.S. Department of Education report Manuel P. Berriozábal is professor of mathematics at the entitled Answers in the Toolbox [6]. One of his University of Texas, San Antonio. His email address is findings was that minority students who enrolled [email protected]. in very rigorous academic high school courses

588 NOTICES OF THE AMS VOLUME 58, NUMBER 4 completed college at a higher rate than their white industry, foundations, local school districts, and counterparts. participating colleges. Since 1979 over 14,000 In 1979, with a grant from the U.S. Department students, mostly Hispanic, have completed at least of Energy and the assistance of Dr. Parker Lamb of one summer in San Antonio PREP. the University of Texas at Austin College of Engi- My challenge to our profession is that we neering, I started the program called the San An- encourage the mathematics departments of our tonio Pre-freshman Engineering Program (SAPREP community and senior colleges to reach out to or PREP) at the University of Texas at San Antonio. the local communities that they serve and to begin In 1976, when I arrived at UTSA, I had observed organized, concentrated efforts, especially in the that only a small number of students in my math- summertime, to identify young, able, and hard- ematics classes were minority, quite surprising because San Antonio is a city nearly 70% minority, working middle school students, particularly local mostly Mexican American. PREP started out as an minority students, and to involve them in struc- eight-week summer mathematics-based academic tured programs such as PREP. We already have enrichment program with the goal of identifying programs such as the high-quality AMS-supported hardworking and able high school students—ini- Young Scholars programs, but they are few and tially, high school juniors and seniors—who indi- serve primarily high school students. I am propos- cated an interest in science and engineering. The ing that outreach begin earlier and that a pipeline program emphasized the development of abstract from middle school to college and even graduate reasoning skills and problem-solving skills in a school be developed especially for students who highly structured teacher-led classroom setting. will ultimately enter college and pursue STEM ma- In the first summer of this eventual three-summer jors at both the undergraduate and graduate lev- program, the major mathematics component of els. I also envision this effort being accomplished the program was basic and its through a consortium of colleges and universities application to theorem proving. Although the pro- with primarily local constituencies working to- gram was open to all students in the San Antonio gether with those with national constituencies. In area, special effort was focused on the recruitment of minority students. My ultimate goal was to es- order to recruit for these programs able students tablish a summer experience for beginning college from low-income families, from which many of our students modeled after the summer mathematics minority students come, it will be important to program of my teacher and mentor, Arnold Ross. offer these students complete financial assistance, It did not turn out that way. I quickly realized including stipends for high school students who that if I was going to have an impact on a large would otherwise need to work during the summer number of local students at an age to make a to assist their families. difference, I needed to reach down to younger What a significant contribution our profes- students who might benefit from the program. sion could make if we prepared our future STEM Arnold himself advised me to continue serving professional workforce to reflect the face of our the younger students rather than emulating his nation. More information about PREP can be found program for older students. By 1984 most of the at http://www.prep-usa.org. program participants were minority (and female) middle school students. References Every summer the PREP office has conducted, [1] Deborah D. Stine and Christine M. Matthews,The and still does, an annual follow-up of former par- U.S. Science and Technology Workforce, Congressional ticipants that has revealed that most college-age Research Service RL 34539, page 10. former participants are indeed attending college [2] Minority Share of S&E and non-S&E Bachelor’s Degrees: and graduating from college, with at least 40% 1990–1996, National Science Foundation, Division of majoring in STEM areas. Today, San Antonio PREP Science Resources Statistics. has been replicated in thirty-five Texas community [3] Minority Share of S&E and non-S&E Master’s Degrees: and senior college campuses and serves nearly 1990–2006, National Science Foundation, Division of 3,500 students annually, most of whom are minor- Science Resources Statistics. ity. Some TexPREP sites have added components [4] Minority Share of S&E and non-S&E Doctoral Degrees to beyond the third year, which include students’ U.S. Citizens: 1990–2006, National Science Foundation, taking college-level STEM courses while still in high Division of Science Resource Statistics. school. Since many minority participants come [5] Bonnie Grossen, The story behind Project Follow from low-income families, some PREP sites offer Through, Effective School Practices 15, no. 1, 1995– students free lunches through the U.S. Department 1996. of Agriculture Summer Food Service Program and [6] Clifford Adelman, Answers in the Toolbox, United provide free transportation. Other program ex- States Department of Education, Office of Educa- penses are paid by local, state, and national public tional Research and Improvement, Washington, pages and private entities: the State of Texas, private 3–7, 2001.

APRIL 2011 NOTICES OF THE AMS 589 Mathematicians and Poets

Cai Tianxin, translated by Robert Berold and Gu Ye

Mathematicians and poets exist in our world as true, yet it stands there basically undeniable. Math- uncanny prophets. The difference between them ematicians work to discover, while poets work to is that poets are thought to be arrogant because create. The painter Degas occasionally wrote son- they tend to be proud and lonely by nature, while nets and once complained to the poet Mallarmé. mathematicians are thought to be unapproachable He said that he had many ideas, in fact too many; because they exist on a transcendent plane. Thus he found it difficult to write. Mallarmé replied, in art and literary circles poets are often consid- “poems are made not with ideas but with words.” ered to be socially inferior to novelists in the same On the other hand, mathematicians work mainly on way that mathematicians are considered socially concepts, combining concepts of the same kind. In inferior to physicists in scientific and technological other words, mathematicians think in an abstract associations. But these things are only superficial. way, while poets think in a concrete way. But again “I’m a failed poet,” the novelist William Faulkner this is not always the case. said humbly in his later years. “Maybe every novel- Both mathematics and poetry are products of ist wants to write poetry first, finds he can’t and imagination. For a pure mathematician, his or then tries the short story, which is the most de- her materials are like lacework, leaves on a tree, manding form after poetry. And failing at that, only a patch of grass, or the light and shade on a per- then does he take up novel writing.” Physicists, by son’s face. In other words, “inspiration”, which comparison, are not so modest. Nevertheless, for Plato denounced as “a mania of poets”, is equally a physicist every increase in knowledge of phys- important to mathematicians. For example, Goethe ics is always guided in two ways, by mathematical fancied that he saw a flash of light when he heard intuition and empirical observation. The art of of his friend Jerusalem’s suicide. He immediately physics is to design experiments in order to derive came up with the outline of The Sorrows of Young the laws of nature. In this process mathematical Werther. He recalled that he “seemed to have writ- intuition is indispensable. In fact, it is easy for ten the book unconsciously”. Another example: mathematicians to switch to studying physics, Gauss, “the prince of mathematics”, wrote to tell a computer science, or economics, just as it is for friend, after solving a problem (symbols of Gauss- poets to turn to writing novels, essays, or plays. ian summation) that had been bothering him for Of course, there are exceptions. years, “Finally, two days ago, I succeeded—not on Mathematics is usually seen as the diametric account of my hard efforts, but by the grace of the opposite of poetry, although there are exceptions Lord. Like a sudden flash of lightning, the riddle here, too. Although the opposition is not always was solved. I am unable to say what the conducting Cai Tianxin is professor of mathematics at Zhejiang thread was that connected what I previously knew University. His email address is [email protected] or with what made my success possible.” [email protected]. Mathematics often appears to be connected to Robert Berold is a South African writer and teaches cre- and interactive with astronomy, physics, and other ative writing at Rhodes University in Grahamstown. His branches of natural science, but it is a completely email address [email protected]. self-referential and vast field of knowledge with Gu Ye is a graduate student majoring in English lit- a reality more enduring than other sciences. It is erature at Zhejiang University. Her email address is like a true language, which not only records and [email protected]. expresses ideas and the process of thinking but

590 NOTICES OF THE AMS VOLUME 58, NUMBER 4 also creates itself through poets and writers. It The language of poets is renowned for its con- could be said that mathematics and poetry are the ciseness. Ezra Pound is praised as a master of the freest intellectual activities of human beings. The concise; no one seems to do better than he in this Hungarian mathematician Paul Turàn maintained regard. But the language of mathematicians is also that “Our mathematics is a strong fortress.” His noted for its conciseness. The British writer Jerome words correspond to Faulkner’s “People will never K. Jerome gave an example, as follows: be destroyed as long as they yearn for freedom,” When a twelfth-century youth fell in love referring to creative writing. he did not take three paces backward, Through years of study and practice, I have gaze into her eyes, and tell her she was come to believe that the process of mathematical too beautiful to live. And if, when he got research is more or less an exercise or an appre- out, he met a man and broke his head— ciation of intelligence. This is perhaps one of the the other man’s head, I mean—then that main reasons for its great charm. I fully understand proved that his—the first fellow’s—girl what the philosopher George Santayana said in his was a pretty girl. But if the other fellow later years: “If my teachers had begun by telling me broke his head—not his own, you know, that mathematics was pure play with presupposi- but the other fellow’s—the other fellow tions, and wholly in the air, I might have become a to the second fellow, that is… good mathematician, because I am happy enough in the realm of essence.” Of course, I cannot rule As he goes on to say, this interminable para- out the possibility that a great thinker can yield to graph would be very succinct if expressed in the intellectual fashions of his times as a man or mathematical symbols, although it would be less a woman can do to fashions in dress. amusing: Compared with other disciplines, mathematics If A broke B’s head, then A’s girl was is often an undertaking for the younger. The Fields a pretty girl; but if B broke A’s head, Medal, the most renowned mathematical prize, then A’s girl wasn’t a pretty girl, but goes only to mathematicians under forty. Riemann B’s girl was. died at forty, Pascal at thirty-nine, Ramanujan at thirty-three, Eisenstein at twenty-nine, Abel at The language of mathematicians is universal. Goethe joked that mathematicians are like the twenty-seven, and Galois at twenty; by the time French, who can translate whatever you say into they died they had all left their deep traces on the their own language and turn it immediately into history of mathematics. Some mathematicians, something totally new. We have been taught that such as Newton and Gauss, lived long lives, but a branch of science is truly developed only when they completed their major work in their youth. it is able to make use of mathematics. In the same Of course, there are exceptions here, too. way, poetry is a common key factor of all the arts. Likewise we can draw up a long list of poets It can be said that every work of art needs “poetic who died young: Pushkin, Lorca, and Apollinaire flavor”. Mozart had a reputation as “the poet of died at thirty-eight, Rimbaud at thirty-seven, Wilde music” and Chopin as “the poet of the piano”. at thirty-four, Mayakovsky at thirty-two, Plath at It’s not difficult to imagine the striking symmetry thirty-one, Shelley and Yesenin at thirty, Novalis at between a beautiful mathematical formula in a 1 twenty-nine, Keats and Petofi at twenty-six, and scientific paper and several brilliant lines of poetry Lautréamont at twenty-four. Whereas if we look at in an essay or a speech. painting, Gauguin, Rousseau, and Kandinsky began Now let’s come back to the proposition stated their artistic careers after they turned thirty. Thus, at the beginning of this essay. Freud said, “Ev- more often than other servants of creation, poets erywhere I go, I find that a poet has been there and mathematicians tend to burn up the flower before me.” This remark was taken up by Breton, of their talent in the midst of their youth. Poets the leader of surrealism, as a golden rule. Nova- may destroy the shapes common to the forms of lis asserted, “Poetry is very similar to prophecy their predecessors in order to renew the form and in its significance. Generally, poems are like the language; mathematicians may be, by the nature intuitions of prophets. Poets—prophets—reveal of their industry, more prone to continuity. Again, the secrets of a strange and wondrous world with there are exceptions. magic lines and images.” Therefore a poet of in- tegrity will inevitably violate the interests of those 1 The Hungarian poet Petofi disappeared in a battle in power. Plato accused poets of being the enemies against the Russian-Austria alliance in 1849. He was of truth and their poetry of spreading mental considered to “have died at the points of the lances of 2 Cossack soldiers” until the end of the nineteenth century, poison. On the other hand, pure mathematics, when Russian researchers found in archives that he had actually been taken to Siberia as a prisoner of war and 2Plato was always precise in his diction. In his last work he died there of tuberculosis in 1856. He would therefore described those who ignored the importance of mathemat- have been thirty-three when he died. ics in the pursuing of ideals as “piggish”.

APRIL 2011 NOTICES OF THE AMS 591 especially modern mathematics, often develops in the American poet Longfellow. It tells of an Indian advance of its time, even in advance of theoretical who made fur mittens: physics. It was more than a full century after the He, to get the warm side inside, invention of Galois’s group theory and Hamilton’s Put the inside (skin side) outside; theory of quaternions that these theories were ap- He, to get the cold side outside, plied to quantum mechanics. In similar situations, Put the warm side (fur side) inside… non-Euclidean geometry was used to describe grav- itational fields, and complex analysis to describe Interestingly, the word topology first appeared electrodynamics. The discovery of conic sections, as Topologie in German, in the work of a student which for over two thousand years was considered of Gauss in 1847, when the concept was known to no more than “the unprofitable amusement of a very few mathematicians. speculative brain”, ultimately found its application Finally, I’m going to raise the question of in Newton’s equation of motion, theory of projec- whether someone can be a poet and a mathema- tile motion, and the law of universal gravitation. tician at the same time. Pascal assures us at the However, more often than not, the work that beginning of his Pensées: “As long as geometri- mathematicians do is not understood by the crowd. cians have good insight, they can be sensitive; as Some people have rebuked them for indulging in long as sensitive people can apply their insight to pointless speculation or being silly and useless geometric principles, they can be geometricians too.” Despite this, historically only the eighteenth dreamers. Lamentably, this is the viewpoint of century Italian mathematician Mascheroni and the learned scholars. For example, Schopenhauer, a nineteenth century French mathematician Cauchy distinguished modern philosopher, acknowledged could possibly be counted as poets, while the twen- poetry as the highest art but described arithmetic tieth-century Chilean poet Parra was a professor of as the lowest activity of the spirit.3 Since the be- mathematics. Perhaps the only one in human his- ginning of the twentieth century, more and more tory who made great contributions in both fields people have come to realize how our times have was Omar Khayyam, the eleventh-century Persian benefited from mathematics. To some extent, who was born four centuries earlier than the ver- however, poets and artists are still in the situation satile Da Vinci. He made his mark in the history they always have been. Perhaps they should con- of mathematics for his geometric solution of cubic sole themselves with Picasso’s words: “People earn equations, and he became known to the world as the title of artists only after they have overcome the author of the Rubáiyát. When the fourteen- innumerable obstacles. Therefore art should be year-old T. S. Eliot came across Edward Fitzgerald’s restricted instead of being encouraged.” English translation of the Rubáiyát at the turn of By coincidence, mathematicians and poets often the twentieth century, he immediately became walk side by side on the frontiers of human civi- enthralled. He recalled the splendor of entering lization. Euclid’s Elements and Aristotle’s Poetics, the world of this magnificent poem and realized, the two most important academic works of ancient after reading those lines full of “dazzling, sweet Greece, were written at almost the same time. They and painful colors”, that he wanted to be a poet. both had what one might call a common belief or attitude consisting, one might say, in an accurate Acknowledgments “imitation” of the outer world. For Euclid, it was the The author is grateful for the referees’ valuable physical-geometrical form of three-dimensional opinions and suggestions, particularly to Professor space; for Aristotle, it was understanding poetics Preda Mihailescu for insightful, precious ideas and as a description of everyday life. The difference is discussions offered during my visit at the Math- that the former was an abstract imitation while the ematisches Institut, Universität Göttingen. They all latter was a concrete one. Poe and Baudelaire, pio- made this paper more readable. neers of modern art, belonged to the same age as Lobachevsky and Bolyai, founders of non-Euclidean geometry. When a group of poets and painters of great talent gathered in Paris, in the 1930s and 1940s, to launch the radical revolution of surreal- ism, some other brilliant minds in the world were working hard in their own way to develop topology, a burgeoning branch of mathematics. Here I want to quote an example, often cited by topologists, which uses a parody of The Song of Hiawatha by

3 This viewpoint of Schopenhauer is completely contrary to that of Plato, who proclaimed that he would drive poets out of his ideal city and that “God is a geometrician”.

592 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2011 Steele Prizes

The 2011 AMS Leroy P. Steele Prizes were pre- such as the Kuznetsov formula and the spectral sented at the 117th Annual Meeting of the AMS theory of Kloosterman sums, are covered for the in New Orleans in January 2011. The Steele Prizes first time in this book. It closes with a discussion of were awarded to Henryk Iwaniec for Mathemati- current research on the size of eigenfunctions on cal Exposition, to Ingrid Daubechies for a Seminal hyperbolic manifolds. By making these tools from Contribution to Research, and to John W. Milnor automorphic forms widely accessible, this book for Lifetime Achievement. has had a tremendous influence on the practice Mathematical Exposition: Henryk Iwaniec of analytic number theory. Topics in Classical Automorphic Forms develops Citation many standard topics in the theory of modular Henryk Iwaniec is awarded the Leroy P. Steele Prize forms in a nontraditional way. Iwaniec’s aim was for Mathematical Exposition for his long record of “to venture into areas where different ideas and excellent exposition, methods mix and interact”. One standout part is both in books and in the treatment of the theory of representation of classroom notes. He is quadratic forms and estimating sizes of Fourier honored particularly coefficients of cusp forms. The breakthrough in for the books Intro- the late 1980s in understanding representations duction to the Spectral by ternary quadratic forms originated with the Theory of Automor- seminal work of Iwaniec, which is described beau- phic Forms (Revista tifully here. Matemática Iberoamer- icana, Madrid, 1995) Biographical Sketch and Topics in Classical Henryk Iwaniec graduated in 1971 from Warsaw Automorphic Forms University, got his Ph.D. the next year, and became (Graduate Studies professor at the Institute of Mathematics of the in Mathematics, 17, Polish Academy of Sciences before leaving for the Henryk Iwaniec American Mathemati- United States in 1983. After taking several visiting cal Society, Providence, positions in the United States (including a long- RI, 1997). These books give beautiful treatments of the theory of automorphic forms from the author’s term appointment at the Institute for Advanced perspective of analytic number theory. They have Study), in January 1987 he was offered a chair as become classics in the field and are now a funda- State Professor at Rutgers, the position mental resource for students. The two books are he enjoys to this day. complementary, with the first presenting the non- Iwaniec’s main interest is analytic number holomorphic theory of Maass forms for GL(2) and theory and automorphic forms. Prime numbers are the latter focusing on holomorphic modular forms. his passion. His accomplishments were acknowl- Introduction to the Spectral Theory of Automor- edged by numerous invitations to give talks at phic Forms begins with the basics of hyperbolic conferences, including the International Congress geometry and takes readers to the frontiers of of Mathematicians. Iwaniec is a member of the Pol- research in analytic number theory. Many topics, ish Academy of Sciences, the American Academy

APRIL 2011 NOTICES OF THE AMS 593 of Arts and Sciences, the National Academy of oscillation, making Sciences, and the Polska Akademia Umiejetnosci. them flexible enough Among several prizes Iwaniec has received are to be used in a variety the Jurzykowski Foundation Award, the Sierpinski of situations. As such, Medal, the Ostrowski Prize, and the Cole Prize in these wavelets (now Number Theory. known as Daubechies Iwaniec teaches graduate students and collabo- wavelets) became ex- rates with researchers from various countries. In tremely popular in 2005 he was honored with the Doctorate Honoris practical signal pro- Causa of Bordeaux University. In 2006 the town cessing (for instance, council of his native city made Iwaniec an Honor- they are used in the ary Citizen of Elblag, a distinction he cherishes JPEG 2000 image com- very proudly. pression scheme). Even nowadays they are still Response Ingrid Daubechies the default, general- I thank the American Mathematical Society and the purpose wavelet family Committee of the Steele Prize for this award. I am of choice to implement in any signal processing very honored and happy. This is a very meaningful algorithm (although for specialized applications, recognition for me because the citation is telling sometimes a more tailored wavelet can be slightly not only about my fascination with the subjects superior). but also about my attention to educating new gen- At the time of this paper, wavelet theory was al- erations of researchers. Modern analytic number ready a booming field, with hundreds of papers de- theory takes ideas from the theory of automorphic voted to wavelet construction, efficient algorithms, forms and gives back new enhanced methods and and so forth. At present the field is more mature results. While more arithmetical aspects of auto- and settled, an effect to which Daubechies’s paper morphic forms are covered relatively well in the significantly contributed by largely “solving” the literature, there is still not a sufficient exposition problem of the best wavelets to use in general of analytic aspects. Hopefully more books will be (and also by giving order to the chaotic explosion written by other specialists that will address simi- of literature). lar topics from many different directions. These In his MathSciNet review of the paper, Hans two books selected by the Committee for the award Feichtinger wrote, “Even before its publication, came out of my teaching graduate courses and giv- the paper had a remarkable impact within applied ing presentations in workshops, so inevitably they analysis, and great interest in wavelet theory has contain some of my favorite ways of handling the been shown from many sides. By the summer of problems. I am glad that my choices and writing 1989 there was already a software package avail- style are well received. If indeed these works do able, running on PCs, which is based on the con- have “influence on the practice of analytic number struction described in this note. This sheds some theory”, I will be most happy. light on the speed with which new mathematical algorithms are brought to work these days and can Seminal Contribution to Research: Ingrid serve to underline the importance of mathematical Daubechies research to applied fields.” Citation The Leroy P. Steele Prize for Seminal Contribu- Biographical Sketch tion to Research is awarded to Ingrid Daubechies Ingrid Daubechies received both her bachelor’s for her paper “Orthonormal bases of compactly and Ph.D. degrees (in 1975 and 1980, respectively) supported wavelets” (Comm. Pure Appl. Math. 41 from the Free University in Brussels, . She (1988), no. 7, 909–996). In this paper Daubechies held a research position at the Free University until constructs the very first examples of families of 1987. From 1987 to 1994 she was a member of the wavelets (rescalings of a single “mother wavelet”) technical staff at AT&T Bell Laboratories, during that are simultaneously smooth, orthonormal, and which time she took leaves to spend six months compactly supported; earlier examples of wavelets (in 1990) at the University of Michigan and two had two out of three of these properties, but not years (1991–93) at Rutgers University. She is now all three at once. The orthonormality makes them at the Mathematics Department and the Program good as a basis to decompose arbitrary signals; in Applied and Computational Mathematics at the smoothness removes edge artifacts and makes Princeton University. wavelet series converge rapidly; and the compact Her research interests focus on the mathemati- support makes them viable for use in actual prac- cal aspects of time-frequency analysis, in particular tical applications. The wavelets also came with wavelets as well as applications. In 1998 she was a parameter that traded off their smoothness elected to the National Academy of Sciences and for the width of their support and amount of became a fellow of the Institute of Electrical and

594 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Electronics Engineers. The American Mathemati- picture of the relation between the topological, cal Society awarded her a Leroy P. Steele Prize combinatorial, and smooth worlds developed by for Mathematical Exposition in 1994 for her book Milnor. Jointly with M. Kervaire, Milnor proved Ten Lectures on Wavelets, as well as the 1997 Ruth the first results showing that the topology of Lyttle Satter Prize. From 1992 to 1997 she was a 4-dimensional manifolds is exceptional by reveal- fellow of the John D. and Catherine T. MacArthur ing obstructions to the realization of 2-dimen- Foundation. sional spherical homology classes by smooth She is a member of the American Academy of embedded 2-spheres. This is one of the founding Arts and Sciences, the American Mathematical results of 4-dimensional topology. Society, the Mathematical Association of America, In this way Milnor opened several fields: singu- the Society for Industrial and Applied Mathematics, larity theory, algebraic K-theory, and the theory and the Institute of Electrical and Electronics Engi- of quadratic forms. Although he did not invent neers. In addition, Daubechies was elected in 2010 these subjects, his work gave them completely new to serve as the next president of the International points of view. For instance, his work on isolated Mathematical Union. singularities of complex hypersurfaces presented a great new topological framework for studying Response singularities and, at the same time, provided a I am delighted and very grateful to receive this rich new source of examples of manifolds with award, especially for this paper. In my work, I try different extra structures. The concepts of Milnor to distill, from extensive contacts with scientists fibers and Milnor number are today among the and engineers, challenging mathematical problems most important notions in the study of complex that nevertheless are still connected to the original singularities. question. When I am lucky, as was the case for this The significance of Milnor’s work goes much be- paper, the answer to the question or the results of yond his own spectacular results. He wrote several the study are not only interesting mathematically books—Morse Theory (Princeton University Press, but also translate into something new and useful Princeton, NJ, 1963), Lectures on the h-Cobordism for the application domain. I also would like to Theorem (Princeton University Press, Princeton, thank Communications in Pure and Applied Math- NJ, 1965), and Characteristic Classes (Princeton ematics, where the paper appeared, for accepting University Press, Princeton, NJ, 1974), among oth- those long tables of coefficients—its impact in ers—that became classical, and several generations engineering would not have been the same without of mathematicians have grown up learning beauti- the tables, at that time a standard feature of papers ful mathematical ideas from these excellent books. on filter constructions in signal analysis. Milnor’s survey “Whitehead torsion” (Bull. Amer. Math. Soc. 72 (1966), no. 3, 358–426) provided an Lifetime Achievement: John W. Milnor entry point for topologists to algebraic K-theory. Citation This was followed by a number of Milnor’s own The 2011 Steele Prize important discoveries in algebraic K-theory and for Lifetime Achieve- related areas: the congruence subgroup theorem, ment is awarded to the computation of Whitehead groups, the intro- John Willard Milnor. duction and study of the functor K2 and higher Milnor stands out from K-functors, numerous contributions to the classi- the list of great math- cal subject of quadratic forms, and in particular ematicians in terms his complete resolution of the theory of symmetric of his overall achieve- inner product spaces over a field of characteristic 2, ments and his influ- just to name a few. Milnor’s introduction of the ence on mathemat- growth function for a finitely presented group ics in general, both and his theorem that the fundamental group of a through his work and negatively curved Riemannian manifold has expo- through his excellent nential growth was the beginning of a spectacular books. His discovery development of the modern geometric group John Milnor of twenty-eight non- theory and eventually led to Gromov’s hyperbolic diffeomorphic smooth group theory. structures on the 7-dimensional sphere and his During the past thirty years, Milnor has been further work developing the surgery techniques for playing a prominent role in development of low- manifolds shaped the development of differential dimensional dynamics, real and complex. His topology beginning in the 1950s. Another of his fa- pioneering work with Thurston on the kneading mous results from this period is a counterexample theory for interval maps laid down the combinato- to the Hauptvermutung: an example of homeomor- rial foundation for the interval dynamics, putting phic but not combinatorially equivalent complexes. it into the focus of intense research for decades. This counterexample is a part of a general big Milnor and Thurston’s conjecture on the entropy

APRIL 2011 NOTICES OF THE AMS 595 monotonicity brought together real and complex recipient, high level of research over a period of dynamics in a deep way, prompting a firework of time, particular influence on the development of a further advances. And, of course, his book Dynam- field, and influence on mathematics through Ph.D. ics in One Complex Variable (Friedr. Vieweg & Sohn, students; (2) Mathematical Exposition: for a book Braunschweig, 1999) immediately became the most or substantial survey or expository research paper; popular gateway to this field. (3) Seminal Contribution to Research: for a paper, The Steele Prize honors John Willard Milnor for whether recent or not, that has proved to be of all of these achievements. fundamental or lasting importance in its field or a model of important research. Each Steele Prize Biographical Sketch carries a cash award of US$5,000. John Milnor was born in Orange, New Jersey, in Beginning with the 1994 prize, there has been a 1931. He spent his undergraduate and graduate five-year cycle of fields for the Seminal Contribu- student years at Princeton, studying knot theory tion to Research Award. For the 2011 prize, the (then a very unfashionable field that has since field was applied mathematics. The Steele Prizes become amazingly fashionable) under the supervi- are awarded by the AMS Council acting on the sion of Ralph Fox. After many years at Princeton recommendation of a selection committee. For the University and the Institute for Advanced Study, 2011 prizes, the members of the selection com- with shorter stays at UCLA and MIT, he has settled mittee were: Peter S. Constantin, , at Stony Brook University, where he is now codi- John E. Fornaess, Barbara L. Keyfitz, Gregory F. rector of the Institute for Mathematical Sciences. Lawler, Richard M. Schoen, Joel A. Smoller, Terence Over the years, he has wandered randomly from C. Tao, and . The list of previ- subject to subject, studying game theory, differ- ous recipients of the Steele Prize may be found ential geometry, algebraic topology, differential on the AMS website at http://www.ams.org/ topology, quadratic forms, and algebraic K-theory. prizes-awards. For the past twenty-five years, his main focus has been on dynamical systems, and particularly on low-dimensional holomorphic dynamical systems. Among his current projects is the preparation of a book to be called Dynamics: Introductory Lectures.

Response It is a particular pleasure to receive an award for what one enjoys doing anyway. I have been very lucky to have had so many years to explore and enjoy some of the many highways and byways of mathematics, and I want to thank the three insti- tutions that have supported and inspired me for most of the past sixty years: Princeton University, where I learned to love mathematics; the Institute for Advanced Study for many years of uninter- rupted research; and Stony Brook University, where I was able to reconnect with students and (to some extent) with teaching. I am very grateful to my many teachers, from Ralph Fox and long ago to Adrien Douady in more re- cent years; and I want to thank the family, friends, students, colleagues, and collaborators who have helped me over the years. Finally, my grateful thanks to the selection committee for this honor.

About the Prize The Steele Prizes were established in 1970 in honor of , , and William Caspar Graustein. Osgood was president of the AMS during 1905–1906, and Birkhoff served in that capacity during 1925–1926. The prizes are endowed under the terms of a bequest from Leroy P. Steele. Up to three prizes are awarded each year in the following catego- ries: (1) Lifetime Achievement: for the cumulative influence of the total mathematical work of the

596 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2011 Conant Prize

David Vogan received the 2011 AMS Levi L. inductively. Next the story shifts to the challenge Conant Prize at the 117th Annual Meeting of the of converting this algorithm into a computer AMS in New Orleans in January 2011. program. Obstacles here include the problem of storing 6 billion polynomials with about 120 bil- Citation lion coefficients, many of them extremely large. A clever idea of —using the Chinese The Levi L. Conant Prize for 2011 is awarded to Remainder Theorem—helps save the day. for his article, “The character table Vogan describes this work with great verve, as for E ” (Notices Amer. Math. Soc. 54 (2007), no. 8 a saga of wildly oscillating emotions. He closes by 9, 1122–1134). The Lie group E was discovered 8 affording us some inkling of the insights afforded in 1887 by Wilhelm Killing in the course of his by this mass of data, as well as the ongoing goals project to determine all of the simple real Lie of the project. In conclusion, groups, whose theory had been initiated shortly he pays touching tribute to the beforehand by Sophus Lie. This project was com- many brilliant mathematicians pleted in the 1890s by Elie Cartan. However, E 8 who have brought their dispa- remained in many ways an “unknown” known Lie rate skills and insights so fruit- group or, at least, a poorly understood one. It has fully to bear on this problem. no clear connection to classical geometry, and its smallest faithful linear representation is the 248- dimensional adjoint representation on its own Lie Biographical Sketch algebra. The mist has cleared slowly in the course David Vogan graduated from the of the twentieth century through the efforts of University of Chicago in 1974 Weyl, Tits, Harish-Chandra, and many others. Yet and earned a Ph.D. under Ber- many open questions remain. tram Kostant at MIT in 1976. He Vogan’s article reports on a recently completed joined the MIT faculty in 1979 © C J Mozzochi, Princeton, NJ. project to determine the set of irreducible unitary and served as department head David Vogan representations of the split real Lie group of type from 1999 to 2004. Since 1996 he has been a fellow of the American Academy of E8. These representations are described by giving a character table. Naively, a character is the trace Arts and Sciences. His research concerns infinite- of a representation and is constant on the conju- dimensional representations of Lie groups; it has gacy classes of the group. This makes sense for been carried out with too many collaborators to finite groups and even for compact groups, but remember, although it is always a pleasure to try. it requires some deep mathematics to give even a He gave an invited address to the International coherent meaning to the phrase “character table Congress of Mathematicians in Berkeley in 1986 and the Hermann Weyl Memorial Lectures at the of split E8”. Part of Vogan’s achievement is to provide the reader a good understanding of what Institute for Advanced Study in the same year. that means, as clarified by the theorems of Harish- Chandra, Langlands, and Knapp and Zuckerman, Response from David Vogan culminating in the fact that the necessary data is I am honored and gratified to receive this recogni- encoded in a finite set of integer matrices. Vogan’s tion, which is for work done by many wonderful next accomplishment is to give the reader insight mathematicians and dear friends. First among into how these integer matrices can be computed. these is Fokko du Cloux, who never allowed any The answer begins with an algorithm of Kazhdan trace of imprecision or sloppy thinking. Our con- and Lusztig for computing intersection homology versations were usually meant to be about my

APRIL 2011 NOTICES OF THE AMS 597 explaining something to him, but most often they ended with his explaining that I did not yet really understand. I am grateful also to all of the great teachers The Institute for Computational and from whose (perfect) examples I learned (imper- fectly) to explain mathematics: Richard Beals and Experimental Research in Mathematics Paul Sally at the University of Chicago, Sigurdur Helgason and Bertram Kostant at MIT, and and Anthony Knapp after I was supposed to Kinetic Theory: Analysis and Computation be done with being a student. September 7, 2011 – December 9, 2011 I am grateful to Dan Barbasch, who for more than thirty years has always known just a bit more Program Description: This semester-long program than me about the problems we both study; he in kinetic theory and computation will provide the has never hesitated to share that knowledge with participants with an introduction to a broad range of anyone who asks. Finally, I am grateful to Jeff Adams, whose math- analytical and computational aspects of kinetic theory. ematical vision and leadership is the heart of the The program will be centered around three broad topics, collaboration whose work I wrote about. He is the for each of which an international workshop will be held. best herder of cats I have ever met.

Workshop 1: About the Prize Vlasov Models in Kinetic Theory The Conant Prize is awarded annually to recog- September 19–23, 2011 nize an outstanding expository paper published in either the Notices of the AMS or the Bulletin of Workshop 2: the AMS in the preceding five years. Established Applications of Kinetic Theory and Computation in 2001, the prize honors the memory of Levi October 17–21, 2011 L. Conant (1857–1916), who was a mathematician Workshop 3: at Worcester Polytechnic University. The prize car- ries a cash award of US$1,000. Boltzmann Models in Kinetic Theory The Conant Prize is awarded by the AMS Council November 7–11, 2011 acting on the recommendation of a selection com- mittee. For the 2011 prize, the members of the Participation: ICERM welcomes applications for selection committee were Georgia Benkart, Jerry long- and short-term visitors. Support for local expenses L. Bona, and Ronald M. Solomon. Previous recipients of the Conant Prize are may be provided. Applications may be submitted at (2001); Elliott Lieb and Jakob any time until the end of the semester program and Yngvason (2002); Nicholas Katz and Peter Sarnak will be considered as long as funds and space remain (2003); Noam D. Elkies (2004); Allen Knutson and available. ICERM encourages women and members of (2005); Ronald M. Solomon (2006); Jef- underrepresented minorities to apply. More information frey Weeks (2007); J. Brian Conrey, Shlomo Hoory, and an application is available online. Nathan Linial, and Avi Wigderson (2008); John W. Morgan (2009); and (2010). About ICERM The Institute for Computational and Experimental Research in Mathematics is a National Science Foundation Mathematics Institute at Brown University in Providence, RI. Its mission is to broaden the relationship between mathematics and computation. http://icerm.brown.edu

Organizing Committee: Francis Filbet, Université de Lyons Irene Gamba, University of Texas Yan Guo, Brown University Chi-Wang Shu, Brown University Walter Strauss, Brown University

598 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2011 Morgan Prize

Maria Monks received the 2011 AMS-MAA-SIAM solving through her MATHCOUNTS team, the Le- Frank and Brennie Morgan Prize for Outstanding high Valley ARML team, and the Math Olympiad Research in Mathematics by an Undergraduate Summer Program. She also began mathematical Student at the Joint Mathematics Meetings in New research as a high school student, writing a paper Orleans in January 2011. Receiving honorable men- on the 3x+1 conjecture and coauthoring another tions were Michael Viscardi and Yufei Zhao. on a conjecture of Erdo˝s and Straus. As an undergraduate, Maria participated in the Citation: Maria Monks Duluth mathematics REU under the direction of Joe Gallian, and she worked with Richard Maria Monks is the winner of the 2011 Morgan Stanley and Mia Minnes at MIT, writing Prize for Outstanding Research in Mathematics a total of five more research papers by an Undergraduate Student. The award is based over the course of her undergraduate on her impressive work in combinatorics and career. She also discovered her pas- number theory, which has appeared in Advances sion for teaching in college; she was in Applied Mathematics, Proceedings of the AMS, a coach of the 2008 USA team for the Electronic Journal of Combinatorics, Discrete Math- Girls’ Math Olympiad in China and ematics, and Journal of Combinatorial Theory, became involved in local educational Series A. programs, such as Girls’ Angle and One of her recommenders wrote, “Although Idea Math. She is a dedicated distance Maria has just finished her bachelor’s degree, her runner, earning All-American honors accomplishments are what you might expect from at the NCAA Cross-Country National someone in the second year of a postdoctoral Championships during her last year position.” Another wrote that her work “reveals a as a varsity athlete at MIT. Maria Monks broad knowledge of relevant methods as well as Maria is currently in a one-year mas- startling insight, and it is in the mainstream of a ter’s program in mathematics at the University of really ‘hot’ area.” Cambridge. She will pursue a Ph.D. at the Univer- Monks is a Churchill Scholar, a Goldwater sity of California, Berkeley, in the fall, where she Scholar, a Hertz Fellow, and an NSF Graduate Re- plans to study combinatorics. search Fellowship recipient. She received the Alice T. Schafer Prize for Women in Mathematics in 2009 and a Morgan Prize Honorable Mention in 2010. Response: Maria Monks She is also an NCAA All-American cross-country I am very honored to have been named the winner runner. She graduated from MIT in 2010. of the 2011 Frank and Brennie Morgan Prize, and I thank the AMS, MAA, and SIAM Morgan Prize Com- Biographical Sketch: Maria Monks mittee for selecting me for this award. Maria Monks grew up in Hazleton, Pennsylvania, I would like to thank the people who have had with her parents and two brothers. Her interest in the most impact on my mathematical education mathematics began in elementary school, when thus far. I thank Joe Gallian for nominating me for her father, Ken Monks, began to home school her this prize and for serving as a wonderful advisor in mathematics. In middle school and high school, at the Duluth REU. I also express my gratitude to she became involved in mathematical problem Ken Ono, Richard Stanley, and Mia Minnes for their

APRIL 2011 NOTICES OF THE AMS 599 help, advice, and mentorship in various research Richard Stanley and Michel Goemans on various projects. Finally, I thank my father, Ken Monks, and problems in combinatorics. In the summer of 2009 the rest of my family for providing a wonderful Yufei attended the Duluth REU directed by Joe Gal- environment in which to grow up and for fostering lian and spent a very productive summer working my interest in mathematics. on problems in additive combinatorics and graph theory. After graduating from MIT, Yufei did a Citation for Honorable Mention: Michael summer internship at Research New Viscardi England working with Henry Cohn on theoreti- The Morgan Prize Committee is pleased to award cal problems in coding theory. Yufei is currently Honorable Mention for the 2011 Morgan Prize for studying at the pursuing Outstanding Research in Mathematics by an Un- a one-year Master of Advanced Study in Mathemat- dergraduate Student to Michael Viscardi. ics. Afterward he plans to return to MIT to start The award recognizes in particular his impres- his Ph.D. in mathematics. sive senior thesis, “Alternate Compactifications of the Moduli Space of Genus One Maps”. Concerning Response: Yufei Zhao his thesis, one recommender wrote, “If this were a I am very honored to receive this recognition, and I doctoral thesis, it would secure an entry-level posi- would like to thank AMS, MAA, and SIAM for select- tion for him at one of the top departments in the ing me for this award. I would like to express my country.” In addition to his mathematical talents, gratitude to my parents for their constant support. Viscardi is an accomplished pianist and violinist. I am indebted to all my teachers and mentors for educating me and furthering my interests in math- Biographical Sketch: Michael Viscardi ematics. There are too many of them to list, but in Michael Viscardi graduated summa cum laude particular, I am grateful to Joe Gallian for running from Harvard in 2010, where he was awarded the an incredible REU program and to Richard Stanley, Thomas T. Hoopes Prize for Outstanding Research Michel Goemans, and Henry Cohn for being won- or Scholarly Work by a Senior and the David Mum- derful mentors and taking the time to supervise ford Mathematics Prize. He is currently finishing me on various projects. And finally, I thank my friends and classmates for creating a wonderfully the Harvard/New England Conservatory five-year supportive environment for doing mathematics. A.B./M.M. joint program in violin performance and will begin his Ph.D. in mathematics at MIT this fall. About the Prize Response: Michael Viscardi The Morgan Prize is awarded annually for out- standing research in mathematics by an under- I want to thank my advisor, Professor Joe Harris, graduate student (or students having submitted for his invaluable guidance, support, and humor joint work). Students in Canada, Mexico, or the throughout the course of this research. United States or its possessions are eligible for Citation for Honorable Mention: Yufei Zhao consideration for the prize. Established in 1995, the prize was endowed by Mrs. Frank (Brennie) The Morgan Prize Committee is pleased to award Morgan of Allentown, Pennsylvania, and carries Honorable Mention for the 2011 Morgan Prize the name of her late husband. The prize is given for Outstanding Research in Mathematics by an jointly by the AMS, the Mathematical Association Undergraduate Student to Yufei Zhao. of America (MAA), and the Society for Industrial The award recognizes his excellent work in and Applied Mathematics (SIAM) and carries a cash combinatorics and number theory. One of his rec- award of US$1,200. ommenders wrote, “Zhao is extraordinarily strong Recipients of the Morgan Prize are chosen by at research, functioning more like an established a joint AMS-MAA-SIAM selection committee. For mathematician than an undergraduate.” Zhao is a the 2011 prize, the members of the selection com- three-time Putnam Fellow and the recipient of a mittee were Georgia Benkart, Anna L. Mazzucato, Gates Cambridge Scholarship. He graduated from Maeve L. McCarthy, Michael E. Orrison, Kannan MIT in 2010. Soundararajan, and Sergei Tabachnikov. Previous recipients of the Morgan Prize are Biographical Sketch: Yufei Zhao Kannan Soundararajan (1995), Yufei Zhao was born in Wuhan, China, and moved (1996), Jade Vinson (1997), Daniel Biss (1998), to Toronto at the age of eleven. In high school Yufei Sean McLaughlin (1999), Jacob Lurie (2000), Ciprian developed his interest in mathematics through Manolescu (2001), Joshua Greene (2002), Melanie competitions. He competed for the Canadian team Wood (2003), Reid Barton (2005), Jacob Fox (2006), three times at the International Mathematics Olym- Daniel Kane (2007), Nathan Kaplan (2008), Aaron piad, where he received a gold medal, and he also Pixton (2009), and Scott Duke Kominers (2010). subsequently coached the team as a Deputy Leader. As an undergraduate at MIT Yufei studied math- ematics and computer science, and he worked with

600 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2011 Satter Prize

Amie Wilkinson received the 2011 AMS Ruth from Berkeley in 1995 under the direction of Lyttle Satter Prize in Mathematics at the 117th Charles Pugh. After serving one year as a Benja- Annual Meeting of the AMS in New Orleans in min Peirce Instructor at Harvard, she moved to January 2011. Northwestern in 1996 where she was promoted to full professor in 2005. She was the recipient of Citation an NSF Postdoctoral Fellowship and has given AMS The Ruth Lyttle Satter Prize in Mathematics is Invited Addresses in Salt Lake City (2002), in Rio de awarded to for her remarkable Janeiro (2007), and at the 2010 Joint Meetings in contributions to the field of ergodic theory of San Francisco. She was also an invited speaker in partially hyperbolic dynamical systems. the Dynamical Systems session at the 2010 ICM in Wilkinson and Burns provided a clean and ap- Hyderabad. She lives in Chicago with her husband plicable solution to a long-standing problem in sta- Benson Farb and their two children. bility of partially hyperbolic systems in the paper “On the ergodicity of partially hyperbolic systems” Response (Annals of Math. (2) 171 (2010), no. 1, 451–489). This is an unexpected honor for which I am very The study of hyperbolic systems was begun in the grateful. As a woman in math, I have certainly faced 1960s by Smale, Anosov, and Sinai; this work was some challenges: shaking the sense of built upon earlier achievements of Morse, Hedlund, being an outsider, coping with occa- and Hopf. The recent papers of Wilkinson, jointly sional sexism, and balancing career with Burns, give what is considered by experts to and family. These difficulties were be the optimal result that unifies much of the deep ameliorated by the support and en- work done by mathematicians during the inter- couragement of numerous individuals vening decades to weaken the strong hypothesis and institutions, beginning with my of hyperbolicity in order to be widely applicable parents, who thought it delightful that while retaining the fundamentals of the associated their older daughter loved math and dynamical behavior. science (and art and cooking). Early Wilkinson has played a central role in the recent guidance from math teachers, espe- major developments in many related areas as well, cially John Benson at Evanston High including making some fundamental advances in School, was invaluable. The people in understanding generic behavior of C1 diffeomor- the Math Department at Northwestern phisms. In addition to her outstanding work with University demonstrated their faith in Amie Wilkinson Burns, Wilkinson works with many coauthors, such me early on and never wavered in their as Avila, Bonatti, Crovisier, Masur, and Viana, with support. Northwestern protected my research time whom she has published many significant results. early on, was flexible in assigning duties later, and A problem on the centralizers of diffeomorphisms promoted me in a timely fashion. Some of this was was stated by Smale more than forty years ago a gamble on Northwestern’s part, one that other and is included in his list of problems for the departments might still be hesitant to make. twenty-first century; the solution in the C1 case I have been educated over the years by a string was provided by Wilkinson in a series of papers of amazing mentors and collaborators, including with Bonatti and Crovisier. those mentioned in the citation. Charles Pugh, Mike Shub, Keith Burns, and Christian Bonatti have Biographical Sketch played a special role; together, they have taught Amie Wilkinson grew up in Evanston, Illinois, me how to think, dream, and write mathematics. received her A.B. from Harvard in 1989 and Ph.D. From early on, Lai-Sang Young (the 1993 Satter

APRIL 2011 NOTICES OF THE AMS 601 Prize winner) has been a role model; her work in dynamics and clarity of exposition has always set the standard. The joint project with Keith Burns mentioned in the citation was an immensely sat- The Institute for Computational and isfying collaboration. Whenever I think that the Experimental Research in Mathematics intricacies of partially hyperbolic dynamics have been largely revealed, a new phenomenon arises to delight and inspire. Complex and I also thank my husband Benson, my best January 30, 2012 – May 4, 2012 friend, mathematical companion, and muse (who occasionally lets me be his muse as well), and my Program Description: Many computational and children Beatrice and Felix, who have forced me graphical tools have been developed for the study of to take a break from mathematics when I needed complex dynamics, tools that have been of immense it the most. value in the development of the complex theory. Among the goals of the program will be the development of About the Prize a comprehensive set of tools for studying p-adic and The Satter Prize is awarded every two years to arithmetic dynamics. This semester-long program will be recognize an outstanding contribution to math- centered around three broad topics, for each of which an ematics research by a woman in the previous five international workshop will be held. years. Established in 1990 with funds donated by Joan S. Birman, the prize honors the memory of Workshop 1: Complex and p-adic Dynamics Birman’s sister, Ruth Lyttle Satter. Satter earned a February 13–17, 2012 bachelor’s degree in mathematics and then joined the research staff at AT&T Bell Laboratories during Workshop 2: Global Arithmetic Dynamics World War II. After raising a family, she received March 12–16, 2012 a Ph.D. in botany at the age of forty-three from the University of Connecticut at Storrs, where she Workshop 3: Moduli Spaces Associated to later became a faculty member. Her research on the Dynamical Systems biological clocks in plants earned her recognition April 16–20, 2012 in the United States and abroad. Birman requested that the prize be established to honor her sister’s Participation: ICERM welcomes applications for commitment to research and to encouraging long- and short-term visitors. Support for local expenses women in science. The prize carries a cash award may be provided. Applications may be submitted at of US$5,000. any time until the end of the semester program and The Satter Prize is awarded by the AMS Coun- will be considered as long as funds and space remain cil acting on the recommendation of a selection available. ICERM encourages women and members of committee. For the 2011 prize, the members of underrepresented minorities to apply. More information the selection committee were Victor W. Guillemin, and an application is available online. Jane M. Hawkins, and Sijue Wu. Previous recipients of the Satter Prize are: Dusa About ICERM: The Institute for Computational and McDuff (1991), Lai-Sang Young (1993), Sun-Yung Experimental Research in Mathematics is a National Alice Chang (1995), Ingrid Daubechies (1997), Science Foundation Mathematics Institute at Brown Bernadette Perrin-Riou (1999), Karen E. Smith University in Providence, RI. Its mission is to broaden the (2001), Sijue Wu (2001), Abigail Thompson (2003), relationship between mathematics and computation. (2005), Claire Voisin (2007), and Laure Saint-Raymond (2009). http://icerm.brown.edu

Organizing Committee: Rob Benedetto, Amherst College Lucien Szpiro, City University Laura DeMarco, University of IL/ of NY Chicago Michael Zieve, University of Mikhail Lyubich, SUNY Stony Michigan Brook Juan Rivera-Letelier, Pontificia Universidad Católica de Chile Joseph Silverman, Brown University

602 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2011 Bôcher Prize

Assaf Naor and Gunther Uhlmann received the Lindenstrauss. He was a postdoctoral researcher 2011 AMS Maxime Bôcher Memorial Prize at the at the Theory Group of Microsoft Research in 117th Annual Meeting of the AMS in New Orleans Redmond, Washington, from 2002 to 2004, and in January 2011. a researcher at the Theory Group from 2004 to 2006. Since 2006 he has been a faculty member at Citation: Assaf Naor the Courant Institute of Mathematical Sciences of The Bôcher Prize is awarded to Assaf Naor for , where he is a professor of introducing new invariants of metric spaces and mathematics and an associated faculty member for applying his new in computer science. He received the Bergmann understanding of the Memorial Award (2007), the European Mathemati- distortion between cal Society Prize (2008), the Packard Fellowship various metric struc- (2008), and the Salem Prize (2008), and he was an tures to theoretical invited speaker at the International Congress of computer science, es- Mathematicians (2010). Naor’s research is focused pecially in the papers on analysis and metric geometry and their interac- “On metric Ramsey tions with approximation algorithms, combinator- type phenomena” (with ics, and probability. Yair Bartal, Nathan Lin- ial, and Manor Mendel, Response: Assaf Naor Annals of Math. (2) 162 I am immensely grateful for the Bôcher Memorial (2005) no. 2, 643–709); Prize. Above all, I wish to thank my collaborators, “Metric cotype” (with especially Manor Mendel, with whom a decade-long Assaf Naor Manor Mendel, Annals collaboration and friendship has resulted in excit- of Math. (2) 168 (2008), ing, sometimes unexpected, discoveries. I thank my no. 1, 247–298); and “Euclidean distortion and the sparsest cut” (with Sanjeev Arora and James advisor Joram Lindenstrauss for his inspiration R. Lee, J. Amer. Math. Soc. 21 (2008), no. 1, 1–21). and encouragement. I am also grateful to Gideon The prize also recognizes Naor’s remarkable work Schechtman for being my unofficial advisor, col- with J. Cheeger and B. Kleiner on a lower bound in laborator, and friend, and for the mentoring and the sparsest cut problem. The bound follows from friendship of Keith Ball. a quantitative version of Cheeger and Kleiner’s In 1964 Lindenstrauss published a seminal beautiful differentiation theorem for Lipschitz paper entitled “On nonlinear projections in Ban- functions with values in L1, which was itself mo- ach spaces”. This paper contains results showing tivated by this application, as envisioned by Naor that Banach spaces exhibit unexpected rigidity and J. R. Lee. phenomena: the existence of certain nonlinear mappings that are quantitatively continuous (e.g., Biographical Sketch: Assaf Naor uniformly continuous or Lipschitz) implies that Assaf Naor was born in Rehovot, Israel, on May 7, certain linear properties are preserved. Subsequent 1975. He received his Ph.D. from the Hebrew Uni- work of Enflo on Hilbert’s fifth problem in infinite versity in Jerusalem under the direction of Joram dimensions and then a beautiful 1976 theorem of Ribe ushered in a new era in metric geometry.

APRIL 2011 NOTICES OF THE AMS 603 Ribe’s theorem in particular showed that local such as nearest neighbor search and clustering, linear properties of Banach spaces are preserved are by definition questions on metric spaces, while under uniform homeomorphisms and can thus metric structures sometimes appear in problems be conceivably formulated in a way that involves that do not have a priori geometry in them, such only distance computations, ignoring entirely the as the structures that arise from continuous re- linear structure. Of course, if this could be done, laxations of combinatorial optimization problems. one could then investigate which metric spaces This general philosophy of embedding theory is have these properties, even if the geometric spaces responsible for many important approximation al- in question have nothing to do with linear spaces gorithms and data structures. The work contained (e.g., Riemannian manifolds, graphs, or discrete in the two other papers that are mentioned in the groups). Since at roughly the same time a deep citation has been partially inspired by these appli- theory of quantitative local linear invariants of Banach spaces was flourishing, this raised the cations: sharp nonlinear Dvoretzky theorems are possibility that this powerful linear theory could applied to online algorithms and data structures; be applied in much greater generality to geometric and new embedding methods manage to asymp- problems that are ubiquitous in mathematics. Un- totically determine (up to lower order terms) the doubtedly many mathematicians noticed the great most nonEuclidean finite subset of L1 (completing potential here, but to the best of my knowledge the 1969 work of Enflo), and at the same time they this program was first formulated in writing by yield the best known approximation algorithm for Bourgain in 1986. the sparsest cut problem with general demands. The 1980s witnessed a remarkable surge of The methods behind these results are in essence activity in this area, yielding several theorems on multiscale analysis, combined with a variety of general metric spaces that are inspired by previ- probabilistic techniques. ously known linear results: these include metric The quantitative structure theory of metric space versions of the theory of Rademacher type, a spaces is flourishing, with important new results nonlinear Dvoretzky theorem, and extension theo- appearing frequently. New algorithmic applica- rems for Lipschitz functions, as well as important tions are constantly being discovered, and deep results such as Bourgain’s embedding theorem connections are being made with harmonic analy- and Bourgain’s metrical characterization of su- sis, group theory, and geometric measure theory. perreflexivity. Key players in this 1980s “golden There is still a lot that we do not know in the origi- age” include Ball, Bourgain, Gromov, Johnson, nal program that is motivated by Ribe’s theorem, Lindenstrauss, Milman, Pisier, Schechtman, and Talagrand. including missing metric invariants that have yet to One reason that might explain why the 1980s be defined. Experience shows that future progress golden age ended is that there were several stub- will be difficult but fruitful, and the fact that many born problems that led to an incomplete theory, talented mathematicians and computer scientists and one needed new definitions of concepts, are working in this field suggests that exciting such as a notion of Rademacher cotype for metric developments are yet to come. I view the Bôcher spaces, in order to proceed. One of the papers that Memorial Prize as a recognition of the achieve- is mentioned in the citation formulated a definition ments of this field and as an encouragement to the of metric cotype and proved several accompany- many researchers who developed the theory thus ing theorems and applications showing that it is far to continue their efforts to uncover the hidden a satisfactory notion of cotype for metric spaces. structure of metric spaces. This work was a result of years of intensive effort and required insights that involve geometry, com- Citation: Gunther Uhlmann binatorics, and harmonic analysis. The Bôcher Prize is awarded to Gunther Uhlmann The quantitative structural theory of metric for his fundamental work on inverse problems spaces also received renewed impetus due to the and in particular for the solution to the Calderón important 1995 paper of Linial, London, and Rabi- problem in the papers “The Calderón problem with novich that exhibited the usefulness of these prob- partial data” (with Carlos E. Kenig and Johannes lems (specifically Bourgain’s embedding theorem) Sjöstrand, Annals of Math. (2) 165 (2007) no. 2, to the design of efficient approximation algorithms for NP-hard problems. The general theme here is 567–591) and “The Calderón problem with partial that a natural way to understand the structure of data in two dimensions” (with Oleg Yu. Imanuvilov a metric space is by representing it as faithfully and Masahiro Yamamoto, J. Amer Math. Soc. 23 as possible as points in a well-understood normed (2010), no. 3, 655–691). The prize also recognizes space, or as a superposition of simpler structures Uhlmann’s incisive work on boundary rigidity with such as trees. The relevance of such problems to L. Pestov and with P. Stepanov and on nonunique- computer science ranges from “obvious” to round- ness (also known as cloaking) with A. Greenleaf, about and surprising: many algorithmic tasks, Y. Kurylev, and M. Lassas.

604 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Biographical Sketch: Gunther Uhlmann kindest person I have ever met—a wonderful role Gunther Uhlmann was born in Quillota, Chile, in model for anybody to follow. I also treasured the 1952. He studied mathematics as an undergradu- friendship I started with Cathleen Morawetz dur- ate at the Universidad de Chile in Santiago, gaining ing my stay at Courant. Most of all I have had the his Licenciatura degree in 1973. He continued his unwavering support of my family, Carolina, Anita, studies at MIT, where and Eric. Without them this would not have been he received a Ph.D. in possible. Carolina would have been so proud. This 1976 under the direc- prize is for her. tion of Victor Guille- About the Prize min. He held postdoc- toral positions at MIT, Established in 1923, the prize honors the memory Harvard, and the Cou- of Maxime Bôcher (1867–1918), who was the So- rant Institute. In 1980 ciety’s second Colloquium Lecturer in 1896 and he became assistant who served as AMS president during 1909–1910. professor at MIT and Bôcher was also one of the founding editors of then moved in 1985 Transactions of the AMS. The original endowment to the University of was contributed by members of the Society. The Washington, where he prize is awarded for a notable paper in analysis was appointed Walker published during the preceding six years. To be Gunther Uhlmann Family Endowed Pro- eligible, the author should be a member of the fessor in 2006. Since AMS, or the paper should have been published in 2010 he has also held the Endowed Excellence a recognized North American journal. The prize is in Teaching Chair at the University of Califor- given every three years and carries a cash award nia, Irvine. Uhlmann received a Sloan Research of US$5,000. Fellowship in 1984 and a Guggenheim Fellowship The Bôcher Prize is awarded by the AMS Council in 2001. Also in 2001 he was elected a correspond- acting on the recommendation of a selection com- ing member of the Chilean Academy of Sciences. mittee. For the 2011 prize, the members of the He has been a fellow of the Institute of Physics selection committee were Alberto Bressan, Reese since 2004. He was elected to the American Acad- Harvey, and David Jerison. emy of Arts and Sciences in 2009 and elected a Previous recipients of the Bôcher Prize are SIAM fellow in 2010. He was an invited speaker G. D. Birkhoff (1923), E. T. Bell (1924), Solomon at ICM in Berlin in 1998 and a plenary speaker at Lefschetz (1924), J. W. Alexander (1928), Marston ICIAM in Zurich in 2007. He was named a Highly Morse (1933), Norbert Wiener (1933), John von Neu- Cited Researcher by ISI in 2004. mann (1938), (1943), A. C. Schaeffer and D. C. Spencer (1948), (1953), Response: Gunther Uhlmann (1959), Paul J. Cohen (1964), I. M. I am greatly honored by being named a corecipi- Singer (1969), Donald S. Ornstein (1974), Alberto P. ent of the 2011 Bôcher Memorial Prize. I would Calderón (1979), Luis A. Caffarelli (1984), Richard like to start by thanking the collaborators who are B. Melrose (1984), Richard M. Schoen (1989), Leon named in the citation; it was a great pleasure to Simon (1994), Demetrios Christodoulou (1999), work with them. I would also like to acknowledge Sergiu Klainerman (1999), Thomas Wolff (1999), my many other collaborators and my graduate Daniel Tataru (2002), Terence Tao (2002), Fanghua students and postdocs who have enriched my life Lin (2002), Frank Merle (2005), and Charles Feffer- both professionally and personally. Many people man, Carlos Kenig, and Alberto Bressan (2008). were very influential in my career. Warren Am- brose made it possible for me to go to graduate school at MIT, and he was a continuous source of support and encouragement, especially in my early years in the United States. Herbert Clemens also helped me to come to the United States, and he has been an example to emulate in my life. My advisor Victor Guillemin taught me so much—he has a contagious enthusiasm for mathematics. Richard Melrose shared with me many times his great insight, and he has been a true friend. I met Alberto Calderón during my graduate studies at MIT; he is my mathematical hero, such an original mathematician. The year I was at Courant, I had the great fortune of meeting Louis Nirenberg. He taught me many things in mathematics and is the

APRIL 2011 NOTICES OF THE AMS 605 2011 Doob Prize

Peter Kronheimer and received in 1987 under the su- the 2011 AMS Joseph Doob Prize at the 117th An- pervision of Michael nual Meeting of the AMS in New Orleans in January Atiyah. After a year as 2011. They were honored for their book Monopoles a junior research fel- and Three-Manifolds (Cambridge University Press, low at Balliol and two 2007). years at the Institute for Advanced Study, Citation he returned to Merton The study of three- and four-dimensional mani- as fellow and tutor in folds has been transformed by the development of mathematics. In 1995 gauge theories adapted from mathematical phys- he moved to Harvard, ics. The appearance of gauge-theoretic invariants where he is now the of four-manifolds led to Donaldson’s discovery of William Caspar Graus- pairs of four-manifolds that were homeomorphic tein Professor of but not diffeomorphic. For three-manifolds, a Peter Kronheimer Mathematics. He is a generalization of Morse theory introduced by Floer recipient of the Förder- gave a home to the solutions of the Yang-Mills preis from the Mathematisches Forschungsinstitut, equations and their topological interpretations. Oberwolfach, and the Whitehead Prize from the In the 1990s Seiberg and Witten developed a more London Mathematical Society. He is a corecipient direct approach to the riches of gauge invariants. (with Tomasz Mrowka) of the Prize The book by Kronheimer and Mrowka presents an from the American Mathematical Society and was ambitious and thorough account of these ideas and elected a fellow of the Royal Society in 1997. Peter their consequences. lives in Newton, Massachusetts, with his wife, The construction of instanton homology by Jenny, and two sons, Matthew and Jonathan. Floer begins with Morse theory and the anti-self- Tomasz Mrowka was born in State College, dual Yang-Mills equation. Substituting the Seiberg- Pennsylvania, and lived in Kalamazoo, Michi- Witten equations leads to three variants of Floer gan, and Amherst, New homology that the authors develop and explain. York, while following To do this they need substantial foundations in his father’s academic analysis, geometry, and topology. Some of this ma- career. He was an un- terial—including basic Morse theory for manifolds dergraduate at MIT with boundary, sharper compactness results, and from 1979 to 1983. In functoriality for Floer theory—appears with details 1983 he entered gradu- for the first time in this book. ate school at the Uni- Three-manifolds are a rich source of geometric versity of California, phenomena, including foliations, contact struc- Berkeley, and studied tures, surgery, and knots. The potency of the with monopole techniques is demonstrated in the final and Robion Kirby. two chapters of the book, in which calculations Taubes moved to and further work are discussed and all of these Harvard in 1985, and phenomena are related to Seiberg-Witten-Floer Tomasz Mrowka Mrowka went along theory. Future researchers interested in manifold from 1985 to 1988 as theory will surely develop these tools further. Their a visiting graduate student, where he also studied apprenticeship will be the book by Kronheimer with John Morgan. He received his Ph.D. from and Mrowka. The authors deserve the Doob Prize UC Berkeley in 1989. After graduate school he for the breadth and depth of their exposition, as held postdoctoral positions at MSRI (1988–89), well as the care with which they make this rich and Stanford (1989–91), and Caltech (1991–92). He technical subject accessible. was promoted to full professor at Caltech in 1992 and remained on the faculty until 1996. He was Biographical Sketch a visiting professor at Harvard (spring of 1995) Born in London, Peter Kronheimer was educated and at MIT (fall of 1995) before returning to MIT at the City of London School and Merton College, as a professor of mathematics in the fall of 1996. Oxford. He obtained his B.A. in 1984 and his D.Phil. He was appointed to the Simons Professorship in

606 NOTICES OF THE AMS VOLUME 58, NUMBER 4 2007, and in 2010 the name of his chair changed include the extensive use of a blown-up space as to the Singer Professor of Mathematics. a setting for gluing theorems and calculations for Mrowka received the National Young Investiga- a type of coupled Morse theory that seems to be tor Grant from the NSF in 1993 and was a Sloan of independent interest. Finally, the book grew Foundation Fellow from 1993 to 1995. In 2007 because we were determined not to rely on the Kronheimer and Mrowka were jointly awarded the assumed understanding that such things could Veblen Prize in Geometry from the AMS. Mrowka be done, but instead to show, wherever possible, is a fellow of the American Academy of Arts and exactly how to do these things. Sciences, class of 2007, and was awarded a Gug- While the Property P conjecture was eventually genheim Fellowship for the 2009 academic year. proved by other means, we are particularly pleased Mrowka works mainly on the analytic aspects of that the results and calculations that went into the gauge theories and applications of gauge theory to manuscript have seen real application. Particu- problems in low-dimensional topology. larly notable was the proof by Cliff Taubes of the Outside mathematics, Mrowka enjoys travel, Weinstein conjecture in dimension three, a result good food, bicycling, swimming, hiking, and most which motivated us in the push to the finish. We of all spending time with his wife also hope that we have written a book whose self- and twins Mario and Sofia. contained treatment of several topics in geometric analysis will be useful to students in nearby fields. Response We would like to thank our families for their We are honored and delighted to hear that we love, support, patience, and understanding dur- have been awarded the Joseph L. Doob Prize. The ing the long writing process. Many thanks to all book we wrote, Monopoles and Three-Manifolds, the mathematicians whose ideas are reflected in evolved out of our wish to better understand the the book and to the staff at Cambridge University construction of monopole . At the Press for working with us as we completed the time that we started writing it in the summer of manuscript. Finally, we thank the American Math- 2000, we had been working for some time on a ematical Society and the selection committee for more ambitious project: we were trying to under- recognizing our work in this way. stand the Floer homology that would arise from the nonabelian monopole equations, with a view to About the Prize using that understanding in a proof of the Property The Doob Prize was established by the AMS in P conjecture for knots. While the substantial extra 2003 and endowed in 2005 by Paul and Virginia difficulties of the nonabelian case remained out of Halmos in honor of Joseph L. Doob (1910–2004). reach, we gradually realized that the usual (abelian) (1916–2006) was Doob’s first Ph.D. monopoles, though generally regarded as well un- student. Doob received his Ph.D. from Harvard in derstood in the folklore of the field, already had 1932 and three years later joined the faculty at many aspects that had never been clearly treated in the University of Illinois, where he remained until the literature. We embarked on writing some notes, his retirement in 1978. He worked in probability with the aim of perhaps producing a short book on theory and measure theory, served as AMS presi- the subject. A year later, we had a manuscript of dent in 1963–1964, and received the AMS Steele about 120 pages, and we reported to the National Prize in 1984. The Doob Prize recognizes a single, Science Foundation that the project was nearly relatively recent, outstanding research book that complete. It remained “nearly complete” while makes a seminal contribution to the research lit- continuing to grow over the next few years, until erature, reflects the highest standards of research it had increased in size by a factor of six. exposition, and promises to have a deep and long- There were several reasons for this growth. term impact in its area. The book must have been One was that the scope of the project naturally in- published within the six calendar years preceding creased as the field continued to develop. New ap- the year in which it is nominated. Books may be plications of Floer homology were found; the work nominated by members of the Society, by members of Ozsváth and Szabó on Heegaard Floer homology of the selection committee, by members of AMS revealed new structures that greatly influenced our editorial committees, or by publishers. The prize approach to the exposition; and the calculation of of US$5,000 is given every three years. several interesting examples came within reach. The Doob Prize is awarded by the AMS Council A second reason for the growth was that we were acting on the recommendation of a selection com- motivated to develop new approaches to several of mittee. For the 2011 prize, the members of the the technical aspects of the theory (perturbations, selection committee were Harold P. Boas, Andrew gluing, and compactness results, among others) Granville, Robin Hartshorne, Neal I. Koblitz, and in order to write a text that would be applicable John H. McCleary. to other, similar problems in geometry. Many of The previous recipients of the Doob Prize are the ideas and techniques that eventually emerged William P. Thurston (2005) and were not known to us when we began: examples and Walter Gubler (2008).

APRIL 2011 NOTICES OF THE AMS 607 2011 Eisenbud Prize

Herbert Spohn received the 2011 AMS Leonard model and a special Eisenbud Prize for Mathematics and Physics at the case of ASEP have the 117th Annual Meeting of the AMS in New Orleans same scaling function in January 2011. for their covariance. This established what Citation physicists call KPZ uni- The Eisenbud Prize for 2011 is awarded to Herbert versality for these two Spohn for his group of works on stochastic growth discrete models. In re- processes: cent work with T. Sasa- 1. “Exact height distribution for the KPZ equa- moto, Spohn both gave tion with narrow wedge initial condition” (with meaning to solutions T. Sasamoto, Nuclear Phys. B 834 (2010), no. 3, of the KPZ equation 523–542). and found the one- 2. “One-dimensional Kardar-Parisi-Zhang equa- point distribution of Herbert Spohn tion: An exact solution and its universality” (with the . As T. Sasamoto, Phys. Rev. Lett. 104, 230–602 (2010)). strange as it sounds, 3. “Scaling limit for the space-time covariance of Sasamoto and Spohn showed that the KPZ equa- the stationary totally asymmetric simple exclusion tion lies in the KPZ universality class! We remark process” (with P. L. Ferrari, Comm. Math. Phys. 265 that this last work was also done independently (2006), no. 1, 1–44). by G. Amir, I. Corwin, and J. Quastel. We note that many of the predictions of Spohn and others are We also cite, outside the six-year window: now being experimentally tested (and verified) in 4. “Scale invariance of the PNG droplet and the the laboratory.2 Airy process” (with M. Prähofer, J. Statist. Phys. 108 There are many questions left to answer regard- (2002), no. 5–6, 1071–1106). ing the Airy process and its universality, but what Stochastically growing interfaces is a subject is clear is that the work of Herbert Spohn has of intense study in both probability theory and opened up many new lines of research in math- nonequilibrium statistical physics. Three of the ematics and physics. most widely studied models that describe the height of a growing interface, h = h(x,t), are (1) the Biographical Sketch asymmetric simple exclusion process (ASEP), an Herbert Spohn is professor of mathematical phys- interacting particle system introduced by F. Spitzer ics at Zentrum Mathematik of the Technische Uni- some forty years ago; (2) the polynuclear growth versität München (TUM). Spohn was born in 1946 model (PNG); and (3) the KPZ equation, a formal at Tübingen, Germany, grandson of the mathema- nonlinear stochastic PDE for the height function tician Konrad Knopp. He earned his Vordiplom in 1 h. The main question is to describe the asymptotic physics at the Technische Universität Stuttgart in properties of the height function h. 1969 and his Diplom and Ph.D. in physics at the In early work with M. Prähofer on the PNG Ludwig-Maximilians-Universität (LMU) München. model, Spohn identified the underlying process After postdoc years at Yeshiva, Princeton, Rutgers, that is expected to describe the fluctuations of h and Leuven and an extended stay at the IHES, Paris, for a large class of growth models; this process in 1982 he joined the statistical physics group they called the Airy process. In work with P. Ferrari, at LMU, headed by Herbert Wagner, as associate Spohn established that in the scaling limit the PNG 2 K. Takeuchi and M. Sano, “Universal fluctuations of 1 The KPZ equation was introduced in the mid-1980s by growing interfaces: Evidence in turbulent liquid crystals” M. Kardar, G. Parisi, and Y.-C. Zhang. (Phys. Rev. Lett. 104, 230–601 (2010)).

608 NOTICES OF THE AMS VOLUME 58, NUMBER 4 professor of solid state physics. In 1998 he became precise statistical fluctuations depend on initial professor of applied probability in conjunction conditions. This feature was not at all anticipated with statistical physics at TUM. but is beautifully confirmed by the recent experi- Spohn received the 1993 Max-Planck Research ments of Takeuchi and Sano. The novel advance Award, jointly with Joel Lebowitz, and the 2011 concerns the KPZ equation in one dimension. Dannie Heineman Prize for Mathematical Physics. Arguably, we obtained for the first time an exact He headed the International Association of Math- time-dependent solution of a nonlinear stochastic ematical Physics. PDE. In the long time limit one finds the same prob- ability distributions as for the stochastic lattice Response growth models, supporting universality. I am deeply honored to receive the 2011 Leonard The analysis of one-dimensional stochastic Eisenbud Prize for Mathematics and Physics. I view growth models in the KPZ class borders at domains this as an appreciation of the work of so many which before were considered to be fairly distinct: physicists and mathematicians on exact solutions random matrix theory, Dyson’s Brownian motion, for growth models in the KPZ universality class. statistical mechanics of line ensembles, directed On the macroscopic scale, matter is commonly polymers in a random medium, combinatorics of organized in equilibrium phases, which in them- tilings, representation theory and Schur functions, selves are spatially homogeneous. Distinct phases are separated by a layer typically a few atomic integrable models and Bethe ansatz, interacting spacings wide. Just think of a water droplet in stochastic particle systems. Such crossroads have contact with its vapor. Such interfaces are of basic often been the source of further advances. scientific and technological interest. In particular I would like to use the occasion to thank Joachim one wants to know how they change in the course Krug, Michael Prähofer, Patrik Ferrari, and Tomo- of time. The corresponding evolution equations hiro Sasamoto. Their deep insights and unfailing have been studied widely in mathematics. To encouragement were instrumental for achieving mention only one prominent example of interface our results. motion: for a droplet immersed in three-space, the local interface velocity is taken to be proportional About the Prize to the local mean curvature, hence the dynamics The Eisenbud Prize was established in 2006 in is motion by mean curvature. The droplet shrinks, memory of the mathematical physicist Leonard but it also may pinch off into two or several pieces. Eisenbud (1913–2004) by his son and daughter- On the mesoscopic scale, still large compared to in-law, David and Monika Eisenbud. Leonard atomic spacings, one observes fluctuations on top Eisenbud, who was a student of Eugene Wigner, of the large-scale motion. The PDE for the motion was particularly known for the book Nuclear Struc- turns into a stochastic PDE of some sort. From a ture (1958), which he coauthored with Wigner. A statistical physics perspective the interest shifts friend of Paul Erd˝os, he once threatened to write a to shape fluctuations, with the hope to discover dictionary of “English to Erd˝os and Erdo˝s to Eng- universal (i.e., to a large extent model independent) lish”. He was one of the founders of the Physics statistical laws. Roughly speaking, this is the main Department at the State University of New York, focus of our research with one twist. We consider Stony Brook, where he taught from 1957 until his an interface which borders a stable against an un- retirement in 1983. His son David was president of stable phase. This imbalance drives the interface the AMS during 2003–2004. The Eisenbud Prize for motion. In a widely cited paper of 1986, Kardar, Mathematics and Physics honors a work or group Parisi, and Zhang proposed a stochastic PDE to of works that brings the two fields closer together. model this particular interface evolution. Thus, for example, the prize might be given for a In 1999 Baik, Deift, and Johansson studied the contribution to mathematics inspired by modern length of the longest increasing subsequence of a developments in physics or for the development of random permutation with the startling result that the fluctuations are governed by the Tracy-Widom a physical theory exploiting modern mathematics distribution, discovered before as the fluctuations in a novel way. The US$5,000 prize will be awarded of the largest eigenvalue of a GUE random matrix in every three years for a work published in the pre- the limit of large N. Shortly later, Johansson noted ceding six years. a related mathematical structure for the single- The Eisenbud Prize is awarded by the AMS Coun- step growth model with wedge initial data (alias cil acting on the recommendation of a selection totally asymmetric simple exclusion process with committee. For the 2011 prize, the members of step initial conditions). We realized that random the selection committee were Barry Simon, Yakov permutations are isomorphic to a driven inter- G. Sinai, and Craig A. Tracy. face model, the polynuclear growth model, and, Previous recipients of the Eisenbud Prize are based on the work of Baik and Rains, we proved Hirosi Ooguri, Andrew Strominger, and Cumrun that, while the scaling exponent is always 1/3, the Vafa (2008).

APRIL 2011 NOTICES OF THE AMS 609 2011 Cole Prize in Number Theory

Chandrashekhar Khare and Jean-Pierre Win- mid-1980s, it implies Fermat’s Last Theorem. tenberger received the 2011 AMS Frank Nelson Serre’s conjecture has inspired much extremely Cole Prize in Number Theory at the 117th Annual important work. In the 1990s Wiles used ideas Meeting of the AMS in New Orleans in January relating to Serre’s conjecture to prove Fermat’s 2011. Last Theorem and much of the Shimura-Taniyama conjecture. However, Serre’s conjecture and the Citation modularity of all odd rank 2 motives over Q The 2011 Frank Nelson Cole Prize in Number still seemed completely out of reach. Serre’s Theory is awarded to and conjecture is essentially a statement about insol- Jean-Pierre Winten- uble Galois groups, which had not been seriously berger for their re- touched in any previous work. In 2004 Khare and markable proof of Wintenberger stunned the community when for the Serre’s modularity first time they found a plausible, and extremely conjecture. In 1973 beautiful, strategy to attack Serre’s conjecture. Jean-Pierre Serre See their paper “On Serre’s conjecture for 2-dimen- made the auda- sional mod p representations of Gal(Q/Q)” (Annals cious and influential of Math. (2) 169 (2009), no. 1, 229–253). They con- conjecture that any tinued to refine their strategy, while at the same irreducible two- time Mark Kisin made important and very original dimensional rep- improvements to the modularity lifting theorems resentation of the on which their strategy relies. Khare first proved absolute Galois the level-one case of Serre’s conjecture in his paper group Gal(Q/Q) that “Serre’s modularity conjecture: The level one case” is odd (in the sense (Duke Math. J. 134 (2006), no. 3, 557–589), and that the determinant then Khare and Wintenberger completed the full Chandrashekhar Khare of complex conju- proof of Serre’s conjecture in their papers “Serre’s gation is −1 and modularity conjecture (I) and (II)” (Invent. Math. not +1) arises from 178 (2009), no. 3, 485–504 and 505–586). modular forms. This conjecture has many Biographical Sketch extremely impor- Chandrashekhar Khare was born in , India, tant consequences: in 1967 and received his B.A. in mathematics from it implies that all the University of Cambridge in 1989 and his Ph.D. odd rank 2 motives from Caltech in 1995, where he worked with Ha- over Q are modular, ruzo Hida at UCLA and Dinakar Ramakrishnan at it implies the Artin Caltech. From 1995 he worked at the Tata Institute conjecture for odd for Fundamental Research in Mumbai. In 2002 he two-dimensional joined the faculty at the University of Utah before representations of moving in 2007 to his current position as profes- Gal(Q/Q) , and, as sor in the mathematics department at UCLA. He Gerhard Frey and received the 2007 Fermat Prize from the Institut Jean-Pierre Wintenberger Serre realized in the Mathématique de Toulouse, was a Guggenheim

610 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Fellow in 2008, and received the Infosys Prize Response from C. Khare 2010 for Mathematical Sciences. He was an invited I am deeply grateful to my parents for the encour- speaker in the Number Theory Section at the In- agement they gave me to indulge in a quixotic ternational Congress of Mathematicians held in pursuit. Hyderabad, India, in August 2010. The institutions I have worked at—TIFR (Mum- Jean-Pierre Wintenberger was born in Neuilly- bai), University of Utah, and UCLA—have all pro- sur-Seine, near Paris, in 1954. He got his first thesis vided very supportive environments at work. My in 1978 and his Thése d’Etat (Habilitation) in 1984 wife, Rajanigandha, and my two children, Arushi in Grenoble, under the supervision of Jean-Marc and Vinayak, create a wonderful atmosphere at Fontaine. He held the position of researcher in home. To all of them a heartfelt thanks! CNRS from 1978 to 1991, first in Grenoble, then in Orsay. He has been a professor at the Université Response from J.-P. Wintenberger de Strasbourg since 1991. He has been a member I have a thought for my parents, who were scien- of the Institut Universitaire de France since 2007, tists and transmitted to me their curiosity, interest, received the Prix Thérèse Gautier from the French and passion for science and research. Academy of Science in 2008, and was an invited I wish to thank mathematicians who particularly speaker in the Number Theory Section at the In- influenced me by their works and personalities: ternational Congress of Mathematicians held in J.-M. Fontaine, my advisor; A. Brumer; J. Coates; Hyderabad, India, in August 2010. L. Illusie; M. Raynaud; and J.-P. Serre. I also wish to thank CNRS and Université de Strasbourg, who Response provided me the privilege of excellent conditions We are truly honored and very happy to be named of work. as corecipients of the 2011 Cole Prize for Number Theory for our work on Serre’s modularity conjec- About the Prize ture. We thank the jury and AMS for this recogni- The Cole Prize in Number Theory is awarded every tion of our work. three years for a notable research memoir in num- The conjecture is a beautifully simple and strik- ber theory that has appeared during the previous ing statement, as summarized in the citation. At five years. The awarding of this prize alternates the time it was made, in the 1970s, it must have with the awarding of the Cole Prize in Algebra, seemed inaccessible. The precision with which it also given every three years. These prizes were was formulated by J.-P. Serre, and the wealth of established in 1928 to honor Frank Nelson Cole consequences he drew from it, attracted the efforts (1861–1926) on the occasion of his retirement as of many people. secretary of the AMS after twenty-five years of ser- Our work relies on the brilliant insights of vice. He also served as editor-in-chief of the Bulletin many mathematicians. The celebrated work of for twenty-one years. The endowment was made by A. Wiles in the 1990s provided a new tool, now Cole and has received contributions from Society called modularity lifting, with which to approach members and from Cole’s son, Charles A. Cole. the conjecture. R. Taylor in the subsequent decade The Cole Prize carries a cash award of US$5,000. added several new insights, proving a potential The Cole Prize in Number Theory is awarded by version of Serre’s conjecture which has had many the AMS Council acting on the recommendation strong consequences, some of which were used in of a selection committee. For the 2011 prize, the our proof of the conjecture. As the citation men- members of the selection committee were Manjul tions, the deeply original work of M. Kisin made Bhargava, Henryk Iwaniek, and Richard L. Taylor. the method of Wiles ever more versatile, and his Previous recipients of the Cole Prize in Number work was crucially used in our proof. Another key Theory are H. S. Vandiver (1931), (1941), H. B. Mann (1946), Paul Erdo˝s (1951), John development that is fundamental to our work is T. Tate (1956), Kenkichi Iwasawa (1962), Bernard the version of modularity lifting theorems proved M. Dwork (1962), James B. Ax and Simon B. Kochen by C. Skinner and A. Wiles. Much of the work in (1967), Wolfgang M. Schmidt (1972), Goro Shimura this area is based on the pioneering work done in (1977), Robert P. Langlands (1982), the 1970s and 1980s by J.-M. Fontaine, H. Hida, (1982), Dorian M. Goldfeld (1987), Benedict H. B. Mazur, K. Ribet, and J.-P. Serre on congruences Gross and Don B. Zagier (1987), Karl Rubin (1992), between modular forms and local and global Galois Paul Vojta (1992), Andrew J. Wiles (1997), Henryk representations. To all these mathematicians we Iwaniec (2002), Richard Taylor (2002), Peter Sarnak are very grateful. (2005), and Manjul Bhargava (2008). Serre’s conjecture, once proved as a culmination of decades of work of many mathematicians, be- comes a first step in linking linear, n-dimensional, finite characteristic representations of absolute Galois groups of number fields to automorphic forms.

APRIL 2011 NOTICES OF THE AMS 611 Mathematics People

in 1994, where he has also been director of the Institute Mok and Phong Receive 2009 of Mathematical Research since 1999. Ngaiming Mok was Bergman Prize a Sloan Fellow in 1984, and he received the Presidential Young Investigator Award in Mathematics in 1985, the Ngaiming Mok of Hong Kong University and Duong H. Croucher Senior Fellowship Award in Hong Kong in 1998, Phong of have been awarded the and the State Natural Science Award in China in 2007. He 2009 Prize. Established in 1988, the prize has been serving on the editorial boards of Inventiones recognizes mathematical accomplishments in the areas Mathematicae, Mathematische Annalen, and other math- of research in which Stefan Bergman worked. The prize ematical research journals in China and in France. He consists of one year’s income from the prize fund. Mok was an invited speaker at the International Congress of and Phong have each received US$12,000. Mathematicians 1994 in Zurich in the subject area of real The previous Bergman Prize winners are: David W. Cat- and complex analysis and served as a core member on the lin (1989), Steven R. Bell and Ewa Ligocka (1991), Charles Panel for Algebraic and Complex Geometry for ICM 2006 in Fefferman (1992), Yum Tong Siu (1993), John Erik Fornaess Madrid. He was a member of the Committee (1994), Harold P. Boas and Emil J. Straube (1995), David E. for ICM 2010 in Hyderabad. Barrett and Michael Christ (1997), John P. D’Angelo (1999), Masatake Kuranishi (2000), László Lempert and Sidney Citation: Duong H. Phong Webster (2001), M. Salah Baouendi and Linda Preiss Roth- The Bergman Prize is awarded to D. Phong for his funda- schild (2003), Joseph J. Kohn (2004), Elias M. Stein (2005), mental contributions to the study of operators related to Kengo Hirachi (2006), and Alexander Nagel and Stephen the d-bar Neumann problem, starting with explicit for- Wainger (2007–2008). On the selection committee for the mulas for the solution and leading in collaborative work 2009 prize were Raphael Coifman, Linda Preiss Rothschild, to development of singular Radon transforms related to and Elias M. Stein (chair). rotational curvature and methods for estimating oscilla- tory integrals, and also, in another series of collaborations, Citation: Ngaiming Mok for obtaining far-reaching results for pseudodifferential The Bergman Prize is awarded to N. Mok for his funda- operators, achieving optimal results for positivity and mental contributions in several complex variables and, in subelliptic eigenvalue problems. particular, in the geometry of Kähler and algebraic mani- folds, and also for his work on the rigidity of irreducible Biographical Sketch: Duong H. Phong Hermitian symmetric spaces of compact type under Kähler Duong Hong Phong was born on August 30, 1953, in deformation, using both analytic and algebraic methods. Nam-Dinh, Vietnam. After high school studies at the Lycée Jean-Jacques Rousseau in Saigon and a year at the École Biographical Sketch: Ngaiming Mok Polytechnique Fédérale in Lausanne, Switzerland, he went After finishing high school in Hong Kong in 1975, Ngaim- to Princeton University, where he obtained both his B.A. ing Mok pursued his undergraduate studies at the Univer- and his Ph.D. degrees. He was an L. E. Dickson Instructor sity of Chicago and then at , obtaining his at the University of Chicago in 1975–77 and a member of M.A. at Yale in 1978 and his Ph.D. at the Institute for Advanced Study in 1977–78. He joined in 1980. He started his career at Princeton University and Columbia University in 1978 and has been there ever was professor at Columbia University and at Université since, serving in 1995–98 as chair of the mathematics de Paris (Orsay) before returning to Hong Kong to take up department. He has held visiting positions at several insti- a chaired professorship at the University of Hong Kong tutions, including the Université de Paris-Sud, the Institute

612 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Mathematics People for Theoretical and Experimental Physics in Moscow, and the Université Pierre et Marie Curie. He was an American Andrews Awarded 2010 Sacks Mathematical Society Centennial Fellow, an Alfred P. Sloan Prize Fellow, an Aisenstadt Chair at the Centre de Recherches Mathématiques in Montreal, and a Distinguished Visiting Uri Andrews of the University of Wisconsin-Madison has Professor at the University of California, Irvine. He was been awarded the 2010 Sacks Prize of the Association for also an invited speaker at the International Congress of Symbolic Logic (ASL). Andrews received his Ph.D. in 2010 Mathematicians in Zurich in 1994. from the University of California, Berkeley, under the supervision of Thomas Scanlon. The Prizes and Awards About the Prize Committee notes that in his thesis, Amalgamation Con- The Bergman Prize honors the memory of Stefan Bergman, structions in Recursive Model Theory, he combines deep best known for his research in several complex variables, methods from model theory and computability to solve as well as the Bergman projection and the Bergman ker- some problems posed by Goncharov that had resisted nel function that bear his name. A native of Poland, he solution by specialists in computability theory. taught at Stanford University for many years and died in The Sacks Prize is awarded for the most outstanding 1977 at the age of eighty-two. He was an AMS member for doctoral dissertation in mathematical logic; it was es- thirty-five years. When his wife died, the terms of her will tablished to honor of Professor Gerald Sacks of MIT and stipulated that funds should go toward a special prize in Harvard for his unique contribution to mathematical logic, her husband’s honor. particularly as adviser to a large number of excellent Ph.D. The AMS was asked by Wells Fargo Bank of California, students. The Sacks Prize consists of a cash award plus the managers of the Bergman Trust, to assemble a com- five years’ free membership in the ASL. mittee to select recipients of the prize. In addition, the Society assisted Wells Fargo in interpreting the terms of —From an ASL announcement the will to ensure sufficient breadth in the mathemati- cal areas in which the prize may be given. Awards are made every one or two years in the following areas: (1) AAAS Fellows for 2011 the theory of the kernel function and its applications in real and complex analysis and (2) function-theoretic Eight mathematicians have been elected as fellows to the methods in the theory of partial differential equations of Section on Mathematics of the American Association for elliptic type with attention to Bergman’s operator method. the Advancement of Science (AAAS) for 2011. They are: Douglas N. Arnold, ; H. T. Banks, —Allyn Jackson North Carolina State University; Donald Burkholder, University of Illinois, Urbana-Champaign; James Carlson, Clay Mathematics Institute; Raúl E. Curto, University of Iowa; Charles W. Groetsch, The Citadel; James (Mac) Anantharaman Awarded 2010 Hyman, Tulane University; Philip C. Kutzko, University Salem Prize of Iowa; Yousef Saad, University of Minnesota; and Ken- neth Stephenson, University of Tennessee, Knoxville. Nalini Anantharaman of the Université Paris-Sud has been awarded the 2010 Salem Prize for her work in the —From an AAAS announcement semiclassical analysis of the Schrödinger equation corre- sponding to quantizations of classically chaotic systems, in particular for her proof that quantum limits in such Cora Sadosky (1940–2010) systems have positive entropy. The prize committee consisted of J. Bourgain, C. Fef- Cora Sadosky, a professor of mathematics at Howard ferman, P. Jones, N. Nikolski, G. Pisier, P. Sarnak, and University and a former president of the Association J.-C. Yoccoz. The Salem Prize is awarded yearly to young for Women in Mathematics who also served on the AMS researchers for outstanding contributions in the field Council in the 1980s and 1990s, died on December 2, of analysis. The previous recipients are N. Varopoulos, 2010. The following short account of her life is taken R. Hunt, Y. Meyer, C. Fefferman, T. Körner, E. M. Nikišin, from the biography that appears on the “Biographies of H. Montgomery, W. Beckner, M. R. Herman, S. B. Boˇckarëv, Women Mathematicians” website created by Larry Riddle B. E. Dahlberg, G. Pisier, S. Pichorides, P. Jones, A. B. of Agnes Scott College (see http://www.agnesscott. Aleksandrov, J. Bourgain, C. Kenig, T. Wolff, N. G. Ma- edu/lriddle/women). karov, G. David, J. L. Journé, A. L. Volberg,´ J.-C. Yoccoz, Cora Sadosky was born on May 23, 1940, in Buenos S. V. Konyagin, C. McMullen, M. Shishikura, S. Treil, Aires, . Her mother, Cora Ratto de Sadosky, K. Astala, H. Eliasson, M. Lacey, C. Thiele, T. Wooley, later became a professor of mathematics at the University F. Nazarov, T. Tao, O. Schramm, S. Smirnov, X. Tolsa, of , and her father, Manuel Sadosky, was a E. Lindenstrauss, K. Soundararajan, B. Green, A. Avila, founding director of the Computer Science Center at the S. Petermichl, A. Venkatesh, B. Klartag, and A. Naor. university. Cora Sadosky entered the University of Buenos Aires at —Salem Prize Committee the age of fifteen with the intention of majoring in physics

APRIL 2011 NOTICES OF THE AMS 613 Mathematics People 2011 New Orleans, LA, Joint Mathematics Meetings Photo Key but switched to mathematics after her first semester. Dur- ing her undergraduate years she became one of the first 13 1 12 14 students of Antoni Zygmund and Alberto Calderón during 2 3 their periodic visits to the University of Buenos Aires from 16 the University of Chicago. After receiving her undergradu- 4 5 6 15 17 ate degree she went to the University of Chicago, where she earned her Ph.D. in 1965 under the direction of Calderón 7 18 19 and Zygmund. At that time she was the only woman Ph.D. student in all the sciences at Chicago. Sadosky then returned to Argentina to marry Daniel 10 21 22 Goldstein, an Argentinean physician she had met while 20 8 9 he was studying molecular biology at Yale University. At 11 23 24 that time opportunities for research and teaching in Ar- gentina were good. What quickly followed, however, were 1. Opening ceremony for the Exhibits. Left to right: turbulent times as a military dictatorship took control Don McClure (AMS), Tina Straley (MAA), George Andrews of the country. Sadosky taught for a year as an assistant (AMS), David Bressoud (MAA), Barbara Faires (MAA), Bob professor of mathematics at the University of Buenos Daverman (AMS). Aires but joined many of her fellow faculty members in 2. JMM participants entering the Exhibits area. a protest resignation after a brutal assault by the police on the School of Sciences. After one semester teaching at 3. AMS-MAA Invited Address speaker, Chuu-Lian Terng. the National University, she was appointed an 4. View of the Mississippi from the New Orleans Convention assistant professor at Johns Hopkins University, where Center. her husband held a postdoctoral position. When Sadosky and her husband returned to Argentina 5. John Milnor receiving Steele Prize for Lifetime Achievement. in 1968, no academic positions were available for her, and 6. Evan O’Dorney (left), winner of the Who Wants to Be a she was forced out of mathematics for several years. Her Mathematician game, and Anthony Grebe, finalist. daughter was born in 1971, and two years later Sadosky re- turned to mathematical research when she began a lengthy 7. Patricia Campbell, winner of AWM Louise Hay Award. collaboration with Mischa Cotlar, who had been her 8. JMM Art Exhibit. mother’s Ph.D. advisor. The next year, however, Sadosky

9. , winner of MAA . and her family were forced to leave Argentina once again because of the social and political unrest. What followed 10. Thomas Mrowka receives the Joseph L. Doob Prize from were positions at the Central University of Venezuela, the AMS President George Andrews. Institute for Advanced Study in Princeton, and finally a 11. Email Center. professorship at Howard University in 1980, where she remained until her retirement. 12. Assaf Naor (left), receiving the AMS Bôcher Prize from Sadosky wrote more than fifty papers in harmonic George Andrews. analysis and operator theory, almost thirty of which were 13. One of the booths in the Exhibits area. coauthored with Cotlar. In 1979 she published a graduate textbook, Interpolation of Operators and Singular Inte- 14. Amie Wilkinson, winner of the AWM Ruth Lyttle Satter Prize. grals: An Introduction to Harmonic Analysis. 15. Mike Breen, host of the Who Wants to Be a Mathematician Cora Sadosky’s mother was a founding member of the game. International Women’s Union in 1945. It is probably not surprising then that Sadosky was herself a strong advocate 16. Ingrid Daubechies, winner of the Steele Prize for Seminal for women in mathematics as well as active in promoting Contribution to Research. the greater participation of African-Americans in math- 17. Contestants in the Who Wants to Be a Mathematician game. ematics. She was president of the Association for Women in Mathematics from 1993 to 1995. Her AMS service in- 18. Ribbon-cutting for Exhibits Opening. cludes two stints on the Council (1987–88 and 1995–98), 19. Invited Address audience. as well as membership on the Committee on Coopera-

20. Gunther Uhlmann, AMS Bôcher Prize winner. tion with Latin American Mathematicians (1990–92), the Committee on the Profession (1995–96), the Committee 21. Henryk Iwaniec receiving the Steele Prize for Mathematical on Science Policy (1996–98), the Committee on Human Exposition from George Andrews. Rights of Mathematicians (1990–96), and the Nominat- 22. AMS Colloquium Lecturer Andrew Lubotzky. ing Committee (2001–03). Sadosky was a Fellow of the American Association for the Advancement of Science. 23, 24. AMS Booth in the Exhibits area. —Allyn Jackson

614 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Mathematics Opportunities

Schauder Medal Award Call for Proposals for Established 2012 NSF-CBMS Regional The Scientific Board of the Juliusz Schauder Centre for Conferences Nonlinear Studies at Nicolaus Copernicus University (NCU) in Toru´n, Poland, has initiated the Honorary Juliusz To stimulate interest and activity in mathematical re- Schauder Medal Award competition. The medal will be search, the National Science Foundation (NSF) intends to given every other year to honor outstanding scientific support up to seven NSF-CBMS Regional Research Confer- achievements in the field of topological methods in non- ences in 2012. A panel chosen by the Conference Board of linear mathematical analysis. The basis for the choice of the Mathematical Sciences will make the selections from the awardee of the medal is original published scientific among the submitted proposals. papers, dissertations, and/or monographs. The awardee Each five-day conference features a distinguished lec- will receive a medal and an expense-paid invitation to turer who delivers ten lectures on a topic of important deliver a lecture at Copernicus University. current research in one sharply-focused area of the math- The first Schauder Medal will be presented in ematical sciences. The lecturer subsequently prepares an 2012. Nominations are welcome and should be sent expository monograph based on these lectures, which is to the prize jury by December 21, 2011. Further in- normally published as a part of a regional conference se- formation is available from Lech Gorniewicz, man- ries. Depending on the conference topic, the monograph ager of the Schauder Center, [email protected]. will be published by the American Mathematical Society, by the Society for Industrial and Applied Mathematics, —From a Schauder Center announcement or jointly by the American Statistical Association and the Institute of Mathematical Statistics. Support is provided for about thirty participants at DMS Workforce Program in the each conference, and both established researchers and interested newcomers, including postdoctoral fellows and Mathematical Sciences graduate students, are invited to attend. The proposal due date is April 15, 2011. For further information on submit- The Division of Mathematical Sciences (DMS) of the Na- ting a proposal, consult the CBMS website, http://www. tional Science Foundation (NSF) welcomes proposals for cbmsweb.org/NSF/2012_call.htm, or contact: Confer- the Workforce Program in the Mathematical Sciences. ence Board of the Mathematical Sciences, 1529 Eighteenth The long-range goal of the program is increasing the Street, NW, Washington, DC 20036; telephone: 202-293- number of well-prepared U.S. citizens, nationals, and 1170; fax: 202-293-3412. permanent residents who successfully pursue careers in the mathematical sciences and in other NSF-supported —From a CBMS announcement disciplines. Of primary interest are activities centered on education that broaden participation in the mathematical sciences through research involvement for trainees at the undergraduate through postdoctoral educational levels. The program is particularly interested in activities that NSF-CBMS Regional improve recruitment and retention, educational breadth, Conferences, 2011 and professional development. The submission period for unsolicited proposals is With funding from the National Science Foundation (NSF), May 15–June 15, 2011. For more information and a list of the Conference Board of the Mathematical Sciences (CBMS) cognizant program directors, go to the website http://www. will hold six NSF-CBMS Regional Research Conferences dur- nsf.gov/funding/pgm_summ.jsp?pims_id=503233. ing the summer of 2011. These conferences are intended to stimulate interest and activity in mathematical research. —From a DMS announcement Each five-day conference features a distinguished lecturer

APRIL 2011 NOTICES OF THE AMS 615 Mathematics Opportunities who delivers ten lectures on a topic of important current has encouraged female undergraduate students to pursue research in one sharply-focused area of the mathematical mathematical careers and/or the study of mathematics at sciences. The lecturer subsequently prepares an exposi- the graduate level. The recipient will receive a cash prize tory monograph based on these lectures. and honorary plaque and will be featured in an article in Support for about thirty participants will be provided the AWM newsletter. The award is open to all regardless for each conference. Both established researchers and of nationality and citizenship. Nominees must be living interested newcomers, including postdoctoral fellows at the time of their nomination. and graduate students, are invited to attend. Information The deadline for nominations is April 30, about an individual conference may be obtained by con- 2011. For details, see www.awm-math.org, tele- tacting the conference organizer. The conferences to be phone 703-934-0163, or email [email protected]. held in 2011 are as follows. May 16–20, 2011: Deformation Theory of Algebras —From an AWM announcement and Modules. Martin Markl, lecturer. North Carolina State University. Organizers: Kailash C. Misra, 919-515-8784, [email protected]; and Thomas J. Lada, 919-515- Project NExT: New Experiences 8773, [email protected]. Conference website: www4. ncsu.edu/~lada/nsfcbms.htm. in Teaching June 18–23, 2011: Ergodic Methods in the Theory of Project NExT (New Experiences in Teaching) is a profes- . Harry Furstenberg, lecturer. Kent State University. sional development program for new and recent Ph.D.’s Organizers: Dmitry Ryabogin, 330-672-9085, ryabogin@ in the mathematical sciences (including pure and ap- math.kent.edu; and Artem Zvavitch, 330-672-3316, plied mathematics, statistics, operations research, and [email protected]. Conference website: www. mathematics education). It addresses all aspects of an kent.edu/math/events/conferences/cbms2011.cfm. academic career: improving the teaching and learning of June 20–24, 2011: Global Harmonic Analysis. Steve mathematics, engaging in research and scholarship, and Zelditch, lecturer. University of Kentucky. Organizers: participating in professional activities. It also provides the Peter Hislop, 859-257-6791, [email protected]; and participants with a network of peers and mentors as they Peter Perry, 859-257-5637, [email protected]. Confer- assume these responsibilities. In 2011 about sixty faculty ence website: www.math.uky.edu/cbms. members from colleges and universities throughout the June 20–24, 2011: Radial Basis Functions: Math- country will be selected to participate in a workshop pre- ematical Developments and Applications. Bengt ceding the Mathematical Association of America (MAA) Fornberg and Natasha Flyer, lecturers. University of summer meeting, in activities during the summer MAA Massachusetts, Dartmouth. Organizers: Saeja Kim, 508- meetings in 2011 and 2012 and the Joint Mathematics 999-8325, [email protected]; Sigal Gottlieb, 508-999-8205, Meetings in January 2012, and in an electronic discussion [email protected]; Alfa Heryudono, 508-999-8316, network. Faculty for whom the 2011–2012 academic year [email protected]; and Cheng Wang, 508-999- will be the first or second year of full-time teaching (post- 8342, [email protected]. Conference website: www. Ph.D.) at the college or university level are invited to apply umassdcomputing.org/conference/rbfcbms2011. to become Project NExT Fellows. July 25–29, 2011: Mathematical Epidemiology with Applications for the 2011–12 Fellowship year will be Applications. Carlos Castillo-Chavez and Fred Brauer, lec- due April 15, 2011. For more information see the Proj- turers. East Tennessee State University. Organizers: Ariel ect NExT website, http://archives.math.utk.edu/ Cintron-Arias, 423-439-7065, [email protected]; projnext/, or contact Aparna Higgins, director, at and Anant P. Godbole, 423-439-5359, godbolea@etsu. [email protected]. Project NExT is a edu. Conference website: www.etsu.edu/cas/math/ program of the MAA. It receives major funding from the cbms.aspx. ExxonMobil Foundation, with additional funding from the August 1–5, 2011: 3-Manifolds, Artin Groups, and Cu- Dolciani-Halloran Foundation, the Educational Advance- bical Geometry. Daniel T. Wise, lecturer. City University ment Foundation, the American Mathematical Society, the of New York. Organizer: Jason Behrstock, 347-661-0835, American Statistical Association, the National Council of [email protected] . Conference web- Teachers of Mathematics, American Institute of Mathemat- site: comet.lehman.edu/behrstock/cbms/. ics, the Association for Symbolic Logic, the W. H. Freeman Publishing Company, Texas Instruments, John Wiley & —From a CBMS announcement Sons, MAA Sections, and the Greater MAA Fund. AWM Gweneth Humphreys —Project NExT announcement Award The Association for Women in Mathematics (AWM) spon- sors the Gweneth Humphreys Award to recognize out- standing mentorship activities. This prize will be awarded annually to a mathematics teacher (female or male) who

616 NOTICES OF THE AMS VOLUME 58, NUMBER 4 For Your Information

event and ensure that students from around the world can Mathematics Awareness continue to demonstrate their passion for mathematics. Month—April 2011 Robbert Dijkgraaf, president of the Royal Netherlands Academy of Arts and Sciences and chair of the IMO 2011 Foundation, said of the gift: “Mathematics is a field in Unraveling Complex Systems which talents can shine at a very young age. This gener- How do epidemics spread, birds flock, and stock markets ous gift of Google will allow the brightest young math- operate? ematicians to show their amazing abilities to the world. Many of these answers fall within the realm of math- The scientific community is grateful for this wonderful ematics. support of Google and the recognition it expresses of the From natural entities such as living cells, insect colo- fundamental importance of mathematics to our society.” nies and whole ecosystems to man-made inventions like Peter Barron, director of External Relations for Google, said: “The International Math Olympiad is an event which power grids, transportation networks and the World Wide demonstrates both the extraordinary abilities of the Web, we see complex systems everywhere. Deciphering students who take part and the value to wider society of the mathematics behind such systems can unravel well- mathematics. We are delighted to be able to support the structured networks and discernible patterns in natural event over the next five years and to encourage excellence and artificial structures. That is the idea behind Math- in mathematics around the world.” ematics Awareness Month, April 2011. Understanding The International Mathematical Olympiad is the world these complex systems cannot only help us manage and championship of secondary school mathematics, designed improve the reliability of such critical infrastructures of to test ingenuity and insight and tax the sharpest minds everyday life, but can also allow us to interpret, enhance in the world. It is held each July at locations around the and better interact with natural systems. Mathematical world. About one hundred nations compete each year. The models can delineate interactions among components of 2011 IMO will be held in the Netherlands. these systems, analyze their spontaneous and emergent Further information can be obtained from the follow- behaviors, and thus help prevent undesirable develop- ing: Peter Barron, Director of External Relations, Google ments while enhancing desirable traits during their ad- (through Mark Jansen, +31615129329); Wim Berkelmans, aptation and evolution. Director of IMO 2011 in the Netherlands (+31 6 53323968, In an effort to improve our understanding of such sys- [email protected]); Robbert Dijkgraaf, President of the tems, the Joint Policy Board of Mathematics has chosen the Royal Netherlands Academy of Arts and Sciences; and theme, “Unraveling Complex Systems” to highlight the role Geoff Smith, University of Bath, U.K., member of the IMO of mathematics in the discipline. The 2011 Mathematics Advisory Board (+44 7941147895, G.C.Smith@bath. Awareness website www.mathaware.org has articles and ac.uk). The official IMO site is http://www.imo-offi- other resources to help explain the math behind such cial.org/, which provides a detailed historical record. diverse systems as our dynamic response to HIV infec- —From an IMO/Google announcement tions to production links that determine product trade between countries.

—Society for Industrial and Applied Mathematics Corrections The introduction to the article “Interview with Abel lau- reate ” described Tate as the recipient of the Google Donation to Support 2009 Abel Prize when he was actually the 2010 Abel Prize winner. IMO In the same article, on page 449, the article refers to “K2 groups of number fields”. It should have said “K groups Google has donated 1 million euros (approximately 2 of number fields”. US$1,370,000) to the Advisory Board of the International The Notices apologizes for these errors. Mathematical Olympiad (IMO) to support the next five an- —Sandy Frost nual International Mathematical Olympiads (from 2011 to 2015). This grant will help cover the costs of this global

APRIL 2011 NOTICES OF THE AMS 617 Reference and Book List

The Reference section of the Notices April 15, 2011: Applications May 1, 2011: Applications for is intended to provide the reader for Project NExT Fellowship for National Academies Christine Mirza- with frequently sought information in the 2011–2012 year. See http:// yan Graduate Fellowship Program an easily accessible manner. New archives.math.utk.edu/ for fall 2011. See http://sites. information is printed as it becomes projnext/, or contact Aparna Hig- nationalacademies.org/PGA/ available and is referenced after the gins, director, at Aparna.Higgins@ policyfellows/index.htm. first printing. As soon as information notes.udayton.edu. See “Mathe- May 1, 2011: Applications for is updated or otherwise changed, it matics Opportunities” in this issue. AWM Travel Grants. See http:// will be noted in this section. April 30, 2011: Nominations for www.awm-math.org/travelgrants. AWM Gweneth Humphreys Award. html#standard. Contacting the Notices May 15–June 15, 2011: Submis- See “Mathematics Opportunities” in The preferred method for contacting sion period for DMS Workforce Pro- this issue. the Notices is electronic mail. The gram in the Mathematical Sciences. May 1, 2011: Applications for editor is the person to whom to send See “Mathematics Opportunities” in May review for National Acad- articles and letters for consideration. this issue. emies Research Associateship Articles include feature articles, me- August 1, 2011: Applications morial articles, communications, Programs. See the National Acad- for August review for National opinion pieces, and book reviews. emies website at http://sites. Academies Research Associateship The editor is also the person to whom nationalacademies.org/PGA/RAP/ Programs. See the National Acad- to send news of unusual interest PGA_050491 or contact Research emies website at http://sites. about other people’s mathematics Associateship Programs, National nationalacademies.org/PGA/RAP/ research. Research Council, Keck 568, 500 Fifth PGA_050491 or contact Research The managing editor is the person Street, NW, Washington, DC 20001; Associateship Programs, National to whom to send items for “Math- telephone 202-334-2760; fax 202- Research Council, Keck 568, 500 Fifth ematics People”, “Mathematics Op- 334-2759; email [email protected]. Street, NW, Washington, DC 20001; portunities”, “For Your Information”, “Reference and Book List”, and “Math- Where to Find It ematics Calendar”. Requests for A brief index to information that appears in this and previous issues of the Notices. permissions, as well as all other AMS Bylaws—November 2009, p. 1320 inquiries, go to the managing editor. The electronic-mail addresses are AMS Email Addresses—February 2011, p. 326 [email protected] in the AMS Ethical Guidelines—June/July 2006, p. 701 case of the editor and notices@ AMS Officers 2008 and 2009 Updates—May 2010, p. 670 ams.org in the case of the managing AMS Officers and Committee Members—October 2010, p. 1152 editor. The fax numbers are 314- Conference Board of the Mathematical Sciences—September 2010, 935-6839 for the editor and 401- p. 1009 331-3842 for the managing editor. IMU Executive Committee—December 2010, page 1488 Postal addresses may be found in the Information for Notices Authors—June/July 2010, p. 768 masthead. Mathematics Research Institutes Contact Information—August 2010, Upcoming Deadlines p. 894 March 31, 2011: Applications for National Science Board—January 2011, p. 77 AMS-Simons Travel Grants for Early- New Journals for 2008—June/July 2009, p. 751 Career Mathematicians. See http:// NRC Board on Mathematical Sciences and Their Applications—March www.ams.org/programs/travel- 2011, p. 482 grants/AMS-SimonsTG. NRC Mathematical Sciences Education Board—April 2011, p. 619 March 31, 2011: Nominations for NSF Mathematical and Physical Sciences Advisory Committee—February the Academy of Sciences for the De- 2011, p. 329 veloping World (TWAS) prizes. See Program Officers for Federal Funding Agencies—October 2010, http://www.twas.org/. p. 1148 (DoD, DoE); December 2010, page 1488 (NSF Mathematics Education) April 15, 2011: Proposals for 2012 Program Officers for NSF Division of Mathematical Sciences—Novem- NSF-CBMS Regional Conferences. See ber 2010, p. 1328 “Mathematics Opportunities” in this issue.

618 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Reference and Book List telephone 202-334-2760; fax 202- MSEB Staff *The Calculus of Selfishness, by 334-2759; email [email protected]. David R. Mandel, Director Karl Sigmund. Princeton University October 1, 2011: Applications for Contact information is: Mathemat- Press, January 2010. ISBN-13: 978- 06911-427-53. AWM Travel Grants. See http:// ical Sciences Education Board, Na- *The Clockwork Universe: Isaac www.awm-math.org/travelgrants. tional Academy of Sciences, 500 Fifth Newton, the Royal Society, and the html#standard. Street, NW, 11th Floor, Washington, Birth of the Modern World, by Edward October 1, 2011: Nominations for DC 20001; telephone 202-334-2353; Dolnick. Harper, February 2011. ISBN- the 2012 Emanuel and Carol Parzen fax 202-344-2210; email: cfeinq@ Prize. Contact Thomas Wehrly, De- 13: 978-00617-195-16. nas.edu; World Wide Web http:// partment of Statistics, 3143 TAMU, *Complexity: A Guided Tour, by www7.nationalacademies.org/ Texas A&M University, College Sta- Melanie Mitchell. Oxford University MSEB/1MSEB_Membership.html. tion, Texas 77843-3143. Press, April 2009. ISBN-13: 978- 01951-244-15. (Reviewed in this November 1, 2011: Applications Book List for November review for National issue.) The Book List highlights books that The Cult of Statistical Significance: Academies Research Associateship have mathematical themes and are How the Standard Error Costs Us Programs. See the National Acad- aimed at a broad audience potentially Jobs, Justice, and Lives, by Stephen T. emies website at http://sites. including mathematicians, students, Ziliak and Deirdre N. McCloskey, nationalacademies.org/PGA/RAP/ and the general public. When a book University of Michigan Press, Febru- PGA_050491 or contact Research has been reviewed in the Notices, a ary 2008. ISBN-13: 978-04720-500-79. Associateship Programs, National reference is given to the review. Gen- (Reviewed October 2010.) Research Council, Keck 568, 500 Fifth erally the list will contain only books Duel at Dawn: Heroes, Martyrs, and Street, NW, Washington, DC 20001; the Rise of Modern Mathematics, by telephone 202-334-2760; fax 202- published within the last two years, though exceptions may be made in Amir Alexander. Harvard University 334-2759; email [email protected]. Press, April 2010. ISBN-13: 978- cases where current events (e.g., the December 21, 2011: Nominations 06740-466-10. (Reviewed November death of a prominent mathemati- for the Schauder Medal. See “Math- 2010.) cian, coverage of a certain piece of ematics Opportunities” in this issue. Euler’s Gem: The Polyhedron For- mathematics in the news) warrant mula and the Birth of Topology, by Mathematical Sciences drawing readers’ attention to older David S. Richeson. Princeton Univer- Education Board, National books. Suggestions for books to sity Press, September 2008. ISBN-13: Research Council include on the list may be sent to 978-06911-267-77. (Reviewed Decem- Jan de Lange, Freudenthal Institute, [email protected]. ber 2010.) The Netherlands *Added to “Book List” since the The Grand Design, by Stephen Keisha M. Ferguson, Pattengill El- list’s last appearance. Hawking and Leonard Mlodinow. ementary School, Ann Arbor, MI Apocalypse When?: Calculating Bantam, September 2010. ISBN-13: Louis Gomez, Northwestern Uni- How Long the Human Race Will Sur- 978-05538-053-76. versity vive, by Willard Wells. Springer Praxis, Here’s Looking at Euclid: A Surpris- Javier Gonzalez, Pioneer High June 2009. ISBN-13: 978-03870-983- ing Excursion through the Astonish- School, Whittier, CA 64. ing World of Math, by Alex Bellos. Sharon Griffin, Clark University The Best Writing on Mathematics: Free Press, June 2010. ISBN-13: 978- Phillip A. Griffiths (chair), Institute 2010, edited by Mircea Pitici. Prince- 14165-882-52. for Advanced Study ton University Press, December 2010. *Hidden Harmonies (The Lives Arthur Jaffe, Harvard University ISBN-13: 978-06911-484-10. and Times of the Pythagorean Theo- rem), by Robert and Ellen Kaplan. Jeremy Kilpatrick, University of The Black Swan: The Impact of the Bloomsbury Press, January 2011. Georgia Highly Improbable, by Nassim Nicho- ISBN-13:978-15969-152-20. Julie Legler, St. Olaf College las Taleb. Random House Trade Pa- The Housekeeper and the Profes- W. James Lewis, University of Ne- perbacks, second edition, May 2010. sor, by Yoko Ogawa. Picador, February braska, Lincoln ISBN-13: 978-08129-738-15. (First 2009. ISBN-13: 978-03124-278-01. (Re- Kevin F. Miller, University of Michi- edition reviewed March 2011.) viewed May 2010.) gan, Ann Arbor Bright Boys: The Making of Infor- How to Read Historical Mathematics, Marge Petit (vice chair), Consul- mation Technology, by Tom Green. by Benjamin Wardhaugh. Princeton tant, Fayston, VT , April 2010. ISBN-13: 978- University Press, March 2010. ISBN-13: Donald Saari, University of Cali- 1-56881-476-6. 978-06911-401-48. fornia, Irvine The Calculus of Friendship: What a Isaac Newton on Mathematical Nancy J. Sattler, Terra State Com- Teacher and Student Learned about Certainty and Method, by Niccolò munity College, Freemont, OH Life While Corresponding about Math, Guicciardini. MIT Press, October 2009. Richard J. Schaar, Texas Instru- by Steven Strogatz. Princeton Uni- ISBN-13: 978-02620-131-78. ments versity Press, August 2009. ISBN-13: Logicomix: An Epic Search for Truth, Frank Wang, Oklahoma School of 978-0691-13493-2. (Reviewed June/ by Apostolos Doxiadis and Christos Science and Mathematics July 2010.) Papadimitriou. Bloomsbury USA,

APRIL 2011 NOTICES OF THE AMS 619 Reference and Book List

September 2009. ISBN-13: 978-15969- Jason Brown. Emblem Editions, April The Strangest Man, by Graham 145-20. (Reviewed December 2010.) 2010. ISBN-13: 978-07710-169-74. Farmelo. Basic Books, August 2009. *Loving + Hating Mathematics: Chal- Perfect Rigor: A Genius and the ISBN-13: 978-04650-182-77. lenging the Myths of Mathematical Life, Mathematical Breakthrough of the Street-Fighting Mathematics: The by and Vera John-Steiner. Century, by Masha Gessen. Houghton Art of Educated Guessing and Oppor- Princeton University Press, January 2011. Mifflin Harcourt, November 2009. tunistic Problem Solving, by Sanjoy ISBN-13: 978-06911-424-70. ISBN-13: 978-01510-140-64. (Re- Mahajan. MIT Press, March 2010. The Math Book: From Pythagoras to viewed January 2011.) ISBN-13: 978-0262-51429-3. the 57th Dimension, 250 Milestones in Pioneering Women in American Survival Guide for Outsiders: How the History of Mathematics, by Clifford Mathematics: The Pre-1940 Ph.D.’s, to Protect Yourself from Politicians, A. Pickover. Sterling, September 2009. by Judy Green and Jeanne LaDuke. Experts, and Other Insiders, by Sher- ISBN-13: 978-14027-579-69. AMS, December 2008. ISBN-13: 978- man Stein. BookSurge Publishing, A Mathematician’s Lament: How 08218-4376-5. February 2010. ISBN-13: 978-14392- School Cheats Us Out of Our Most Fas- Plato’s Ghost: The Modernist Trans- 532-74. cinating and Imaginative Art Form, by formation of Mathematics, by Jeremy Symmetry: A Journey into the Pat- Paul Lockhart. Bellevue Literary Press, Gray. Princeton University Press, Sep- terns of Nature, by Marcus du Sau- April 2009. ISBN-13:978-1-934137- tember 2008. ISBN-13: 978-06911- toy. Harper, March 2008. ISBN: 978- 17-8. 361-03. (Reviewed February 2010.) 00607-8940-4. (Reviewed February 2011.) Mathematicians: An Outer View Probabilities: The Little Numbers Symmetry in Chaos: A Search for of the Inner World, by Mariana Cook. That Rule Our Lives, by Peter Olofs- Pattern in Mathematics, Art, and Na- Princeton University Press, June 2009. son. Wiley, March 2010. ISBN-13: 978- ture, by Michael Field and Martin ISBN-13: 978-06911-3951-7. (Reviewed 04706-244-56. August 2010.) Golubitsky. Society for Industrial *Problem-Solving and Selected Top- and Applied Mathematics, second Mathematicians Fleeing from Nazi ics in Number Theory in the Spirit Germany: Individual Fates and Global revised edition, May 2009. ISBN-13: of the Mathematical Olympiads, by 978-08987-167-26. Impact, by Reinhard Siegmund- Michael Th. Rassias. Springer, 2011. Schultze. Princeton University Press, Teaching Statistics Using Base- ISBN-13: 978-1-4419-0494-2. July 2009. ISBN-13: 978-06911- ball, by James Albert. Mathematical Proofs from THE BOOK, by Martin 4041-4. (Reviewed November 2010.) Association of America, July 2003. Aigner and Günter Ziegler. Expanded A Motif of Mathematics: History ISBN-13: 978-08838-572-74. (Re- fourth edition, Springer, October and Application of the Mediant and viewed April 2010.) 2009. ISBN-13: 978-3-642-00855-9. the Farey Sequence, by Scott B. What’s Luck Got to Do with It? The Pythagoras’ Revenge: A Math- Guthery. Docent Press, September History, Mathematics and Psychology ematical Mystery, by Arturo San- 2010. ISBN-13:978-4538-105-76. of the Gambler’s Illusion, by Joseph galli. Princeton University Press, May Mrs. Perkins’s Electric Quilt: And Mazur. Princeton University Press, 2009. ISBN-13: 978-06910-495-57. Other Intriguing Stories of Mathemati- July 2010. ISBN: 978-069-113890-9. (Reviewed May 2010.) cal Physics, Paul J. Nahin, Princeton University Press, August 2009. ISBN-13: Recountings: Conversations with 978-06911-354-03. MIT Mathematicians, edited by Joel Naming Infinity: A True Story of Segel. A K Peters, January 2009. ISBN- Religious Mysticism and Mathemati- 13: 978-15688-144-90. cal Creativity, by Loren Graham and Roger Boscovich, by Radoslav Jean-Michel Kantor. Belknap Press Dimitric (Serbian). Helios Publishing of Harvard University Press, March Company, September 2006. ISBN-13: 2009. ISBN-13: 978-06740-329-34. 978-09788-256-21. Nonsense on Stilts: How to Tell Sci- The Shape of Inner Space: String ence from Bunk, by Massimo Pigliucci. Theory and the Geometry of the Uni- University of Chicago Press, May verse's Hidden Dimensions, by Shing- 2010. ISBN-13: 978-02266-678-67. Tung Yau (with Steve Nadis). Basic (Reviewed in this issue.) Books, September 2010. ISBN-13: Numbers Rule: The Vexing Math- 978-04650-202-32. (Reviewed Febru- ematics of Democracy, from Plato to the ary 2011.) Present, by George G. Szpiro. Princeton The Solitude of Prime Numbers, University Press, April 2010. ISBN-13: by Paolo Giordano. Pamela Dorman 978-06911-399-44. (Reviewed January Books, March 2010. ISBN-13: 978- 2011.) 06700-214-82. (Reviewed September The Numerati, by Stephen Baker. 2010.) Houghton Mifflin, August 2008. ISBN- Solving Mathematical Problems: A 13: 978-06187-846-08. (Reviewed Personal Perspective, by Terence Tao. October 2009.) Oxford University Press, September Our Days Are Numbered: How 2006. ISBN-13: 978-0199-20560-8. (Re- Mathematics Orders Our Lives, by viewed February 2010.)

620 NOTICES OF THE AMS VOLUME 58, NUMBER 4 A MERICAN MATHEMATICAL SOCIETY

Leroy P. Steele Prizes

The selection committee for these prizes requests nominations for consideration for the 2012 awards. Further information about the prizes can be found in the November 2009 Notices, pp. 1326–1345 (also avail- able at http://www.ams.org/prizes-awards).

Three Leroy P. Steele Prizes are awarded each year in the following cat- egories: (1) the Steele Prize for Lifetime Achievement: for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) the Steele Prize for Mathematical Exposition: for a book or substantial sur- vey or expository-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 2012 the prize for Seminal Contribution to Research will be awarded for a paper in geometry/topology.

Nominations with supporting information should be submitted to the Secretary, Robert J. Daverman, American Mathematical Society, 1403 Circle Drive, Department of Mathematics, University of Tennessee, Knoxville TN 37996-1320. Include a short descrip- tion of the work that is the basis of the nomination, including complete bibliographic citations. A curriculum vitae should be included. The nominations will be forwarded by the Secretary to the prize selection committee, which will make final decisions on the awarding of prizes.

Deadline for nominations is May 31, 2011. Call for Nominations SECRETARY

Position

The American Mathematical Society is seeking candidates for the position of Secretary, one of the most important and infl uential positions within the Society. The Secretary partici- pates in formulating policy for the Society, participates actively in governance activities, plays a key role in managing committee structures, oversees the scientifi c program of meetings, and helps to maintain institutional memory.

The fi rst term of the new AMS Secretary will begin February 1, 2013, with initial appointment expected in Fall 2011 in order that the Secretary-designate may observe the conduct of Society business for a full year before taking offi ce.

All necessary expenses incurred by the Secretary in performance of duties for the Society are reimbursed, including travel and communications. The Society is prepared to negotiate a fi nancial arrangement with the successful candidate and his/her employer in order that the new Secretary be granted suffi cient release time to carry out the many functions of the offi ce.

Qualifi cations

The Secretary should be a research mathematician and must have substantial knowledge of Society activities. Although the AMS Secretary is appointed by the Council for a term of two years, candidates should be willing to make a long-term commitment, for it is expected that the new Secretary will be reappointed for subsequent terms pending successful performance reviews.

Duties of the offi ce include:

Organizing and coordinating the Council and its committees. Serving as ex offi cio member of the Council, the Executive Committee, the Agenda and Budget Committee, the Liaison Committee, the Long Range Planning Committee, the Committee on Meeting and Conferences, the Committee on the Profession, and the Committee on Publications. The Secretary also serves as a non-voting member of the Committee on Education and the Committee on Science Policy. Working closely with the President to coordinate and administer the activities of committees. Overseeing, together with the Associate Secretaries, the scientifi c program of all Society meetings.

Applications

A Search Committee with as chair has been formed to seek and review applications. Persons wishing to be considered or to make a nomination are enthusiastically encouraged to inform

AMS Secretary Search Committee c/o Robert J. Daverman Department of Mathematics University of Tennessee Knoxville, TN 37996-1320

For full consideration, nominations and supporting documentation should be received before April 15, 2011.

AMS-Secretary-ad.indd 1 12/15/10 3:33 PM A MERICAN MATHEMATICAL SOCIETY AMS Award for Mathematics Programs Th at Make a Diff erence

Deadline: September 15, 2011

Th is award was established in 2005 in response to a recommendation from the AMS’s Committee on the Profession that the AMS compile and publish a series of profi les of υ programs that: + 1. aim to bring more persons from underrepresented backgrounds into some portion of the pipeline beginning at the undergraduate level and leading to advanced degrees in σ mathematics and professional success, or retain them once in the pipeline; 2. have achieved documentable success in doing so; and 3. are replicable models.

Preference will be given to programs with signifi cant participation by underrepresented 2 minorities. Two programs are highlighted annually.

Nomination process: Letters of nomination may be submitted by one or more individuals. Nomination of the writer’s own institution is permitted. Th e letter should describe the specifi c program(s) for which the department is being nominated as well as the achievements that make the program(s) an outstanding success, and may include any ancillary documents which support the success of the program. Th e letter of nomination should not exceed two pages, with supporting documentation not to exceed three more pages. Up to three supporting letters may be included in addition to these fi ve pages.

Send nominations to: Programs Th at Make a Diff erence c/o Ellen Maycock φ American Mathematical Society 201 Charles Street Providence, RI 02904 ζ or via email to [email protected] Recent Winners: η 2010: Department of Computational and Applied Mathematics (CAAM), Rice University; Summer Program in Quantitative Sciences, Harvard School of Public Health

2009: Department of Mathematics at the University of Mississippi; Department of Statistics at North Carolina State University.

2008: Summer Undergraduate Mathematical Science Research Institute (SUMSRI), εγ Miami University (Ohio); Mathematics Summer Program in Research and Learning (Math SPIRAL), University of Maryland, College Park.

2007: Enhancing Diversity in Graduate Education (EDGE), Bryn Mawr College and τ Spelman College; and Mathematical Th eoretical Biology Institute (MTBI), Arizona State University.

progs-that-make-ad-diff-ad.indd 1 1/28/11 8:33 AM Mathematics Calendar

April 2011 graduate students. This conference is supported by the National Science Foundation. * 16 Herbert Federer Memorial Conference, Brown University, Provi- Information: http://www.shsu.edu/~ldg005/combinatexas. dence, Rhode Island. Description: This one-day conference will feature six lectures rel- * 18–22 4th annual conference on Polynomial Computer Algebra evant to the work of Herbert Federer. 2011, The Euler International Mathematical Institute (EIMI), St. Pe- Speakers: William Allard, Wendell Fleming, Robert Hardt, Jean tersburg, Russia. Taylor, Brian White, and William Ziemer. There will be a banquet Description: The Conference will be devoted to modern polynomial algorithms in Computer Algebra which are gaining importance in in the evening. various applications of science as well as in fundamental researches. Information: http://artin.math.brown.edu/federer/. Topics: Groebner bases, Combinatorics of monomial orderings, * 16–17 CombinaTexas 2011, Sam Houston State University, Hunts- Differential bases, Involutive algorithms, Computational algebraic ville, Texas. Geometry, D-modules polynomial differential operators, Paralleliza- Description: CombinaTexas is an annual regional conference on tion of algorithms, Algorithms of tropical mathematics, Quantum Combinatorics, Graph Theory, and Computing. It is dedicated to the computing cryptography, Tropical manifolds, Matrix algorithms, enhancement of both the educational and the research atmosphere Complexity of algorithms and other. of the community of combinatorialists and graph theorists in Texas Information: http://www.pdmi.ras.ru/EIMI/2011/pca/ and surrounding states. The special focus of CombinaTexas 2011 index.html. will be algebraic combinatorics. * 25–29 The Kervaire Invariant and Stable Homotopy Theory, ICMS, Invited speakers: Federico Ardila (San Francisco State University), Edinburgh, Scotland. Chris Godsil (University of Waterloo), Gregg Musiker (University of Description: At the 2009 Atiyah80 ICMS Workshop, Hill, Hopkins, Minnesota), Michael Orrison (Harvey Mudd College), Rosa Orellana and Ravenel announced a proof that there do not exist manifolds (Dartmouth College), Bernhard Schmidt (Nanyang Technological with Kervaire invariant 1 in dimensions 2k − 2 for any k ≥ 8. This University), and Catherine Yan (Texas A&M University). Combina- solves the longstanding open Kervaire invariant problem (except Texas 2011 will also include several contributed talks by faculty, for dimension 126, the only dimension which now remains open), postdocs, and graduate students; and a poster session by under- which has been of central importance in homotopy theory for over

This section contains announcements of meetings and conferences in the mathematical sciences should be sent to the Editor of the Notices in of interest to some segment of the mathematical public, including ad care of the American Mathematical Society in Providence or electronically hoc, local, or regional meetings, and meetings and symposia devoted to [email protected] or [email protected]. to specialized topics, as well as announcements of regularly scheduled In order to allow participants to arrange their travel plans, organizers of meetings of national or international mathematical organizations. A meetings are urged to submit information for these listings early enough complete list of meetings of the Society can be found on the last page to allow them to appear in more than one issue of the Notices prior to of each issue. the meeting in question. To achieve this, listings should be received in An announcement will be published in the Notices if it contains a call Providence eight months prior to the scheduled date of the meeting. for papers and specifies the place, date, subject (when applicable), and The complete listing of the Mathematics Calendar will be published the speakers; a second announcement will be published only if there are only in the September issue of the Notices. The March, June/July, and changes or necessary additional information. Once an announcement December issues will include, along with new announcements, references has appeared, the event will be briefly noted in every third issue until to any previously announced meetings and conferences occurring it has been held and a reference will be given in parentheses to the month, year, and page of the issue in which the complete information within the twelve-month period following the month of those issues. appeared. Asterisks (*) mark those announcements containing new or New information about meetings and conferences that will occur later revised information. than the twelve-month period will be announced once in full and will In general, announcements of meetings and conferences carry only not be repeated until the date of the conference or meeting falls within the date, title of meeting, place of meeting, names of speakers (or the twelve-month period. sometimes a general statement on the program), deadlines for abstracts The Mathematics Calendar, as well as Meetings and Conferences of or contributed papers, and source of further information. If there is any the AMS, is now available electronically through the AMS website on application deadline with respect to participation in the meeting, this the World Wide Web. To access the AMS website, use the URL: http:// fact should be noted. All communications on meetings and conferences www.ams.org/.

624 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Mathematics Calendar

40 years. The principal objective is to disseminate the methods used Description: The scientific and organization committees of SETIT in the solution of the Kervaire invariant conjecture, and the use of have decided to postpone the conference to May 12–15, 2011. So the equivariant and motivic methods more generally. It is also hoped submission date is reopened to February 15, 2011. to assess the prospects for using the methods to prove structural Information: http://www.setit.rnu.tn. results about stable homotopy, and to report on progress which will * 13–15 Connections in Geometry and Physics: 2011 (GAP 2011), have taken place by the time of the meeting. Fields Institute for Research in Mathematical Sciences, Toronto, On- Information: http://www.icms.org.uk/workshops/ tario, Canada. kervaire. Description: Each year, the format of GAP combines three separate May 2011 but related themes in geometry and physics. This year’s themes are: advances in Floer Theory, geometric flows, and the AdS/CFT * 2–13 Pedagogical school on “Knots and links: from theory to ap- correspondence. plications”, Centro di Ricerca Matematica “”, Collegio Principal speakers: Michael Anderson (Stony Brook), Octav Cornea Puteano, Piazza dei Cavalieri 3, 56124, Pisa, Italy. (Montréal), Robin Graham (Washington), François Lalonde (Montréal), Description: This pedagogical school wants to offer an introduction John Lott (UC Berkeley), Robert McCann (Toronto), Tom Mrowka to the subject, aiming at covering in a brainstorming 2-week period (M.I.T.), André Neves (Imperial), Natasa Sesum (Rutgers), Pedro Vieira some of the fundamental topics in both pure and applied contexts. (Perimeter), Katrin Wehrheim (M.I.T.), Xi Yin (Harvard). Starting from the basic concepts of the theory of knots and links Organizers: Marco Gualtieri (Toronto), Spiro Karigiannis (Water- up to the forefront of topics of modern research, from knot poly- loo), Ruxandra Moraru (Waterloo), Rob Myers (Perimeter), McKenzie nomials to aspects of hyperbolic geometry, from tangles and braids Wang (McMaster). to the role of invariants in dynamical and biological systems, the Information: http://www.math.uwaterloo.ca/~gap. school will offer a unique opportunity to doctorate students as well as young researchers and mid-career academics to be exposed to * 18–22 2011 Georgia Topology Conference: Symplectic Topology some of the most interesting open problems of current research. and Floer Theory, University of Georgia, Athens, Georgia. Lecturers: Slavik Jablan, Univ. Belgrade, Republic of Serbia; Louis Description: Funding is available for participation by graduate stu- H. Kauffman, Univ. Illinois at Chicago, USA; Sergei Matveev, Che- dents and recent Ph.D.s. lyabinsk State Univ., Russia; Kenneth C. Millett, Univ. California at Speakers: Peter Albers (Purdue), Strom Borman (University of Chi- Santa Barbara, USA; Carlo Petronio, Univ. Pisa, Italy; Renzo L. Ricca, cago), Jacqueline Espina (UC Santa Cruz), Viktor Ginzburg (UC Santa Univ. Milano-Bicocca, Italy; Mauro Spera, Univ. Verona, Italy; De Witt Cruz), Doris Hein (UC Santa Cruz), Richard Hind (Notre Dame), Sonja Sumners, Florida State Univ., USA. Hohloch (Stanford), (IAS), Sikimeti Ma’u (Columbia), Contact: [email protected]. (This event is part of the in- Al Momin (Purdue), Yong-Geun Oh (Wisconsin), Yasha Savelyev tensive research period “Knots and Applications”). (Massachusetts), Claude Viterbo (Ecole Polytechnique), Weiwei Wu Information: http://www.crm.sns.it/event/203/. (Minnesota). Information: http://math.uga.edu/~topology. * 8–12 Conference on Algebra and Applications in honour of Prof. Said Sidki on the occasion of his 70th birthday, Caldas Novas, June 2011 Goias, Brazil. * 2–4 IMA Hot Topics Workshop: Uncertainty Quantification in In- Description: This conference will bring together many of the leading dustrial and Energy Applications: Experiences and Challenges, mathematicians to report on recent developments of broad interest Institute for Mathematics and its Applications (IMA), University of and to point the way for exciting directions for future research. In Minnesota, Minneapolis, Minnesota. this way we plan to honor the significant contributions of Prof. Said Description: The workshop will bring together industrial scientists, N. Sidki on the occasion of his 70th Birthday. lab scientists, and university-based researchers to share UQ state-of- Topics: Group theory; ring theory; and alge- the-art application experience, best practices, and future challenges braic geometry; associative and non-associative algebras; number in diverse applications and from different sectors, including aero- theory; applications. space and automotive applications, engine design, energy (nuclear, Invited Speakers: (University of California, USA); wind, solar, etc.) and global climate change. Presentations on aca- Rostislav Grigorchuk (Texas A&M University, USA); Donald Pass- demic UQ research and progress will open opportunities to transfer man (University of Wisconsin-Madison, USA); George Glauberman results for application problem solutions. Special focus will be given (University of Chicago, USA); Laurent Bartholdi (University of Got- to exploit the joint potential of data-driven statistical approaches tingen, Germany); Vladimir Nekrashevich (Texas A&M University, and model-based methodology. Moderated discussion sessions will USA); Thomas Muller (Queen Mary College, University of London, catalyze joint research activities and knowledge transfers. UK); Zoran Sunik (Texas A&M University, USA). Information: http://www.ima.umn.edu/2010-2011/ Information: http://www.mat.ufg.br/algebra. SW6.2-4.11/. * 11–14 41st Barrett Memorial Lectures—Mathematical Relativity, * 3–8 XIIIth International Conference on Geometry, Integrability University of Tennessee, Knoxville, Tennessee. and Quantization, Sts. Constantine and Elena Resort near Varna, Description: This longstanding lecture series at the University of Bulgaria. Tennessee focuses this year on recent developments in mathemati- Description: This conference is a continuation of meetings on Ge- cal general relativity. The format consists of three series of survey ometry and Mathematical Physics which took place in Bulgaria– lectures, six plenary lectures, and 30-minute talks by new Ph.D.s Zlatograd (1995) and Varna (1998-2010). “Geometry” in the title and postdocs. refers to modern differential geometry of real and complex mani- Survery lecture speakers: Igor Rodnianski, , Rob- folds with some emphasis on curves, sigma models and minimal ert Wald. surface theory; “Integrability” to either the integrability of complex Plenary lectures: Lydia Bieri, Hugh Bray, Mihalis Dafermos, Greg structures or classical dynamical systems of particles, soliton dy- Galloway, Jim Isenberg, Marcus Khuri. namics and hydrodynamical flows presented in geometrical form; Information: http://www.math.utk.edu/barrett/. and “Quantization” to the transition from classical to quantum me- * 12–15 SETIT, Sfax University, Tunisia. chanics expressed in geometrical terms.

APRIL 2011 NOTICES OF THE AMS 625 Mathematics Calendar

Aim: To bring together experts in classical and modern differen- Focal Theme: Emerging Interfaces of Physical Sciences and Technol- tial geometry, complex analysis, mathematical physics, and related ogy. The College of Engineering Studies, UPES, in collaboration with fields to assess recent developments in these areas and to stimulate the International Academy of Physical Sciences, Allahabad, is orga- research in related topics. nizing a 3-day international conference (CONIAPS-XIII) on “Emerging Information: http://www.bio21.bas.bg/conference/. Interfaces of Physical Sciences and Technology”. Call for Papers: Original contributions on any topics related to all of * 6–8 Abelian Varieties & Galois Actions, Faculty of Mathematics the engineering field and other applied sciences: Physics, Chemistry, and Computer Sciences, the Adam Mickiewicz University, Poznan´, Mathematics, Statistics, Computer Science, Earth Sciences (Geophys- Poland. ics, Geology & Geography) as well as topics related to Applications Main topics: Will cover some aspects of: Arithmetics of abelian va- of Physical Sciences to Biosciences (Biophysics, Bioinformatics, Bio- rieties, Iwasawa theory and structure, L-functions, chemistry, Biomathematics etc.) are invited for presentation. Galois representations, fundamental groups, algebraic K -theory. Deadline for Abstract: Each abstract with a maximum of 200 words Organizers: Wojciech Gajda (Poznan´), Cornelius Greither may be sent to: [email protected] with registration form (Monachium), Sebastian Petersen (Monachium). on or before March 30th, 2011. Information: http://avga.wmi.amu.edu.pl. * 16–17 8th Canadian Student Conference on Quantum Informa- * 6–15 11th Canadian Summer School on Quantum Information, tion, Centre de Villégiature de Jouvence, Québec, Canada. Centre de Villégiature de Jouvence, Québec, Canada. Description: An official academic activity for which students will Description: This summer school is the 11th edition of a highly receive 3 graduate-level credits from the University of Sherbrooke. successful series of schools, with previous editions held in Calgary, Participants of the student conference will receive 1 graduate level Montreal, Toronto, Vancouver, and Waterloo. It follows the tradition credit. These credits can be accounted for the student’s graduate of educating young researchers (prospective and current graduate program, depending on regulations of their host institution. students, as well as postdocs) on the rapidly-evolving field of quan- Information: http://www.crm.umontreal.ca/QI11/index_e. tum information science and brings together the world’s experts php. from different areas. Information: http://www.crm.umontreal.ca/QI11/. * 20–23 Complex Analysis and Potential Theory in honour of Paul M. Gauthier and Kohur Gowrisankaran, Centre de recherches * 7–11 Finite Groups and Their Automorphisms, Bog˘aziçi University, mathématiques, Université de Montréal, Montréal, Canada. Istanbul, Turkey. Description: Complex Analysis and Potential Theory have always Description: This five-day workshop aims to bring together lead- lived and thrived in symbiosis, and our aim is to bring together spe- ing mathematicians and active researchers working on the theory of cialists from both areas to foster further cooperation and exchange groups in order to exchange ideas, present new results, and identify of ideas and to find new research perspectives. With about 25 plenary the key problems in the field, especially but not exclusively, on the lectures, given by some of the most established specialists in the relationship between a finite group and another one acted upon by fields, as well as shorter talks in parallel sessions, this conference the first. There will be seven minicourses, several invited talks, a will also provide researchers with a forum to present some of the limited number of contributed talks and a poster session. main topics of current research, report on the latest developments Information: http://istanbulgroup.metu.edu.tr/. in these areas, and be exposed to an overview of Complex Analysis * 13–17 Cluster Algebras and Lusztig’s Canonical Basis, University and Potential Theory. of Oregon, Eugene, Oregon. Information: http://www.crm.math.ca/Complex11/index_e. Description: The goal of this workshop is to understand the state- php. ment that the cluster monomials on the coordinate ring of G/N are an important subset of the dual of Lusztig’s canonical basis. The work- July 2011 shop will be aimed at graduate students and postdocs, with most * 1–3 (NEW DATE) The 4th Congress of the Turkic World Mathemati- of the talks given by the participants. We do not expect any of the cal Society (TWMS), Baku, Azerbaijan. (Jan. 2011, p. 84) participants to be experts in all of the subjects that are represented Description: The aim of the Congress is to provide a forum where in this workshop. Rather, we hope to bring together participants with scientists and mathematicians from academia and industry can meet diverse backgrounds, and to weave these backgrounds together into to share ideas of latest research work in all branches of pure and a coherent picture through a combination of lectures and informal applied mathematics. discussion sessions. The workshop will be led by David Speyer. Information: http://www.twmsc2011.com/. Information: http://pages.uoregon.edu/njp/cluster. * 1–4 8-International Conference on the Computer Analysis of html. Problems of Science and Technology, Dushanbe, Tajikistan. * 13–17 Workshop on Moving in Geometry, Centre de recherches Themes: Mathematical aspects of computer sciences, computer mathématiques, Université de Montréal, Québec, Canada. problems, and information security. Description: Brought to maturity by Élie Cartan, the method of mov- Requirements for registration: (As usual for international confer- ing frames has been in the mathematical landscape for more than ences), all documents must be presented no later than May 15, 2011, a century. From the Frenet-Serret frame to Cartan’s “repère mobile” to the address: 734042, Dushanbe, Str. 17, Tajikistan. The invitations and beyond, moving-frame techniques have proven indispensable and program of the conference will be sent out on June 1, 2011. in the study of symmetries, invariants, and other intrinsic proper- Questions may be directed to: [email protected]. The conference ties of geometrical objects. Explicit applications of moving-frame fee, including conference publications, informational reports, and techniques range from classical differential geometry to integrable postage is US$100. To apply to participate in the conference, please systems, and on toward control theory and computer vision. send text of talks (in duplicate), up until May 15, 2011, to the above Information: http://www.crm.math.ca/Moving11/index_e. address and an electronic version to: [email protected]. php. Information: http://www.yunusi.com. * 14–16 CONIAPS-XIII: 13th Conference of the International Acad- * 1–29 Non-equilibrium Statistical Mechanics, Centre de recherches emy of Physical Sciences, University of Petroleum and Energy Stud- mathématiques, Université de Montréal, Pavillon André-Aisenstadt, ies, Dehradun, India. Montréal, Québec, Canada.

626 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Mathematics Calendar

Description: This school is a second part of the joint semester conference organizers Erik Tou ([email protected]) or Dominic “Frontiers in Mathematical Physics” organized by the Université de Klyve ([email protected]) for more information. Cergy-Pontoise, and McGill University and the CRM. The first part of Information: http://www.eulersociety.org. the semester will be held at Cergy in May 2011. July 1–10 will be a * 25–August 12 International Seminar and Workshop on Weak Chaos, concentration period devoted to scientific interaction and collabora- Infinite Ergodic Theory, and Anomalous Dynamics, Max Planck In- tion. Jan Derezinski will give a special 15-hour course on Quantum stitute for the Physics of Complex Systems, Dresden, Germany. Electrodynamics, July 11–15, 3 hours per day. The summer school Description: Weak chaos refers to systems exhibiting zero Lyapu- lectures will take place July 18–22 and July 25–29. There will be nov exponents, meaning that the separation of nearby trajectories is 3-hour mini-course lectures and invited talks. The lectures will be weaker than exponential. Still, the dynamics is typically very irregu- accessible to graduate students. About 2/3 of the lectures will focus lar. Rigorous mathematical results about such systems have recently on open problems and overall state of the art regarding various re- been obtained by infinite ergodic theory, which is an extension of or- search directions in non-equilibrium statistical mechanics. The other dinary ergodic theory to dynamical systems with non-normalizable lectures will deal with the state of the art developments in other areas measures. These theoretical concepts predict novel nonequilibrium of statistical mechanics. physical properties in form of anomalous dynamics, which can be Information: http://www.crm.umontreal.ca/Mechanics11/ tested in experiments. The purpose of this conference is to initi- index_e.php. ate cross-disciplinary collaborations between physicists working on * 3–8 Completely Integrable Systems and Applications—ESF-EMS- both the deterministic and the stochastic aspects of weakly chaotic ERCOM Conference, Erwin Schrödinger Institute, Vienna, Austria. systems and anomalous dynamics, and mathematicians being active Description: The scope of the conference includes Completely Inte- in the relevant branches of dynamical systems and ergodic theory. grable Systems (mostly PDEs and systems of ODEs) and related sub- Information: http://www.pks.mpg.de/~wchaos11/. jects such as Random Matrices, Whitham and Seiberg-Witten theory, * 25–August 12 IMA Participating Institution Summer Graduate Orthogonal Polynomials, Processes in Combinatorial Probability that Program Topological Methods in Complex Systems, University of are asymptotically described by Integrable Systems and Exactly Solv- Pennsylvania, Philadelphia, Pennsylvania. able Interacting Particle Systems modeling nonequilibrium phenom- Description: The University of Pennsylvania will be the host of the ena. We will particularly stress questions of “universality” appearing Institute for Mathematics and its Applications (IMA) Summer Graduate in both random matrices and semiclassical limits of integrable sys- Program in Mathematics. The course will concentrate on topological tems and “non-self-adjoint” problems. We will also address the newly methods in complex systems. Topological methods are generating evolving area of boundary problems for integrable PDEs. a revolution in the understanding and management of data in large Information: http://www.esf.org/conferences/11369. systems ranging from robotics, dynamics, sensors, materials, biology, * 3–16 41st Probability Summer School, Saint-Flour, France. communications, and vision. Such methods, inspired by a century’s Description: This summer school is intended for Ph.D. students, worth of development in algebraic and geometric topology, have the teachers, and researchers who are interested in probability theory, virtue of being qualitative and robust under perturbations. Of par- statistics, and in applications of these techniques. Three courses will ticular utility is the local-to-global nature of topological invariants, a be given: Itai Benjamini: Isoperimetric inequalities, invariance and ran- feature of increasing relevance in large systems with distributed or dom processes; Emmanuel Candes: The power of convex relaxation: modular constraints. the surprising stories of compressed sensing and matrix completion; Information: http://www.ima.umn.edu/2010-2011/PISG7. Gilles Schaeffer: Enumerative and bijective combinatorics for random 25-8.12.11/. walks, trees and planar maps. The participants will also have the op- portunity to give short lectures. August 2011 Information: http://math.univ-bpclermont.fr/stflour/. * 1–5 2011 CBMS-NSF Conference: 3-Manifolds, Artin Groups and * 4–8 2011 Taiwan International Conference on Geometry: Special Cubical Geometry, CUNY Graduate Center, New York City, New York. Lagrangians and Related Topics, Department of Mathematics, Na- Keynote speaker: Daniel Wise (McGill University, Canada). tional Taiwan University, Taipei, Taiwan. Additional speakers: (Univ. Calif., Berkeley), Description: This conference will start a series of bi-yearly interna- (Brandeis), Cornelia Drutu (Oxford), Nathan Dunfield (Univ. Illinois, tional conferences on differential geometry in Taiwan. An important Urbana-Champaign), Mark Hagan (McGill), Chris Hruska (Univ. Wis- area in geometry will be specified as the main theme each time. Our consin, Milwaukee), Michael Kapovich (Univ. Calif., Davis), Yair Min- purpose is to create a discussion and interaction platform in the cho- sky (Yale), Mark Sapir (Vanderbilt), Zlil Sela (Hebrew University), Eric sen area, and at the same time to foster future cooperations and in- Swensen (Brigham Young). troduce new people into the field. Additional short courses may also Financial support: From the NSF, especially for students and early be arranged around the same time as the conference. career researchers. We intend to cover the local expenses for many Topic: Special Lagrangians and Related Topics will include special such applicants. Lagrangians, Lagrangian mean curvature flow, J -holomorphic curve Information: Please check the website to apply for support. To be techniques for Lagrangians, and the calibrated geometries in general. considered for support, your application must be submitted by April The tentative topic for 2013 is “Geometry and General Relativity”. 15th, 2011. Please direct all correspondence to: cubulate.nsf- Information: http://www.tims.ntu.edu.tw/workshop/ [email protected]. Default/index.php?WID=111. Organizers: Jason Behrstock (CUNY Lehman & GC), Abhijit Cham- panerkar (CUNY Staten Island); http://comet.lehman.cuny. * 25–27 Euler Society 2011 Conference, Carthage College, Kenosha, edu/behrstock/cbms/. Wisconsin. Description: The 10th Annual Euler Society Conference will be held * 8–12 AIM Workshop: Relating test ideals and multiplier ideals, this year in a new venue, in Kenosha, WI. Talks are welcomed discuss- American Institute of Mathematics, Palo Alto, California. ing any aspect of Euler’s life, work, and influence. Also welcome are Description: This workshop, sponsored by AIM and the NSF, will discussions from the larger world of eighteenth century mathematics be devoted to the connection between two prominent and distinct and science. Talks can be given in either 20- or 50-minute formats. means of measuring singularities: the multiplier ideal in complex Deadline: Abstract deadline is June 15. Reduced conference fees algebraic geometry, and the test ideal in positive characteristic com- are available for those with no institutional support. Contact mutative algebra.

APRIL 2011 NOTICES OF THE AMS 627 Mathematics Calendar

Information: http://www.aimath.org/ARCC/workshops/ on special geometric structures and their applications in differen- testideals.html. tial and algebraic geometry, theoretical physics and string theory. http://www.fmi.uni-sofia.bg/ivanovsp/ * 22–December 21 Multiscale Numerics for the Atmosphere and Information: MathPhys2011.html. Ocean, Isaac Newton Institute for Mathematical Sciences, Cambridge, . October 2011 Description: Numerical models of the atmosphere and ocean have * 4–6 Sixth International Workshop—Meshfree Methods for Partial proved to be immensely valuable forecasting tools for short time- Differential Equations, Universitätsclub Bonn, Bonn, Germany. scale weather and longer time-scale seasonal and climate predic- Description: While contributions in all aspects of meshfree and tion. As the decades pass, these models have been improving due to particle methods are invited, some of the key topics to be featured increased computing power, improved modelling of the dynamics, are: Application of meshfree, particle, generalized/extended finite improved parametrisation of sub-grid scale processes and improved element methods, e.g., to multiscale problems, problems with mul- use of observations. These modelling improvements may be slowing tiple discontinuities and singularities, problems in high-dimensions, and further large increases in computing power will almost certainly coupling of meshfree methods, finite element methods, particle emerge from heterogenous computing architectures configured in methods, and finite difference methods, parallel computation in even more massively parallel machines. If we are unable to exploit meshfree methods, mathematical theory of meshfree, generalized these new opportunities in high-performance computing, our cur- finite element, and particle methods, fast and stable domain integra- rent models and codes risk becoming obsolete. tion methods, enhanced treatment of boundary conditions, identifi- Organizers: Dr. D. Ham, Dr. M. Piggott, Dr. T. Ringler, Dr. H. Weller cation and characterization of problems where meshfree methods and Dr. N. Wood. have clear advantage over classical approaches. Information: http://www.newton.ac.uk/programmes/AMM/. Sponsor: Sonderforschungsbereich 611, Universität Bonn. September 2011 Information: http://wissrech.ins.uni-bonn.de/meshfree. * 3–9 10th International Conference on Geometry and Applications, * 11–14 Mal’tsev Meeting, Sobolev Institute of Mathematics, Novosi- Geometrical Society “Boyan Petkanchin”, Sofia, Bulgaria. birsk, Russia. Description: 10th International Conference on Geometry and Ap- Description: Mal’tsev Meeting is an annual conference on algebra, plications is organized from the Geometrical Society “Boyan Pet- mathematical logic, and applications organized by Sobolev Insti- kanchin” in Bulgaria. The following fields are included: differential tute of Mathematics and Novosibirsk State University. In 2011 the geometry, finite geometries, computer methods in geometry, algebra meeting is dedicated to the 60th birthday of Sergei Goncharov. The and analysis, education in the school and university by computers, programme of the conference will consist of invited talks and con- didactic of mathematics. tributions in sections. Information: Please contact: Prof. Dr. Grozio Stanilov; stanilov@ Main topics: Include computability theory, theoretical computer fmi.uni-sofia.bg and Chavdar Lozanov; lozanov@fmi. science, mathematical logic, group theory, ring theory, universal uni-sofia.bg. algebra, and related areas of mathematics. Information: http://www.math.nsc.ru/conference/mal- * 11–17 14th International Conference on Functional Equations meet/11/index.html. and Inequalities, Mathematical Research and Conference Center, Bedlewo (near Poznan), Poland. * 31–November 4 AIM Workshop: Geometry of large networks, Description: The International Conference on Functional Equations American Institute of Mathematics, Palo Alto, California. and Inequalities—ICFEI has been organized by the Institute of Math- Description: This workshop, sponsored by AIM and the NSF, is de- ematics of the Pedagogical University of Cracow since 1984. The voted to geometric models of large networks. It intends to bring to- conference is devoted to functional equations and inequalities, their gether mathematicians, computer scientists, and engineers. applications in various branches of mathematics and other scien- Information: http://aimath.org/ARCC/workshops/ tific disciplines, as well as related topics. The 14th ICFEI is included largenetworks.html. in the programme of the Stefan Banach International Mathematical November 2011 Center of the Polish Academy of Sciences. Information: http://mat.up.krakow.pl/icfei/14ICFEI/. * 15–17 International Seminar on the Application of Science and Mathematics 2011, University Tun Hussein Onn Malaysia, Kuala * 18–24 8th International Conference on Function Spaces, Differ- Lumpur Convention Centre, Kuala Lumpur, Malaysia. ential Operators, Nonlinear Analysis (FSDONA-2011), Tabarz/ Description: ISASM 2011 aims to provide an international forum for Thuringia, Germany. researchers to present and discuss recent advances and new tech- Description: This meeting will continue the series of previous suc- niques in Science and Mathematics and its Applications. To bring cessful FSDONA-conferences held in Finland, Czech Republic, and together researchers and scientists in promoting and enhancing Germany: Sodankylä-88, Friedrichroda-92, Paseky-95, Syöte-99, Teis- research collaboration among local and international participants. tungen-01, Milovy-04, -08. It is our intention to stimulate in- Information: http://k-utech.com.my/isasm2011/. ternational collaboration, and to promote the interaction of function spaces, PDE and computational mathematics in unifying efforts. This December 2011 time the focus will lie on the theory of function spaces and its ap- * 17–18 The International Symposium on Biomathematics and Ecol- plications to various fields of mathematics like: PDE’s (existence of ogy: Education and Research (BEER-2011), University of Portland, solutions and regularity theory), spectral theory of differential and Portland, Oregon. integral operators, approximation and computational mathematics, Description: The main objective of this meeting is to provide a forum nonlinear analysis, inverse problems. for researchers, educators, students and industries to exchange http://fsdona2011.uni-jena.de/. Information: ideas, to communicate and discuss research findings in the fields * 19–26 Conference on Geometric Structures in Mathematical Phys- of mathematics, biology, ecology and statistics. ics, Albena, Bulgaria. Topics: Biomathematics, Mathematics, Biology, Ecology, Biostatistics. Description: The purpose of the conference is to bring together Organizers: Olcay Akman, Hannah Callender, Timothy Comar, Ste- physicists and mathematicians working in related areas of geometry, ven A. Juliano. geometric analysis and theoretical physics. The main focus will be Information: http://www.biomath.ilstu.edu/beer.

628 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Mathematics Calendar

January 2012 some measure of invariance under continuous deformation. Dynami- cal evolution is then subject to the topological constraints that express * 9–July 6 Semantics and Syntax: A Legacy of , Isaac New- this invariance. A basic common problem is to determine minimum ton Institute for Mathematical Sciences, Cambridge, United Kingdom. energy structures (and routes towards these structures) permitted by Description: In several mathematical areas of Theoretical Computer such constraints; and to explore mechanisms, e.g., diffusive, by which Science, we perceive a distinction between research focusing on sym- such constraints may be broken. Workshops: A number of workshops bolic manipulation of language and structures (independent of mean- will take place during the programme. For full details please see: ing) and research dealing with interpreted computational meaning of http://www.newton.ac.uk/events.html. structures. In mathematical logic, the distinction is known as syntax Organizers: Professor K. Bajer (Warsaw), Professor T. W. Kephart (symbolic manipulation) versus semantics (interpreted structures). (Vanderbilt), Professor Y. Kimura (Nagoya), Professor H. K. Moffatt This distinction recurs in many research areas, often under different (Cambridge) and Professor A. Stasiak (Lausanne). (and sometimes incompatible) names. For research in these fields, Information: http://www.newton.ac.uk/programmes/TOD/. both views are important and fundamental for gaining full under- standing of the formal issues involved. This programme will bring * 23–August 17 Spectral Theory of Relativistic Operators, Isaac New- together researchers from both sides of the syntax-semantics divide. ton Institute for Mathematical Sciences, Cambridge, United Kingdom. We shall focus on four mathematical areas bordering computer sci- Description: Relativistic operators are used to model important ence: logic, complexity, cryptography, and randomness. physical systems which include transport properties of graphene, and Organizers: Dr. A. Beckmann, Professor S. B. Cooper, Professor B. relativistic quantum field theory. This meeting will focus on the fol- Löwe, Professor E. Mayordomo, and Professor N. Smart. lowing areas of current research interest in such operators applied to Information: http://www.newton.ac.uk/programmes/SAS/. mathematical physics. 1. For classical (one-particle) Dirac operators, current topics of interest include the Weyl-type theory, dissolution of * 17–19 ACM-SIAM Symposium on Discrete Algorithms (SODA12), eigenvalues of corresponding relativistic systems into resonances, as- The Westin Miyako, Kyoto, Japan. ymptotics of the spectral function and spectral concentration as well Description: This symposium focuses on research topics related to as the role of the mass term of Dirac operators. 2. Stability of matter efficient algorithms and data structures for discrete problems. In ad- and asymptotic behaviour of the ground state energy for relativistic dition to the design of such methods and structures, the scope also many-particle systems. 3. The interaction of photons with fast mov- includes their use, performance analysis, and the mathematical prob- ing (relativising) electrons, positrons, and photons. lems related to their development or limitations. Performance analy- Organizers: Professor M. Brown (Cardiff), Professor M. J. Esteban ses may be analytical or experimental and may address worst-case (Ceremade), Dr. K. M. Schmidt (Cardiff) and Professor H. Siedentop or expected-case performance. Studies can be theoretical or based on (Munich). data sets that have arisen in practice and may address methodologi- Information: http://www.newton.ac.uk/programmes/SRO/. cal issues involved in performance analysis. Information: http://www.siam.org/meetings/da12/. February 2012 * 13–17 The 10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2012), The University of New South Wales, Sydney, NSW, Australia. Description: MCQMC is a biennial conference devoted to Monte Carlo and quasi-Monte Carlo methods and their interactions and applica- tions. (In brief, quasi-Monte Carlo methods replace the random choices that characterize the Monte Carlo method by well chosen determinis- tic choices.) For more information, click on the “Background” tab on the web site. This will be the first MCQMC conference to be held in the southern hemisphere. (Northerners may like to be reminded that February is summertime in Sydney!). Plenary speakers: P. Del Moral, M. Giles, F. J. Hickernell, A. Hinrichs, M. Lacey, K. Mengersen, A. Neuenkirch, A. B. Owen, L. Plaskota, E. Platen. To receive further announcements please go the web site, click on the “mailing list” tab, and sign up. The web site includes a call for special sessions. Information: http://www.mcqmc2012.unsw.edu.au/.

The following new announcements will not be repeated until the criteria in the next to the last paragraph at the bottom of the first page of this section are met.

July 2012 * 16–December 21 Topological Dynamics in the Physical and Bio- logical Sciences, Isaac Newton Institute for Mathematical Sciences, Cambridge, United Kingdom. Description: The programme is intended to stimulate interaction between applied mathematicians, biologists and physicists who fre- quently encounter dynamical problems that have some explicit or implicit topological content. We use the term “topological” to convey the idea of structures, e.g., knots, links or braids in 3D, that exhibit

APRIL 2011 NOTICES OF THE AMS 629 New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to http://www.ams.org/bookstore-email.

Algebra and Algebraic contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include Geometry quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The Fonction Zêta book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded des Hauteurs des cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate Variétés Toriques non nontrivial examples in toric geometry. Déployées Readers of this book should be familiar with the material covered in David Bourqui, I.R.M.A.R., basic graduate courses in algebra and topology, and to a somewhat Rennes, France lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a This item will also be of interest to those useful reference for graduate students and researchers who are working in number theory. interested in algebraic geometry, polyhedral geometry, and toric varieties. Contents: Introduction; Tores algébriques; Hauteurs sur une Contents: Basic theory of toric varieties: Affine toric varieties; variété torique et fonction zêta associée; Calcul des transformées de Projective toric varieties; Normal toric varieties; Divisors on toric Fourier et expression intégrale de la fonction zêta des hauteurs; varieties; Homogeneous coordinates on toric varieties; Line bundles Évaluation de l’intégrale dans le cas arithmétique; Évaluation on toric varieties; Projective toric morphisms; The canonical divisor de l’intégrale dans le cas fonctionnel; Bibliographie; Index des of a toric variety; Sheaf cohomology of toric varieties; Topics in toric notations; Index des définitions. geometry: Toric surfaces; Toric resolutions and toric singularities; Memoirs of the American Mathematical Society, Volume 211, The topology of toric varieties; Toric Hirzebruch-Riemann-Roch; Number 994 Toric GIT and the secondary fan; Geometry of the secondary fan; The history of toric varieties; Computational methods; Spectral April 2011, 151 pages, Softcover, ISBN: 978-0-8218-4936-1, 2010 sequences; Bibliography; Index. Mathematics Subject Classification: 11G35, 11G50, 14M25, 11M41, Graduate Studies in Mathematics, Volume 124 Individual member US$46.20, List US$77, Institutional member US$61.60, Order code MEMO/211/994 June 2011, approximately 858 pages, Hardcover, ISBN: 978-0-8218- 4819-7, LC 2010053054, 2010 Mathematics Subject Classification: 14M25, AMS members US$76, List US$95, Order code GSM/124 Toric Varieties David A. Cox, Amherst College, MA, John B. Little, College of the Holy Cross, Worcester, MA, and Henry K. Schenck, University of Illinois at Urbana-Champaign, IL

Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters

630 Notices of the AMS Volume 58, Number 4 New Publications Offered by the AMS Iwasawa Theory, Applications Projective Modules, and Modular Representations BioMath in the Schools Ralph Greenberg, University of Margaret B. Cozzens and Fred Washington, Seattle, WA S. Roberts, Rutgers University, Piscataway, NJ, Editors This item will also be of interest to those working in number theory. Even though contemporary biology and mathematics are inextricably linked, high school biology and mathematics courses Contents: Introduction; Projective and quasi-projective modules; have traditionally been taught in isolation. Σ0 Projectivity or quasi-projectivity of XE (K∞); Selmer atoms; But this is beginning to change. This The structure of Hv (K∞,E); The case where ∆ is a p-group; volume presents papers related to the Other specific groups; Some arithmetic illustrations; Self-dual integration of biology and mathematics representations; A duality theorem; p-modular functions; Parity; in high school classes. More arithmetic illustrations; Bibliography. The first part of the book provides the rationale for integrating Memoirs of the American Mathematical Society, Volume 211, mathematics and biology in high school courses as well as Number 992 opportunities for doing so. The second part explores the development and integration of curricular materials and includes April 2011, 185 pages, Softcover, ISBN: 978-0-8218-4931-6, 2010 responses from teachers. Mathematics Subject Classification: 11G05, 11R23; 20C15, 20C20, Individual member US$49.80, List US$83, Institutional member Papers in the third part of the book explore the interconnections US$66.40, Order code MEMO/211/992 between biology and mathematics in light of new technologies in biology. The last paper in the book discusses what works and what doesn’t and presents positive responses from students to the integration of mathematics and biology in their classes. Analysis This item will also be of interest to those working in general interest. Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes Differential Forms on 1–7 were co-published with the Association for Computer Wasserstein Space and Machinery (ACM). Contents: The rationale for high school BioMath: F. S. Roberts, Infinite-Dimensional Why BioMath? Why now?; E. Jakobsson, The interdisciplinary scientist of the 21st century; L. J. Heyer and A. M. Campbell, Hamiltonian Systems Teaching bioinformatics and genomics: An interdisciplinary Wilfrid Gangbo and Hwa Kil Kim, approach; N. H. Fefferman and L. M. Fefferman, Mathematical Georgia Institute of Technology, macrobiology: An unexploited opportunity in high school Atlanta, GA, and Tommaso education; A. Nkwanta, D. Hill, A. Swamy, and K. Peters, Counting RNA patterns in the classroom: A link between molecular biology Pacini, Scuola Normale Superiore, and enumerative combinatorics; Curriculum materials and teacher Pisa, Italy training/development: M. B. Cozzens, New materials to integrate biology and mathematics in the high school curriculum; K. M. Contents: Introduction; The topology on M and a differential Gabric, The awakening of a high school biology teacher to calculus of curves; The calculus of curves, revisited; Tangent and the BioMath connection; L. J. Morris, C. Long, and J. Kissler,A cotangent bundles; Calculus of pseudo differential 1-forms; A beginning experience: Linking high school biology and mathematics; symplectic foliation of M; The symplectic foliation as a Poisson K. G. Herbert and J. H. Dyer, Integrating interdisciplinary science structure; Review of relevant notions of differential geometry; into high school science modules through a preproinsulin example; Bibliography. M. C. Rogers and D. S. Yuster, Insights from math-science Memoirs of the American Mathematical Society, Volume 211, collaboration at the high school level; Topics, course changes, Number 993 and technology: H. Scheintaub, E. Klopfer, M. Scheintaub, and E. Rosenbaum, Complexity and biology—bringing quantitative April 2011, 77 pages, Softcover, ISBN: 978-0-8218-4939-2, 2010 science to the life sciences classroom; J. Malkevitch, Distance Mathematics Subject Classification: 35Qxx, 49-XX; 53Dxx, 70-XX, and trees in high school biology and mathematics classrooms; Individual member US$39.60, List US$66, Institutional member M. E. Martin, Mathematical biology: Tools for inquiry on the US$52.80, Order code MEMO/211/993 Internet; E. S. Marland and M. E. Searcy, The calculus cycle: Using biology to connect discrete and continuous modeling in calculus; C. Mullins and D. W. Cranston, Research at ASMSA based on the DIMACS BioMath program; Evaluation of how integration of biology/mathematics works: A. E. Weinberg and L. Albright, Integrating biology and mathematics in high school classrooms.

April 2011 Notices of the AMS 631 New Publications Offered by the AMS

DIMACS: Series in Discrete Mathematics and Theoretical Neumann boundary conditions; C. A. Stuart, Bifurcation and Computer Science, Volume 76 decay of solutions for a class of elliptic equations on RN ; S. de Valeriola and M. Willem, Existence of nodal solutions for some April 2011, approximately 254 pages, Hardcover, ISBN: 978-0-8218- nonlinear elliptic problems; F. Robert, Admissible Q-curvatures 4295-9, 2010 Mathematics Subject Classification: 00A06, 00A08, under isometries for the conformal GJMS operators. 92F05, 97-XX, 93A30, 00A69, 92-XX, 05Cxx, 68Uxx, AMS members Contemporary Mathematics, Volume 540 US$41.60, List US$52, Order code DIMACS/76 May 2011, 259 pages, Softcover, ISBN: 978-0-8218-4907-1, LC 2010049092, 2010 Mathematics Subject Classification: 35B50, Differential Equations 35J15, 35J20, 35J25, 35J60, 35J70, 35P30, 46E35, 49Q20, 58J05, AMS members US$71.20, List US$89, Order code CONM/540

Nonlinear Elliptic Partial Differential Equations Denis Bonheure, Université Libre de Bruxelles, Belgium, Elliptic Partial Mabel Cuesta, Université du Differential Equations Littoral, Calais, France, Enrique Second Edition J. Lami Dozo, Université Libre de Bruxelles, Belgium, Peter Qing Han, University of Notre Takáˇc, Universität Rostock, Dame, IN, and Fanghua Lin, Germany, and Jean Van Courant Institute, New York Schaftingen and Michel Willem, University, NY Université Catholique de Louvain, Louvain-la-Neuve, Belgium, Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of Editors the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful This volume contains papers on semi-linear and quasi-linear and extraordinary useful material. I have used it as a textbook at elliptic equations from the workshop on Nonlinear Elliptic Partial both graduate and undergraduate levels which is possible since Differential Equations, in honor of Jean-Pierre Gossez’s 65th it only requires very little background material yet it covers an birthday, held September 2–4, 2009 at the Université Libre de enormous amount of material. In my opinion it is a must read for Bruxelles, Belgium. all interested in analysis and geometry, and for all of my own PhD The workshop reflected Gossez’s contributions in nonlinear elliptic students it is indeed just that. I cannot say enough good things PDEs and provided an opening to new directions in this very active about it—it is a wonderful book. research area. Presentations covered recent progress in Gossez’s — favorite topics, namely various problems related to the p-Laplacian operator, the antimaximum principle, the FuˇcíkSpectrum, and This volume is based on PDE courses given by the authors at the other related subjects. This volume will be of principle interest to Courant Institute and at the University of Notre Dame, Indiana. researchers in nonlinear analysis, especially in partial differential Presented are basic methods for obtaining various a priori estimates equations of elliptic type. for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. This item will also be of interest to those working in analysis. The equations considered in the book are linear; however, the Contents: J. Mawhin, Partial differential equations also have presented methods also apply to nonlinear problems. principles: Maximum and antimaximum; B. Ruf, On the Fuˇcík This second edition has been thoroughly revised and in a new spectrum for equations with symmetries; B. Kawohl, Variations chapter the authors discuss several methods for proving the on the p-Laplacian; V. Bouchez and J. Van Schaftingen, Extremal existence of solutions of primarily the Dirichlet problem for various functions in Poincaré-Sobolev inequalities for functions of bounded types of elliptic equations. variation; P. Bousquet and P. Mironescu, An elementary proof of an inequality of Maz’ya involving L1 vector fields; D. G. Costa and Titles in this series are co-published with the Courant Institute of C. Li, Homoclinic type solutions for a class of differential equations Mathematical Sciences at New York University. with periodic coefficients; J. Giacomoni, J. Hernández, and Contents: Harmonic functions; Maximum principles; Weak A. Moussaoui, Quasilinear and singular systems: The cooperative solutions: Part I; Weak solutions: Part II; Viscosity solutions; case; P. Drábek, R. F. Manásevich, and P. Takáˇc, Manifolds of Existence of solutions; Bibliography. critical points in a quasilinear model for phase transitions; L. Leadi and H. R. Quoirin, Weighted asymmetric problems for an Courant Lecture Notes, Volume 1 indefinite elliptic operator; F. Obersnel and P. Omari, Multiple April 2011, 147 pages, Softcover, ISBN: 978-0-8218-5313-9, LC non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation; M. Perez-Llanos and J. D. Rossi, Limits 2010051489, 2010 Mathematics Subject Classification: 35-01, 35Jxx, as p(x) → ∞ of p(x)-harmonic functions with non-homogeneous AMS members US$24.80, List US$31, Order code CLN/1.R

632 Notices of the AMS Volume 58, Number 4 New Publications Offered by the AMS Sturm-Liouville General Interest Operators and Applications Understanding Revised Edition Numbers in Vladimir A. Marchenko, Verkin Elementary School Institute for Low Temperature Physics and Engineering, Mathematics Kharkov, Hung-Hsi Wu, University of California, Berkeley, CA The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems. This is a textbook for pre-service Besides the basic results on the structure of the spectrum and the elementary school teachers and for eigenfunction expansion of regular and singular Sturm-Liouville current teachers who are taking problems, it is in this domain that one-dimensional quantum professional development courses. By emphasizing the precision scattering theory, inverse spectral problems, and the surprising of mathematics, the exposition achieves a logical and coherent connections of the theory with nonlinear evolution equations first account of school mathematics at the appropriate level for become related. The main goal of this book is to show what can be the readership. Wu provides a comprehensive treatment of all achieved with the aid of transformation operators in spectral theory the standard topics about numbers in the school mathematics as well as in their applications. The main methods and results in this curriculum: whole numbers, fractions, and rational numbers. area (many of which are credited to the author) are for the first time Assuming no previous knowledge of mathematics, the presentation examined from a unified point of view. develops the basic facts about numbers from the beginning and The direct and inverse problems of spectral analysis and the inverse thoroughly covers the subject matter for grades K through 7. scattering problem are solved with the help of the transformation Every single assertion is established in the context of elementary operators in both self-adjoint and nonself-adjoint cases. The school mathematics in a manner that is completely consistent with asymptotic formulae for spectral functions, trace formulae, and the basic requirements of mathematics. While it is a textbook for the exact relation (in both directions) between the smoothness pre-service elementary teachers, it is also a reference book that of potential and the asymptotics of eigenvalues (or the lengths school teachers can refer to for explanations of well-known but of gaps in the spectrum) are obtained. Also, the applications of hitherto unexplained facts. For example, the sometimes-puzzling transformation operators and their generalizations to soliton concepts of percent, ratio, and rate are each given a treatment theory (i.e., solving nonlinear equations of Korteweg-de Vries type) that is down to earth and devoid of mysticism. The fact that a are considered. negative times a negative is a positive is explained in a leisurely and The new Chapter 5 is devoted to the stability of the inverse problem comprehensible fashion. solutions. The estimation of the accuracy with which the potential Contents: Whole numbers: Place value; The basic laws of operations; of the Sturm-Liouville operator can be restored from the scattering The standard algorithms; The addition algorithm; The subtraction data or the spectral function, if they are only known on a finite algorithm; The multiplication algorithm; The long division interval of a spectral parameter (i.e., on a finite interval of energy), is algorithm; The number line and the four operations revisited; obtained. What is a number?; Some comments on estimation; Numbers in Contents: The Sturm-Liouville equation and transformation base b; Fractions: Definitions of fraction and decimal; Equivalent operators; The Sturm-Liouville boundary value problem on the half fractions and FFFP; Addition of fractions and decimals; Equivalent line; The boundary value problem of scattering theory; Nonlinear fractions: further applications; Subtraction of fractions and equations; Stability of inverse problems; References. decimals; Multiplication of fractions and decimals; Division of fractions; Complex fractions; Percent; Fundamental Assumption AMS Chelsea Publishing, Volume 373 of School Mathematics (FASM); Ratio and rate; Some interesting April 2011, approximately 404 pages, Hardcover, ISBN: 978-0-8218- word problems; On the teaching of fractions in elementary school; 5316-0, 2010 Mathematics Subject Classification: 34A55, 34B24, Rational numbers: The (two-sided) number line; A different view 35Q51, 47E05, 47J35, AMS members US$54.90, List US$61, Order of rational numbers; Adding and subtracting rational numbers; code CHEL/373.H Adding and subtracting rational numbers redux; Multiplying rational numbers; Dividing rational numbers; Ordering rational numbers; Number theory: Divisibility rules; Primes and divisors; The Fundamental Theorem of Arithmetic (FTA); The Euclidean algorithm; Applications; Pythagorean triples; More on decimals: Why finite decimals are important; Review of finite decimals; Scientific notation; Decimals; Decimal expansions of fractions; Bibliography; Index. June 2011, approximately 542 pages, Hardcover, ISBN: 978-0-8218- 5260-6, LC 2010053021, 2010 Mathematics Subject Classification: 97-01, 00-01, 97F30, 97F40, 97F80, AMS members US$63.20, List US$79, Order code MBK/79

April 2011 Notices of the AMS 633 New Publications Offered by the AMS

Geometry and Topology of hyperbolic 3-manifolds; D. D. Long and A. W. Reid, Fields of definition of canonical curves. Contemporary Mathematics, Volume 541 Interactions Between May 2011, approximately 262 pages, Softcover, ISBN: 978-0-8218- 4960-6, 2010 Mathematics Subject Classification: 57Mxx, 32Qxx, Hyperbolic Geometry, 60Gxx, 16Txx, 17Bxx, 81Rxx, 81Txx, 11Sxx, 14Txx, AMS members Quantum Topology US$71.20, List US$89, Order code CONM/541 and Number Theory Abhijit Champanerkar, College Q-Valued Functions of Staten Island, CUNY, Staten Revisited Island, NY, Oliver Dasbach, Louisiana State University, Baton Camillo De Lellis, University Rouge, LA, Efstratia Kalfagianni, of Zurich, Switzerland, and Michigan State University, East Emanuele Nunzio Spadaro, Lansing, MI, Ilya Kofman, College University of Bonn, Germany of Staten Island, CUNY, Staten Contents: Introduction; The elementary Island, NY, Walter Neumann, theory of Q-valued functions; Almgren’s Barnard College, Columbia extrinsic theory; Regularity theory; University, New York, NY, and Intrinsic theory; The improved estimate of the singular set in 2 dimensions; Neal Stoltzfus, Louisiana State Bibliography. University, Baton Rouge, LA, Memoirs of the American Mathematical Society, Volume 211, Editors Number 991 This book is based on a 10-day workshop given by leading experts in April 2011, 79 pages, Softcover, ISBN: 978-0-8218-4914-9, 2010 hyperbolic geometry, quantum topology and number theory, in Mathematics Subject Classification: 49Q20, 35J55, 54E40, 53A10, June 2009 at Columbia University. Each speaker gave a minicourse Individual member US$39.60, List US$66, Institutional member consisting of three or four lectures aimed at graduate students US$52.80, Order code MEMO/211/991 and recent PhDs. The proceedings of this enormously successful workshop can serve as an introduction to this active research area in a way that is expository and broadly accessible to graduate students. Mathematical Physics Although many ideas overlap, the twelve expository/research papers in this volume can be grouped into four rough categories: (1) different approaches to the Volume Conjecture and relations between the main quantum and geometric invariants; Combinatorics and (2) the geometry associated to triangulations of hyperbolic Physics 3-manifolds; Kurusch Ebrahimi-Fard, (3) arithmetic invariants of hyperbolic 3-manifolds; Universidad de Zaragoza, , (4) quantum invariants associated to knots and hyperbolic Matilde Marcolli, California 3-manifolds. Institute of Technology, Pasadena, The workshop, the conference that followed, and these proceedings CA, and Walter D. van Suijlekom, continue a long tradition in quantum and geometric topology of Radboud University Nijmegen, bringing together ideas from diverse areas of mathematics and physics, and highlights the importance of collaborative research in The Netherlands, Editors tackling big problems that require expertise in disparate disciplines. This book is based on the mini-workshop Renormalization, Contents: H. Murakami, An introduction to the volume conjecture; held in December 2006, and the conference Combinatorics and T. Dimofte and S. Gukov, Quantum field theory and the Physics, held in March 2007. Both meetings took place at the volume conjecture; R. M. Kashaev, R-matrix knot invariants Max-Planck-Institut für Mathematik in Bonn, Germany. and triangulations; S. Garoufalidis, Knots and tropical curves; Research papers in the volume provide an overview of applications S. Baseilhac, Quantum coadjoint action and the 6j-symbols of of combinatorics to various problems, such as applications to Hopf U sl ; S. Garoufalidis, What is a sequence of Nilsson type?; D. Futer q 2 algebras, techniques to renormalization problems in quantum and F. Guéritaud, From angled triangulations to hyperbolic field theory, as well as combinatorial problems appearing in the structures; F. Luo, Triangulated 3-manifolds: From Haken’s context of the numerical integration of dynamical systems, in normal surfaces to Thurston’s algebraic equation; J. S. Purcell, An noncommutative geometry and in quantum gravity. introduction to fully augmented links; G. S. Walsh, Orbifolds and commensurability; W. D. Neumann, Realizing arithmetic invariants In addition, it contains several introductory notes on renormalization Hopf algebras, Wilsonian renormalization and motives.

634 Notices of the AMS Volume 58, Number 4 New AMS-Distributed Publications

Contents: C. Brouder and F. Patras, One-particle irreducibility with initial correlations; F. Brown, Multiple zeta values and periods: From moduli spaces to Feynman integrals; F. Chapoton and New AMS-Distributed A. Frabetti, From quantum electrodynamics to posets of planar binary trees; G. H. E. Duchamp and C. Tollu, Sweedler’s duals and Schützenberger’s calculus; L. Foissy, Primitive elements of the Publications Hopf algebra of free quasi-symmetric functions; R. Friedrich,A Renormalisation Group approach to Stochastic Lœwner Evolutions; J. M. Gracia-Bondía, On the causal gauge principle; M. Gubinelli, Abstract integration, combinatorics of trees and differential equations; R. Holtkamp, Rooted trees appearing in products Analysis and co-products; A. Iserles, Magnus expansions and beyond; T. Krajewski and P. Martinetti, Wilsonian renormalization, differential equations and Hopf algebras; E. Kraus, Algebraic analysis of non-renormalization theorems in supersymmetric Entropy of field theories; D. Kreimer, Not so non-renormalizable gravity; Meromorphic Maps D. Manchon, Renormalised multiple zeta values which respect quasi-shuffle relations; F. Menous, Formulas for the and Dynamics of Connes-Moscovici Hopf algebra; A. Mestre and R. Oeckl, Hopf algebras and the combinatorics of connected graphs in quantum Birational Maps field theory; A. Lundervold and H. Munthe-Kaas, Hopf algebras of Henry De Thélin, Université formal diffeomorphisms and numerical integration on manifolds; Paris 13, Villetaneuse, France, D. Oriti, A combinatorial and field theoretic path to quantum gravity: The new challenges of group field theory; S. Paycha, and Gabriel Vigny, Laboratoire Noncommutative formal Taylor expansions and second quantised Amiénois de Mathématique regularised traces; A. Rej, Motives: An introductory survey for Fondamentale et Appliquée, physicists; W. van Suijlekom, Combinatorics and Feynman graphs Amiens, France for gauge theories; F. Vignes-Tourneret, Multi-scale analysis and non-commutative field theory. The authors study the dynamics of meromorphic maps for a Contemporary Mathematics, Volume 539 compact Kahler manifold X. More precisely, they give a simple criterion that allows them to produce a measure of maximal entropy. May 2011, 465 pages, Softcover, ISBN: 978-0-8218-5329-0, LC They can apply this result to bound the Lyapunov exponents. 2010048012, 2010 Mathematics Subject Classification: 81T15, The authors then study the particular case of a family of generic 65D30, AMS members US$104, List US$130, Order code CONM/539 birational maps of Pk for which they construct the Green currents and the equilibrium measure. They use for that the theory of super-potentials. They show that the measure is and gives Rearranging no mass to pluripolar sets. Using the criterion they get that the measure is of maximal entropy. This implies finally that the Dyson-Schwinger measure is hyperbolic. Equations A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Karen Yeats, Simon Fraser Orders from other countries should be sent to the SMF. Members of University, Burnaby, BC, Canada the SMF receive a 30% discount from list. Contents: Introduction; Entropy of meromorphic maps; Dynamics Contents: Introduction; Background; of birational maps of Pk; Super-potentials; Bibliography. Dyson-Schwinger equations; The first recursion; Reduction to one insertion Mémoires de la Société Mathématique de France, Number 122 place; Reduction to geometric series; January 2011, 98 pages, Softcover, ISBN: 978-2-85629-302-7, 2010 The second recursion; The radius of Mathematics Subject Classification: 37Fxx, 32H04, 32Uxx, 37A35, convergence; The second recursion as a differential equation; Bibliography; Index. 37Dxx, Individual member US$37.80, List US$42, Order code SMFMEM/122 Memoirs of the American Mathematical Society, Volume 211, Number 995 April 2011, 82 pages, Softcover, ISBN: 978-0-8218-5306-1, 2010 Mathematics Subject Classification: 81T18, Individual member US$39.60, List US$66, Institutional member US$52.80, Order code MEMO/211/995

April 2011 Notices of the AMS 635 New AMS-Distributed Publications

Projections in Several Contents: Introduction; The space V3; Captures and counting; The resident’s view; Fundamental domains; Easy cases of the main Complex Variables theorem; The hard case: Final statement and examples; Proof of the Chin-Yu Hsiao, University of hard and interesting case; Open questions; Bibliography. Cologne, Germany Mémoires de la Société Mathématique de France, Number 121 December 2010, 139 pages, Softcover, ISBN: 978-2-85629-301-0, This work consists two parts. In the first part, the author studies 2010 Mathematics Subject Classification: 32A25, 32V05, 32V20, completely the heat equation method 32W30, 58A14, Individual member US$37.80, List US$42, Order of Menikoff-Sjöstrand and applies it to code SMFMEM/121 the Kohn Laplacian defined on a compact orientable connected CR manifold. He then gets the full asymptotic expansion of the Szeg˝oprojection for (0, q) forms when the Levi form is non-degenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for (0, 0) forms. The author’s main tools are Fourier integral operators with complex valued phase Melin and Sjöstrand functions. In the second part, the author obtains the full asymptotic expansion of the Bergman projection for (0, q) forms when the Levi form is non-degenerate. This also generalizes a result of Boutet de Monvel and Sjöstrand for (0, 0) forms. He introduces a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and applies the heat equation method of Menikoff and Sjöstrand to this operator. He obtains a description of a new Szeg˝o projection up to smoothing operators and gets his main result by using the Poisson operator. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Contents: Introduction; Part I. On the singularities of the Szegö projection for (0, q) forms; Part II. On the singularities of the Bergman projection for (0, q forms; Bibliography. Mémoires de la Société Mathématique de France, Number 123 January 2011, 136 pages, Softcover, ISBN: 978-2-85629-304-1, 2010 Mathematics Subject Classification: 32A25, 32V05, 32V20, 32W30, 58A14, Individual member US$37.80, List US$42, Order code SMFMEM/123

A Fundamental Domain for V3 Mary Rees, , England

The author describes a fundamental domain for the punctured Riemann surface V3,m which parametrises (up to Möbius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three and the second critical point is not mapped in m iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to , of all dynamics in all type III hyperbolic components in V3, and gives indications of a topological model for V3, together with the hyperbolic components contained in it. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

636 Notices of the AMS Volume 58, Number 4 Classified Advertisements

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MISSISSIPPI ments, summaries of research plans and algorithms and improving existing ones; teaching philosophy, a completed AMS research and develop new mathemati- MISSISSIPPI STATE UNIVERSITY Standard Cover Sheet (http://ww.ams. cal models; perform computations and org/employment), and three letters of apply mathematical analysis; validate Mathematics and Statistics recommendation. One reference letter algorithms and mathematical models; Faculty Position in Mathematics or should address the applicant’s teach- write reports and presentations concern- Mathematics Education ing. In addition, a cover letter should ing algorithms. Applicant must obtain a passing score on a programming test Applications are invited for one antici- be submitted through mathjobs.org pated tenure-track position in mathemat- addressed to: Chair, Mathematics Search uniformly administered to all applicants ics or mathematics education, pending Committee, Department of Mathematics for the position. availability of funds. The anticipated and Statistics, P.O. Drawer MA, Mississippi If interested, please mail resume to: start date will be August 16, 2011, or State, MS 39762. All applicants must also On Time Systems, Inc. later. The department offers the Ph.D. complete the online Personal Information 1850 Millrace Drive, Ste. 1 degree in mathematical sciences and is Data Form located at: http://www.jobs. Eugene, OR 97403 highly research oriented. Requirements msstate.edu. Select “Create Application” Attn: Ms. Buchanan include a doctoral degree in mathematics and choose “Personal Information Data 000029 education or an area of mathematical sci- Form”. Review of applications will begin ences, demonstrated success or a strong upon approval of funding and will con- potential for research and a commitment tinue until position is filled. Mississippi UTAH to effective undergraduate and graduate State University is an AA/EOE. teaching. Mississippi State University is 000028 the largest university in Mississippi and BRIGHAM YOUNG UNIVERSITY a land-grant Carnegie Doctoral/Research- Mathematics Department extensive institution. Further information OREGON Applications are invited for a visiting pro- about the department may be found at: fessorship at any level (assistant, associ- http://www.msstate.edu/dept/math. ate, or full professor) in the Department of Salary is competitive and commensurate ON TIME SYSTEMS, INC. Mathematics at Brigham Young University. with qualifications. Research Scientist The department has a strong commitment All supporting material should be sub- We are currently hiring for the position to undergraduate education and has a mitted electronically through http:// of Research Scientist for our company in substantial doctoral program. Candidates www.mathjobs.org/jobs. Supporting Eugene, Oregon. The successful candidate will be evaluated on evidence of excel- materials should include a detailed ré- will be responsible for solving computa- lent teaching and potential to make sig- sumé with detailed research accomplish- tional problems through developing new nificant research contributions. Review of

Suggested uses for classified advertising are positions available, books or August 2011 issue–May 27, 2010; September 2011 issue–June 28, 2011; October lecture notes for sale, books being sought, exchange or rental of houses, 2011 issue–July 28, 2011; November 2011 issue–August 30, 2011. and typing services. U.S. laws prohibit discrimination in employment on the basis of color, age, The 2011 rate is $3.25 per word. No discounts for multiple ads or sex, race, religion, or national origin. “Positions Available” advertisements from institutions outside the U.S. cannot be published unless they are the same ad in consecutive issues. For an additional $10 charge, an- accompanied by a statement that the institution does not discriminate on nouncements can be placed anonymously. Correspondence will be these grounds whether or not it is subject to U.S. laws. Details and specific forwarded. wording may be found on page 667 (vol. 56). Advertisements in the “Positions Available” classified section will be set Situations wanted advertisements from involuntarily unemployed math- with a minimum one-line headline, consisting of the institution name above ematicians are accepted under certain conditions for free publication. Call body copy, unless additional headline copy is specified by the advertiser. toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or 401-455-4084 Headlines will be centered in boldface at no extra charge. Ads will appear worldwide for further information. in the language in which they are submitted. Submission: Promotions Department, AMS, P.O. Box 6248, Providence, There are no member discounts for classified ads. Dictation over the Rhode Island 02940; or via fax: 401-331-3842; or send email to [email protected]. AMS location for express delivery packages is telephone will not be accepted for classified ads. 201 Charles Street, Providence, Rhode Island 20904. Advertisers will be Upcoming deadlines for classified advertising are as follows: May billed upon publication. 2011 issue–February 28 2011; June/July 2011 issue–April 28, 2011;

APRIL 2011 NOTICES OF THE AMS 637 Classified Advertisements AMERICAN MATHEMATICAL SOCIETY applications will begin on February 1, RIO DE JANEIRO COLLECTED PAPERS OF 2011, and will continue until the posi- tion is filled. For information about how PUC-RIO (Catholic University of Rio de JOHN to apply, as well as further information Janeiro) about the department and the position, Department of Mathematics go to: http://www.math.byu.edu/home/ Assistant Professorship MILNOR hiring/. Five Volume Set Qualifications: A Ph.D. in mathemat- The Mathematics Department of PUC-Rio ics; demonstrated excellence in teaching; has an opening for an assistant profes- sorship starting in August of 2011. This This set is highly recommended desire and ability to teach undergraduate and graduate level mathematics, including is a tenure-track contract, initially lasting to a broad mathematical audi- courses at the freshman level; desire and one year but renewable (see “Hiring pro- cedures” below). The university expects ence, and, in particular, to ability to conduct high-quality mathemati- young mathematicians who candidates to have a consistent record of cal research. Duties and responsibilities: high-level research and also to be qualified will certainly benefi t from their Teaching mathematics at the undergradu- for undergraduate and graduate teaching. acquaintance with Milnor’s ate and graduate level, including courses PUC-Rio offers a stimulating research- mode of thinking and writing. at the freshman level; conducting and oriented environment with plenty of The volumes in this set have directing independent research; collabo- academic freedom and excellent work- rating with other faculty at the university; been organized by subject. ing conditions, such as a pleasant, well- supporting university, college, and depart- located, and safe campus; administrative Set: Hardcover; List US$361; AMS mem- ment goals and missions. flexibility; research fellowships funded by bers US$288.80; Order code MILNORSET BYU is an Equal Employment Oppor- the university as well as by the Brazilian government; and a Bachelors program that Items contained in this set are tunity Employer. Preference is given to historically attracts excellent students. also available for individual sale: qualified candidates who are members in good standing of the affiliated church, Pay is internationally competitive (cur- The Church of Jesus Christ of Latter-day rently around $50,000 plus benefits per I. Geometry year, keep in mind that living expenses in Saints. John Milnor Brazil are relatively low). 000030 Language Skills: Knowledge of Por- A publication of Publish or Perish, Inc. tuguese is highly desirable as courses 1994; 295 pages; Hardcover; ISBN: 978-0- are usually taught in Portuguese; in the 914098-30-0; List US$59; AMS members CHILE absence of that, knowledge of a similar US$47.20; Order code MILNOR/1 Romance language (e.g., Spanish, French, or Italian) is a plus. A successful candidate II. The Fundamental Group PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE who is not ready to teach in Portuguese John Milnor by August 2011 will be given an English- Departamento de Matemticas language courseload during the August- Collected Works, Volume 19; 1995; 302 November 2011 semester under the con- pages; Hardcover; ISBN: 978-0-8218-4875- The Department of Mathematics invites dition that he or she commits to learning 3; List US$65; AMS members US$52; applications for three tenure-track posi- sufficient Portuguese by March 2012. Order code CWORKS/19.2 tions at the assistant professor level Full details of the submission process beginning either March or August 2012. and hiring procedures can be found by III. Differential Topology Applicants should have a Ph.D. in math- John Milnor clicking on the “New Faculty Position” ematics, proven research potential either link in the Announcements section of Collected Works, Volume 19; 2007; 343 in pure or applied mathematics, and our homepage at: http://www.mat. pages; Hardcover; ISBN: 978-0-8218-4230- a strong commitment to teaching and puc-rio.br/pagina.php?id=anuncios. 0; List US$69; AMS members US$55.20; research. The regular teaching load for Candidates must send the required Order code CWORKS/19.3 assistant professors consists of three documents to [email protected]. one-semester courses per year, reduced br by April 15, 2011. IV. Homotopy, Homology to two during the first two years. The an- 000023 and Manifolds nual salary will be US$45,000 (calculated John McCleary, Editor at the current exchange rate of 500 chilean pesos per dollar). Collected Works, Volume 19; 2009; 368 pages; Hardcover; ISBN: 978-0-8218-4475- Please send a letter indicating your main 5; List US$79; AMS members US$63.20; research interests, potential collaborators Order code CWORKS/19.4 in our department (http://www.mat. puc.cl), detailed curriculum vitae, and V. Algebra three letters of recommendation to: and T. Y. Lam, Editors Director Departamento de Matem- Collected Works, Volume 19; 2010; 425 ticas, pages; Hardcover; ISBN: 978-0-8218-4876- Pontificia Universidad Catolica de 0; List US$89; AMS members US$71.20; Chile, Order code CWORKS/19.5 Av. Vicua Mackenna 4860 Santiago, Chile; fax: (56-2) 552-5916; email: [email protected] For full consideration, complete applica- tion materials must arrive by June 30, www.ams.org/bookstore 2011. 000027

638 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Meetings & Conferences of the AMS

IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http://www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting.

Combinatorial Representation Theory, Cristina Bal- Worcester, lantine, College of the Holy Cross, and Rosa Orellana, Dartmouth College. Massachusetts Combinatorics of Coxeter Groups, Dana C. Ernst, Plym- outh State University, and Matthew Macauley, Clemson College of the Holy Cross University. Complex Analysis and Banach Algebras, John T. An- April 9–10, 2011 derson, College of the Holy Cross, and Alexander J. Izzo, Saturday – Sunday Bowling Green State University. Computability Theory and Applications, Brooke Ander- Meeting #1070 sen, Assumption College. Eastern Section Dynamics of Rational Systems of Difference Equations Associate secretary: Steven H. Weintraub with Applications, M. R. S. Kulenovic and O. Merino, Uni- Announcement issue of Notices: February 2011 versity of Rhode Island. Program first available on AMS website: March 10, 2011 Geometric and Topological Problems in Curvature, Program issue of electronic Notices: April 2011 Megan Kerr and Stanley Chang, Wellesley College. Issue of Abstracts: Volume 32, Issue 3 Geometry and Applications of 3-Manifolds, Abhijit Champanerkar and Ilya Kofman, College of Staten Island, Deadlines CUNY, and Walter Neumann, Barnard College, Columbia For organizers: Expired University. For consideration of contributed papers in Special Ses- Geometry of Nilpotent Lie Groups, Rachelle DeCoste, sions: Expired Wheaton College, Lisa DeMeyer, Central Michigan Univer- For abstracts: Expired sity, and Maura Mast, University of Massachusetts-Boston. History and Philosophy of Mathematics, James J. Tatter- The scientific information listed below may be dated. sall, Providence College, and V. Frederick Rickey, United For the latest information, see www.ams.org/amsmtgs/ States Military Academy. sectional.html. Interactions between Dynamical Systems, Number The- ory, and Combinatorics, Vitaly Bergelson, The Ohio State Invited Addresses University, and Dmitry Kleinbock, Brandeis University. Vitaly Bergelson, Ohio State University, Ergodic Ramsey Mathematical and Computational Advances in Interfa- theory: Dynamical systems at the service of combinatorics cial Fluid Dynamics, Burt S. Tilley, Worcester Polytechnic and number theory. Institute, and Lou Kondic, New Jersey Institute of Tech- Kenneth M. Golden, University of Utah, Title to be an- nology. nounced. Mathematics and Climate, Kenneth M. Golden, Univer- Walter D. Neumann, Columbia University, What does a sity of Utah, Catherine Roberts, College of the Holy Cross, complex surface really look like? and MaryLou Zeeman, Bowdoin College. Natasa Sesum, University of Pennsylvania, Title to be Modular Forms, Elliptic Curves, L-functions, and Number announced. Theory, Sharon Frechette and Keith Ouellette, College of the Holy Cross. Special Sessions New Trends in College and University Faculty Engage- Celestial Mechanics, Glen R. Hall, Boston University, and ment in K–12 Education, Jennifer Beineke, Western New Gareth E. Roberts, College of the Holy Cross. England College, and Corri Taylor, Wellesley College.

APRIL 2011 NOTICES OF THE AMS 639 Meetings & Conferences

Number Theory, Arithmetic Topology, and Arithmetic Special Sessions Dynamics, Michael Bush, Smith College, Farshid Hajir, Advances in Modeling, Numerical Analysis and Compu- University of Massachusetts, Amherst, and Rafe Jones, tations of Fluid Flow Problems, Monika Neda, University College of the Holy Cross. of Nevada, Las Vegas. Physically Inspired Higher Homotopy Algebra, Thomas Computational Algebra, Groups and Applications, Ben- J. Lada, North Carolina State University, and Jim Stasheff, jamin Fine, Fairfield University, Gerhard Rosenberger, University of North Carolina, Chapel Hill. Random Processes, Andrew Ledoan, Boston College, University of Hamburg, Germany, and Delaram Kahrobaei, and Steven J. Miller and Mihai Stoiciu, Williams College. City University of New York. The Algebraic Geometry and Topology of Hyperplane Discrete Dynamical Systems in Graph Theory, Combi- Arrangements, Graham Denham, University of Western natorics, and Geometry, Eunjeong Yi and Cong X. Kang, Ontario, and Alexander I. Suciu, Northeastern University. Texas A&M University at Galveston. Topics in Partial Differential Equations and Geometric Extremal Combinatorics, Jozsef Balogh, University of Analysis, Maria-Cristina Caputo, University of Arkansas, California San Diego, and Ryan Martin, Iowa State Uni- and Natasa Sesum, Rutgers University. versity. Topological, Geometric, and Quantum Invariants of Flow-Structure Interaction, Paul Atzberger, University 3-manifolds, David Damiano, College of the Holy Cross, of California Santa Barbara. Scott Taylor, Colby College, and Helen Wong, Carleton Geometric Group Theory and Dynamics, Matthew Day, College. , and Joel Louwsma, California Institute of Undergraduate Research, David Damiano, College of Technology, and Andy Putman, Rice University. the Holy Cross, Giuliana Davidoff, Mount Holyoke College, Geometric PDEs, Matthew Gursky, Notre Dame Univer- Steve Levandosky, College of the Holy Cross, and Steven sity, and Emmanuel Hebey, Université de Cergy-Pontoise. J. Miller, Williams College. Knots, Surfaces and 3-manifolds, Stanislav Jabuka, Swa- tee Naik, and Chris Herald, University of Nevada, Reno. Lie Algebras, Algebraic Transformation Groups and Las Vegas, Nevada Representation Theory, Andrew Douglas and Bart Van University of Nevada Steirteghem, City University of New York. Multilevel Mesh Adaptation and Beyond: Computational April 30 – May 1, 2011 Methods for Solving Complex Systems, Pengtao Sun, Uni- Saturday – Sunday versity of Nevada, Las Vegas, and Long Chen, University of California Irvine. Meeting #1071 Nonlinear PDEs and Variational Methods, David Costa Western Section and Hossein Tehrani, University of Nevada, Las Vegas, Associate secretary: Michel L. Lapidus and Zhi-Qiang Wang, Utah State University. Announcement issue of Notices: February 2011 Partial Differential Equations Modeling Fluids, Quansen Program first available on AMS website: March 17, 2011 Jiu, Capital Normal University, Beijing, China, and Jiahong Program issue of electronic Notices: April 2011 Wu, Oklahoma State University. Issue of Abstracts: Volume 32, Issue 3 Recent Advances in Finite Element Methods, Jichun Li, Deadlines University of Nevada, Las Vegas. For organizers: Expired Recent Developments in Stochastic Partial Differential For consideration of contributed papers in Special Ses- Equations, Igor Cialenco, Illinois Institute of Technology, sions: Expired and Nathan Glatt-Holtz, Indiana University, Bloomington. For abstracts: Expired Set Theory, Douglas Burke and Derrick DuBose, Uni- versity of Nevada, Las Vegas. The scientific information listed below may be dated. Special Session in Arithmetic Dynamics, Arthur Baragar, For the latest information, see www.ams.org/amsmtgs/ University of Nevada, Las Vegas, and Patrick Ingram, Uni- sectional.html. versity of Waterloo. Special Session on Computational and Mathematical Invited Addresses Finance, Hongtao Yang, University of Nevada, Las Vegas. Elizabeth Allman, University of Alaska, Evolutionary Topics in Modern Complex Analysis, Zair Ibragimov, trees and phylogenetics: An algebraic perspective. California State University, Fullerton, Zafar Ibragimov, Danny Calegari, California Institute of Technology, Urgench State University, and Hrant Hakobyan, Kansas Stable commutator length in free groups. State University. Hector Ceniceros, University of California Santa Bar- bara, Immersed boundaries in complex fluids. Tai-Ping Liu, Stanford University, Hilbert’s sixth prob- lem.

640 NNOTICESOTICES OFOF THETHE AMSAMS VOLUME 58, NUMBER 4 Meetings & Conferences

Gerhard Rosenberger, Passau University and Hamburg Ithaca, New York University, Germany. Multivariable Operator Theory (Code: SS 13A), Ronald G Douglas, Texas A&M University, and Rongwei Yang, State University of New York at Albany. September 10–11, 2011 Parabolic Evolution Equations of Geometric Type (Code: Saturday – Sunday SS 4A), Xiaodong Cao, Cornell University, and Bennett Chow, University of California, San Diego. Meeting #1072 Partial Differential Equations of Mixed Elliptic-Hyper- Eastern Section bolic Type and Applications (Code: SS 3A), Marcus Khuri, Associate secretary: Steven H. Weintraub Stony Brook University, and Dehua Wang, University of Announcement issue of Notices: June 2011 Pittsburgh. Program first available on AMS website: July 28, 2011 Representations of Local and Global Groups (Code: SS Program issue of electronic Notices: September 2011 12A), Mahdi Asgari, Oklahoma State University, and Birgit Issue of Abstracts: Volume 32, Issue 4 Speh, Cornell University. Deadlines Set Theory (Code: SS 2A), Paul Larson, Miami University, Ohio, Justin Moore, Cornell University, and Ernest Schim- For organizers: Expired merling, Carnegie Mellon University. For consideration of contributed papers in Special Ses- Species and Hopf Algebraic Combinatorics (Code: SS sions: May 24, 2011 6A), Marcelo Aguiar, Texas A&M University, and Samuel For abstracts: July 5, 2011 Hsiao, Bard College. Symplectic Geometry and Topology (Code: SS 5A), Tara The scientific information listed below may be dated. Holm, Cornell University, and Katrin Wehrheim, M.I.T. For the latest information, see www.ams.org/amsmtgs/ sectional.html. Invited Addresses Winston-Salem, Mladen Bestvina, University of Utah, Title to be an- nounced. North Carolina Nigel Higson, Pennsylvania State University, Title to be announced. Wake Forest University Gang Tian, Princeton University, Title to be announced. September 24–25, 2011 Katrin Wehrheim, Massachusetts Institute of Technol- ogy, Title to be announced. Saturday – Sunday

Special Sessions Meeting #1073 Analysis, Probability, and Mathematical Physics on Frac- Southeastern Section tals (Code: SS 10A), Luke Rogers, University of Connecti- Associate secretary: Matthew Miller cut, Robert Strichartz, Cornell University, and Alexander Announcement issue of Notices: June 2011 Teplyaev, University of Connecticut. Program first available on AMS website: August 11, 2011 Difference Equations and Applications (Code: SS 1A), Program issue of electronic Notices: September 2011 Michael Radin, Rochester Institute of Technology. Issue of Abstracts: Volume 32, Issue 4 Gauge Theory and Low-dimensional Topology (Code: SS 7A), Weimin Chen, University of Massachusetts-Amherst, Deadlines and Daniel Ruberman, Brandeis University. For organizers: Expired Geometric Aspects of Analysis and Measure Theory For consideration of contributed papers in Special Ses- (Code: SS 9A), Leonid Kovalev and Jani Onninen, Syracuse sions: June 7, 2011 University, and Raanan Schul, State University of New For abstracts: August 2, 2011 York at Stony Brook. Geometry of Artihmetic Groups (Code: SS 11A), Mladen The scientific information listed below may be dated. Bestvina, University of Utah, and Ken Brown, Martin For the latest information, see www.ams.org/amsmtgs/ Kassabov, and Tim Riley, Cornell University. sectional.html. Kac-Moody Lie Algebras, Vertex Algebras, and Related Topics (Code: SS 14A), Alex Feingold, Binghamton Uni- Invited Addresses versity, and Antun Milas, State University of New York Benjamin B. Brubaker, Massachusetts Institute of at Albany. Technology, Title to be announced. Mathematical Aspects of Cryptography and Cyber Se- Shelly Harvey, Rice University, Title to be announced. curity (Code: SS 8A), Benjamin Fine, Fairfield University, Allen Knutson, Cornell University, Title to be an- Delaram Kahrobaei, City University of New York, and nounced.

APRIL 2011 NOTICES OF THE AMS 641 Meetings & Conferences

Seth M. Sullivant, North Carolina State University, Title Extremal and Probabilistic Combinatorics (Code: SS to be announced. 5A), Stephen Hartke and Jamie Radcliffe, University of Nebraska-Lincoln. Special Sessions Quantum Groups and Representation Theory (Code: Algebraic and Geometric Aspects of Matroids (Code: SS SS 2A), Jonathan Kujawa, University of Oklahoma, and 1A), Hoda Bidkhori, Alex Fink, and Seth Sullivant, North Natasha Rozhkovskaya, Kansas State University. Carolina State University. Applications of Difference and Differential Equations to Biology (Code: SS 2A), Anna Mummert, Marshall University, and Richard C. Schugart, Western Kentucky University. Salt Lake City, Utah University of Utah

Lincoln, Nebraska October 22–23, 2011 University of Nebraska-Lincoln Saturday – Sunday

October 14–16, 2011 Meeting #1075 Friday – Sunday Western Section Associate secretary: Michel L. Lapidus Meeting #1074 Announcement issue of Notices: August 2011 Central Section Program first available on AMS website: September 8, 2011 Associate secretary: Georgia Benkart Announcement issue of Notices: August 2011 Program issue of electronic Notices: October 2011 Program first available on AMS website: September 1, 2011 Issue of Abstracts: Volume 32, Issue 4 Program issue of electronic Notices: October 2011 Issue of Abstracts: Volume 32, Issue 4 Deadlines For organizers: March 22, 2011 Deadlines For consideration of contributed papers in Special Ses- For organizers: Expired sions: July 5, 2011 For consideration of contributed papers in Special Ses- For abstracts: August 30, 2011 sions: June 28, 2011 For abstracts: August 23, 2011 The scientific information listed below may be dated. The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ For the latest information, see www.ams.org/amsmtgs/ sectional.html. sectional.html. Invited Addresses Invited Addresses Graeme Milton, University of Utah, Title to be an- Lewis Bowen, Texas A&M University, Title to be an- nounced. nounced. Lei Ni, University of California San Diego, Title to be Emmanuel Candes, Stanford University, Title to be an- announced. nounced (Erdo˝s Memorial Lecture). Igor Pak, University of California Los Angeles, Title to Alina Cojocaru, University of Illinois at Chicago, Title be announced. to be announced. Michael Zieve, University of Michigan, Title to be an- Monica Visan, University of California Los Angeles, nounced. Title to be announced.

Special Sessions Special Sessions Association Schemes and Related Topics (Code: SS 1A), Commutative Algebra (Code: SS 3A), Chin-Yi Jean Sung Y. Song, Iowa State University, and Paul Terwilliger, Chan, Central Michigan University, and Lance E. Miller, University of Wisconsin, Madison. University of Utah. Asymptotic Behavior and Regularity for Nonlinear Evo- Geometric, Combinatorial, and Computational Group lution Equations (Code: SS 4A), Petronela Radu and Lorena Theory (Code: SS 1A), Eric Freden, Southern Utah Univer- Bociu, University of Nebraska-Lincoln. sity, and Eric Swenson, Brigham Young University. Coding Theory (Code: SS 7A), Christine Kelley and Judy Walker, University of Nebraska-Lincoln. Geometric Evolution Equations and Related Topics. Dynamic Systems on Time Scales with Applications (Code: SS 2A), Andrejs Treibergs, University of Utah, Salt (Code: SS 3A), Lynn Erbe and Allan Peterson, University Lake City, Lei Ni, University of California, San Diego, and of Nebraska-Lincoln. Brett Kotschwar, Arizona State University.

642 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Meetings & Conferences

ematical Association of America, annual meetings of the Port Elizabeth, Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the Republic of South winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Africa Associate secretary: Michel L. Lapidus Nelson Mandela Metropolitan University Announcement issue of Notices: October 2011 Program first available on AMS website: November 1, 2011 November 29 – December 3, 2011 Program issue of electronic Notices: January 2012 Tuesday – Saturday Issue of Abstracts: Volume 33, Issue 1

Meeting #1076 Deadlines First Joint International Meeting between the AMS and the For organizers: April 1, 2011 South African Mathematical Society. For consideration of contributed papers in Special Ses- Associate secretary: Matthew Miller sions: To be announced Announcement issue of Notices: June 2011 For abstracts: September 22, 2011 Program first available on AMS website: Not applicable Program issue of electronic Notices: Not applicable Issue of Abstracts: Not applicable Honolulu, Hawaii Deadlines University of Hawaii For organizers: Expired For consideration of contributed papers in Special Ses- March 3–4, 2012 sions: To be announced Saturday – Sunday For abstracts: To be announced Western Section Associate secretary: Michel L. Lapidus The scientific information listed below may be dated. Announcement issue of Notices: March 2012 For the latest information, see www.ams.org/amsmtgs/ Program first available on AMS website: To be announced internmtgs.html. Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Invited Addresses Mark J. Ablowitz, University of Colorado, Title to be Deadlines announced. For organizers: August 3, 2011 James Raftery, University of Kwazulu Natal, Title to For consideration of contributed papers in Special Ses- be announced. sions: To be announced Daya Reddy, University of Cape Town, Title to be an- For abstracts: To be announced nounced. Peter Sarnak, Princeton University, Title to be an- nounced. Robin Thomas, Georgia Institute of Technology, Title Tampa, Florida to be announced. University of South Florida Amanda Weltman, University of Cape Town, Title to be announced. March 10–11, 2012 Saturday – Sunday Southeastern Section Boston, Associate secretary: Matthew Miller Announcement issue of Notices: To be announced Massachusetts Program first available on AMS website: To be announced Program issue of electronic Notices: March 2012 John B. Hynes Veterans Memorial Conven- Issue of Abstracts: To be announced tion Center, Boston Marriott Hotel, and Boston Sheraton Hotel Deadlines For organizers: August 10, 2011 January 4–7, 2012 For consideration of contributed papers in Special Ses- Wednesday – Saturday sions: To be announced Joint Mathematics Meetings, including the 118th Annual For abstracts: To be announced Meeting of the AMS, 95th Annual Meeting of the Math-

APRIL 2011 NOTICES OF THE AMS 643 Meetings & Conferences

The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Rochester, New York sectional.html. Rochester Institute of Technology Special Sessions September 22–23, 2012 Discrete Models in Molecular Biology (Code: SS 1A), Ales- Saturday – Sunday sandra Carbone, Université Pierre et Marie Curie and Labo- Eastern Section ratory of Microorganisms Genomics, Natasha Jonoska, Associate secretary: Steven H. Weintraub University of South Florida, and Reidun Twarock, Uni- Announcement issue of Notices: To be announced versity of York. Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Washington, District Issue of Abstracts: To be announced Deadlines of Columbia For organizers: February 22, 2012 For consideration of contributed papers in Special Ses- George Washington University sions: To be announced For abstracts: To be announced March 17–18, 2012 Saturday – Sunday Eastern Section New Orleans, Associate secretary: Steven H. Weintraub Announcement issue of Notices: To be announced Louisiana Program first available on AMS website: To be announced Program issue of electronic Notices: March 2012 Tulane University Issue of Abstracts: To be announced October 13–14, 2012 Saturday – Sunday Deadlines Southeastern Section For organizers: August 17, 2011 Associate secretary: Matthew Miller For consideration of contributed papers in Special Ses- Announcement issue of Notices: To be announced sions: To be announced Program first available on AMS website: To be announced For abstracts: To be announced Program issue of electronic Notices: October 2012 Issue of Abstracts: To be announced Lawrence, Kansas Deadlines For organizers: March 13, 2012 University of Kansas For consideration of contributed papers in Special Ses- sions: To be announced March 30 – April 1, 2012 For abstracts: To be announced Friday – Sunday Central Section Associate secretary: Georgia Benkart Akron, Ohio Announcement issue of Notices: To be announced Program first available on AMS website: To be announced University of Akron Program issue of electronic Notices: To be announced October 20–21, 2012 Issue of Abstracts: To be announced Saturday – Sunday Central Section Deadlines Associate secretary: Georgia Benkart For organizers: To be announced Announcement issue of Notices: To be announced For consideration of contributed papers in Special Ses- Program first available on AMS website: To be announced sions: To be announced Program issue of electronic Notices: To be announced For abstracts: August 31, 2011 Issue of Abstracts: To be announced

Deadlines For organizers: March 22, 2012 For consideration of contributed papers in Special Ses- sions: To be announced For abstracts: To be announced

644 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Meetings & Conferences San Diego, California Ames, Iowa San Diego Convention Center and San Iowa State University Diego Marriott Hotel and Marina April 27–28, 2013 January 9–12, 2013 Saturday – Sunday Central Section Wednesday – Saturday Associate secretary: Georgia Benkart Joint Mathematics Meetings, including the 119th Annual Announcement issue of Notices: To be announced Meeting of the AMS, 96th Annual Meeting of the Math- Program first available on AMS website: To be announced ematical Association of America, annual meetings of the Program issue of electronic Notices: April 2013 Association for Women in Mathematics (AWM) and the Issue of Abstracts: To be announced National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), Deadlines with sessions contributed by the Society for Industrial and For organizers: September 27, 2012 Applied Mathematics (SIAM). For consideration of contributed papers in Special Ses- Associate secretary: Georgia Benkart sions: To be announced Announcement issue of Notices: October 2012 For abstracts: To be announced Program first available on AMS website: November 1, 2012 Program issue of electronic Notices: January 2012 The scientific information listed below may be dated. Issue of Abstracts: Volume 34, Issue 1 For the latest information, see www.ams.org/amsmtgs/ sectional.html. Deadlines Special Sessions For organizers: April 1, 2012 For consideration of contributed papers in Special Ses- Operator Algebras and Topological Dynamics (Code: SS 1A), Benton L. Duncan, North Dakota State University, sions: To be announced and Justin R. Peters, Iowa State University. For abstracts: To be announced Chestnut Hill, Alba Iulia, Romania June 27–30, 2013 Massachusetts Thursday – Sunday First Joint International Meeting of the AMS and the Ro- Boston College manian Mathematical Society, in partnership with the “Simion Stoilow” Institute of Mathematics of the Romanian April 6–7, 2013 Academy. Saturday – Sunday Associate secretary: Steven H. Weintraub Eastern Section Announcement issue of Notices: To be announced Associate secretary: Steven H. Weintraub Program first available on AMS website: Not applicable Announcement issue of Notices: To be announced Program issue of electronic Notices: Not applicable Program first available on AMS website: To be announced Issue of Abstracts: Not applicable Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines For organizers: To be announced Deadlines For consideration of contributed papers in Special Ses- For organizers: September 6, 2012 sions: To be announced For consideration of contributed papers in Special Ses- For abstracts: To be announced sions: To be announced For abstracts: To be announced Riverside, California University of California, Riverside November 2–3, 2013 Saturday – Sunday Western Section Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced

APRIL 2011 NOTICES OF THE AMS 645 Meetings & Conferences

Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: April 1, 2014 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses- sions: To be announced Deadlines For abstracts: To be announced For organizers: April 2, 2013 For consideration of contributed papers in Special Ses- sions: To be announced Seattle, Washington For abstracts: To be announced Washington State Convention & Trade Center and the Sheraton Seattle Hotel Baltimore, Maryland January 6–9, 2016 Baltimore Convention Center, Baltimore Wednesday – Saturday Joint Mathematics Meetings, including the 122nd Annual Hilton, and Marriott Inner Harbor Meeting of the AMS, 99th Annual Meeting of the Math- ematical Association of America, annual meetings of the January 15–18, 2014 Association for Women in Mathematics (AWM) and the Wednesday – Saturday National Association of Mathematicians (NAM), and the Joint Mathematics Meetings, including the 120th Annual winter meeting of the Association of Symbolic Logic, with Meeting of the AMS, 97th Annual Meeting of the Math- sessions contributed by the Society for Industrial and Ap- ematical Association of America, annual meetings of the plied Mathematics (SIAM). Association for Women in Mathematics (AWM) and the Associate secretary: Michel L. Lapidus National Association of Mathematicians (NAM), and the Announcement issue of Notices: October 2015 winter meeting of the Association for Symbolic Logic, with Program first available on AMS website: To be announced sessions contributed by the Society for Industrial and Ap- Program issue of electronic Notices: January 2016 plied Mathematics (SIAM). Issue of Abstracts: Volume 37, Issue 1 Associate secretary: Matthew Miller Announcement issue of Notices: October 2013 Deadlines Program first available on AMS website: November 1, 2013 For organizers: April 1, 2015 Program issue of electronic Notices: January 2013 For consideration of contributed papers in Special Ses- Issue of Abstracts: Volume 35, Issue 1 sions: To be announced For abstracts: To be announced Deadlines For organizers: April 1, 2013 For consideration of contributed papers in Special Ses- Atlanta, Georgia sions: To be announced For abstracts: To be announced Hyatt Regency Atlanta and Marriott At- lanta Marquis January 4–7, 2017 San Antonio, Texas Wednesday – Saturday Joint Mathematics Meetings, including the 123rd Annual Henry B. Gonzalez Convention Center and Meeting of the AMS, 100th Annual Meeting of the Math- Grand Hyatt San Antonio ematical Association of America, annual meetings of the Association for Women in Mathematics (AWM) and the January 10–13, 2015 National Association of Mathematicians (NAM), and the Saturday – Tuesday winter meeting of the Association of Symbolic Logic, with Joint Mathematics Meetings, including the 121st Annual sessions contributed by the Society for Industrial and Ap- Meeting of the AMS, 98th Annual Meeting of the Math- plied Mathematics (SIAM). ematical Association of America, annual meetings of the Associate secretary: Georgia Benkart Association for Women in Mathematics (AWM) and the Announcement issue of Notices: October 2016 National Association of Mathematicians (NAM), and the Program first available on AMS website: To be announced winter meeting of the Association of Symbolic Logic, with Program issue of electronic Notices: January 2017 sessions contributed by the Society for Industrial and Ap- Issue of Abstracts: Volume 38, Issue 1 plied Mathematics (SIAM). Associate secretary: Steven H. Weintraub Deadlines Announcement issue of Notices: October 2014 For organizers: April 1, 2016 Program first available on AMS website: To be announced For consideration of contributed papers in Special Ses- Program issue of electronic Notices: January 2015 sions: To be announced Issue of Abstracts: Volume 36, Issue 1 For abstracts: To be announced

646 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Meetings and Conferences of the AMS

Associate Secretaries of the AMS Eastern Section: Steven H. Weintraub, Department of Math- Western Section: Michel L. Lapidus, Department of Math- ematics, Lehigh University, Bethlehem, PA 18105-3174; e-mail: ematics, University of California, Surge Bldg., Riverside, CA [email protected]; telephone: 610-758-3717. 92521-0135; e-mail: [email protected]; telephone: Southeastern Section: Matthew Miller, Department of Math- 951-827-5910. ematics, University of South Carolina, Columbia, SC 29208-0001, Central Section: Georgia Benkart, University of Wisconsin- e-mail: [email protected]; telephone: 803-777-3690. Madison, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706-1388; e-mail: [email protected]; telephone: 608-263-4283.

The Meetings and Conferences section of the Notices 2013 gives information on all AMS meetings and conferences January 9–12 San Diego, California p. 645 approved by press time for this issue. Please refer to the page Annual Meeting numbers cited in the table of contents on this page for more April 27–28 Ames, Iowa p. 645 detailed information on each event. Invited Speakers and June 27–30 Alba Iulia, Romania p. 645 Special Sessions are listed as soon as they are approved November 2–3 Riverside, California p. 645 by the cognizant program committee; the codes listed are needed for electronic abstract submission. For some 2014 meetings the list may be incomplete. Information in this January 15–18 Baltimore, Maryland p. 646 issue may be dated. Up-to-date meeting and conference Annual Meeting information can be found at www.ams.org/meetings/. 2015 January 10–13 San Antonio, Texas p. 646 Meetings: Annual Meeting 2011 2016 April 9–10 Worcester, Massachusetts p. 639 January 6–9 Seattle, Washington p. 646 April 30–May 1 Las Vegas, Nevada p. 640 Annual Meeting September 10–11 Ithaca, New York p. 641 2017 September 24–25 Winston-Salem, North January 4–7 Atlanta, Georgia p. 646 Carolina p. 641 Annual Meeting October 14–16 Lincoln, Nebraska p. 642 October 22–23 Salt Lake City, Utah p. 642 Important Information Regarding AMS Meetings November 29– Port Elizabeth, Republic p. 643 Potential organizers, speakers, and hosts should refer to December 3 of South Africa page 100 in the January 2011 issue of the Notices for general information regarding participation in AMS meetings and 2012 conferences. January 4–7 Boston, Massachusetts p. 643 Abstracts Annual Meeting Speakers should submit abstracts on the easy-to-use interac- March 3–4 Honolulu, Hawaii p. 643 tive Web form. No knowledge of is necessary to submit March 10–11 Tampa, Florida p. 643 an electronic form, although those who use may submit March 17–18 Washington, DC p. 644 abstracts with such coding, and all math displays and simi- March 30–April 1 Lawrence, Kansas p. 644 larily coded material (such as accent marks in text) must September 22–23 Rochester, New York p. 644 be typeset in . Visit http://www.ams.org/cgi-bin/ October 13–14 New Orleans, Louisiana p. 644 abstracts/abstract.pl. Questions about abstracts may be October 20–21 Akron, Ohio p. 644 sent to [email protected]. Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated.

Conferences: (see http://www.ams.org/meetings/ for the most up-to-date information on these conferences.) June 12–July 2, 2011: Mathematics Research Research Communities, Snowbird, Utah. (Please see http://www.ams.org/ amsmtgs/mrc.html for more information.) July 4–7, 2011: von Neumann Symposium on Multimodel and Multialgorithm Coupling for Multiscale Problems, Snowbird, Utah. (Please see http://www.ams.org/meetings/amsconf/symposia/symposia-2011 for more information.) July 24–29, 2011: Conference on Applied Mathematics, Modeling, and Computational Science, Waterloo, Canada (held in cooperation with the AMS).

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Modelling in Healthcare Dynamical Systems and Population The Complex Systems Modelling Group (CSMG), Persistence The IRMACS Center, Simon Fraser University, Hal L. Smith and Horst R. Thieme, Arizona State University, Burnaby, BC, Canada Tempe, AZ Applied mathematics conveyed in plain language A self-contained treatment of persistence theory, employing to illustrate the process of modeling in healthcare many examples to illustrate use of the theoretical results and its potential uses Graduate Studies in Mathematics, Volume 118; 2011; 405 pages; 2010; 218 pages; Hardcover; ISBN: 978-0-8218-4969-9; List US$69; AMS Hardcover; ISBN: 978-0-8218-4945-3; List US$75; AMS members US$60; members US$55.20; Order code MBK/74 Order code GSM/118

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April 2011 Volume 58, Number 4

Deformation of Bordered Surfaces and Convex Polytopes What if Newton saw an apple as just an apple? page 530

Taking Math to Heart: Challenges in Cardiac Electrophysiology page 542

Remembering Paul Malliavin page 550

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