Berkeley

Newsletter A newsletter of the mment of and Center for Pure and Applied Mathematics at the Unlve~lyof Callfomla, Beklq m9 VOL XIV. NO. i

MESSAGE FROM THE CHAIR

Dear UC Berkeley Math Department alumni and other friends, There's a lot to report abut the two years which have passed since the last edi- tion of this newsletter.

ALUMNI NEWS CHAIRAm WEIN~IN In the last newsletter, I asked readers to report on what they had ken doing. I got just a few replies, which I'll summarize here. I hope that more of you will send news for the next issue. Clark Abrahams (BA 1973) is Director for Fair Banking at $AS Institute in North Carolina and came out in 2W8 with a book on the timely subject of "Fair Lending Compliance, Intelligence and Implications for Credit Risk Management9'*Frederic Alzofon (PhD 1956, the last student of Griffith Evans !) has had a career span- ning "thirty years in industry and more than twenty years in 'retirement"', including enginwring work in the aerospace industry, teaching relativity at San Jose State, and a 2003 book on radiation scattering. Janet Muny (BA 1975) wrote, "Though I did not go on as so many profiled in the newsletter, I do munt Math as a special gift. It has been a ballast throughout my weer in business for which I am profoundly grateful." I was person- tcminued on =2) FIFTY YEARS OF LEWY'S COUNTEREXAMPLE

by Maciej Zworski

Just over fifty years ago, Hans Lewy [2] - in Berkeley since 1935 - published one of the nmst famous papsin the histq of pddifferential equations -and it was ow4 pages long. 1 will &scribe the result he obtained and review the history of th pbkrn b created. For a &tailed discussion and tefe~ncesI suggest [3] and [I]. Let us wnsi&r the simplest partid difkrential equation:

where V # 0. He= P denotes the space of infitdy dif- ferentiable wmpkx valued hctions. Can we always solve it locally? (cmtlmd on page 3) ally pleased to learn that I "caught" Janet as her Math 1A FACULTY HONORS AND AWARDS teacher in the Fall of 1971 and led her to major in math, though it was not part of her original plan. 200712008 Academic Year It has been a prticularly god time for the Class of 1978. Previously, I reportcd in this space that Andrew Fire (BA 1978) was awarded the 2006 Nobel Prize in UCB PhD Henrique Bursztyn and Miller Fellow Physiology of Medicine. Since then, the 2007 Sundance Mar& Crainic were the first 2 recipients of the Film Festival's Special Jury hzefor Documentary has Andrk Lichnerowicz prize in Poisson Geometry, been awarded to "No End in Sight," written, directed, awarded in July 2008 at the conference3Poisson and prduced by Charles Ferguson (BA 1978). The film, 2008, in Lausanne, Switzerland. which tells the story behind the US involvement in Iraq, also won a string of Film Critics' awards. Andrew and Charles were our Commencement speakers in 2007 and 2008, respectively. Professor Ed Frenkel was the first recipient of the FRIENDS OF BERKEJXY MATH Chair of Excellence of the Mathematical Sciences Foundation of Paris. The Friends of Berkeley Mathematics is an informal organization created in Fall 20'7 to bring together supporters and other friends of our Department. We have so far had two meetings, in October 207 and Professor Jenny Harrison was elected to the FXQi November 2008. Each meeting included short talks on Foundation, or the Foundational Questions Institute, their research by some of our faculty, lunch and/or din- a non-federal agency affiliated with the Templeton ner, informal discussion among faculty, students, and Foundation. Friends on issues of current concern to the llepartment, and attendance at a special lecture. In 2m7, the lecture was Ken Ribet's colloquium talk on Serre's Modular- ity Conjecture. This was recorded (with the help of Will Hearst, one of our Friends) and is available for view- Associate Professor Olga Holtz was awarded one ing on the F0RA.tv website at http://fora.tv/search- of the 10 European Mathematical Society Pnzes video?q=ribet&x=O&y=O. In 2008, the lecture was for 2008 at the meeting of thc EMS in Amskrdam, by Peter Jones of Yale University, in the annual Serge 2008. She was also elected to the Jhge Akademie hng Undergraduate Jicture series. The title was "Di- of Germany 2008-20 13. vide and Conquer"; you can find an abstract of this talk (in which fractals played a central role) at http://math. ~rkeley.edu/index.php?module=announce&ANN~user~ op=view &ANN-id= 137. Professor Richard Karp is this year's winner of We plan to continue these meetings each fall. Please the Kyoto Prize in Advanced Technology. contact us if you are interested in prticipating. A mWVICE-CHAIR POSITION Professor Beresford Parktt shared the 2008 Hans Since the Spring of 2008, Ken Rikt has wcupied the Schneider Prize in Linear Algebra with Cleve Moler newly-created position of Vice-Chair for Development. (who created Matlab). Together with the Director of the Center for Pure and Applied Mathematics (currently Craig Evans), the Vice- Chair is responsible for cmrdinating efforL5 to secure outside support for our research and instructional activi- Professor James Sethian was elected to the Na- ties. These responsibilities include planning the annual tional Acadcmy Engineefing meetings of the Friends of Berkeley Mathematics and of - 2008. overseeing the production of this Newsletter. Ken is also the chair of the new Distinguished J~cturesCommittee. NEW FACULTY Professor Bend Stw-mfels was elected as VP of the Americm Mathcmalical Society for the perid Ian Ago], who was appinted on July 1,2006, arrived 2/1/2008- 1/31/2011. + from thc UniversiQ of Illinois at Chicago in the Fall of 2007, after a year of leave. He works in low-dimensional geometry and toplogy. Although Ian was largely occu- pied during his first semester here with the MSRI pro- grams on Geometric Group Theory and on Teichmuller

2 Berkeley MUNewsletter Spring 2009 m Following tt~work of Schwariz, Hhmnpis (1954) and MaI- grange (19551 shwed that, for my nmtrivial polynomial P((], the^ exists a dktrihtion E such that

This quickly shms that P(D)u = f is locally solvable. A smalI digmsim about tk notation DZj= (l/i)&, .This last oprator is the quanthd momntum:

This is why we write our opmbrs as Solving an equation locally at xo, mans finding u such that heequation k satisfied in a mightmhood of m. We can de- Reasonable functions, not just plynomials in 6, can b qum- or even ask simply for sow weaker form of a solutio~~ tized as operators,

Becaw of the theorems of Cauchy -balevskaya (local soh- ability in the red analytic case) and of MalgmngeAhznpise (lmd solvability for constant c~fficientsand with smoolh, or dislributional,right handside) it was widely hlieved that all op- Suppose that q(x) â R. Then we can locally stmighkn V erators satisfying some simple nodegeneracy conditions were. locdly solvable. It cam as a great surprk when fifty yem ago Hans Lewy, he^ in Berkelq, showed that far

Nwsupp~~ that the cmficients adthe right hand side in V = Dm + iDz2+ i(xl + ix2)Da the equation Vu = f has rn solutions near any pint for most f E Cm. He arrived at the equation above in his work on =em1 complex vaiables. He cculd not prove existenm of a solution and was audacious encugh to see that it could hd tn am ma1 analytic, that is the at every point their Taylor series have om. converge in =me neighbourhood. Akdy in the 1% centuq Cauchy and Kmalwslaya gave us Iml d dytic dutims. Swn after, Hmmder provided a gemral and beautifully In kt, the Cauchy-KovakvskayaTheorem gives lwal analytic clear explanation. For P(x,D ) we &he the principal symhl: solution to my mble@al We~ntialequation:

Then a mcessaty wnditian for solvability near xo is given as follows One cmdition to guarantee it is that for some E Nn,l@l = m, we have ap(x) # 0. What about the cass of smmth cdcient~aj(x) and nm- Here, Ifl g} is a principal object of classical mechanics, tk homogeneities f 7 Poisson brackek Sm& functions m a more abstract notion than analytic fundom, md om standard example is a fimctim equal to e-'iX fax > OandtoOforx 2 0. Theyplayar5leintbedefini- lion of generalized functions, or distributims, which are heir dud space- The themy of tk latter was developed by Laurent Schwm who, in bewmds of hisNknberg spoken a k Schwa& Memd Meeting, "changed the way we, analysts, think about hnctims". A famous example is the d 'Yunction" defined in physics by

As m ekmnt of the dud of Cm it assims- to & the numk .., , Spring 2009 Berkeley M- Newsletter 3 POSTDOCTORAL FACULTY AND FELLOWS

Maw0 Aldi is a Visiting Lek-Heng Lim is a Visit- lege, hndon. His areas of Wisconsin-Madison. His Assistant Professor who ing Assistant l%ofessor who research inkrest lie in the fields of interest include received his PhD in 2007 received his PhD in 2007 interaction between num- , main- from Northwestern Univer- from Stanford Univer- krtheory, representation ly Gromov- Witten theory, sity. His fields of interest sity. His fields of interest theory and geometry. quantum cohomology, and include Algebraic and include numerical analysis mirror symmetry. geometric applicatioiis of and applied math. string theory. Alvaro Pelayo is an NSF Fellow who received his Raman Sanyal is a Miller Robert Martin is a PhD in 2007 from the Uni- Research Fcllow who Thvor Bruen is a NSERC NSLRC Fcllow who versity of Michigan. His rcceived his PhD in 2008 fellow who received his received his PhD in 2008 areas of interest 11e in ge- from Technical Univer- PhD in 2007 from McGill from the University of Wa- ometry, integrable systems, sity, Berlin. His research University. His fields of terloo. His fields of interest and group actions. interests lie in discrete interest include computa- are in the areas of sampling geometry and toplogy. tional biology, computa- theory, applied harmonic tional phylogenetics, and analysis, sampling theory Jan Reimann is a Visiting evolutional biology. on manifolds, de Branges Assistant Professor who Joshua Sussan is a Visit- spaces, I-Iilkrt spaces of received his PhD in 204 ing Assistant F'rofessor ai~alyhcfunctions, self-ad- from the University of who received his PhD Calder Qaemer is a NSF joint extensions of sym- Heidelberg. His fields of in 2007 from Yale. His fellow who rccc~vedhis metric operators, spectral interest include recursion fields of interest include PHDin 2007, University of theory, physics on curved theory, theories of algo- representation theory, low Pennsylvania. His fields of manifolds, and quantum rithms. dimensional toplogy, and interest include non-com- gravity. categorification. mutative geometry as it pertains to mathematical Brendon Rhoade is a physics. Chang Pang Mok is a NSI-' Fellow who received Mauricio Velasco is a Visiting Ass isIan1 Professor his Phll ~n2038 from the Visiting Assistant Profes- who rcceivcd his PhD in University of Minnesota. sor who received his PhD h&t~phW~QU&W 2007 from Haward IJniver- HIS research interests are in 2007 from Cornell a Millcr Rcsearch Fcllow sity. 111s fields of interest in the area of algebraic University. His fields of who received his PhD in include special values of com binatorical representa- interest include commuta- 2005 from MIT. His fields L-functions, and the ap- tion theory. tive algebra, combinator- of interest lie in thc area of plications of theta-lifting in ics, and algebraic geom- algebraic topology and it!! arithmetic geomety. etry. application to problems in Daniel Rokrtz is a geometry and malhemati- lM,W fellow who re- cal physics. Martin Olbermann is a ceived his 1'hD in 2006 Konrad Waldorf is a Visiting Assistan1 Professor from IWTH Aachen Fedor Lye11Research who received his PhD in University. His fields of fellow who received his Sebastian Herr received 2007 from the University interest lie in the areas of PhD in 20'7 from the his PhD in 2006 from of I-Ieidelberg. His fields of asymptotic behavior of University of Hamburg. Univcrsitat Dortmund. His interest include algebraic, ~stelnuovo-Mumfordreg- His areas of research fields of interest lie in the differential and geometric ularity using Janet bases, interest are topology and area of partial diiTerentia1 toplogy commutative algebra, and quantum field theory. equations. algebraic geomcty.

Brett Parker is a Visiting Christian Zickert is a David Hi11 is a Visit- Assistant Professor who Mamw Roper is a Miller Visiting Assistant Profes- ing Assistant Professor received his PhD in 2005 Rcsearch Fcllow who sor who received his PhD who recc~vedhis PhD in from Stanford University. received his PhD in 20'7 in 2008 from the Colum- 2007 from the University His fie!& of interest are from . bia University. His areas of Oregon. His fields of in the areas of symplectic His areas of inkrest are ap- of research interest are interest include representa- toplogy and holomorphic plied math, fluid dynamics, hyperbolic manifolds, low tion theory and Lie theory, cunles. and bio-mechanics. dimensional toplogy, and using techniques also from knot theory. combinatorics and the theory of symmetric func- Alexander Pauh is a Michael Rose is a Visiting tions. Visiting Assistant Professor Assistant Professor who who received his PhD in received his PhD in 20'7 2008 from Imperial Col- from the University of

4 Berkeley MUNewsletter Spring 2009 The Berkeley Math Circle - reated the thwry of hgrauge multipliers (without king aware High Expectations in Years of it!) just for the purpose of solving one problem. Ahead The sessions are delivered by a number of mathematicians, prominent hgh school teachers, and problem solvers from by David Wertkimm,BMCAssismt; around the Bay Area, including a number of UC Berkeley Ivan Mdk, BMC Associate Direc- faculty. For those who are interested in the topics of recent ses- tor; and Zvez&linu Sm&.ova, BMC sions, here are some samples from the fall semester: Director Monovatiants, culminating with the famous "Conway's checkersyy,by F!rogram Director Zvezdelina S&va (MIUS College and UC Berkeley). Gmbinamics, by Paul Zutz (University of San Francisco). Fmbenius Problems, by Mathas Beck (San Francisco State University).

Every professional and academic will be the btto recog- Rubik's Cuh and Grow Thm.u. bv u Tom Davis (founder of nize that the desire for someone to learn and to actually love silim Graphcs). what they leam is vital for success. And almost every person The Mmty, by Quan Ian (Office of the UC hsident). will agree that to see that &ice to learn in children is even Gmbinamial Geometry,- by- FhcoAdla (San Fmcisco greater, because at that youug age their ambition can be nur- State University) tured and molded into something that will make them excel. Histom of Comulex Numh. bv RobHhshome (UC hdevery Tuesday evening from 6 to 8 pm in Evans Hall on the UC Berkeley campus, 50 to 60 of those children come to participate in the Berkeley Math Circle (BMC)* The spmg semester dlhold a number of exciting lectures at the BMC from UC Berkeley F!rofessors Elwyn Berlekamp, The BMC is a program that givea kids with a love for math- Vera Sergamva,Alexander G~vental,and Bjom P-IL as ematics a place to come and listen to some of the top mathe- well as a numk of other instructom from the larger Bay Area, maticians In their field talk about various topics. In the past ten from San Jose all the way to Davis. years during wbich the program has existed, the BMC has had exfraordinary sums. Several of the students who have come BMC students have been able to understand and apply to the circle have moved on to become the bat aud the bright- the concqts that most pqlewould msi&r to lx way too est mathematiciaus, physicists, and businessmen in the nation. complicated even for undergraduates.A number of our circlers Among them, four have gone on to the Inkmatid Math- dlhave covered so much advanced mathematics, to name ematical Olympiad: Gabriel Card, Tiankai Liu, Oaz Nir, aud a few basica like red aualysis, abstract algebra, toplogy and Yi Sun. Joining last year, but already gaining naiional fame, is algebmc topology, and numk thwry, hat by the time they Evan O'Dorney from Dauville. Evan won as an 8th grader the enter college, they will lx cmquemg graduate-level courses. Bay Area Math Olympiad (BAMO), rauked nationally very hgh in the USA Math Olympiad (USAMO), aud championed Math catuptitions are usually the htand primary goal the 2007 Scrips National S lling Bee. Those who watched associated by the general public with math circles. Even Evan's win are not likely to rorget his brave words on national though the BMC's main purpose is to deepen and expand the television and his eloquent and passionate &scription of his mathematical horiwns of talented young students and prepare love for math and music. and encourage them to mtinue with careers in mathematical research or research in other related sciences and disciplines, From its kginning, the Berkeley Math Circle has been sup- we are proud that a won&r€ "by-product"of the ciders' ported by the UC Berkeley Math Deparbnent, the Mathemati- training is their inevitable domination in various mathematical cal Sciences Research Institute (MSRI), and jwrents' contribu- contests, Olympiads and other events. In the recent past, half tions. of the USA team at the International Mathematical Olympiad in Washington D.C. 2001, oiginated from the Berkeley Math As far as classroom demographics go, this is likely the most Circle (that year, USA tied with Russia for the second place diverse classraom any one of the kids has been or will ever among &Itcm~es). We expect great results from our Olym- be in. Students range in de fmelementary all the way pians in the next few years, further growth of their mathemati- through their last ORfi school, and come from over cal knowledge, and a deepunderstanding of and appreciation the larger Bay Area. There is an almost even split between for the huty of mathematics. male and female, as well as a broad mix of ethuicities. If you would like to cdbuteto the Berkeley Math Circle There is also a broad range of experience among the BMC with a session, leam more about the circle, enroll a cbild or students, as evidenced by the Cide's MonMy Contests. Each mendit to someme else, or attend the Circle as a student month, Associate &tor Ivan Matic gives the students 5 helper, please visit the BMC's website: http:l/mathcircle. problems to solve on their own for 4 weeks. Then the contests hkeley.edu. You may also contact Director Zvezdelina are graded and the top scores receive math books as prizes. Stankova ([email protected]), or Associate h- The results are more than unexpected. This year, for example, tor Ivan Matic ([email protected]),with any questions. two girls tied for 6rst place in the 6rst contest of the year: 6th We always welcome fresh ideas, constructive input, and new grader Sahana Vasu&van and 1lth grader Jennifer Thompson. lecturers. Everyone at the Circle - administrators, parents, and The students come up with a huge number of different ideas espcially the kids - is very grateful for the support, help, and for solving the problems, and it is a great pleasure to look at hospitality that the UC Berkeley Mathematics Department has the solutions they are discovering. Quite frequently the BMC provided over the last 10 years. + instructors are amazed by the non-staudard and innovative paths that these middle and high school students develop for solving the problems; we have witnessed studentsycomplete discoveries aud utilization of various topics in combinatorics and linear algebra, all on their own. Evan O'Domey once ree

Spring 2009 Berkeley M- Newsletter 5 (Chair ^s Letter, continuedfrom page 2) Theory and Kleinian Groups, he was already spending lots of time in Evans Hall, meeting Visiting Faculty with colleagues and students. Robert Bryant of Duke University arrived in OUTSTANDING the Summer of 2007 to take up the positions of GRADUATE Director of MSRI (75% time) and Professor in our Department (25% time). (At the same time, STUDENT outgoing Director has returned INSTRUCTOR full-time to the Department.) With his appoint- ment, and that of John Lott (see below), our AWARDS Department is back on track toward regaining the eminence in differential geometry which it Bruce Driver visited from UC enjoyed in the days of S.-S. Chern. San Diego. His fields of interest include geometric analysis asso- Olga Holtz, who originally arrived as a Mor- ciated to heat kernel and Wie- re) Assistant Professor in 2004, was appointed Jomy Joseph Alappattu as an Associate Professor, effective July 1, ner measures on path and loop Adam Booth spaces of Riemannian manifolds. 2007. For the first couple of years of her ap- Taiyo Inoue pointment, she has been dividing her time Ivan Matic between Berkeley and the Technical University Arturo Rat-Waldron of Berlin, where she holds a prestigious Sophia Irwin Kra is emeritus at SUNY, Janak Ramakrishnan Kowalewskaja Fellowship. Among Olga's Stony Brook. He is, currently, Daren Rhea broad research interests are numerical linear the Executive Director of Math. Anthony Varilly algebra, matrix and operator theory, enumera- for America, a non-profit organi- Martin Vito Cruz tive combinatorics, and compressive sensing. zation working to improve math David Zywina John Lott of the University of Michigan was education in secondary public appointed on July 1,2008, as a Professor and schools in the . arrived here in January, 2009. John is a differ- His areas of research interests ential geometer and geometric analyst who was are complex analysis, Kleinian deeply involved in the activities surrounding groups and deformations of com- Perelman's solution of the Poincark Conjecture plex structures. but whose work covers a wide swath of geo- metric analysis, from transportation problems Meghan Anderson to index theory. Richard Melrose visited from Khalilah Beal MIT. His research interests lie Jennifer Berg Per-Olof Persson arrived this Fall from MIT in the areas of topology, partial Stephen Canon as an Assistant Professor. He is interested in Kiril Datchev high-order methods for fluid and solid me- differential equations, analysis, chanics, in particular techniques based on the and microlocal analysis. Jeffrey Doker Daniel Erman Discontinuous Galerkin method. Between his Master's degree from Lund in his native Swe- Erica Isaacson den, and his PhD from MIT, he worked for sev- Emily Peters eral years as a Senior Development Engineer at Ivan Penkov visited from Cinna Wu the software company Comsol. Jacobs University. His fields of Guoliang Wu interest include representations of finite and infinite dimensional ENDOWED CHAIRS Lie algebras, generated Har- In 2007, the William and Flora Hewlett Foun- ish-Chandra modules, represen- dation gave UC Berkeley a gift of $113 million tations of Lie superalgebras and to support the creation of 100 new endowed Lie super-groups, geometry of supermanifolds, homogeneous chairs. Half of each chair is funded from this superspaces, super 11-modules, 0uift, with the other half coming from a match- homogeneous ind-spaces and ing contribution from another source. I am pleased to report that the family of James geometry of ind-varieties. * Simons has provided funding for the James H. Simons Chair in Mathematics and the Chern- Simons Chair in Mathematical Physics. These will be the first endowed chairs in our Depart- ment and will be held for five-year terms. We expect to appoint the holders of these chairs in

(continued on page 12)

6 Berkeley ~svtkemabwNewsletter Spring 2009 talks. In our experience this makes a hig difference in helping those attending to get some real understanding out of the talk.

Apart from its two-hour format, the main seminar is a trach- tional research seminar, with talks on current work and a new topic each week. We also run at least one instructional seminar each semester, with several weeks or the whole semester de- voted to studying a single topic in depth. Talks in the instruc- tional seminars are given by the participants. Sometimes one of the faculty members in our group organizes an instructional seminar, but recently they have for the most part been orga- nized by the students themselves. RESEARCH TRAINING GROUP Finally, each year at or just after the end of the spring se- by Mark Haiman & Kolya Reshetikhin mester, we hold a week-long workshop-style conference with participation by people both from outside and from on campus. Each workshop features two lecture series by distinguished We--Mark Haiman and Kolya Reshetikhin--are the orgauiz- invited speakers, plus talks by the participants. The 2005 work- ers of a Research Training Group (RTG, for short) that brings shop concentrated on representations of non-commutative al- together a number of us within the Mathematics Department gebras, with Victor Gmzburg and Claudio Procesi as the main whose research interests involve Representation Theory, speakers. The themes of the 2006 workshop were lie algebras Geometry and Combinatorics. The other faculty members af- and Coxeter groups, with main speakers Alexei Borodm and filiated with the group are Richard Borcherds, Edward Frenkel, Alexander Goncharov. Sasha Givental, Martin Olsson, Vera Serganova, Bemd Stur- rnfels, Constantm Teleman, and Joe Wolf. Alien Knutson was While the RTG grant provides some travel support for semi- also a member until he moved to UC San Diego. The chair has nar speakers and workshop participants, the bulk of its funds asked us to give an introduction here to what our group is and go to fellowship support for graduate students and postdocs. It what it does. has also supported postdoctoral fellows. Included are Dagan Karp, now in his third year with us and second on the RTG Our RTG is supported by a grant from the National Sci- (Dagan had an NSERC fellowship for his first year), Beth ence Foundation program called "Enhancing the Mathematical Samuels, who to our great sorrow died of cancer during her Sciences Workforce in the 21st Century." This same NSF pro- second year, David Hill and Joshua Sussan, currently in their gram awards the perhaps more familiar VIGRE grants. Unlike first year, and Michael Rose, who joined us in Fall 2008 after a VIGRE, however, which supports undergraduate, graduate, year at UBC. and postdoctoral education across a whole department, the smaller RTG grants support activities by a group of faculty, The 2008-09 academic year will be the fifth and final year graduate students and postdocs organized around a common of the current RTG grant. With luck we hope to renew it and research interest. continue our activities at full strength. In any event, whether or not we succeed in renewing the grant, we will try to sustain The common interest in our case is the interactions of rep- the sense of community that our activities have engendered by resentation theory, geometry and combinatorics, an area which keeping at least the RTG research and instructional seminars has been highly active in recent years. Among the subjects alive in the future. ¥ encompassed by or having close contact with this area are &G Moody algebras and their q-analogs (quantum groups), finite and infinite-dimensional flag varieties and Schubert varieties, Schubert calculus, Hecke algebras and Cherednik algebras, bzhh-hsztig polynomials, Macdonald polynomials, toric varieties, quiver varieties, Hilbert schemes and other moduli spaces, Gromov-Witten invariants, mirror symmetry, integra- GRATEFUL THANKS TO OUR FRIENDS ble systems, and the geometric Langlands correspondence. The Department of Mathematics extends heartfelt thanks to As one can imagine, young researchers need to become all our donors over the past years for their generous support. conversant with a lot of mathematics in order to prepare for Our donors have contributed to the strength and vitality of our work in these areas. The activities of the RTG are directed students and the Department. The list of our donors from 19% towards helping them do so. The group's central activity is our on can be found on the departmental website at http://rnath. seminar, which meets for two hours each week and features berkeley-daboutussifts-donors.htm1. talks by a mix of outside speakers and local faculty, grad stu- dents and postdocs. Ordinarily we have just one speaker each We apologize if we have omitted anyone. Please do let us know week. We encourage our speakers to use the two hours to keep if that is the case. A special thanks to all our donors who wish the pace leisurely, pause frequently to answer questions, and to remain anonymous. incorporate lots of background material and context in their Spring 2009 Berkeley ~a^iwmttCfNewsletter 7 MANAGER'S REPORT the Center for Pure and Applied Mathematics. Judie was the Assistant I1 for the Mathematics Diagnostic Testing Pro'ect for the last three years. Congratulations to Judie on ier promotion. On November 5,2007, Korinne Reyes joined our staff as the Purchasingrnravel Assistant I11 replacing Stepha- nie Caselli. Korinne had worked for 5 months as an Accounting Assistant at UC Irvine and before that she Faculties Update was an Accounting Clerk at the Alameda County Public Health Department. Korinne graduated from UC River- The second phase of the 7th floor renovation project was side in 2005 with a BA in Political Science. completed in August 2008. The finished space includes a computer lab, and the desperately needed addition of On December 5,2007, Jacqueline Bonds was hired to two seminar rooms, a conference room and an interactive work as the Assistant I1 for MDTP to replace Judie Welch lounge for faculty and graduate students. The Math- Filomeo. Jacqueline graduated from Emery High School ematical and Computational Biology lab located in 748 in Emeryville, California in 2004. She has worked as a Evans, and conference room located in 762 Evans, were Student Loan Counselor, a Retail Store Manager and a completed in January. Bank Teller. Fond Farewell On January 14,2008, Elma Serrano joined the Center for Pure and Applied Mathematics as our new Research The Spring 2007 semester began with the retirement of Center Manager replacing Bernice Gangale. Elma brings Bernice Gangale from the Center for Pure and Applied 10 years of UC Berkeley experience to this position. She Mathematics. Bernice began her UC BerkeleylMath- worked as a Contracts and Grants Administrator for the ematics career twenty-nine years ago as the Graduate Department of Plant and Microbial Biology and has a BA Assistant, and became the Contracts and Grants Admin- in Accounting from Holy Angel University. istrator for the Center for Pure and Applied Mathematics in 1982. Bernice collaborated successfully with the full On June 2,2008, Lynn Greene joined the department range of faculty, staff, students and visitors. as our new EventslPayroll Specialist. Lynn has a BA in Art History and Photography from Syracuse University In June 2007, Linda Jarvis retired from the Chair's and 12 years of work experience including bookkeep- office. Linda was the Academic Personnel and Chair's ing, events planning, pastry chef, human resources and Assistant in the Department of Mathematics for 9 years. pay roll. Previous to joining us, she worked for the Department of Molecular and Cell Biology, the Department of Zoology, We are delighted to welcome all of our new staff mem- and the School of Education. bers! On September 9,2007, Stephanie Caselli died from an I'd also like to take this opportunity to acknowledge the inoperable malignant brain tumor. Stephanie worked in efforts of all of our hardworking staff that have held this the Math Department as the Purchasing/Travel Assistant operation together during our severe staffing shortage. for 8 years. We will always remember her. In view of the current budget climate, we are especially grateful to have these vacancies filled. Staff Awards The Mathematics Department has one of the finest staffs on campus. In 2006-07, Marsha Snow, Kathy Santos and Henri- etta Williams each received $250 Spot Awards for their commendable service during the year. Changing Places and New Faces In May 2007, Mary Pepple was nominated by the math During the last year and a half six staff personnel chang- staff and affirmed by the Berkeley Staff Assembly with es have taken place in the Department an Excellence in Management Award. The theme for the 2006-2007 Excellence in Management Award was On September 26,2007, Kim Oyler was hired as the "Leading by Mentorship and Example". Academic Personnel Analyst and Chair's Administrator to replace Linda Jarvis. Kim has 9 years of Academic In 2007-08, Michael Kim, Nancy Palmer and Barbara Personnel experience with Environmental Science Policy Peavy each received Campus $250 Spot Awards recog- Management and 18 years of UC Berkeley experience. nizing their noteworthy service during the year. On October 1,2007, Judie Welch Filomeo was hired to In February 2008, Barbara Peavy was selected by the replace Eva Wong as the PayrollITravel Assistant I11 for Chancellor's Advisory Committee to receive a 2008 (conrinued on page 13) 8 Berkeley ~eudwwwtwNewsletter Spring 2009 (Fifty Years, continuedfrom page 3) This "explains" nonsolvability as A has a kernel and hence A- is not onto. A conjecture on a necessary and sufficient condition for local solvability of any pseudodifferential equation, P(x,Db = f, and we do not have solvability near any point x E Rg. was formulated by Nirenberg and Tieves in 1970. It involved One conceptual explanation of Htinnander's bracket condi- the mysterious condition 'S: tion uses normal forms in the sympkctic category. A locally Suppose that for C 3 a # 0, defined n : R" x Rn 4 Rn x Rn,is catled symplectic if it preserves the symplectic form u):

Condition '& requires that on Re(ap) = 0, Im ap does not change sign from - to +, along ori- ented trajectories of hfap). A classical theorem of Darboux states that The condition is illustrated in Fig3 where we show the plots of Im(up) as functions of the parameter on the integral curves of H6rman&r (197 1) and Duistennaat-Hormander (1 972) showed HRt(ac).On the left is a change of sign banned by condition a, that this normal form can be implemented on the level of oper- and on the right an allowed change of sign. The most difficult ators by quantizing nonlinear sy rnplectic transformations using case is that of smooth functions with infinitely many zeros. Fourier integral operators:

Hence, any real principal type operator is solvable (just as for vector fields). But how to consider functions and operators which are only local as function of (x, t),position and momentum? Functions of x have structure in phase space (x,$) and an example of that is given in Fig2 On the left are the real and imaginary parts of a The conjecture can be motivated by considering a simple of x, and on the right the plot where function density indicating model: the function concentrates in the (x, ^) space, that is in position and frequency. This particular example arising in physics (first {hDt + ia(t))u(t)= f (t), f (t)= 0 for It) > T. resonant state of the Rckart barrier) was computed by D. Bindel (2006 Ph.D. from Berkeley in computer science), and the phase When can we solve this equation without introducing exponen- space density plot waas provided by L. Demanet {who despite tial gmwtti in I/h? Here, as before, h = I/&, and an exponen- being at Stanford helps us out occasionally!). tial growth prevents us from taking inverse Fourier transforms (in the analytic case such exponential growth can still be OK). This type of localization in phase space allows us to make The answer, which can be obtained by Math 1B methods is: sense of the local equivalence of operators. Pseudodifferen- when a(t)does not change sign from - to +. rial operators, a(x,D), which generalize differential operators condition P(x,D) by allowing more general functions a, are essential for A stronger condition is the P: thai Im ap does not change sign on integral curves of Haelap). For operators with complex p(x,^) and {p,p} # 0 as in Since toe principal symbol, p(x of a differential operator is the nonsolvability condition, a normal form was provided by a check Duistennaat-Sjostrand (1973): either odd or even in 4 it is not difficult to that P ^=^> Q for differential operators. near $2 Ã 0. If we take Fourier transform in x2, write h = Solvability on a compact subset,K, of R" means that 1/[2, we obtain annihilation and creation operators:

is verified in a neighbourhwd of K by some distribution u for any / in a subspace of finite codimeosion in Cm (this means all functions near any point, but this more general statement is needed for a formulation with a set K). In this sense the Nirenberg-Tteves Conjecture can be form- laced as follows: for principal type operators (i-e. for which dp# 0.Hp#4+wknp=0), Q in a neighbourhood of K <==^ solvability at K. Nirenberg and Treves (1970) proved invariance of condition ^f : (conrifoind on page 10) Spring 2009 Berkeley ~atkww^cfNewsletter 9 /

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(FibYears, continued frompage 9) cipal type. Berkeley's math colloquium (Spring 2002) was one it is not at all obvious that the property of p demanded by of the first places where he lectured on this work. does not change when we multiply p by a nonvanishing com- Many techniques developed in the investigation of solvabil- plex function (solvability does not). In particular it is enough to ity are now useful in mathematical physics, study of nonlin- check @ for a = 1and a = i so that it is more easily verifiable. ear waves, spectral instability, and other branches of mathemat- They also proved (1971) that the condition P is sufficient for ics. Among current faculty in our department, Mike Christ and operators with real analytic coefficients. Daniel Tatani have made contributions to solvability, in the case In 1978 Moyer (Ph.D. from Berkeley Math in 1964) for di- of multiple characteristics and principally normal operators re- mension two. and in 1981 HGrrnander for all dimensions, showed spectively. My own connection to the subject comes through that the pseudospectra in the semiclassical limit [4].

solvability at K =^ @ in a neighbourhood of A". Beak and Feffennan ( 1974) pushed the uncertainty principle to a limit in phase space analysis and showed that [I] N. Dencker, The resolution of the Nirenberg-Treves conjechire. Ann. of Math. 1W2006). 40544. P in a neighbourhood of K =+ solvability at K. [2] H. Lewy,An example of u smooth linear partial differentid eqmition with- mit solution, Ann. of Math. 66( 1957). 155-1 58. This settled the problem for differential operators. In many sit- [3] L. Honnander, On the sd~wbilityof pseudoiliffemfid equntions, In: uations the solvability problem for pseudodifferential operators Structure of solutionsof differential equations (KataWKyoto, 1995). Riier appears. but it remained open. Edge, NJ:World Sci. Publishing, 1996, 183-213. [4] M. Zworski. Numerical linear ul,qebru and solvability of pirticd differen- In dimension two, there arc many simplifications: from the tuit equations, Comm.Math. Phy s. 229(20B2). 293-307. semiclassical point of view (with h = 1/.5 playing the role of the Planck constant). the problem is essentially one dimen- sional. In that case, Nicolas Lemer proved the sufficiency in 1988 and a simple proof was also provided by Hart Smith (who obtained his RSc. from Berkeley in math in 1984). Nils Dencker concluded this long quest by publishing in 2006 the proof of sufficiency of I& for solvability of operators of prin-

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Spring 2009 Berkeley ~.ahkesWLwtiaNewsletter 11 (Chair's Leasr, continuedfrom page 6) the tropics," gives some hint of the ematics majors. the near future and look forward to subject, which stems from the lack the creation of further such posi- of Coriolis forces near the equator. The upper division program has tions. As Majda explained, this phcnom- been enriched by the addition of enon has important implications several new courses, including: SPECIAL LECTURES for weather prediction and climate change in mid-latitudes. Math 116 - Cryptography Stop by our Department and you'll Math 127 - Mathematical and find a busy calendar of seminars The Chern lectures this year will be Computational Methods in Molecu- that have been organized by faculty given by Richard Taylor of Harvard lar Biology members and graduate students. University. Taylor is known for his Math 136 - Incompleteness and Traditionally, seminar amounce- pioneering work on connections Undecidability ments are posted near the eleva- between automorphic representa- Math 143 - Elementary Algebraic tors on the 9th floor. Increasingly, tions and Galois representations. Geometry however, the best place to find The Taylor-Wiles systems that the schedule of the week's talks is underpinned the proof of Fermat's We also have two new courses becoming the Department's web Last Theorem in 1994 have played which are still under the "Experi- site. Daily, weekly and monthly a central role in subsequent devel- mental" rubric of Math 191. One is calendars are available from the opments in the subject. Finally, the in Mathematical Problem Solving, home page http://math.berkeley. Tarski lectures will be delivered by an outgrowth of the Putnam Exam edu/; meanwhile, information about the eminent model-theorist Anand preparation class. The other is Proj- a particular seminar (or the depart- llllay, who moved recently to the ects in Mathematics, an opportunity mental colloquium) is best obtained University of Leeds from the Uni- for students to work in teams on by clicking the "Events, Seminars, versity of Illinois. open-ended research problems. Etc." link on the home page. A NEW NAME At the graduate level, new addi- In addition to seminar schedules, tions include: you'll find links to the "Annual In 1971, the College of Letters and events" -- the special lecture series Science was divided for administra- Math 203 - Asymptotic Analysis in that we have developed over the tive purposes into four Divisions, Applied Mathematics years. During each academic year, one of which was the Division of Math 239 - Algebraic Statistics we host the Bowen lectures, the Physical Sciences. The first Dean Lang lecture, the DiPerna lecture, of this Division was Cal Moore EVANS HALL the Chern lectures, and the Tarski (who report1; that in the early days, lectures. The links for each of these Divisional Deans had limited Much has happened to make Evans the series explain the origin of the authority, but that over time as the Hall a more attractive and func- series and list of lecturers. College decentralized and flattened tional place. The Big Clean was its structure, the Divisional Deans followed by the repainting of all The first special lecture of the gained much more authority). Last the corridors in Evans Hall, and the academic year was the Serge Lang year, with the full support of Di- renovation of Room 939. The large undergraduate lecture; this year's visional Dean Mark Richards, we space in the middle of the 7th floor, speaker was Peter Jones, who was succeeded in having our Division which old-timers may remember as Serge's colleague at Yale. Peter renamed the Division of Mathe- the site of graduate student carrels, spoke on the "Oivide and Con- matical and Physical Sciences. Our and more recent Evans denizens re- quer" philosophy of dividing hard proposal for this change, originally call as the scene of a computer lab problems into smaller pieces and promoted by Moore when he was and a vast storage space for old fur- then assembling the results at the Department chair, gained quick ap- niture, has been completely reno- end. Quite a few of his examples proval from the Berkeley Campus, vated. A corridor was built through had to do with fractals - which not surprisingly since Mathematics the middle of the space, which was also appeared in the Bowen lec- and Statistics make up about half of then divided into six rooms. We tures by Hillel Furstenberg of the the Division by virtually any mea- now have a new computer lab, two Hebrew University of Jerusalem! sure. (The other three departments new seminar rooms, a conference Indeed, the theme of those lectures are Astronomy, Earth and Planetary room, and a lounge for impromptu was in fact the role of ergodic Sciences, and Phyics.) discussions, all of these open since theory in the geometry of frac- the beginning of Fall Semester tals. Next on the calendar is the THE CURRICULUM 2008. The sixth space, opened annual DiPerna lecture, which is in January, is the Laboratory for held late in January each year on In keeping with the growing im- Computation in Biology. This a Thursday afternoon. This year's portance of discrete mathematics in activity, run by Steve Evans, Lior speaker was Andy Majda from the science and engineering (as well as Pachter, and Bernd Sturmfels, will Courant Institute, who was our within mathematics itself), we have have ongoing funding for gradu- colleague at Berkeley from 1978 made Math 55, Discrete Mathemat- ate students and postdocs, some to 1984. The title of Majda's talk, ics, a required course for both the it provided through the generous "Applied and theoretical challenges Mathematics and Applied Math- support of donors David Desjardins for multi-scale hyperbolic PDEs in (continued on page J3)

12 Berkeley ~svtkemabwNewsletter Spring 2009 (Chair'sLetter, continuedfront page 12) (Manager's Report, continuedfrom page 81 (PhD 2002 with Elwyn Berlekamp) Chancellor's Outstanding Staff and his wife Nancy Blachman. The Award. As each of you knows, IN MEMORIAM remainder of the funding for the Barbara has done a remarkable job 7th floor project came from savings developing the student services during the term of a more frugal programs over the last 10 years. chairman, Ted Slaman, and from a Recognition of her ongoing exem- David Gale contribution from Dean Richards. plary service was long overdue! (192 1-2008) This Spring, the Chancellor's Advi- THANKS sory Committee has selected Faye Yeager to receive a 2009 Chan- Murray Protter During my chairmanship, I have cellor's Outstanding Staff Award. (1918-2008) been assisted by a great team of Faye has worked tirelessly to sup- Vice-Chairs --- Ken Ribet (Devel- port the needs of the Mathematics opment), Maciej Zworski (Faculty Department for the last 20 years. Affairs), Mike Christ and Dan She is our Webmaster and faculty Tataru (Graduate Affairs), Bjorn technical manuscript specialist John Stallings Poonen, Jamie Sethian, and John and the sole provider of this kind ( 1935-2008] Steel (Undergraduate Affairs). of support for the entire faculty Some of these Vice-Chairs will including 55 faculty, 33 emeriti and be continuing under the coming 20 post doctorate fellows and visit- (in July 2009) old'new regime of ing scholars. Needless to say, Faye Ted Slaman (2009- 10) and Hugh has an enormous workload. The Wodin (2010- 13). I would like to quality of her work is consistently thank all of them, as well as all of excellent. In January 2008, her the Department staff led by Mary husband was diagnosed with a life Pepple, for their contributions to threatening illness. Faye insisted Mathematics at UC Berkeley. Fi- that working through her husband's nally, I would like to thank all those medical treatments would be best who have supported the Berkeley for all involved. She purchased a Math Department over the past laptop, and for the last year, has years. Especially, but not only, in maintained a full-time alternative times when the State of California work schedule while caring for her is cutting back on its support, these husband. Faye's perseverance and contributions by alumni and other courage has been an inspiration friends are essential in keeping this for all of us. Her dedication and one of the world's leading places to commitment to the Department do and learn mathematics. + of Mathematics and the Berkeley campus community is noteworthy, and well deserving of a Chancel- lor's Outstanding Staff Award. *

Editor: Thomas Brown Copy Editors: Barbara Peavy, Mary Pepple, Faye Yeager Photographer: George Bergman Contributors: Haruko Bruce, Mark Haiman, Ivan Matic, Barbara Peavy, Mary Pepple, Nicolai Reshetikhin, Kenneth Ribet, Zvezdelina Stankova, Alan Weinstein, David Wertheimer, Maciej Zworski.

Spring 2009 Berkeley ~atkemcvbiaNewsletter 13