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Managing Editors: This electronic-only journal publishes research announcements (up to about 10 Keith Burns journal pages) of significant advances in all branches of mathematics. A research Svetlana Katok announcement should be designed to communicate its contents to a broad mathematical audience and should meet high standards for clarity as well as Editors: mathematical content. Stuart S. Antman Dmitri Burago All papers are reviewed, and the entire Editorial Board must approve the accep­ Krzysztof Bu rdzy tance of any paper. Since an electronic journal does not have to fill a certain Tobias H. Colding of pages, absolute rather than relative standards for quality will be Brian Conrey applied. Walter Craig Marc Culler NOW WITH A NEW PUBLISHER Robert L. Griess Volumes 1 - 13 of Electronic Research Announcements were published by the Boris Hasselblatt American Mathematical Society from 1995 to mid-2007. Beginning with Volume E. Kenig 14, 2007, this journal is published by the American Institute of Mathematical Janos Kollar Sciences as Electronic Research Announcements in Mathematical Sciences. It is Alex Lubotzky available free of charge as a to the mathematical community. Barry.Mazur Walter Neumann The American Institute of Mathematical Sciences (AIMS) is an organization promot­ Alexander Ol'shanskii ing research in mathematical sciences via its journals and conferences. AIMS Leonid Polterovich journals are internationally recognized as being among top academic publica­ Klaus Schmidt tions. Further information about AIMS can be found at www. a imSci ences. org. Alejandro Uribe Guido L. Weiss ADVANTAGES OF ELECTRONIC RESEARCH ANNOUNCEMENTS Sylvia M. Wiegand W. Hugh Woodin • Quick and convenient submission of a PDF file via the web submission form http://www.math.psu.edu/era/submission-form.php. ; • All papers are reviewed, and recommendations by the editor are put before ~he ADVISORY BOARD . I editorial board for final approval. Usually a decision will be communicated to ~ he Ronald L. Graham author within 2 months of submision, and the editorial board is committed to David Kazhdan rendering a decision within 5 months in all cases. • Articles are posted on ERA website as soon as they are accepted and processed. Don B. Zagier • It is expected that papers with full proofs will be later published elsewhere. • ERA features "Comments" that may be attached to the articles at any ; they may contain errata, future references, computer programs, etc. NEW f6 FORTHCOMING from Birkhiiuser

A History of Mathematical Olympiad Selecta Mathematica, KLEINER, York University, Toronto, ON, Canada Challenges New (SM) This presentation provides an account of the Second Edition Selecta Mathematica, New Series (SM) is a peer intellectual lineage behind many of the basic concepts, TITU ANDREESCU, University of Texas at Dallas, Richardson, reviewed journal addressed to a general audience results, and theories of abstract algebra. A major TX; RAZVAN GELCA, Texas Tech University, lubbock, TX of research and graduate theme in this book is the rise of abstract algebra in Review of the First Edition: students. Topics in pure and will attempts to solve classical problems, and a context "This f book j is ... much more than just another be considered. Selecta is global in philosophy and from which the reader may gain a deeper appreciation collection of interesting, challenging problems, but is flexible in its acceptance of individual styles. The of the mathematics involved. Mathematics instructors, instead organized specifically for learning ... I strongly diversity of the editorial board guarantees the algebraists, and historians of science will find the recommend this book for anyone interested in creative coexistence in the journal of a wide variety of pure and a valuable reference. problem-solving in mathematics .... It has already applied topics in the mathematical language in which the subjects are conveyed. 2007/APPROX. 200 PP., 10 ILLUS./SOFTCOVER taken up a prized position in my persona/, and ISBN 978-0-8176-4684-4/$49.95 (TENT.) is bound to provide me with many hours of intellectual New Editors-in-Chid (MIT, Cambridge, pleasure." -THE MATHEMATICAL GAZETTE USA) DAVID KAZHDAN (Hebrew Univ., Jeruselum, Israel) (Univ. Chicago, USA) The Legacy of Mario Pieri in Hundreds of beautiful, challenging, and instructive Founding Editor.· ISRAEL M. GELFAND (Rutgers, USA) and Arithmetic problems from algebra, geometry, trigonometry, combinatorics, and number theory from numerous Editorial Board Members: ELENA A. MARCHJSOTTO, State University, mathematical competitions and journals have been Vladimir I. Arnol'd (Steklov Mathematical Institute) Northridge, CA; JAMES T. SMITH, San Francisco State selected and updated for this second edition. University. San Francisco, CA Roman Bezrukavnikov (MIT) Historical insights and asides are presented to Haim Brezis (Rutgers University) The Italian Mario Pieri (1860-1913) had stimulate further inquiry. Featuring enhanced (Rutgers University) a significant, yet somewhat underappreciated impact motivation for advanced high school and beginning (lnstitut des Hautes Etudes Scientifiques) on several branches of mathematics, particularly on the college students, as well as instructors and Olympiad (lAS, Princeton) development of algebraic geometry and the foundations coaches, this text can be used for creative problem­ Matthew Emerton (Northwestern University), Athanassios S. Fokas (Imperial College of Science) of mathematics in the years around the turn of the 20th solving courses, for professional teacher development Phillip A. Griffiths (lAS, Princeton) century. This book comprehensively examines Pieri's seminars and workshops, for self-study, or as a resource for training for mathematical competitions. Mikhael Gromov (lnstitut des Hautes Etudes Scientifiques) life, mathematical work and influence in such diverse Lars Hesselholt (MIT) fields as mathematical , algebraic geometry, Henryk lwaniec (Rutgers University) 2007/APPROX. 320 PP./SOFTCOVER number theory, inversive geometry, intersection theory, Anton Kapustin (Caltech) ISBN 978-0-8176-4528-1/$29.95 (TENT.) mathematical analysis, vector analysis, and differential (lnstitut des Hautes Etudes geometry. The work is a combination of engaging Scientifiques) exposition, little-known historical notes, exhaustive (MIT) ABeginner's Guide to Angus Macintyre (Queen Mary ) references and an excellent . Graph Theory Ciprian Manolescu (Columbia University) Dusa McDuff, (SUNY at Stony Brook) 2007 /XVIII, 502 PP., 67 ILLUS./HARDCOVER Second Edition ISBN 978-0-8176-3210-6/$129.00 (Princeton) W.O. WALLIS, Southern Illinois University, Carbondale, IL Tetsuji Miwa (University of ) Review of the First Edition: Alexander Postnikov (MIT) Fuchsian Reduction ''Altogether the book gives a comprehensive introduction Vladimir Retakh, (Rutgers University) Applications to Geometry, Cosmology Igor Rodnianski () to graphs, their theory and their application ... The use of (Princeton University) and Mathematical the text is optimized when the exercises are solved. The (Royal Institute of Technology, Stockholm) SATYANAD KICHENASSAMY, Universite de Reims obtained skills improve understanding of graph theory Gigliola Staffilani (MIT) Champagne-Ardenne, Reims, as well ... It is very useful that the solutions of these Richard Thomas (Imperial College, London) exercises are collected in an appendix." Viktor Vassiliev (Steklov Mathematical Institute) Developed in the 1990s for semilinear wave equations, Robert Lee Wilson (Rutgers University) -SIMULATION NEWS EUROPE Fuchsian reduction research has grown in response (lAS, Princeton) to those problems in pure and applied mathematics This concisely written textbook is intended for an Christopher Woodward (Rutgers University) where numerical computations fail. This method has introductory course in graph theory for undergraduate (Max-Planck lnstitut flir Mathematik) multiple applications in soliton theory, Einstein's mathematics majors or advanced undergraduate and Boris Zilber (Oxford) equations and cosmology, stellar models, graduate students from the many fields that benefit ISSN 1022-1824 (Printed edition), collapse, conformal geometry and combustion. Most from graph-theoretic applications, including computer lSSN 1420-9020 (Electronic Edition) chapters contain a problem section and notes with science, engineering, chemistry, management science, 1 VOLUME PER YEAR, 4 ISSUES PER VOLUME references to the literature, and thus can be used as a social science, and resource planning. The second edition text in graduate courses in PDEs and/or algebra, or as includes new chapters on labeling and communications a resource for researchers in the . networks and small worlds, as well as expanded beginner's material in the early chapters, including more 2007/APPROX. 316 PP./HARDCOVER examples, exercises, hints and solutions to key problems. ISBN 978-0-8176-43S2-2/$149.00 (TENT.) 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May 2007 Communications

619 WHATIS.. .anlnfinite Swindle? Valentin Poenaru

626 If Had Been Japanese Bill Casselman P' • •• 631 Furstenberg and Smale 626 Receive 2006-2007 Wolf Prize 633 2007 Award for an Exemplary Program or Achievement in a Features Mathematics Department 634 Mathematics Programs 592 How to Generate Random Matrices from the That Make a Difference Classical Compact Groups 637 jumping Ship: Francesco Mezzadri Board Resigns Allyn jackson Random generators are used in applications in physics, applied mathematics, engineering, and other 640 2007 JPBM fields. The author discusses an algorithim, which can Communications Award easily be implemented using standard packages, to generate random matrices from the 641 MAA Prizes Presented in classical compact groups. New Orleans 643 AWM Prizes Presented in New Orleans 606 The Vision, Insight, and Influence of Commentary Steve Batterson Many contemporary mathematicians are somewhat 589 Opinion: Should journals aware of Oswald Veblen's mathematical legacy as1a Compensate Referees? researcher and mentor. Less well known tq,day Michael Fried are his achievements in shaping many of the pro.res- ! 623 ProjectOrigami: sional aspects of mathematics now taken for granted -' Activities for Exploring The author reviews this important history. Mathematics- A Book Review Reviewed by Helena Verrill

629 Letters to a Young Mathematician- A Book . Review Reviewed by Emma Carberry Notices Departments of the American Mathematical Society About the Cover ...... 636 Mathematics People ...... 644 EDITOR: Andy Magid johnson Receives AAAS Mentoring Award, Yi Ni Receives ASSOCIATE EDITORS: AIM Five-Year Fellowship, National Academy of Engineering Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert]. Daverman, Susan Friedlander, Robion Elections. Kirby, Steven G. Krantz, Mark Saul, Usa Traynor SENIOR WRITER and DEPUTY EDITOR: Mathematics Opportunities ...... 645 Allyn Jackson NSF Postdoctoral Research Fellowships; Domestic Nuclear MANAGING EDITOR: Sandra Frost Detection Office/NSF Academic Research Initiative ; NSF-CBMS CONTRIBUTING WRITER: Elaine Kehoe Regional Conferences, 2007; Call for Proposals for 2008 PRODUCTION ASSISTANT: Muriel Toupin NSF-CBMS Regional Conferences; Inter national Mathematics for University Students; News f rom the IMA; News PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Murphy, Lori Nero, Karen Ouellette, Donna from /PAM; International Dobrushin Foundation. Salter, Deborah Smith, Peter Sykes ADVERTISING SALES: Anne Newcomb Reference and Book list...... 648

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Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Opinion

community, often lack serious analysis. I know some (edi­ Should Journals tors and referees) don't expect this of a referee. Yet, without such, refereeing is rife for abuse. "Except for what the referee learns, and that would be Compensate Referees? preciously little from many referee reports, he or she has a Publications are central to advancement in science, and all thankless task for the ." research communities rely heavily on refereeing to insure P.S.: Journal B never responded to my discussion of the integrity of the literature. With growing community compensation! pressures and expansion of the mathematical enterprise, we Comments of five respondents: All my correspondents might ask what refereeing does for mathematics, how well agree refereeing is a service relevant for those publishing it goes and why, and what are its special needs and difficul­ papers. Yet, so is administration. Still, not all are appropri­ ties? That's a lot, so I limit myself to these points. ate for it. Quality refereeing is a skill implying for • Quality referees have valuable skills, neither adequately considered analysis, and ability to surmise topics outside acknowledged nor appreciated. one's own published work. That skill is scarce; its upkeep • Compensating referees might improve their standards requires practice. and timeliness. #1 suggested a need for differentiating editor and referee For example, here's a typical request to me from a jour­ tasks. "There are referees, editors, publishers, authors, and nal (with a large, no, huge editorial staff). Following that is except maybe for the last, their roles have changed and may my response (with a P.S. for the reader). I asked five col­ no longer be well-defined." Most respondents mentioned leagues-experienced in these tasks- to comment on my many mathematicians play several roles in turn. interchange. I then compare my comments with theirs. A Certainly, I agree!: Those huge editorial boards ought to two-question postscript seeks answers on who contributes edit! Editors should initiate the process with partial referee­ most significantly to editing and refereeing. ing. The anonymous referee should be for nontrivial tasks Journal Message: [Salutation from editorial staff-not that represent refined expertise. #1 also had a peeve: "It an editor] "We would greatly appreciate your serving as a is annoying ... to read that I should do the job in so many reviewer for the enclosed paper (pdf-file-title): [Article Title] weeks." [Author A] which has been submitted to [Journal B]. #2 and #5 add that refereeing tasks are onerous if done "We have asked several referees but unluckily they all declined to do the job. We really hope you could take a look seriously, and those tasks don't fall equally on all. at this paper. Please complete your review and send it to us All agree on calling some publishers to task. #2 example: by May 31, 2006 [gives one month for an extremely techni­ "Many journals are .. . expensive, though both their content cal 23-page paper]. Send your review to [e-mail]. If you are and ... aspects of processing it, are typically done for free. not able to referee the paper, please, if possible, suggest an ... as T£X has virtually ended the role publishers [had once] alternate referee." [Request I acknowledge receipt] in enhancing ... papers. So, what do publishers give for what My response to journal B: "Author A writes technical­ they get?" papers requiring savvy refereeing. Even expert referees will #3 and #4 see editorial compensation as an equally tire of the same author's papers. This gives quality math­ important topic: "For most journals, there is little compen­ ematicians like Author A, with little political base, serious sation. [They included other comments on what specific problems getting into print. journals provide to editors, though one wouldn't amaze ''I'm asked to referee roughly one paper a month. (One one's university with such remuneration.]" year I had 4 7 refereeing requests, from many different #5 was most emphatic that referees should do something journals. That dropped when I slowed my response rate.) about lack of compensation. "Why do mathematicians ... ref­ These requests are often far from topics in which I publish eree without compensation? .. .because they find the paper papers. Too, it is my experience that over 50% of papers at hand interesting and want to study it? Don't most have (yes, tough topics, but .. .) have very serious (not typo or more important papers they should read first? Are they oversight) errors. This reflects poorly on referees. flattered to be asked to judge? Is it a fair trade for having "Why referees, rather than authors? Others don't report their papers refereed? such an error rate. Yet, about 1/4 of these papers have an­ "[Many-he lists some] journals make and can other referee: I see unreported errors directly. afford to pay their referees. But, why should they do [that] "Journals should recognize there are situations, including if mathematicians ... referee for free?" this, that require top-notch refereeing, and they should pay Final Questions: Which mathematicians do editors see as for it. I suspect slightly-above-token pay might go far. Maybe the major contributors to refereeing, and by what criteria? even develop a cadre of referees who establish expertise on How many mathematicians regularly do serious refereeing handling difficult cases. Compensation could help the whole and how do they feel about it? community: Journals would not over-rely on referees biased - Michael Fried toward papers of their acolytes. University of California, Irvine, Emeritus "Referees' reports, maybe with a slim list of typos, Montana State University, Billings and what they think is the author's significance to the mfri4@aol .com

MAY 2007 NOTICES OF THE AMS 589 letters to the Editor

Matlhematical Gaps included much other published ma­ remark on the comment by Harvard In a letter on page 6 of the January terial on the Ricci flow to make a president Larry Summers calling for 2007 issue, castigated complete exposition of the proof. research on the innate differences the mathematics community for "not None of this amounts to asserting between men and women to see if it policing [itself) very well" in the Perel­ that there is a gap. In addition, we might explain the difference between man affair. I will limit myself to stated that we had to substitute sev­ the achievements of men and women addressing only one of the many eral arguments by our own "because in the sciences. points Birman tried to make. Birman we were unable to comprehend these Perhaps Birman and Summers do took issue with the editors-in-chief original arguments of Perelman." In not know that a great deal of re­ of the Asian journal of Mathematics other words, since we were not sure search has been done that shows the for allowing Cao and Zhu to assert in whether our arguments were the extent and of unconscious their paper that there were "gaps in same as what Perelman had in his bias against women which results Perelman's proof". mind, we were responsible for what in discriminatory treatment. For an A gap, by general consensus, means we wrote, and as such we made that excellent exposition of this research non-trivial but missing logical steps statement. It does not imply in any I recommend Why So Slow, by Virginia leading from one claim to the next. By way that we have doubt whether Valian, published by the MIT Press. this definition, Perelman's papers are Perelman knew how to prove them. At this time we need action to full of gaps because, for more than We are sure that Perelman has bet­ reverse discriminatory treatment. two years after their appearances, the ter arguments in these places, as the When that has been accomplished, whole world waited with bated breath results he stated are stronger than if inequality of achievement still ex­ for the definitive judgment of experts what we proved. ists we might then appropriately on whether the missing steps could We hope this message will clarify call for further research on the dif­ indeed be filled in. The existence of the misunderstanding of Professor ferences between the of men and women. For now, the issue is gaps in Perelman's paper was thus Birman. to acknowledge discrimination and obvious to one and all. work to end it. I do not wish to give the impres­ -Huai-Dong Cao sion that I am questioning the math­ Lehigh University [email protected] - Charity Hirsch ematical merits of Perelman's contri­ cbhirsch@tds. bution to the solution of the Poincare . Yes, he fully deserves -Xi-Ping Zhu (Received January 12, 2007) the . However, I do not Zhongshan University, China believe that the way he presented [email protected] his work to the world is a model that others should emulate. Why did Bir­ (Received January 19, 2007) man choose to ignore this aspect of Perelman's work? Event Should Not Have Been Canceled - H. Wu Shame on the AMS for canceling the University of California at Berkeley Special Event on the Poincare Con­ [email protected] .edu jecture and Geometrization at the 2007 Annual Meeting in New (Received January 15, 2007) Orleans. The reason given is, "It be­ came apparent that the continuing controversy was undermining this No Assertion of Gaps special event." But that controversy Corrections to George In the January 2007 issue of the is precisely why the event should not Dantzig memorial article Notices, Joan Birman wrote in her have been canceled! letter that our joint paper, which The photo caption on page 359 appeared in the June 2006 issue of - jonathan Sandow in the March 2007 issue of the Asian journal of Mathematics, "as­ Princeton University Notices incorrectly identified the serted that there were gaps in Perel­ [email protected] pictured event as the "National man's proof." We do not know how Medal of Honor ceremony, 1971". Professor Birman drew her conclu­ (Received January 8, 2007) Instead, the caption should have sion, as nowhere in our paper did we read "National Medal of Science make such an assertion. The phrase ceremony, 1975". "complete" in the title of our paper End Discrimination First The photograph on page 345 refers to the facts that Perelman's The interview in the January 2007 should have been credited to Ed article contains the phrase "sketch" issue of the Notices with Joan Birman Souza, Stanford News Service. in several places and we completed was very interesting. I would like, -Sandy Frost these sketches, and also that we however, to take issue with Birman's

590 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Zurich, Switzerland I C IAM July 16-20, 2007

The AMS is proud to participate in the 6th International Congress on Industrial and Applied Mathematics (I ClAM) in ZUrich, Switzerland, July 16-20, 2007. The AMS booth will include books for sale at a special discounted price, complimentary copies of Mathematical Moments translated in several languages, and information about MathSciNet and other AMS programs and services. I' J See you in Zurich! How to Generate Random Matrices from the Classical Compact Groups Francesco Mezzadri

ince Theory (RMT) was be able to implement in a code of only a few lines. introduced by Wishart [17] in 1928, it has This is achieved in the section "A Correct and Ef­ found applications in a variety of areas of ficient ". Secondly, we discuss in detail Sphysics, pure and applied mathematics, the main ideas, which turn out be fairly general probability, , and engineering. and quite interesting, behind this algorithm. A few examples-far from being exhaustive­ AnN x N unitary matrix U = (u1k) is defined include: analytic number theory, combinatorics, by the relation U* U = UU* = I, which in terms of graph theory, multivariate statistics, nuclear the matrix elements reads physics, chaos, , N N statistical , structural , and I u j kukl = I UkJUki = 811 and wireless telecommunications. The reasons for the k=l k=l (1) ever growing success of RMT are mainly. two. N N Firstly, in the limit of large matrix dimension the I UjkUkl = I UjkU!k = Oj[, statistical correlations of the spectra of a family, k=l k=l or ensemble, of matrices are independent of the where U* = (uj k) is the conjugate that defines the ensemble, of U, i.e., ujk = UkJ· In this article we will use but depend only on the invariant properties of the symbol - to denote complex conjugation, in such a distribution. As a consequence random order to distinguish it from *, which is reserved matrices turn out to be very accurate models for a for the conjugate transpose of a matrix. The large number of mathematical and physical prob­ constraints (1) simply state that the columns lems. Secondly, RMT techniques allow analytical (rows) of a unitary matrix form an orthonormal computations to an extent that is often impossible in ([N _ The U(N) of unitary matrices to achieve in the contexts that they are modelling. forms a compact Lie group whose real dimension This predictive ability of RMT is particularly is N 2 ; it is then made into a probability by powerful whenever in the original problem there assigning as a distribution the unique are no natural parameters to average over. invariant under group multiplication, known as Although the advantage of using RMT lies in . Such a probability space is often the possibility of computing explicit mathematical referred to as Circular Unitary Ensemble (CUE). and physical quantities analytically, it is some­ Usually the correct ensemble to model a par­ necessary to resort to numerical simulations. The purpose of this article is twofold. Firstly, we ticular situation depends on the symmetries of provide the reader with a simple method for gener­ the problem. Ensembles of unitary matrices are ating random matrices from the classical compact constructed in two steps: we first identify a groups that most mathematicians-not necessari­ U c U(N) by imposing further restrictions on U; ly familiar with computer programming - should then we assign to U a probability measure with the appropriate invariant properties. As well as U(N), Francesco Mezzadri is a Lecturer in Applied Mathemat­ we will discuss how to generate random matrices ics at the University of Bristol, United Kingdom. His email from the orthogonal O(N) and unitary symplec­ address is f. mezzad ri @b ri s to l . ac . u k. tic USp(2N) groups with probability distributions

592 NOTICES OF THE AMS VOLUME 54, NUMBER 5 given by the respective unique invariant mea­ not have any other symmetry. From general con­ sures. We shall also consider the two remaining siderations (see Mehta [9] p. 36) we know that U Dyson circular ensembles [2], namely the Circular is always conjugate by a unitary transformation Orthogonal Ensemble (COE) and Circular Symplec­ to a symmetric matrix. Therefore, we can always tic Ensemble (CSE). Other symmetric appear choose U so that in the applications [18], but we will not concern (3) U = U1, ourselves with them. Writing an algorithm to generate random uni­ where U 1 denotes the transpose of U. Since there tary matrices that is both correct and numerically are no other symmetries in the problem, this is stable presents some pitfalls. The reason is that the only constraint that we can impose. Therefore, the conditions (1) imply that the matrix elements the appropriate matrices that model this physical are not independent and thus are statistically cor­ system should be symmetric. Let us denote by 0 related. The main ideas discussed in this article the set of unitary symmetric matrices. If U E 0 it are centered around the QR decomposition and go can be proved (see Meta [9] p. 499) that it admits back to Wedderburn [16], Heiberger [5] (corrected the representation by Tanner and Thisted [15]), Stewart [14], and Dia­ (4) U = WW1 , WE U(N). conis and Shahshahani [1]. However, the technical This factorization is not unique. Let O(N) be the literature may be difficult to access for a reader group of real matrices 0 such that 001 = 0 10 = without a background in or I and set W' = WO. By definition we have statistics, while the implementation of such tech­ niques is elementary. Another method discussed (5) u = W'W' 1 = wootwt = WW1• in the literature involves an explicit representation This statement is true also in the opposite direc­ of the matrix elements of U in terms of N 2 inde­ tion: if WW1 = W' W'1 there exists an 0 E O(N) pendent parameters (Euler angles) [19], but it does such that W' = WO. Therefore, 0 is isomorphic to not seem to be equally efficient or convenient. the left coset space of O(N) in U(N), i.e.,

(6) 0 ~ U(N)/O(N). Some Examples and Motivations Before discussing how to generate random matri­ Since a measure space with total equal to ces it is helpful to give a few examples that show one is a probability space, in what follows we shall how they appear in the applications. use the two terminologies interchangeably. An en­ In all the information about semble of random matrices is defined by a matrix an isolated physical system at a given time to is space and a probability measure on it. We have contained in a state vector l/Jo belonging to a found the former; we are left to identify the latter. J{- in general infinite dimensional. Haar measure, which will be discussed in detail in The time evolution of l/Jo, i.e., its dynamics, is de­ the section "Haar Measure and Invariance", pro­ termined by a unitary U. In other words, vides a natural probability distribution on U(N); at a time t > to, l/Jo has evolved into "natural" in the sense that it equally weighs differ­ ent regions of U (N), thus it behaves like a uniform (2) tfJ = UtfJo. distribution. From the factorization (4) the proba­ The fact that U is unitary guarantees that II tfJ II = bility distribution on U(N) induces a measure on II l/Jo II = 1, which is required by the probabilistic 0. As a consequence, if W is Haar distributed the interpretation of quantum mechanics. resulting measure on 0 will be uniform too. In If the dynamics is complicated- as in heavy nu­ the section "A Group Theoretical Interpretation", clei or in quantum systems whose classical limits we shall see that such a measure is the unique are characterized by chaotic dynamics-writing probability distribution induced by Haar measure down an explicit expression for U may be hope­ on 0. Therefore, it provides a natural choice to less. Therefore, we can attempt to replace U by model a time reversal invariant quantum system. a random operator and check if the predictions The space 0 together with this measure is the COE that we obtain are consistent with the empirical ensemble. observations. It is also reasonable to simplify the If a quantum system does not have any sym­ problem even further and replace U by a random metry, then th~re are no restrictions on U(N), and unitary matrix of finite, but large, dimension. Then the natural choice of probability distribution is the main question is: What are the matrix space Haar measure. This is the CUE ensemble. If the and the probability distribution that best model system is invariant under time reversal and has our system? a half-integer , then the appropriate ensem­ In physics the symmetries of a problem are ble is the CSE. The matrix space of the CSE is often known a priori, even if the details of the the c U(2N) whose elements admit the dynamics remain obscure. Now, suppose that our representation system is invariant under time reversal but does (7) U = - W]W1 ], W E U(2N),

MAY 2007 NOTICES OF THE AMS 593 where complex matrices Z = (z1k); the matrix elements are independent identically distributed (i.i.d.) stan­ (8) dard normal complex random variables. In other words, the probability density (p.d.f.) of From the factorization (7) the probability distri­ z1k is bution on U(2N) induces a measure on S. As previously, such a measure is fixed by assigning (12) Haar measure to U(2N). The setS is isomorphic to a coset space too. The By definition the matrix entries are statistically in­ unitary symplectic group USp(2N) is the subgroup dependent, therefore the joint probability density of U(2N) whose elements obey the relation function u.p.d.f.) for the matrix elements is (9) S]S1 = ]. 1 N 2 P(Z) = ------;;;2 n e-lzjkl Therefore, the matrix U in equation (7) does not TT j,k ~ l change if we replace W with W' = WS, where (13) S E USp(2N). Similarly, if Wand W' are such that 2 = rr~ z exp (- i I ZJk 1 ) (10) j,k~l U = -W]W 1] = - W']W'1], W, W' E U(2N), = ~ exp (- Tr Z* Z) . then W'W- 1 E USp(2N). Therefore, TT 2 (11) S :::=: U(2N)/USp(2N). Since P(Z) is a probability density, it is normal­ ized to one, i.e., The probability distribution of the CSE is the unique induced on the coset (14) f P(Z) dZ = 1, space (11) by Haar measure on U(2N). JcN2 From equations (4) and (7) all we need to where dZ = Il~k ~ l dx1kdYJk and z1k = x1k + iYJk · generate random matrices in the CUE, COE, and The j.p.d.f. P(Z) contains all the statistical infor­ CSE ensembles is an algorithm whose output is mation on the Ginibre ensemble. Haar distributed unitary matrices. The rest of Since <[Nx N :::=: <[N2 , we will use the two notations this article will concentrate on generating ran­ according to what is more appropriate for the dom matrices from all three classical compact context. Thus, we can write groups U(N), O(N), and USp(2N) with probability distributions given by the respective Haar mea­ (15) dJJG(Z) = P(Z) dZ sures. These groups are not only functional to and think of dJ1G as an infinitesimal volume or constructing matrices in the COE and CSE, but are measure in <[N2. Iff : <[Nx N - <[Nx N, we say that also important ensembles in their own right. In­ dJJG is invariant under f if deed, the work of Montgomery [11], Odlyzko [12], Katz and Sarnak [6], Keating and Snaith [7, 8], (16) dJJG(f(Z)) = dJ1G(Z). and Rubinstein [13] has shown beyond doubt that Lemma 1. The measure of the Ginibre ensemble is the local statistical properties of the Riemann zeta invariant under left and right multiplication of Z function and other L-functions can be modelled by by arbitrary unitary matrices, i. e., the characteristic polynomials of Haar distributed (17) random matrices. Over the last few years the pre­ dJ1G(UZ) = dJ1G(ZV) = dJ1G(Z), U, V E U(N). dictive of this approach has brought about impressive progress in analytic number theory Proof First we need to show that P(UZ) = P(Z); that could not have been achieved with traditional then we must prove that the Jacobian of the map techniques. (See [10] for a collection of review (18) z ,_. uz articles on the subject.) (seen as a transformation in <[N2) is one. Since by Haar Measure and lnvariance definition U* U = I, we have 1 Since the algorithm we shall discuss is essentially P (U Z) = ------;;;2 exp (- Tr Z* U* U Z) based on the invariant properties of Haar measure, (19) TT in this section we introduce the main concepts that 1 = ------;;;2 exp (- Tr Z* Z) = P (Z). are needed to understand how it works. We never­ TT theless begin with another ensemble: the Ginibre Now, the map (18) is isomorphic to ensemble. Besides being a simpler illustration of (20) X = U Ell .. · Ell U. the ideas we need, generating a matrix in the Gini­ bre ensemble is the first step toward producing a Ntimes random unitary matrix. It follows immediately that X is a N 2 x N 2 uni- The space of matrices for the Ginibre ensemble tary matrix, therefore ldet XI 1. The proof of is GL(N, ([) , the set of all the invertible N x N right invariance is identical. D

594 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Because the elements of a unitary matrix are It turns out that if the entries of Z are i.i.d. stan­ not independent, writing an explicit formula for dard complex normal random variables, i.e., if Z the infinitesimal volume of U(N) is more belongs to the Ginibre ensemble, then Q is dis­ complicated than for the Ginibre ensemble. An tributed with Haar measure (see Eaton [3), p. 234, N x N unitary matrix contains 2N2 real for a proof). Unfortunately, the implementation of and the constraints (1) form a system of N 2 re­ this algorithm is numerically unstable. However, al equations. Therefore, U(N) is isomorphic to a we may observe that 2 2 2 N -dimensional embedded in IR\ N . Such (23) Z = QR, a manifold is compact and has a natural group structure that comes from matrix multiplication. where R is upper-triangular and invertible. In oth­ Thus, an infinitesimal volume element on U(N) er words, the Gram-Schmidt algorithm realizes will have the form the QR decomposition. This factorization is widely used in numerical analysis to solve linear least (21) dJ1{U) = m((){1, ... , (){N z)d(){1 · · · d(){Nz, squares problems and as the first step of a par­ where (){ 1, ... , (){Nz are local coordinates on U(N). ticular eigenvalue algorithm. Indeed, every linear Every compact Lie group has a unique (up to an algebra package has a routine that implements arbitrary constant) left and right invariant mea­ it. In most cases, however, the algorithm adopt­ sure, known as Haar measure. In other words, if ed is not the Gram-Schmidt orthonormalization we denote Haar measure on U(N) by dJ1H{U), we but uses the Householder reflections, which are have numerically stable. (22) Because of this simple observation, at first one dJ1H(VU) = dJ1H{UW) = dJ1H{U), V, WE U(N). might be tempted to produce a matrix in the Ginibre ensemble and then to use a black box Although an explicit expression for Haar measure QR decomposition routine. Writing such a code on U(N) in terms of local coordinates can be is straightforward. For example, if we choose the written down (see Zyczkowski and Kus [19) for SciPy library in Python, we may implement the a formula), we will see that in order to generate following function: matrices distributed with Haar measure we only need to know that it is invariant and unique. from scipy import ,, Haar measure normalized to one is a natural def wrong_distribution(n): choice for a probability measure on a compact "'"'A Random matrix with the wrong group because, being invariant under group mul­ distribution""" tiplication, any region of U(N) carries the same z = (randn(n,n) + lj*randn(n,n))/ weight in a group average. It is the analogue of the sqrt(2.0) uniform density on a finite interval. In order to q,r = linalg.qr(z) understand this point consider the simplest exam­ return q ple: U(1). It is the set {eie} of the complex numbers with modulo one, therefore it has the topology Unfortunately, as Edelman and Rao observed [4), of the unit circle § 1 . Since in this case matrix multiplication is simply addition mod 2rr, U(1) is the output is not distributed with Haar measure. It is instructive to give an explicit example of this isomorphic to the group of translations on § 1 . A probability density function that equally weighs phenomenon. A unitary matrix can always be diagonalized in any part of the unit circle is the constant density U(N). Therefore, its eigenvalues {eie1 , ..• , eieN} lie p(8) = 1/(2rr). This is the standard Lebesgue on the unit circle. A classical calculation in RMT measure, which is invariant under translations. Therefore, it is the unique Haar measure on U(1). (see Mehta [9) pp. 203-205) consists of computing Note that it is not possible to define an "un­ the statistical correlations among the arguments The simplest correlation function to determine biased" measure on a non-compact manifold. For ej. is the density of the eigenvalues p(8), or-as example, we can provide a finite interval with a sometimes it is called-the one-point correlation. constant p.d.f. p(x), but not the whole real line !Rl., Since Haar measure is the analogue of a uniform since the e'oo p(x)dx would diverge. distribution, each set of eigenvalues must have the same weight, therefore the normalized eigenvalue The QR Decomposition and a Numerical density is Experiment 1 By definition the columns of an N x N unitary (24) p(e) = 2rr · matrix are orthonormal vectors in <[N . Thus, if we take an arbitrary complex N x N matrix Z of full It is important to point out that equation (24) does rank and apply Gram-Schmidt orthonormalization not mean that the eigenvalues are statistically to its columns, the resulting matrix Q is unitary. uncorrelated.

MAY 2007 NOTICES OF THE AMS 595 Testing (L4) numencally is very simple. we 0.18rr---~-~--~--~-~--~ generated 10,000 random unitary matrices us­ xxXxxxxxxx x xxxxxxxxx ing wrong_di st ri buti on (n). The density of the 0.16f.------"------.:c_x_;:.x ______eigenvalues of such matrices is clearly not constant 0.14 (Figure l(a)). Figure 1(b) shows the histogram of X X the spacing distribution, which deviates from the 0.12

theoretical prediction too. This statistic is often 0.10 plotted because it encodes the knowledge of all p(O) the spectral correlations and is easy to determine 0.08

empirically. For unitary matrices it is defined as 0.06 follows. Take the arguments of the eigenvalues and order them in ascending order: 0.04 (25) 0.02

0.00 The normalized distances, or spacings, between -3 -2 -1 consecutive eigenvalues are (a) Eigenvalue density N (26) SJ = 2rr (eJ+l- eJ), j = 1, ... ,N. The spacing distribution p(s) is the probability density of s. (For a discussion on the spacing distribution see Mehta [9] p. 118.) It is worth emphasizing that the QR decompo­ sition is a standard routine. The most commonly known mathematical software packages like Mat­ lab, Mathematica, Maple, and SciPy for Python p(s) essentially use a combination of found in LAP ACK routines. Changing software would not alter the of this numerical experiment.

A Correct and Efficient Algorithm

What is wrong with standard QR factorization X X routines? Where do they differ from the Gram­ Schmidt orthonormalization? Why is the proba­ (b) Spacing distribution bility distribution of the output matrix not Haar measure? Figure 1. Empirical histograms of the density The main problem is that QR decomposition is of the eigenvalues and of the spacing distributions compared with the theoretical not unique. Indeed, let Z E GL(N, () and suppose that Z = QR, where Q is unitary and R is invertible curves for the CUE. The data are computed and upper-triangular. If from the eigenvalues of ten thousand 50 x 50 random unitary matrices obtained from the (27) routine wrong_distribution(n). 1 A = (e'e ·• ) = diag ( e'e1, ... , e'eN) , e'eN then is dictated only by the performance and stabil­ (28) Q' = QA and R' = A- 1R ity of the code. For our purposes, however, the subset of U(N) x T(N), in which the output of are still unitary and upper-triangular, respectively. the QR decomposition is chosen, is fundamental Furthermore, and we need to pay particular attention to it. It is (29) Z = QR = Q'R'. convenient from a mathematical point of view to Therefore, the QR decomposition defines a multi­ introduce a variation of the mapping (30), which valued map is not only single-valued but also one-to-one. In (30) QR: GL(N, () ~ U(N) x T(N), this way we will not have to refer all the time where T(N) denotes the group of invertible upper­ to a specific algorithm. Indeed, the idea is that triangular matrices. we should be able to alter the output of a QR In order to make the mapping (30) single-valued, decomposition routine without even knowing the we need to specify the algorithm that achieves the algorithm implemented. factorization. In most applications such a choice We first need

596 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Lemma 2. Equation (29) implies (28), where A E Theorem 1. Suppose that the map (32) satisfies A(N) and A(N) denotes the group of all unitary the hypothesis (33). Then it decomposes the mea­ diagonal matrices (27). sure (15) of the Ginibre ensemble as Proof Equation (29) can be rearranged as (36) djlc(Z) = dJ1H(Q) X dJ1[( N) (}') .

(31) Q - 1Q' =RR'-1 • Proof We have Since U(N) and T(N) are groups, both sides of (37a) dJJc(UZ) = dJJc(Z) by lemma 1 equation (31) must belong to U(N) n T(N). By definition the inverse of a unitary matrix U is its (37b) = dp(UQ, y) = dp(Q, y) by equation (33) conjugate transpose and the inverse of an upper­ triangular matrix is upper-triangular. Therefore, (37c) = dpH(Q) x dJ1r cN) (y) by the uniqueness if a matrix is both unitary and upper-triangular it of Haar measure. must be diagonal, i.e., A(N) = U(N) n T(N). D D This lemma suggests that, more naturally, in­ The choice of the of representatives that stead of the QR factorization (30) we should we made coincides exactly with outcome of the consider a one-to-one map Gram-Schmidt orthonormalization. The output of (32) QR: GL(N, ([) ~ U(N) x f(N), standard QR decomposition routines are such that if Z- (Q,R) then UZ- (Q',R') with Q' * UQ where f(N) = T(N)/A(N) is the right coset space and R' R. Therefore, the corresponding map (32) of A(N) in T(N). We construct (32) as follows: we * does not obey (33) and theorem 1 does not hold. first define it on a class of representatives of f(N) We can now give a recipe to create a random using the QR factorization; then we extend it to the unitary matrix with distribution given by Haar whole f(N). However, since the QR decomposition measure. is not unique, there is a certain degree of arbi­ trariness in this definition. We need to find a map (1) Take anN x N complex matrix Z whose en­ under which the measure of the Ginibre ensemble tries are complex standard normal random induces Haar measure on U(N). The main tool to variables. achieve this goal is the invariance under group (2) Feed Z into any QR decomposition routine. multiplication of Haar measure and its unique­ Let (Q,R), where Z = QR, be the output. ness. Thus, our choice of the decomposition (32) (3) Create the following must be such that if rn lru l (33) Z- (Q, :y) then UZ- (UQ, :y) (38) A= ( with the same :y E f(N) and for any U E U(N). This property implies that left multiplication of Z where the r1Js are the diagonal elements by a unitary matrix reduces, after the decompo­ of R. sition, to the left action of U(N) into itself. But (4) The diagonal elements of R' = A - 1 R are lemma 1 states that always real and strictly positive, therefore (34) the matrix Q' = QA is distributed with Haar measure. for any U E U(N). As a consequence, if the map (32) satisfies (33) the induced measure on The corresponding Python function is: U(N) will be invariant under left multiplication from sci py import ,., too and therefore must be Haar measure. def haar_measure(n): How do we construct the map (32)? A class of """A Random matrix distributed with representatives of r (N) can be chosen by fixing the Haar measure""" arguments of the elements of the main diagonal z = (randn(n,n) + lj*randn(n,n))/ of R E T(N). Let us impose that such elements sqrt(2. 0) all be real and strictly positive. Using (28) we can q,r = linalg.qr(z) uniquely factorize any Z E GL(N, ([) so that the d = diagonal(r) main diagonal of R has this property. It follows ph = d/absolute(d) = that if Z = QR, then q multiply(q,ph,q) return q (35) UZ = UQR, U E U(N). If we repeat the numerical experiment discussed This QR decomposition of U Z is unique within the in the section "The QR Decomposition and a Nu­ chosen class of representatives of r (N). Therefore, merical Experiment", using this routine, we obtain the resulting map (32) obeys (33). Finally, we arrive the histograms in Figure 2, which are consistent at with the theoretical predictions.

MAY 2007 NOTICES OF THE AMS 597 where 1 is the identity and h , i2, i3 are the quaternion units; they obey the algebra (40) if= ii = i ~ = hi2i3 = -1. We can also define the conjugate of q, (41) q = a · 1 - bh - ci2 - di3, as well the norm p(8) (42) llqll 2 = qq = qq = a 2 + b2 + c2 + d 2. When c = d = 0, !HI reduces to ( and q is simply the complex conjugate of q. In analogy with ~N and (N -provided we are careful with the fact that multiplication in !HI is not commutative- we can study the space IHJ N. e Elements in IHIN are N- q = (ql, ... , qN ). The (a) Eigenvalue density bilinear map N (43) (p, q) = I pjqj, p, q E IHI N, j = l is the analogue of the usual Hermitian inner prod­ uct in (N, and the norm of a quaternion vector is simply N 2 2 p(s) (44) llqll = (q, q) = I llqj ll . j = l Similarly, GL(N, IHI) is the group of all the N x N invertible matrices with quaternion elements. The quaternion units admit a representation in terms of the 2 x 2 matrices (45a)

h = o.), (b) Spacing distribution (~ ~), - I ~) 0 Figure 2. Empirical histograms of the density and e3 = ( i 0i) , of the eigenvalues and of the spacing distributions compared with the theoretical where curves for the CUE. The data are computed (45b) 1 ,_.. h h ,_.. e1, i2 ,_.. e2 and i3 ,_.. e3. from the eigenvalues of ten thousand SOx 50 random unitary matrices output of the Thus, q = a · 1 + bh + ci2 + di3 is mapped into the function haar_measure(n). complex matrix

(46a) A = ( -~w ~)z where z =a+ ib and w = c + id. In addition The Unitary Symplectic Group USp(2N) Up to now we have only considered U (N). The dis­ (46b) _q >-+ A * -_ (z- -w) . cussion for O(N) is identical, except that the input w z matrix of the QR decomposition routine must be Equations (46) generalize to an arbitrary N x N real. Unfortunately, however, for USp(2N) there quaternion matrix Q, which can be represented in are no black box routines that we can use, and we terms of a 2N x 2N complex matrix Q using the must put more effort into writing an algorithm. decomposition The algebra of unitary symplectic matrices (47) Q ,_.. Q = Qo®h + Q1 ®e1 +Q2 0 e2 + Q3 ®e3, can be rephrased in terms of Hamilton's quater­ where Q0 , Q 1, Q2, and Q3 are arbitrary N x N mons; it is convenient for our purposes to use real matrices. Proceeding in the same fashion, if this formalism. A quaternion q E !HI is a linear Q E GL(N, IHI) we define its conjugate transpose combination Q* = (qj k) by setting qjk = qkj· The symplectic group Sp(N) is the subset of (39) q = a· 1 + bh + ci2 + di3, a,b,c,d E ~. GL(N, IHI) whose matrices satisfy the identity S* S =

598 NOTICES OF THE AMS VOLUME 54, NUMBER 5 SS* = 1. Because of the analogy between U(N) and The important fact that we need to pay attention Sp(N), the latter is sometimes called the hyper­ to is that unitary group and is denoted by U(N, IHI). The (54) .Q* ,__. Q *, usefulness of the quaternion algebra lies in if and only if the coefficients of the quaternion Theorem 2. The groups Sp(N) and USp(2N) are units are real numbers. This is a straightforward isomorphic, i.e. consequence of the representation (46a). (48) USp(2N) ::= Sp(N). Let S E USp(2N) be the complex representa­ tion of the quaternion matrix S, but assume that Proof It is convenient to replace the skew­ S* is not mapped into S*. It is still true, however, symmetric matrix] in the definition (9) with that 0 1 (55) S* ,__. - nsr n, - 1 0 because equation (50) is only a consequence of matrix manipulations. But since Sis unitary sym­ (49) = I® ez. plectic S* = - Q sr Q, which is a contradiction. 0 0 1 -1 0 The algebra of Sp(N) is the generalization to This substitution is equivalent to a permutation Hamilton's quaternions of the algebra of U(N). of the rows and columns of], therefore it is sim­ Therefore, it is not surprising that the discussion ply a conjugation by a unitary matrix. in the section "A Correct and Efficient Algorithm" is not affected by replacing GL(N, () and U(N) We first prove that if S E Sp(N), then its com­ with GL(N, IHI) and Sp(N) respectively. Thus, since plex representation S belongs to USp(2N). By Sp(N) and USp(2N) are isomorphic, USp(2N) and equation (47) S* is mapped to Sp(N) have the same Haar measure dJlH· In par­ (50) 56 ® h - Sf ® e1 - S~ ® ez - S£® e3 = -Q srQ, ticular, we can introduce the quaternion Ginibre which follows from the identities ensemble, which is the set GL(N, IHI) equipped with (51) the probability density (A®B)t = At®Bt and (A®B)(C®D) = AC®BD, 1 (56) P(Z) = ----wz exp (- Tr Z* Z) = TT and from the algebra (40) of the quaternion units. As a consequence, 2 rrz1Nz exp ( - JtN ll zJkll ) · (52) SS* ,__, -snsrn =I. 1 Therefore, the matrix S is symplectic. Combining Quaternion matrices can be factorized by the equations (46b) and (50) gives QR decomposition too: for any Z E GL(N, IHI) we can always write (53) - nsrn = S*, (57) z = .Q'R , thus S E USp(2N). where .Q E Sp (N) and 'R is an invertible and We now need to show that if S E USp(2N) then upper-triangular quaternion matrix. Now, let it is the representation of a matrix S E Sp(N). This statement follows if we prove that S admits (58) a decomposition of the form (47), where 50 , 51 , 52 , A(N, IHI) = {A E T(N, IHI) I A = diag (q1, .. . , qN ), and 53 must be real N x N matrices. If this is true, then the argument of the first part of the proof llqJII = 1, J = 1, .. . ,N}, can simply be reversed. where T(N, IHI) is the group of invertible upper­ Let us allow the coefficients a, b, c, and d in triangular quaternion matrices. Furthermore, let the definition (39) to be complex numbers. The f(N, IHI) = T (N, IHI) I A (N, IHI) be the right coset definitions of conjugate quaternion (41) and con­ space of A(N, IHI) in T(N, IHI). We have thefollowing jugate transpose of a quaternion matrix, however, remain the same. The matrices (45a) form a ba­ Theorem 3. There exists a one-to-one map sis in c zxz . Therefore, any 2 x 2 complex matrix (59) .Q'R: GL(N, IHI) - Sp(N) x f(N, IHI) can be represented as a linear combination of [z, e1, ez, and e3. Thus, any matrix Q E ( ZNx ZN ad­ such that mits a decomposition of the form (47), but now (60) Z ,__. (.Q , ;y) and 'UZ ,__. ('U.Q,;y), Qo, the matrices Q1, Qz, and Q3 are allowed to be where ;y = A(N, IHI)'R. Furthermore, it factorizes complex. In other words, Q is always the repre­ the measure dJlc of the Ginibre ensemble as sentation of a quaternion matrix .Q, but in gener­ al the quaternion units have complex coefficients. (61) d11c(Z) = dJlH(.Q) X dJ1nN,D·l) (}').

MAY 2007 NOTICES OF THE AMS 599 We leave proving these generalizations as an exercise for the reader.

Householder Reflections Theorem 3 provides us with the theoretical tools to generate a random matrix in USp(2N). Howev­ er, when we implement these results in computer y code, we need to devise an algorithm whose out­ put satisfies the condition (60). The first one that v comes to one's mind is Gram-Schmidt orthonor­ malization. But given that black box routines for quaternion matrices do not exist on the market and that we are forced to write the complete code ourselves, we may as well choose one that is nu­ merically stable and that as it turns out, requires the same effort. The most common algorithm that Figure 3. Householder reflection in 1Rl. 2• achieves the QR decomposition uses the House­ holder reflections. For the sake of clarity, we will discuss this method for O(N); the generalizations is by construction an orthogonal matrix. In equa­ to U(N) and Sp(N) are straightforward. tions (67) and (68) Hm denotes the block matrix Given an arbitrary vector v E IRl. m, the main idea of the Householder reflections is to construct a simple orthogonal transformation Hm (dependent (69) Hm = c N-m Hm) ' on v) such that where Hm is defined in equation (62). (62) The matrix Hm is constructed using elementary geometry. Consider a vector in the , v = = (1, E !Rl.m. where e1 0, .. . , 0) For any real matrix (x1, x2 ), and assume, for simplicity, that x1 > 0. X = (XJk), HN is determined by replacing v in Furthermore, let ii denote the unit vector along equation (62) with the first column of X. The the interior bisector of the angle <:/> that v makes product HNX will have the structure with the x1-axis, i.e., * (70) (63) * HNX~ ("'~ I). where e1 is the unit vector along the x1-axis. The reflection of v along the direction of ii is * llvll e1 (see Figure 3). Reflections are distance­ where preserving linear transformations, therefore their representations in an orthonormal basis are or­ (64) rn = II vii = ~I~= l x]l· thogonal matrices. In this simple example it can Then, define the matrix be constructed from elementary linear algebra:

(71) H 2 (v) =-I + 2iiii1. Finally, we obtain (72)

It is worth noting that H 2 depends only on the where direction of v and not on its modulus. Thus we (66) HN- l(v')v' = llv'll e1. can rewrite equation (72) as (73) H2 = e1, In this case v ' is the (N - !)-dimensional vector (v)v obtained by dropping the first element of the sec­ where v = v/ ll v ll. ond column of the matrix (63). We proceed in this The generalization to arbitrary dimensions is fashion until the matrix straightforward. For any vector v E IRl. m, the House­ holder reflection is defined as (67) R = ffJi2... HN- lHNX (74) Hm(v) = + (I - 2iiii1), isupper-triangularwithdiagonal entries rn, ... , rNN · The product where (68) (75)

600 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Furthermore, we have definitions of the Householder reflections. A suit­ (76) Hm Cv)v = e1. able choice for U(N) is (83) Hm(v) = -e-ie (I- 2iiii*) . How do we choose the sign in the right-hand side of equation (75)? From a mathematical point The unit vector ii is of view such a choice is irrelevant: in both cases A v +eieel (84) Hm (v) maps v into a vector whose only compo­ u = llv + eieelll' nent different from zero is the first one. However, numerically it can be important. The square of the where v = (xl, ... , Xm) E em and xl = eie lxll· The denominator in (75) is matrix Hm (v) is unitary and (85) Hm (v)v = e1. (77) llv ± llvll e1l l2 = 2 II vii ( llvll ± X1) , Note that the introduction of e;e in equations (8 3) where x1 is the first component ofv. If x1 is compa­ and (84) takes into account both the potential can­ rable in magnitude to llvll and negative (positive) cellations and the correct values of the arguments and we choose the plus (minus) sign, then the term of the diagonal elements of the upper-triangular (78) matrix R: equation (85) implies that all the rjjS are real and strictly positive. could be very small and cancellations with signif­ For Sp (N) we have icant round-off errors may occur. Therefore, the Householder transformation to be implemented (86) Hm (v) = - q (I - 2ih1*) , in computer code should be with

(79) Hm(V) =- sgn(x ) (I - 2iilY), A v + qe1 1 (87) where u = llv + qe1ll ' where v = (x , ... , Xm ) E !Him and x = q llx 11. Also A v + sgn(x1 )e1 1 1 1 (80) u = llv + sgn(xd e1ll" in this case The additional factor of sgn(x1) in the right­ (88) hand side of equation (79) assures that there is no ambiguity in the sign of the right-hand A Group Theoretical Interpretation side of equation (76). In turn, it guarantees that We now know how to generate random matrices all the diagonal elements of the upper-triangular from any of the classical compact groups U(N), matrix R are positive. This is not the definition O(N), and Sp(N). In order to achieve this goal, we of Householder reflection used in standard QR have used little more than linear algebra. However decomposition routines. Usually, simple and convenient this approach is (after all linear algebra plays a major role in writing most (81) H;,(v) = I - 2iiiit, numerical algorithms), it hides a natural group with the same ii as in (75). Therefore, theoretical structure behind the Householder re­ (82) flections, which was uncovered by Diaconis and Shahshahani [1]. Indeed, generating a random ma­ As a consequence, the signs of the diagonal ele­ trix as a product of Householder reflections is only ments of R are random. This is the reason why the one example of a more general method that can output of black box QR decomposition routines be applied to any finite or compact Lie group. Our must be modified in order to obtain orthogonal purpose in this section is to give a flavor of this matrices with the correct distribution. perspective. For the sake of clarity, as before, we Besides being numerically stable, this algorithm will discuss the orthogonal group O(N); the treat­ has another advantage with respect to Gram­ ment of U(N) and Sp(N) is, once again, Schmidt orthonormalization. In most applications identical. of numerical analysis Q need not be computed The need for a more general and elegant ap­ explicitly, only Qw does, where w is a specific proach arises also if one observes that there is vector. Generating all the Householder reflections one feature of the QR decomposition that may is an 0 (N2 ) process and computing HNw requires not be entirely satisfactory to a pure mathemati­ 0 (N) operations- it just evaluates the scalar prod­ cian: Why in order to generate a random point uct (ii, w). SuccessivelymultiplyingHN, ... , H1 into on an N(N - 1) /2-dimensional manifold- O(N) w is an 0 (N2 ) process. Therefore, it takes in total in this case- do we need to generate N 2 random 0 (N2 ) operations to compute Qw. Instead, Gram­ numbers? It does not look like the most efficient Schmidt orthonormalization is an 0 (N3 ) process. option, even if it is a luxury that can be easily However, if Q is explicitly needed, computing the afforded on today's . product (68) requires 0 (N3 ) operations too. We will first show how the key ideas that we The generalizations to U(N) and Sp(N) are want to describe apply to finite groups, as in this straightforward. The only differences are in the setting they are more transparent. Suppose that

MAY 2007 NOTICES OF THE AMS 601 we need to generate a random element g in a finite Proof. Suppose we construct 0 E O(N) distribut­ group rN. In this context, if rN has p elements, ed with Haar measure by factorizing a matrix X in uniform distribution simply means that the prob­ the Ginibre ensemble as described in the section ability of extracting any g ErN is 1/ p. In addition, "Householder Reflections". The random matrix 0 . we assume that there exists a chain of subgroups is the product of Householder reflections (68) and offN: each factor HmCvm) is a function of the unit vec­ tor Vm E §m-1 only. We need to show that such (89) VmS are independent and uniformly distributed in 1 In practical situations it is often easier to generate §m- form= 1, ... ,N. a random element {J in a smaller subgroup, say At each step in the construction of the upper­ triangular matrix (67), the matrix multiplied by fm - 1 E [ N, than in [N itself; we may also know how to take a random representative gm in the left the m-th Householder reflection, i.e., coset Cm = fm/fm -1. Now, write the decomposition (96) Xm = Hm (vm) · · · HN-1 (VN-1 )HN (vN)X, (90) is still in the Ginibre ensemble. All its elements are, therefore, i.i.d. normal random variables. This Once we have chosen a set of representatives of is a consequence of the invariance Cm, an element g E fm is uniquely factorized as (97) dJ-1c(OX) = dJ-1c(X), 0 E O(N), g = gm {J, where gm E Cm. If both gm and {J are uniformly distributed in Cm and fm- 1 respectively, of the measure of the Ginibre ensemble. Now, then g is uniformly distributed in fm. Vm = (X1, ... , Xm) iS COnstructed by taking the We can apply this algorithm iteratively start­ m-th dimensional vector Vm obtained by drop­ ing from f1 and eventually generate a random ping the first N- m elements of the (N- m + 1)-th element in rN. In other words, we are given the column of Xm. The components of Vm are i.i.d. decomposition normal random variables. It follows that the p.d.f. ofvm is fN~CNX···XC2Xf1. (91) 1 m An element g E rN has a unique representation as (98) P(Vm) = ~ n exp ( -x]) TT j=1 a product - 1 ( 1 ( (92) - rrm /2 exp - ~L.. xj2) - rrm /2 exp -llvmll 2) . ]=1 where gm is a representative in Cm. If the gms are uniformly distributed in Cm so is g in rN. This is Since P(vm) depends only on the length of Vm, known as the subgroup algorithm [1]. and not on any angular variable, the unit vector 1 This technique applies to random permutations Vm = Vm/ llvmll is uniformly distributed in §m- and is statistically independent of vk for k m. of N letters. The chains of subgroups is * D (93) {Id} c 52 c ... c SN, Theorem 4 is more transparent than relying on where Sm is the m-th symmetric group. Other the QR decomposition, which seems only a clever examples include generating random positions of technical trick. If nothing else, the counting of Rubik's cube and random elements in GL(N, 1Fp). the number of degrees of freedom matches. In where 1Fp is a finite field with p elements. fact, the dimension of the unit sphere §m-1 is For O(N) the decompositions (91) and (92) m - 1. Thus, the total number of independent real are hidden behind the factorization (68) in terms parameters is of Householder reflections. Indeed, the subgroup N algorithm for O(N) is contained in (99) I (m -1) = N(N2-1). m=1 Theorem 4. Letv1, ... , VN be uniformly distribut­ Why is theorem 4 the subgroup algorithm for 0 1 ed on § , ... , §N- respectively, where O(N)? As we shall see in theorem 5, the factoriza­ (94) tion (95) is unique-provided that we restrict to §m-1 = {vm = (x1·····Xm) E ~ml I7=1x] = 1} the definition (79) of the Householder reflections. This means that is the unit sphere in ~m. Furthermore, let Hm (v) be the m-th Householder reflection defined in equa­ (100) O(N) ~ §N-1 X··· X §1 X 0(1), tion (79). The product where (95) o = HN (vN)H N- 1(v N- Il · · · H2 (v2 lH1 (vd (101) 0(1) ~ §0 = {-1, 1}. is a random orthogonal matrix with distribution If we proceed by induction, we obtain given by Haar measure on 0 (N). (102) O(N) = §N-1 x O(N- 1).

602 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Therefore, a matrix 0 E O(N) admits a unique Corollary 1. Let dJio(N) and dJio (N- 1) be the Haar representation as measures on 0 (N) and 0 (N -1) respectively. Then

(103) (ll2) dJiO (N ) = dJi §N- 1 X dJiO (N - 1) 1 where where dJi §N -1 is the uniform measure on §N- 1 . What is the meaning of dJi §N -1? Given that we are dealing with uniform random variables, it is quite natural that we end up with the uniform measure. In this case, however, it has a precise group theoretical interpretation. Left multiplica­ and 0 E 0 (N - 1). A consequence of theorem 4 tion of the right-hand side of equation (103) by is that if vN is uniformly distributed in §N- 1 and 0' E O(N) induces a map on the coset space: 0 is distributed with Haar measure on O(N - 1), (ll3) o 'HN(v)n = HN(v')n'n = HN(v')n". then 0 is Haar distributed too. The final link with the subgroup algorithm is given by Since the decomposition (103) is unique the trans­ formation v ~ v' is well defined. This map can be Theorem 5. The left coset space of O(N- 1) in easily determined. A coset is specified by where O(N) is isomorphic to §N- 1 , i.e., e1 is mapped, therefore

(105) O(N) / O(N - 1) ~ §N- 1. (ll4) O'HN(v)e1 = O'v = v' = HN(v'le1. A complete class of representatives is provided by If vis uniformly distributed on the unit circle so is the map 1 HN : §N- 1 - O(N), v' = Ov. Thus, dJi §N -1 is the unique measure on the coset space 0 (N) I 0 (N -1) invariant under the left (106) HN(v) = {;Nsgn(x1) (I- 2uu.t) ifv*e1, action of O(N). Its uniqueness follows from that ifv=e1, of Haar measure and from the factorization (112). Corollary 1 is a particular case of a theorem where that holds under general hypotheses for topologi­ (107) cal compact groups. Indeed, let r be such a group, B a closed subgroup and C = r / B. Furthermore, and x1 is the first component ofv. let dJir, dJic and dJix be the respective invariant measures, then Proof The group of N x N matrices Q defined in (ll5) equation (104) is isomorphic to O(N- 1). Since (108) Acknowledgements O(N- 1) can be identified with the subgroup of This article stems from a lecture that I gave at the O(N) that leave e 1 invariant, i.e., summer school on Number Theory and Random (109) O(N- 1) = {0 E O(N) I Oe1 = ed. Matrix Theory held at the University of Rochester in June 2006. I would like to the thank the organiz­ Now, if two matrices 0 and 0' belong to the same ers David Farmer, Steve Gonek, and Chris Hughes coset, then for inviting me. I am also particularly grateful to (llO) Brian Conrey, David Farmer, and Chris Hughes for the encouragement to write up the content of this and vice versa. In other words, cosets are speci­ lecture. fied by where e1 is mapped. Furthermore, since II Oe111 = 1, we see that they can be identified with the points in the unit sphere. Finally, the References map (106) is one-to-one and is such that (1] P. DIACONIS and M. SHAHSHAHANI, The sub­ (lll) group algorithm for generating uniform random variables, Prob. Eng. Inf Sc. 1 (1987), 15-32. Therefore, HN spans a complete class of represen­ [2] F. M. DYSON, The threefold way. Algebraic structure tatives. D of symmetry groups and ensembles in quantum mechanics,]. Math. Phys. 3 (1962), 1199-1215. Incidentally, theorem 4 implies [3] M. L. EATON, Multivariate Statistics: A Approach, Wiley and Sons, , NY, 1983. [4] A. EDELMAN and N. R. RAo, Random matrix theory, 1 The Householder reflections defined in equation (79) are Acta Num. 14 (2005), 233-297. not continuous at e1 . Indeed, it can be proved that there (5] R. M. HEIBERGER, Algorithm AS127. Generation of is no continuous choice of coset representatives. In the random orthogonal matrices, App. Stat. 27 (1978), section "Householder Reflections", this distinction was su­ 199-206. perfluous: ifv is randomly generated, the probability that v = ae1 is zero.

MAY 2007 NOTICES OF THE AMS 603 [6] N. M. KATz and P. SARNAK, Random matrices, Frobe­ The Mind of the nius eigenvalues, and monodromy, Amer. Math. Mathematician Soc. Colloquium Publications, 45, Amer. Math. Soc., MICHAEL FITZGERALD Providence, Rl, 1999. AND lOAN JAMES (7] ]. P. KEATING and N. C. SNAITH, Random matrix Internationally famous mathemati­ theory and !;;'(1 / 2 +it), Commun. Math. Phys. 214 cian loan James and accomplished (2000), 57-89. psychiatrist Michael Fitzgerald [8] __ , Random matrix theory and L-functions at explore the behavior and personality s = 1/ 2, Commun. Math. Phys. 214 (2000), 91-110. traits that tend to fit the profile of a [9] M. L. MEHTA, Random matrices, Elsevier, San Diego, mathematician. CA, 2004. $30.00 hardcover [10] Recent perspectives in random matrix theory and number theory, LMS Lecture Note Series, 322, Equations from God (F. Mezzadri and N. C. Snaith, eds.), Cambridge Pure Mathematics and Victorian Faith University Press, Cambridge, 2005. DANIEL]. COHEN [1 i] H. L. MONTGOMERY, The pair correlation of zeros of the zeta function, Analy tic Number Theory: Proc. Daniel Cohen captures the origins of Symp. Pure Math. (St. Louis, MO, 1972), vol. 24, the rebirth of abstract mathematics Amer. Math. Soc., Providence, Rl, 1973, pp. 181-93. in the intellectual quest to rise above [1 2] A. M. ODLYZKO, The 1020 -th zero of the Riemann common existence and touch the zeta function and 70 million of its neighbors, mind of the deity. 1989, h tt p: //www .dt c .umn.edu / ~ odlyzko/ fohns Hopkins Studies in the : Ronald Calinger, Series Editor unpubli s hed/ index.html. $50.00 hardcover [1 3] M. RUBINSTEIN, Low-lying zeros of L-functions and random matrix theory, Duke Math. ]. 109 (2001), THE JOHNS HOPKINS UNIVERSITY PRESS 147-181. 1-800-537-5487 • www.press.jhu.edu [14] G. W. STEWART, The efficient generation of random orthogonal matrices with an application to con­ dition estimators, SIAM]. Num. A nal. 17 (1980), 403-409. [15] M. A. TANNER and R A. THISTED, A remark on

A\IFRI{;:\J.\: 1\'lArHE,\P.TJC u SocrET1' AS 12 7. Generation of random orthogonal matrices, App. Stat. 31 (1982), 190-192. Announcing the AMS Polo Shirt! [16] R. W. M. WEDDERBURN, Generating random rota­ tions, Research report, Rothamsted Experimental • Navy blue with white embroidery Station (1975). • 100% Egyptian cotton [1 7] ]. WISHART, The generalised product moment dis­ tribution in samples from a normal multivariate population, 20A (1 928), 32- 52. [18] M. R. ZIRNBAUER, Riemannian symmetric super­ spaces and their origin in random-matrix theory, ]. Math. Phys. 37 (1996), 4986- 5018. (19] K. ZYCZKOWSKI and M. KUS, Random unitary matrices, ]. Phys. A : Math. Gen. 27 (1994), 4235-4245.

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Princeton University Press 800-777-4726. Read excerpts at math.press.princeton.edu The Vision, Insight, and Influence of Oswald Veblen Steve Batterson

en Oswald Veblen assumed the proof of the . It was Veblen residency of the American Math­ who made Poincare's brilliant algebraic topol­ matical Society in 1923, there ogy concepts accessible to the field's pioneers. Wwas little support for mathemati­ Among the students supervised by Veblen were al research in the United States. R. L. Moore, Alonzo Church, and]. H. C. Whitehead, University teaching loads varied from nine hours who went on to become leaders in the development to fifteen hours per week and higher. The birth of of their respective specialities. the National Science Foundation was over a quarter I first came to appreciate Veblen's significance century in the future. Research when I began research for a book on the origins grants and postdoctoral positions of the Institute for Advanced Study. In discuss­ (with little or no teaching) did not ing my project with others it became evident that exist for mathematicians. Veblen's contributions were largely unknown to As a professor at Princeton the mathematicians of today. Oswald was fre­ University, Veblen himself was a quently confused with his more famous uncle, member of a premier mathemat­ economist Thorstein Veblen. The objective of this ics department. Yet just four of article is to review Oswald Veblen's overall influ­ Veblen's colleagues were actively ence as mathematician, mentor, and advocate. I engaged in research, and, among am grateful to Michele Benzi and Albert Lewis for them, James Alexander, Einar their suggestions. Hille, and worked with no office space. Hille, Mathematical Ascendance as an instructor, was assigned Oswald Veblen was of Norwegian descent. His teaching duties that included two mother Kirsti Hougen was born in Norway while sections of trigonometry. Instruc­ his father, Andrew, and Uncle Thorstein were tors at Columbia and Michigan first generation Americans from Wisconsin. The Oswald Veblen, 1924• taught twelve and sixteen hours per week respectively [1], [2], [3]. Hougen and Veblen families settled on nearby Veblen took up the task of farms in Minnesota. Kirsti and Andrew married improving the circumstances for research math­ in 1877 [4], [5]. ematicians. His success in this endeavor was On June 25, 1880, Oswald Veblen was born in but one facet of a career that had an enormous Decorah, Iowa, where his father was teaching at impact on mathematics and the profession. For Luther College. Shortly afterward Andrew began example, due entirely to Veblen's initiative the graduate study at Johns Hopkins. In 1883 the fam­ young scholars Kurt Go del and ily returned to Iowa for Andrew to teach physics were recruited for academic opportunities in the and mathematics at the University of Iowa. There United States. As a mathematician Veblen was Oswald was educated from elementary school no slouch. His work included the first rigorous through an undergraduate degree at the university in 1898. Two years later he received a second B.A. Steve Batterson is associate professor of mathematics and from Harvard. at Emory University. His email address In 1900 Veblen began graduate school at the is sb@mathcs. emory. edu. . He arrived without any

606 NOTICES OF THE AMS VOLUME 54, NUMBER 5 financial support, "but with a hope that something recommended Veblen. Veblen might turn up." It did. Another student with a fel­ and Eisenhart were joined by lowship dropped out during his first quarter, and Gilbert Bliss and J. W. Young as Veblen was awarded the US$320 stipend. [4/2/38, the first mathematics precep­ 5/2/38 Oswald Veblen to John Howe, from Oswald tors [1]. Veblen Papers, Manuscript Division, Library of The selections were superb. Congress] Financial considerations aside, Chicago Each preceptor was both a rising was a wise choice for graduate study in mathemat­ scholar and leader. Veblen and ics. The university, begun just eight years earlier, Eisenhart would work with Fine in was in the midst of a historic period in the edu­ engineering Princeton's advance­ cation of American mathematicians. Among the ment. Bliss and Young remained rising generation of leaders to receive Ph.D.s there for three years before moving at this time were Leonard Dickson (1896), Gilbert on. Bliss returned to Chicago and Bliss (1900), Veblen (1903), R. L. Moore (1905), and together with Dickson became G. D. Birkhoff (1907). Each would later serve a term the department's second genera- E. H. Moore, 1902. as president of the AMS. tion of leaders. Young eventually The head of the Chicago department was E. H. settled at Dartmouth where he Moore who was one of the most highly regarded initiated a transformation toward mathematicians in the country. Among Moore's research. was 's recent work on the At Princeton Veblen began foundations of geometry. Veblen attended Moore's to collaborate with Young on fall1901 seminar covering this topic. Hilbert had . Their work employed the undefined terms of point, line, and culminated in a classic text on plane, to devise a scheme of 20 . Questions the subject. The first volume soon arose over the independence of these axioms. appeared in 1910, with a second In the seminar Moore identified and sharpened the volume slated for the following independence deficiencies. Veblen was inspired year. The scheduling was opti­ to go further in his 1903 thesis. Creating a frame­ mistic. The authors were sepa­ work based on just the two undefined elements of rated, and Young was moving point and order, Veblen proposed twelve axioms from the chair duties at Kansas to for Euclidean geometry. He showed the axioms the the headship at Dartmouth. were independent and that they were satisfied by Veblen, who had taken on other Henry Fine, 1911. a system that was essentially unique [6]. projects as well, finished the sec­ Veblen remained at Chicago for two notable ond volume on his own in 1917. postdoctoral years. During this period he polished Some indication of the enduring his thesis for publication, proved a major theorem, of these books is given by began a , and joined his advisor E. H. a mid-2006 lSI Web of Knowledge Moore in directing the thesis research of a newly citation search, which recorded arrived student. The student was R. L. Moore and twenty-seven hits among articles the collaborator a young Scottish algebraist, Joseph published since 2000. Wedderburn. The theorem was the long accepted The year 1905 marks the be­ Jordan Curve Theorem for which previous proofs ginning of Princeton's ascen­ were unsatisfactory. Veblen was well prepared to dance to world class mathematics go out on his own. standing. Veblen's fingerprints It was at this time, in 1905, that Princeton Uni­ immediately appear all over the versity President Woodrow Wilson embarked on a university's successful efforts to program to upgrade the university. A key element identify and recruit promising of his plan was the creation of a large number of young talent. Prior to the com­ junior faculty positions to reduce class size and pletion of his first year, Veblen G. D. Birkhoff, 1913. individualize undergraduate education. These wrote to R. L. Moore to sound out "preceptors" were to teach traditional classes as his in a position. Moore well as provide one-on-one supervision to upper became the fifth preceptor, before going to North­ class students. Selection of the mathematics pre­ western University and then to the University of ceptors was in the hands of mathematician and and the University of Texas. With Dean of the Faculty Henry Fine. the departures of Moore, Bliss, and Young, two Fine recognized an opportunity to change the outstanding mathematicians came to Princeton direction of a department that, aside from himself as replacements in 1909. Both had overlapped and the young instructor Luther Eisenhart, had with Veblen at Chicago. They were his collabora­ no commitment to research. When Fine contacted tor Wedderburn and G.D. Birkhoff, a later E. H. E. H. Moore to solicit nominations, Moore strongly Moore student.

MAY 2007 NOTICES OF THE AMS 607 Birkhoff was just twenty-five when his appoint­ of mathematical research. According to longtime ment began at Princeton. Frequent walks there with Princeton mathematician Albert Tucker, Veblen the four-year-older Veblen cemented a lifelong was sufficiently concerned over the subject's personal and professional friendship. It would standing that he advised Alexander to obtain his have been fascinating to listen in on these conver­ Ph.D. in a more fashionable area [7]. In 1915 Alex­ sations, for the interaction undoubtedly stimulated ander published two important papers. His thesis, subsequent developments at this historic time under T. H. Gronwall, was on univalent functions. in the emergence of mathematical research in In another article Alexander established that the America. Birkhoff and Veblen were pursuing dif­ homology of a 3-dimensional polyhedron was in­ ferent programs of Poincare. With Birkhoff it was dependent of its . that would lead to his seminal Veblen's standing as a mathematician grew proof of the existence of fixed points for certain steadily at Princeton. An offer from Yale contrib­ area preserving maps of the annulus. Veblen be­ uted to his 1910 promotion to the rank of full came interested in Poincare's conceptualization of professor. At that time, one of the most prestigious (known then as analysis situs) recognitions for an American mathematician was which, to this point, was largely undeveloped. an invitation to deliver the Colloquium Lectures. Birkhoff remained at Princeton for three years and These opportunities arose every few summers then went to Harvard where he earned acclaim as when two prominent scholars were selected to the foremost American mathematician. showcase their work in a series of talks to the en­ Poincare's publications on analysis situs ap­ tire American Mathematical Society. The lectures peared over the period 1892-1905. Jean Dieudonne were closely followed and much discussed. provided the following assessment: When Veblen was invited to deliver the 1916 Colloquium Lectures, he might well have selected As in so many of his papers [Poincare] gave free rein to his imaginative pow­ the established topic of projective geometry. In­ stead Veblen took the decision to proselytize the ers and his extraordinary "intuition", less respected subject of analysis situs. Over six which only very seldom led him astray; lectures he developed the notions of Betti number, in almost every section is an original torsion, the fundamental group, and the topologi­ idea. But we should not look for precise cal classification problem. definitions, and it is often necessary to As was customary with the Colloquium Lec­ guess what he had in mind by interpret­ tures, Veblen revised and expanded his presenta­ ing the context. For many results, he tion into a monograph that the AMS published simply gave no proof at all, and when he endeavored to write down a proof in 1922. In the preface Veblen stated his goal to provide "a systematic treatment of the elements of hardly a single argument does not raise Analysis Situs", but cautioned that it should not be doubts. The paper is really a blueprint for future development of entirely new viewed as "a definitive treatment. For the subject ideas, each of which demanded the cre­ is still in such a state that the best welcome that ation of a new technique to put it on a can be offered to any comprehensive treatment is to wish it a speedy obsolescence." The words sound basis. [Jean Dieudonne, A History were prophetic. Through Veblen a generation of of Algebraic and mathematicians was introduced to the concepts of 1900-1960, Birkhauser (1989) p. 17] algebraic topology. The ideas quickly gained trac­ In view of these obstacles it is not surprising tion, and new developments followed. However, that the concepts of homology and fundamental even after Lefschetz' more modern discussion of group remained fallow for nearly a decade. Ve­ the subject was published in 1930, Veblen's book blen, a remarkable connoisseur of mathematics continued to serve as an important (and user­ and mathematicians, appreciated the value of friendlier) primer. Poincare's ideas and saw the need to establish a The long lag between Veblen's Colloquium solid foundation upon which they could be com­ Lectures and their appearance in print coincided municated to other mathematicians. Veblen set with the onset of World War I. The war contributed himself to the task, recruiting a brilliant young to the publication delay in two ways. The AMS Princeton graduate student, James W. Alexander, underwent financial difficulties that set back its to join him in the effort. Their combinatorial book program. Second, Veblen shifted his intel­ formulation of homology for polyhedra, deriving lectual and physical energy to the war effort. the Betti numbers, torsion coefficients, and their This career move illustrated his modus operandi. properties, appeared in the 1913 Annals of Math­ Veblen was a visionary with an unusual capacity ematics. The notion of homology groups was still for implementation and realization of his ideas. In a dozen years in the future. this case he wanted to utilize his skills to benefit Even with this introduction to the subject, al­ the army. That Veblen lacked both military and gebraic topology remained out of the mainstream applied mathematics experience was no deterrent

608 NOTICES OF THE AMS VOLUME 54, NUMBER 5 to him. He would find a way to make a substantive His forceful personality, tall frame, and tweed contribution. In 1917 Veblen was commissioned as suits made him an effective advocate. However, it an army reserve captain and went through basic was Veblen's faith in America's potential and his training. He then assumed charge of experimental international per- ballistics at the new in spective that would [8]. shape his actions. Veblen took up the task of producing range Veblen had been tables for new weapons. These tables were de­ quick to broaden signed for use by artillery officers in the field. The his experience be­ objective was to specify values for the distance yond his midwest­ a shell would travel, presented as a function of ern upbringing. In independent variables such as the angle of the the summer of 1905 cannon barrel and the amount of charge. There he traveled through is a long mathematical history to the problem of Europe to pursue configuring artillery to reach a target, accounting contacts that arose for drag and other factors [9]. At Aberdeen Veblen out of his thesis was to conduct test fires and then utilize the data research. The ap­ to calculate the range table. pointment at Princ­ Neither the existing theory nor the Aberdeen eton, in the fall, infrastructure were adequate to produce satis­ coincided with the factory tables. The mathematical, logistical, and hiring of British sci­ physical obstacles that arose are recounted in [8] entists James Jeans and [9]. Veblen overcame the difficulties with a and Owen Richard­ considerable combination of ingenuity and ad­ son. Richardson Veblen in U.S. Army uniform. ministrative skill. Along the way he built up his had already done staff by recruiting young mathematicians such as the work that would Joseph Ritt and . Moreover, Veblen later be recognized with a . worked closely with the army office of theoretical When Richardson's sister, Elizabeth, visited from ballistics, headed by University of Chicago as­ England, Veblen met the woman he would marry tronomer Forest Moulton, to create the needed in 1908. The marriage lasted until Veblen's death numerical methods. In [9], Goldstine portrays the in 1960. There were no children. Owen returned World War I ballistics project as a crucial step in to England in 1914, giving Veblen a connection to the development of computational mathematics high level European science. and the computer. The Veblens spent the fall term of 1913 tour­ Veblen devoted two years to the army. Return­ ing the mathematical centers of Europe, including ing to Princeton in 1919, he went to work on the Oslo, Gottingen, and . Sylow, Mittag-Leffler, colloquium book manuscript. In that same year Klein, and Schwarz were among the nineteenth Veblen was inducted into the National Academy of century legends whom Veblen met personally. Sciences. At the age of thirty-nine he had reached At each stop he keenly observed the scientific the pinnacle of American scholarship. His intel­ culture, particularly noting local frameworks for lectual development, over the first two decades of promoting mathematical interaction and for facili­ the twentieth century, occurred amidst an overall tating the communication of new results. Veblen advance of American mathematics. For the first was intrigued by how American mathematicians time United States mathematicians were gaining measured up to the higher status Europeans. He the respect of their European counterparts [6]. concluded that the United States was generally At the same time, mathematical scholarship was competitive with the Gottingen faculty, excepting becoming more prevalent in the United States. Un­ Hilbert who was in a class by himself. This judg­ like in the previous century, researchers were to be ment, weighing one German department against found, though certainly not in any abundance, at an entire country, may have been influenced by most quality universities. Despite these advances chauvinism. neither the American academy nor government There were fundamental differences in the Euro­ had done much to support mathematical scholar­ pean and American educational systems that made ship. Teaching and service loads remained high, for an uneven playing field. Especially significant particularly in comparison to those for professors to Veblen was the huge elementary service course at European institutions. Veblen was well aware of burden that fell upon United States mathematics these discrepancies. faculty. In Europe the syllabi for these courses were covered in the secondary schools. Their university Advocate for Mathematics faculty, unencumbered by high teaching loads, Membership in the National Academy gave Veblen were free to focus on research and advanced stu­ the standing to represent research mathematicians. dents. Veblen's vision was for American research

MAY 2007 NOTICES OF THE AMS 609 mathematicians to enjoy similar circumstances. It ply no precedent for extramural support in the was a laudable goal, but how could a single math­ United States. Veblen was asking philanthropy ematician make a difference? Veblen would find an and government to begin perceiving mathemati­ opportunity with his appointment to the National cal research as essential to the national interest. Research Council (NRC), the operating arm of the The cases for physics and chemistry had been National Academy of Sciences. based on the importance of these subjects to the The NRC was composed of representatives from military, medicine, and industry. Attempting the universities, private laboratories, government, and same argument for mathematics posed problems. professional societies. The NRC's original mission Range tables were less compelling than explosives was to coordinate the American scientific contri­ and gases. Any medical justification needed to bution to World War I, but its political roots went pass the scrutiny of Simon Flexner. Nevertheless, deeper. In the early twentieth century astronomer Veblen succeeded by making the indirect argument George Hale sought to upgrade the profile of the that mathematics was to chemistry and physics National Academy of Sciences. Hale's ambitions as chemistry and physics were to medicine. In for the Academy became part of a power struggle the environment of the NRC, Veblen's diplomatic over the direction of United States science policy. skills may have served him as well as his logic. It These issues were set aside when war appeared on is unclear whether the same analogy would have the horizon. At this time, in 1916, Hale conceived worked if put forward by another mathematician the NRC. Among those joining Hale's team on the with the initiative to act. Council was the director of the Rockefeller Insti­ When Veblen brought about the first research tute for Medical Research, Simon Flexner. grants for American mathematicians, he was serv­ After the war, the NRC's mission was unclear. ing both as president of the AMS and as one of the Meanwhile the debate resumed over how to chan­ Society's representatives on the NRC. During its nel science to serve a national interest that had previous thirty-five years the AMS had advanced been redefined by war. NRC members were among American mathematics through an agenda of orga­ urging that an infusion of funds for basic nizing meetings and publishing research. Veblen's research in physics and chemistry was needed to 1923-1924 presidency marked an expansion of maintain American strength. Flexner's endorse­ AMS activity into the realm of advocacy. This move, ment influenced the to while largely the product of a new president's vi­ consider adopting this new cause. For some time sion, was also driven by financial necessity. Since the foundation grappled with whether to create the beginning of World War I, increasing publica­ a pure science laboratory or to enhance existing tion costs had stressed the AMS finances. Print­ university operations. The outcome, in 1919, was ing the Bulletin and Transactions had driven the a new mechanism for scientific support. Rather budget to five figures. Even with a 1920 boost in than allocate funds to laboratories, the Rockefeller dues to US$6 and a successful membership drive, Foundation underwrote a program of postdoctoral revenues were insufficient to keep up with the fellowships, in physics and chemistry, that was to rising costs. be administered by the NRC. The politically en­ The 1923 incorporation of the AMS provided gaged NRC gained new influence and a connection new financial flexibility. To address the bud­ to the Rockefeller resources [10], [11]. get shortfall, Veblen appointed Harvard math­ Although mathematics had been entirely out­ ematician Julian Coolidge to lead an endowment side the consideration for research support in campaign. A US$100,000 goal was established. pure science, the AMS was represented on the NRC. Coolidge raised one fourth of this sum through When a new mathematics appointee was needed, solicitations of AMS members. Late in 1923 Ve­ Veblen's war work and National Academy mem­ blen joined Coolidge in bringing the appeal to the bership made him an choice. Veblen joined private sector. A simultaneous campaign was un­ the NRC in 1920. As a member of the Executive dertaken to increase awareness of the importance Committee and the Division of Physical Sciences, of mathematics to civilization. Out of this two­ he became an insider in discussions on national pronged approach came the creation of the annual science policy. It was a role that suited him well. In Lecture on mathematics and 1923 Veblen became chair of the NRC Division of its applications [4]. Physical Sciences. Within a few months he single­ Coolidge and Veblen worked hard to win over handedly persuaded the NRC, Flexner, and the chief executives. However, the tax system was not Rockefeller Foundation to expand the postdoctoral yet configured to promote corporate donations. program to include mathematics. When industrial officers expressed reluctance Given the abundance of postdoctoral oppor­ to authorize outright contributions, Veblen and tunities in mathematics today, it is difficult to Coolidge devised a variety of inducements. The comprehend the magnitude of Veblen's break­ most successful was the patron membership, a through in the funding of mathematical research. forerunner of today's institutional version, in Consider the obstacles he faced. There was sim- which companies paid annual premiums and then

610 NOTICES OF THE AMS VOLUME 54, NUMBER 5 designated selected employees for gratis AMS of a US$3.5 million plan for the sciences. The Gen­ memberships [12]. eral Education Board received the overall proposal The outcome of the endowment drive was with favor, but questioned the institute aspect. mixed. The US$55,000 raised in pledges fell well During the final Rockefeller-University negotia­ short of the goal, but the patron memberships tions, the mathematics institute was deleted from added an uncapitalized US$4,000 in revenue to the plan. Princeton was awarded a US$1 million the annual budget. In addition Veblen secured sig­ challenge grant toward the adjusted goal of US$3 nificant subventions from the NRC and Rockefeller million. Foundation to assist, over a period of years, with Fine played a crucial role in raising the US$2 publication expenses. The bottom line was that the million Princeton obligation. Alumnus Thomas budget problems were temporarily resolved. Jones, a long time friend of Fine, and Jones' niece While working with Coolidge on behalf of the Gwethalyn endowed four chairs in the sciences. AMS, Veblen independently launched his most Veblen became the first Henry Fine Research Pro­ ambitious attempt at advocacy for mathematical fessor of Mathematics, a position with no formal research. After the funding was in place for the teaching duties. postdoctoral fellowships, he sent new appeals to From the new science endowment the Princeton his NRC and Rockefeller connections. Veblen de­ mathematics department gained an annual sti­ scribed the plight of research mathematicians in pend to program for research. Out of these funds the United States as follows: Over the prior quarter Wedderburn, Alexander, and century American universities had come to select each received salary supplements to reduce their for mathematical scholarship, but had failed to teaching loads. The homegrown Alexander had make accommodations to cultivate it. The cur­ joined the faculty in 1915 and gone on to obtain rent situation was that research mathematicians, fundamental results in topology. Lefschetz arrived although in demand, had little time for research from Kansas in 1924 for a visiting position that after completion of their teaching and service was made permanent the following year. With obligations. Even so, American mathematics had Lefschetz, Alexander, and Veblen, Princeton was a made enormous progress. In Europe conditions world center for the emerging subject of algebraic were much better. The teaching load of nine hours topology. The stimulating environment attracted at Harvard contrasted with three hours at the Col­ European scholars Pavel Alexandroff and Heinz lege de France. Hopf for productive visits in 1927-1928. Veblen suggested that the foundations carry Mathematics at Princeton had come a long way support for mathematics further. He portrayed the NRC postdoctoral fellowships as a vital first since Veblen's arrival in 1905. The department step in providing research opportunities to prom­ stood with Harvard as the two leading mathematics ising young scholars. "What remains to do is to institutions in the United States. Yet infrastructure find a way of assuring the continuance of their remained essentially nonexistent. Fine and Eisen­ research to men who have proved their ability." hart, as deans of science and of the faculty, had [6/10/24 Oswald Veblen to Vernon Kellogg and offices in an administration building. The rest of to , from Oswald Veblen Papers, the department operated out of a small portion of Manuscript Division, Library of Congress] Veblen the physics building that consisted of an office for proposed two solutions. The first was the creation Veblen, a library, and, according to Lefschetz, one of a mathematics institute where research, rather room for "everything else". Wedderburn, Alexan­ than teaching, was the primary business. The sec­ der, and Lefschetz worked at home [1]. ond proposal was the endowment of a number of The lack of physical space limited interaction research professorships in which recipients would among the mathematicians. Veblen was a strong remain at their universities and have their salaries believer in the notion of a community of scholars. subsidized to reduce teaching loads. He had observed the rich cultures at Gottingen and While Veblen initially pitched a generic institute other European institutions. With planning under that could stand alone or be incorporated as part way for a mathematics building at the University of of a university, developments at his own university Chicago, Veblen pushed for Princeton to construct soon led him to customize the proposal. In 192 5 a home for its department. Any prospect of moving President John Grier Hibben mounted a major fund forward on the project required the endorsement drive for Princeton. The needs of the sciences were of Fine, who had responsibility for the interests to be presented in a request to the General Educa­ of other subjects as well. Outside funding was the tion Board of the Rockefeller Foundation. Veblen surest route to the head of the queue. By the fall lobbied for inclusion of a mathematics institute, of 1928 some hope existed that the Rockefeller targeted at applications of the department's Foundation might finance the addition of a math­ strengths in analysis situs and geometry. ematics wing onto the physics laboratory. This was With the support of Fine and Eisenhart, Veblen the situation as Veblen left for Oxford as part of a succeeded in having his institute included as part year long exchange with G. H. Hardy.

MAY 2007 NOTICES OF THE AMS 611 Joining Hardy at Princeton was Veblen saw an opportunity, through Fine Hall, who had accepted the Thomas Jones research to realize crucial elements of his mathematics professorship in . Weyl was institute. Comfortable offices, a first class library, on leave from his chair in Zurich, retaining the a modern lecture hall, and meeting places would option to return, which he would. However the bring together scholars for mathematical study presence for 1928-1929 of, arguably, the world's and discussion in the manner that he had so ad­ two leading mathematicians, Hardy and Weyl, was mired in Europe. Veblen quickly took over the plan­ a coup for both Princeton and ning of the new building, incorporating features the United States. Max Mason from Oxford, and seeking to create the ambiance was impressed when he visited of Gottingen. Many of the measures were unusual Princeton in November, just a for the United States at this time. Undergraduate few months after taking charge education received no consideration. The scope of of the Rockefeller Foundation Fine Hall was restricted to research and advanced science program. Mason was an instruction in mathematics and mathematical American mathematician who physics. Each member of the research faculty had been a student of David in these subjects received an office. A spacious Hilbert at Gottingen. library, common room, and professors' room In December 1928 Veblen were designed to promote study and interaction. received a letter from Fine pre­ Policies of daily afternoon teas and 24-hour access dicting that Mason's Rockefeller would make the first two venues into magnets for group "will give us the Math­ mathematical activity. ematics Building." [11/ 28/28 While Fine Hall proved itself to be a building of Henry Fine to Oswald Veblen, exceptional , the most striking feature was from Oswald Veblen Papers, its opulence. Lavish oak paneling, carved figures, Manuscript Division, Library and fireplaces were incorporated throughout. It of Congress] A few weeks later was all Veblen's doing. He believed that mathemati­ Hermann Weyl (right) with Fine was dead, hit from behind cal research was a high calling and that scholars David Hilbert, mid-1920s. by an automobile while riding deserved comfort and consideration. Veblen his bicycle at dusk. Shortly after injected himself into every detail of the design, Fine's death Thomas Jones an­ including the selection of each piece of quality fur­ nounced a further beneficence, niture and even down to the placement of electrical his intention to donate a new sockets and choice of trash baskets. The furniture mathematics building as a me­ budget was over US$26,000. Nearly US$8,000 was morial to Fine.· Money was no allocated for rugs. Fine Hall opened in 1931 setting longer an issue. The wealthy a new standard for American mathematical accom­ Jones intended to place the modations. A large number of graduate students, mathematics department in a postdoctoral fellows, and visitors passed through home that would stand as a its hallways, later lobbying their own universities tribute to his friend Dean Fine. to adopt aspects of its design. Rather than a wing added onto physics, Fine Hall would be a Veblen Shapes the Institute for Advanced separate building with a corri­ Study dor connecting it to physics. In the summer of 1930, as Veblen oversaw the The next step was to up­ planning for Fine Hall, a front page article in the grade the plans that already New York Times caught his attention. A New Jer­ Abraham Flexner. existed for the physics wing. sey family had donated US$ 5 million to endow Wedderburn was serving as the creation of an institution devoted entirely to the liaison between the depart­ research and graduate education. The Institute for ment and the architects. The frugal Wedderburn Advanced Study (IAS) was to be located in the vicin­ had been well placed when the budget and scale ity of Newark and be directed by Abraham Flexner. were limited, but he was ill suited to act on the Veblen was acquainted with Flexner through his possibilities opened by Jones' generosity. Alex­ brother Simon. When Veblen first pitched his ander wrote to Veblen asking him to intervene: mathematics institute in 1924, Simon referred "The only thing between us and really homelike him to Abraham who was then a key figure at the headquarters is Wedderburn's rather puritanical General Education Board. Nothing had come from attitude. He acts, at times, as if he felt there was Veblen's follow up. However, a few months prior something just a little immoral about material to the appearance of the Times piece, Abraham had comforts." [1 2/28/28 James Alexander to Oswald initiated a seemingly innocuous correspondence Veblen, from Oswald Veblen Papers, Manuscript about the state of research in the United States. Division, Library of Congress] Now Veblen sent Abraham a short congratulatory

612 NOTICES OF THE AMS VOLUME 54, NUMBER 5 note in which he proposed Princeton as the site Flexner had not contemplated more than one for the new institute. (For further details and refer­ or two professors for any school. He allowed ences on the lAS see [14].) mathematics to reach three only when extraordi­ Abraham Flexner's ambition was to elevate nary opportunities arose to obtain Weyl and the scholarship in the United States. His plan was to Einstein. After making the initial hires recruit a small number of world class scholars and in the School of Mathematics, Flexner expected to provide them with ideal conditions to conduct re­ move on to economics and then to other subjects. search and train a few Ph.D. students. Flexner was He explained to Veblen that no more mathemat­ interested in obtaining only the very best research­ ics faculty were possible at the present time. Ve­ ers. Subject area was less important. Over the next blen temporarily pulled back, but he would soon year Flexner traveled throughout the United States resume his campaign for expansion. Meanwhile and Europe for consultations. Although he himself Flexner searched, in vain, for an economist. Al­ had no knowledge of mathematics, an interview though Veblen's persistent advocacy generated with G. D. Birkhoff made a profound impression. occasional , time and again he persuaded Flexner decided then on mathematics as the first Flexner to go with one more outstanding mathema­ lAS program (or school). He would delay making tician. By 1935 the School of Mathematics faculty Birkhoff an offer until further elements were in consisted of Veblen, Einstein, Weyl, Alexander, place. John von Neumann, Morse, and a one-year visitor, One of the most delicate matters was the se­ . lection of a location for the lAS. The founders, Von Neumann's path to the lAS, through Princ­ Louis Bamberger and his sister Carrie Fuld, were eton University, illustrates Veblen's connoiseur­ set on Newark where they had made their fortune ship and his resourcefulness in building up the two in the department store business. Flexner was at­ institutions. Veblen, along with Eisenhart and Fine, tracted to the aesthetic and university resources of always kept an eye out for young mathematical tal­ Princeton. Moreover, if space could be rented for ent, both at home and abroad. Veblen took notice Birkhoff in Fine Hall, then, together with the uni­ of von Neumann about the time of his Ph.D. from versity mathematicians, the lAS would become a in 1926. Two years later they met at the leading international center in its initial discipline. International Congress of Mathematicians where As Flexner moved quietly behind the scenes to Veblen broached the possibility of von Neumann coming to Princeton on an international fellowship. explore a Princeton relationship, he was naturally An opportunity at Hamburg halted von Neumann's drawn to the support and discretion of Veblen. consideration of Princeton. When Weyl resigned Flexner worked out a Fine Hall arrangement, main­ from the Thomas Jones chair the following year, taining lAS autonomy, that he was eventually able Veblen began thinking of the twenty-six-year-old to sell to Bamberger and Fuld. von Neumann as a possible successor. Veblen pro­ In the midst of the diplomacy in Princeton, an posed to Eisenhart that von Neumann be given a important meeting took place between Veblen and visiting position as a tryout. Von Neumann came Flexner. It occurred late in 1931, several months to Princeton for the spring term of 1930 and then prior to an offer being made to Birkhoff. Flexner began splitting his time between Princeton and confidentially disclosed to Veblen his intention Berlin. The Jones chair remained vacant. to begin the lAS with mathematics and Birkhoff. The opportunity for von Neumann at the lAS Veblen informed Flexner that a recent letter from also arose out of Weyl's career (in)decisions. Hermann Weyl indicated that Weyl might be mov­ Throughout 1932 Weyl continued to waffle on able to the United States. Following this exchange the lAS offer, prompting Bamberger to question of inside information Veblen became Flexner's his character and suitability for an appointment. most influential advisor. Flexner defended Weyl, taking the risk of alienat­ Flexner's first offers went to Birkhoff, Weyl, and ing his benefactor. Late in 1932, Weyl appeared to . When Birkhoff decided to remain have decided on the IAS. Veblen then persuaded at Harvard, Flexner turned to Veblen as a replace­ Flexner that the younger American, James Alex­ ment. Weyl entered a long period of indecision ander, would add desirable balance to the School over whether to resign the chair in which he had of Mathematics. recently succeeded Hilbert at Gottingen. Veblen The appointments of Weyland Alexander were and Einstein accepted in June 1932. Veblen im­ on the docket for approval at the lAS trustees mediately went to work at realizing his institute meeting in mid-January. During the week prior to through Flexner's. Along with his acceptance Ve­ the meeting, Weyl sent three telegrams, alternately blen enclosed an elaborate plan for the School of accepting, withdrawing, and accepting again the Mathematics. His list of names to be considered for lAS position. The second telegram caused Flexner additional professorships consisted of Lefschetz, to face the unpleasant prospect of explaining Alexander, , , , Weyl's peculiar behavior to a skeptical Bamberger Alexandroff, and Emmy Noether. and the board. Veblen seized the opportunity to

MAY 2007 NOTICES OF THE AMS 613 push for another mathematician. He knew that von superior faculty devoted to research. Outside pres­ Neumann wanted a permanent, full-time position sures forced Gilman to revise his program and in the United States. Veblen urged Flexner to sub­ include undergraduates. stitute von Neumann for Weyl. Flexner was one of these undergraduates dur­ Flexner weighed his decision carefully. With von ing the early years of Johns Hopkins. Out of this Neumann already in Princeton, a quick acceptance experience he adopted Gilman as a hero. When was practically assured. Flexner liked the idea of Flexner began to formulate his own higher educa­ giving the trustees the von Neumann good news to tion dogma, the Gilman influence pervaded his mitigate the Weyl bad news. The one reservation thinking. Among Flexner's strongest beliefs was for Flexner was his genuine deference to Eisenhart that undergraduate and graduate education were who was moving to hire von Neumann for one of incompatible. Only Ph.D. candidates were in the the university vacancies created by Veblen and student plans at the founding of the lAS. In the Alexander. Then Flexner learned of Weyl's third fall of 1932 Flexner went one step further. He telegram and everything suddenly changed. An eliminated all degree study, replacing it with a offer to von Neumann would not only weaken the new advanced educational class consisting of university, but it would require creation of a fifth freshly minted Ph.D.s. These "students" were to be position in the School of Mathematics. Despite selected by individual lAS professors from whom Veblen's best efforts, Flexner decided to present they would receive further research mentoring. just Weyl and Alexander to the trustees. The lAS training was intended to strengthen the Two days after his appointment was approved, students' preparation for launching their own in­ Weyl sent a fourth telegram dependent research programs as rookie faculty. with another withdrawal. The switch from graduate to postdoctoral edu­ Flexner then moved quickly cation occurred shortly after Veblen began promot­ to secure von Neumann. The ing another European discovery, Kurt Godel, for a lAS would open in 1933 year-long appointment. Flexner had at first balked with a faculty of Veblen, Ein­ at consideration of the twenty-six-year-old logi­ stein, Alexander, and von cian who was three years beyond his Ph.D. Godel, Neumann. As for Weyl, his however, fit the revised profile of a promising erratic actions were taken pre-faculty scholar. Flexner penciled in Godel as in the midst of a nervous Veblen's (and the lAS's) first student. breakdown. The timing was Veblen had his own vision for a mathematics catastrophic. Less than a institute: To bring together an exceptional group month after Weyl spurned of scholars and provide them with an opportu­ the lAS, Hitler became chan­ nity to interact and the freedom to concentrate cellor of Germany. Although on research. The differences with Flexner, aside Weyl himself was Aryan, his from the number of personnel, were subtle. Ve­ wife was Jewish. Their fam­ blen wanted a continuum of age and experience, ily was doomed. in contrast to Flexner's polar student-professor During his recovery Weyl scheme. Moreover, Veblen regarded everyone, forthrightly poured out his even new Ph.D.s, as independent scholars. While feelings and fears in letters he happily followed up on Flexner's suggestion of to the supportive Veblen. inviting Godel, Veblen had no intention of acting When Weyl regained his as a supervisor. John von Neumann. health and discovered the In January 1933 Veblen decided to push the state of German society, he envelope on postdoctoral scholars. He proposed to appealed to Veblen for reconsideration. This time Flexner that the lAS host one-year visits by young Flexner was easy to convince. The problem was mathematicians who already held positions at Bamberger, whose objections Flexner overcame. Harvard, Johns Hopkins, and Chicago. Since these The School of Mathematics got its fifth professor. faculty were too junior to be eligible for sabbati~ Veblen would also be instrumental in the hiring of cals, Veblen suggested that their salaries be split Marston Morse who completed the first generation between the home university and the lAS. Flexner of lAS mathematics faculty. was negative. While he was anxious to advance the Despite his role in assembling an extraordinary development of rising faculty, he expected the lAS collection of scholars, Veblen's greatest lAS legacy to exert its role while the scholar was still a free may be his shaping of its postdoctoral policy. What agent. Supporting the research of a faculty member follows is a brief description of how this crucial from a wealthy university struck Flexner as a one­ lAS component evolved. Flexner's vision for the sided arrangement in which the lAS was effectively lAS was inspired by Daniel Coit Gilman's original making a subsidy to the other institution. 1875 plan for Johns Hopkins University. Gilman This reaction of the well meaning Flexner conceived a graduate (only) university built around indicates how radical Veblen's proposal was for

614 NOTICES OF THE AMS VOLUME 54, NUMBER 5 the time. Veblen defended his ground. He argued States. By this time Veblen's own mathematical that all parties would benefit, with lAS personnel interests had shifted. He was working on differ­ gaining stimulation from the interaction. Flexner ential geometry, influenced in part by Einstein's agreed to try out the program on an experimental discoveries in relativity. T. Y. Thomas pursued basis. The salaries of Adrian Albert and Egbertus this thread. van Kampen were shared with Chicago and Johns Another distinguished Veblen student was the Hopkins. British topologist J. H. C. Whitehead. They became Twenty other postdoctoral scholars joined acquainted during Whitehead's study at Oxford Go del, Albert, and van Kampen at the lAS in 193 3- in 1928. Veblen was then in residence through 1934. Several were NRC fellows or Europeans on the exchange with G. H. Hardy. Whitehead was so a Rockefeller Foundation program. Others found inspired by Veblen's lectures that he transferred their own means of support. Together with the to Princeton where he completed his Ph.D. in 1932. faculty and graduate students, Fine Hall quickly Out of their collaboration Veblen and Whitehead attained the standing and atmosphere that had so introduced the modern definition of a differen­ impressed Veblen at Gottingen. Whenever Flexner tiable manifold. Whitehead later returned to Ox­ passed through the building he marveled at the ford. His contributions to algebraic topology were mathematical activity. The postdoctoral student recognized by election to the Royal Society. scheme was quietly abandoned. Flexner began providing Veblen with an annual Statesman of Mathematics budget to support a class of visiting mathemati­ The success of the lAS enhanced Veblen's already cians (who became known as members). The funds considerable prestige. Among American math­ went a long way. Veblen and his colleagues needed ematical contemporaries, only G. D. Birkhoff and few incentives to attract the best to Princeton. Leonard Dickson stood as high. While his two In the second year of lAS operation the member E. H. Moore siblings had the better , the roster numbered in the thirties and included more combination of Veblen's scholarship, associations, senior scholars such as Georges Lemaitre, Joseph and leadership gave him enormous stature in the Walsh, and . United States' mathematical community. Under Over the past seventy-five years the School of the prevailing hiring practices of the day, depart­ Mathematics model has influenced the creation of ments frequently identified their job candidates by numerous year-long research positions throughout soliciting names from prominent outside scholars. the world. It would be easy to regard these oppor­ Veblen was a key node in this "old boy network". tunities, along with the sixty or so current annual For some time Veblen used his influence to lAS mathematics memberships, as simply a natural assist Princeton graduates and NRC postdoctoral outgrowth of Veblen's initiative and vision in the fellows obtain suitable positions. When Hitler and early 1930s. Were it not for Veblen, however, the the Nazis began to cleanse Jewish mathematicians lAS program might well have been too short-lived from German universities, Veblen immediately took for emulation. In the later 1930s severe financial on a daunting challenge. He joined the Emergency exigencies forced difficult choices on Flexner. To Committee in Aid of Displaced Foreign Scholars, keep the lAS solvent and complete its two other becoming a point person for the relocation of schools he reluctantly attempted to redirect money mathematicians to the United States. Through his out of the mathematics members budget. Each year European contacts Veblen kept abreast of the latest Veblen tenaciously battled to protect the endan­ victims, and then pressed colleagues in American gered program. Cuts were made, but the essential mathematics departments to consider some form character was maintained. of adoption. Advocating, in the 1930s, on behalf There was one unfortunate by-product of the of European Jewish refugees was a delicate matter. lAS postdoctoral program. Its establishment The was under way and jobs were prematurely phased out a notable career in the scarce for American mathematicians. In addition, mentoring of graduate students. Even so, Veblen elements of anti-Semitism and xenophobia were inspired a remarkable number of influential math­ prevalent on many campuses. Veblen managed ematicians who went on to careers in diverse spe­ to navigate this minefield while skillfully parlay­ cialities. R. L. Moore and Alexander were leaders ing his connections with private foundations and in point set and algebraic topology respectively. American mathematicians to create new opportu­ In the 1920s Veblen supervised the theses of two nities for Europeans. students, Alonzo Church and T. Y. Thomas, who Two examples, Richard Courant and Richard would follow Moore and Alexander into the Na­ Brauer, provide some flavor of the range of these tional Academy of Sciences. Church, a Princeton activities. Courant was the director of the Math­ undergraduate, was nurtured by Veblen to stay on ematics Institute at Gottingen while the thirty­ for graduate school and to complete a thesis in two-year-old Brauer held a junior position at foundations [15]. He went on to become a central Konigsberg. Both were dismissed in 1933. Veblen figure in the development of logic in the United and Flexner made personal appeals on Courant's

MAY 2007 NOTICES OF THE AMS 615 behalf. These efforts eventually led to Courant's United States. Neugebauer estimated that an an­ consequential placement at New York University. nual subvention of US$20,000 would be required. A temporary position for Brauer developed out To understand the scale consider that in 1937 total of a more general Veblen call that reached Leon AMS disbursements, including all journals and Cohen at the University of Kentucky. Cohen had publications, were under US$33,000. In addition, recently completed an NRC fellowship at Princeton. there were other infrastructure demands of start­ Grateful for his own career support from Veblen, ing up such a personnel-intensive venture. Cohen managed to bring Brauer to Kentucky. The Veblen was determined to begin an American following year Brauer went to the lAS as Weyl's journal with abstracts of international mathemati­ assistant. Local funds for both Courant and Brauer cal publications. He discussed the problem with were supplemented by the Emergency Committee AMS secretary R. G. D. Richardson and other promi­ in Aid of Displaced Foreign Scholars [16]. nent mathematicians. A plan took shape to bring That Veblen was able to succeed Neugebauer to the United States to edit a new peri­ in these humanitarian endeavors odical sponsored by the AMS. Placing Neugebauer, was likely what earned him the the world's foremost authority on the history of unusual appellation statesman mathematics, in an American university seemed of mathematics. Indeed, Veblen's doable. However, the journal start-up funds were bold diplomacy created manifold well beyond the reach of scientific organizations pathways that decisively improved and universities. The expectation was that at least the plight of mathematicians and five years of operation were required before the elevated American research. A deficit would reach a level the AMS could manage. final illustration is provided by his The only solution was a large grant from a wealthy role in the birth of Mathematical philanthropic organization, but few mathemati­ Reviews. cians had any entree to the Rockefeller Foundation The first issue of Mathemati­ or Carnegie Corporation. cal Reviews appeared in January By the end of November, Veblen had laid the 1940. The notion of an American plans before Carnegie president Frederick Keppel journal of mathematical abstracts and Rockefeller Director of Natural Sciences War­ dates back to at least the 1920s. ren Weaver. Given the disastrous economic events Veblen was among the early advo­ of the past decade, both foundations were already R. G. D. Richardson. cates, but, at the time, the publish- inundated with requests from worthy causes. It is ing venture was infeasible for the amazing that within two months Veblen learned AMS. The need for an abstracting that Keppel was on board to back a US$66,000 journal was filled in 1931 by Berlin publisher Ju­ grant. Meanwhile Richardson had secured a posi­ lius Springer with the creation of Zentralblatt fii.r tion for Neugebauer at Brown University. Mathematik und ihre Grenzgebiete under the able On a parallel track, Veblen and Richardson lob­ supervision of Otto Neugebauer [17], [16]. bied the AMS leadership to formally sanction the From his 1933 dealings on refugee mathema­ journal abstracts project. In late December 1938 ticians, Veblen was quick to anticipate future the AMS Council approved the idea in principle, conflicts between and Zentralblatt. Neuge­ establishing a special committee. The committee's bauer left Gottingen in 1934 when his political charge was to investigate, and, "in case it is deemed views, rather than his religious ancestry, made his wise," to proceed with the journal on a five-year situation impossible. For the next several years trial. To avoid the appearance of a conflict of in­ Neugebauer edited Zentralblatt from his new base terest with Zentralblatt, Veblen was deliberately in Copenhagen. Late in 1938 events unfolded rap­ excluded from membership on the committee. idly. Springer removed the Italian Jewish mathema­ Nevertheless, he served as an influential consul­ tician, Tullio Levi-Civita, from the editorial board. tant, constantly pushing to bring the project to Neugebauer questioned this action, only to receive fruition. a new decree that emigres be excluded from re­ Some mathematicians were reluctant to impinge viewing the work of German authors. Neugebauer on Springer's domain. The dismissal of Levi-Civita then resigned as editor of Zentralblatt. was attributed to outside political pressure. Sub­ Veblen learned of these developments on No­ stantial good will remained toward Springer for vember 7, 1938, and sprang into action. He imme­ his contributions in scientific publishing. Even so, diately coordinated with Courant, Tamarkin, and world events cast doubt over how much control he Hardy to resign their own associate editorships en would have over future policy. The deliberations masse. The act of protest left unresolved the sud­ took place as Hitler was conquering Czechoslo­ den need for an objective international journal of vakia. mathematical abstracts. Two years earlier Veblen Out of deference to Springer, the AMS commit­ had had the foresight to ask Neugebauer to outline tee waited until a personal meeting was possible. a budget for launching such an undertaking in the After several delays, Springer's representative

616 NOTICES OF THE AMS VOLUME 54, NUMBER 5 arrived in the United States in May. The discussions Were it not for Veblen, however, extramural were unsatisfactory, and the final decision was funding for basic research in mathematics would taken to proceed with . Over not have begun when it did. His diplomacy per­ the summer the Rockefeller Foundation and Carn­ suaded the NRC and private foundations of their egie Corporation awarded grants of US$60,000 interest in supporting mathematical research. and US$12,000 respectively. Veblen continued to It is unlikely that anyone else would have filled negotiate with the American Philosophical Society Veblen's role between the world wars. Today, for additional funds. the debate continues over mathematics' share Without Veblen's vision, influence, and initia­ of the federal research budget. Oswald Veblen tive, Mathematical Reviews would not have come gave mathematicians the standing to participate, into existence, let alone in such a short time. singlehandedly championing the cause for the first Birkhoff, who opposed the concept, was the only twenty years. other mathematician in a position to obtain the funding. While Richardson and Neugebauer were Sources important players, Veblen was the impetus behind The sources for this article included records drawn the project. from the following archival collections: Veblen's forceful actions could arouse strong American Mathematical Society Records, John feelings. Despite their differences on Mathematical Hay Library, Brown University Reviews and European immigration, Birkhoff and R. L. Moore Papers, 1898-1974, Archives of Veblen remained close friends. Others reacted dif­ American Mathematics, University Archives, Uni­ ferently. Among those with animus toward Veblen versity of Texas at Austin were Solomon Lefschetz, ]. Robert Oppenheimer, G. D. Birkhoff Papers, Ar­ and Abraham Flexner. Flexner's feelings were chive understandable. As Veblen was working on Math­ Archives of the Institute for Advanced Study ematical Reviews he joined other faculty and trust­ Oswald Veblen Papers, Manuscript Division, ees at the lAS to the seventy-three-year-old Library of Congress director into retirement. Although the coup had widespread support, Flexner blamed much of his References trouble on Veblen. Hostile attitudes of Lefschetz [1] WILUAM AsrRAY, The Emergence of Princeton as a World and Oppenheimer toward Veblen were reported by Center for Mathematical Research, 1896-1939, A Cen­ lAS faculty of the time. tury of Mathematics in America, Part II, (Peter Duren, ed.), Amer. Math. Soc, 1988. Overall, Veblen was widely liked and respected [2] EDGAR LoRCH, Mathematics at Columbia During Ado­ in the scholarly community. Some indication of a lescence, A Century of Mathematics in America, Part mathematician's standing can be obtained from the III, (Peter Duren, ed.), Amer. Math. Soc., 1989. programs of the International Congress of Math­ [3] RAYMOND WILDER, Reminiscences of Mathematics at ematicians. In 1920 and 1924 Leonard Dickson Michigan, A Century of Mathematics in America, Part became the first American to deliver two plenary III, (Peter Duren, ed.), Amer. Math. Soc., 1989. lectures. After Birkhoff was selected to speak in [4] RAYMOND ARCHIBALD, A Semicentennial History of the 1928, the Bologna organizers asked Gilbert Bliss of American Mathematical Society 1888-1938, Amer. the University of Chicago to choose another Ameri­ Math. Soc., 1938. can. Bliss picked Veblen. Eight years later Birkhoff [5] , Oswald Veblen, A Century ofMath· and Veblen repeated the roles in Oslo. Birkhoff ematics in America, Part II, (Peter Duren, ed.), Amer. Math. Soc., 1988. died before the next Congress which was held in [6] and DAVID ROWE, The Emergence of the 1950. This gathering, in Cambridge, was the first in American Mathematical Community 1876-1900:].]. the United States. Veblen received the distinction Sylvester, , and E. H. Moore, Amer. Math. of serving as president of the meeting. Ten years Soc., 1994. later he died at his summer home in Maine. [7] ALBERT TUCKER interview On "The Mathemat­ Given the ahistorical nature of mathematicians, ics Community at Princeton Before 1930" in The it is not surprising that appreciation for Veblen Princeton Mathematics Community in the 1930s: has evaporated since his passing. Courses do not An Oral History Project, on the Web at http: I I cover topics such as the first proof of the Jordan infosharel . princeton.edullibrariesl Curve Theorem, how algebraic topology entered firestonelfinding_aidslmathorallmath.html. the mainstream, or the evolution of the notion [8] DAVID ALAN GRIER, Dr. Veblen takes a uniform: Math­ ematics in the first world War, American Mathematical of manifold. More modern developments are pre­ Monthly 108 (2001), 922-931. sented, and the pioneers' names are lost, unless [9] , The Computer from Pascal to von they happen to stick to the theorem or definition. Neumann, Princeton University Press, 1972. In Veblen's case, an argument could be made that [10] NATHAN REINGOLD, The Case of the Disappearing Labo­ his discoveries were so ripe that others would have ratory, Science, American Style, Rutgers University come along and obtained similar results. Press, 1991.

MAY 2007 NOTICES OF THE AMS 617 [ll) ALEXI AssMUs, The creation of postdoctoral fellow­ ships and the siting of American scientific research, "Each and every paragraph Minerva 31 (1993), 151-183. [12) LOREN BUTLER FEFFER, Oswald Veblen and the capi­ is worth reading ... " talization of American mathematics, ISIS 89 (1998), 474-497. [13) , Some Leaders in American Math­ ematics: 1891-1941, The Bicentennial Tribute to American Mathematics 1776-1976, Math. Assoc. Pursuit of. Amer., 1977. [14) STEVE BATTERSON, Pursuit of Genius: Flexner, Einstein, Genius and the Early Faculty at the Institute for Advanced Study, A K Peters, 2006. Flexner, Einstein, and the Early Faculty [15) ALO NZO CHURCH interview in The Princeton at the Institute for Advanced Study Mathematics Community in the 1930s: An Oral History Project, on the Web at http: I I Steve Batterson infosharel.princeton.edullibrariesl firestonelfinding_aidslmathorallmath.html. (16) NATHAN REINGOLD, Refugee Mathematicians, 1933- 1941, Science, American Style, Rutgers University Press, 1991 pp. 249-283. [17) EvERETT PITCHER, A History of the Second Fifty Years, American Mathematical Society, 1939-1988, Vol. I, Amer. Math. Soc., 1988, 69-89.

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618 NOTICES OF THE AMS VOLUME 54, NUMBER 5 H A T I S an Infinite Swindle? Valentin Poenaru

The infinite comes with magic and power. Here is a nice geometric example of what an infinite process can do. Start with a closed surface S and a simple closed curve that is homotopically nontrivial, C c S. Let f be a hyperbolic diffeomor­ phism of S. Bill Thurston has shown that, when the infinite sequence of iterates of f is applied to C, i.e., when one goes to fC, f 2 C, [3C, ... then, as n gets larger and larger, the strands of rnc gather more and more into parallel sheaves, until in the limit there appears a foliation :f with a transverse Lebesgue type measure. In a situation like the one just described, the final pattern reveals itself more and more as n Figure 1. grows, like in a usual convergent infinite computa­ tion. An infinite swindle is also an infinite iterative process of sorts, but one in which the grand final pattern is never, even partially, visible at any finite stage; it only reveals itself at the bitter end, out of the blue, as if by magic. We will start with a simple example. Consider two solid tori T1 and Tz, with T1 embedded in the interior of T2 , like in Figure 1; T1 is the blue torus, and T2 is the gray torus. Where T1 links with itself, there are two choices, and we have chosen one. Next, embed T2 into a third solid torus T3 , just like T1 c Tz (see Figure 2), and iterate indefinitely

Figure 2. 00 (1) The U Tn is an open 3-manifold called n ~ l the Whitehead manifold Wh 3. Wh3 is contractible, Valentin Poenaru is professor emeritus of mathematics at although if you stop the sequence (1) at any finite the Universite de Paris-Sud, Orsay. His email address is stage, you get a non-simply connected object. And valpoe@hotmail :com. it is not homeomorphic to R 3, since it fails to be

MAY 2007 NOTICES OF THE AMS 619 simply connected at infinity. This venerable object a lot of technicalities have been pushed under was discovered by Henry(]. H. C.) Whitehead, more the rug. We now have a composition law among than seventy years ago, as a counterexample to bounded n- that (up to his own wrong proof for the Poincare Conjecture. at least) is commutative and associative. More­ Now consider two standard smooth of over, the standard DIFF n-ball Bn acts as a unit, dimensions n - 1 and n, with a smooth embed­ i.e., Mn#Bn = Mn. ding between them, sn- 1 _!__. sn. The image i sn- 1 splits sn into two smooth compact submanifolds, called Schoenflies ·balls. The question of what a Schoenflies ball really is (or, equivalently, what the pair (Sn, i sn- 1 ) looks like) is the celebrated . (Notice that I have studiously and deliberately put myself in a DIFF, i.e., smooth, setting and, in this paper we will go TOP, i.e., purely topological rather than DIFF, only when forced to do so.) One should be aware that a lot of funny things might happen as soon as the dimension n is three or more. For instance, in the mid-1920s, for n = 3, ]. W. Alexander showed, via a rather tedious ar­ gument, that Schoenflies 3-balls are standard. At the same time, he also came up with his famous "Alexander's horned sphere", a reminder that in dimensions strictly higher than two, in the ab­ sence of some additional local conditions beyond mere continuity (and such conditions are largely fulfilled in the smooth case), things can get very Figure 3. If sn - 1 is embedded into in a wild. Do not always trust your intuition. sn non-standard way, are the components X and In a related context, consider the following Y of its copies of Bn? "easy" lemma. In some arbitrary dimension, call it m, take a smooth embedding Bm __!_. sm; the pair (Sm, jBm) is then standard. This may look deceptively quite similar to what we are after, i.e., the pair (Sn, isn- 1 ). But our "easy" lemma, stated above, does not require any breathtaking new idea, only good solid technology. The next advance came only in the late 1950s, with the revolutionary work of . At the time, what people expected was a painful climb up the ladder of increasing dimensions, from 3 to 4, from 4 to 5, and so on. Barry's work, handling all the dimensions at once, came like a thunder­ bolt and was also a psychological revolution that, together with other developments, paved the way for what came next in high-dimensional topology. We need to introduce now the concept of con­ nected sum of n-manifolds with boundary. Let Mf, M~ be compact manifolds that are connected and have connected boundaries. An (n - 1)-ball, together with embeddings (2) oMf ::> Bn- 1 c oM~. is also given. There are some orientation issues involved here, which we will skip in a cavalier man­ Figure 4. To obtain X#Y, pull each of X and Y ner. One can make sense ofthe union of MP andM~ into themselves, then glue together along a 1 along Bn- as a new connected n-manifold denoted copy of Bn - 1• Since Bn - 1 x B1 is homeomorphic MT # M~. Up to diffeomorphism, this "connected to F = Bn, the "easy" lemma implies that X#Y sum" is independent of the precise set-up (2). is homeomorphic to Bn. All this should be intuitively clear, but of course

620 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Let us take one step further, to an infinite a very unambiguous recipe for its construction, at sequence Mf, M~, Mr , ... and replace (2) by the cas­ least as a topological space, by gluing together in cade oMr :) Er-1 c oM~ :) Eg-1 c oMr :) E3 -1 c a proper manner infinitely many compact pieces . . . . , with Eg -1 c aMg - Er-1, and so on. Then It also has an unambiguous smooth structure, as it turns out. The relation between (9) and (7) is not (3) Mr#M~#M!j#M;# ... unlike the one between the old Greek pun, in which is an unambigously well-defined non-compact Epimenides the Cretan says that "all Cretans are smooth n-manifold, with non-empty boundary liars", and Godel's incompleteness theorem. with a single end. Try this same game now for the Secondly, and more seriously, you may wonder special case when all the Mt's are copies of the what happens with the discrepancy DIFF versus standard n-ball En. It should not be very hard to TOP, in (7) versus (8). Here is what goes on (I will prove that what (3) becomes, in this special case, not come back to the simpler cases n .:s; 3, where is everything is as it should be). (4) En#En#En # ... =En - { p E oEn}. In all dimensions n ;::: 6 Smale's h- theorem allows us to replace the TOP in (8) by 1 sn, So, let us go back to sn- -.!.... which splits DIFF. To do that for n = 5 we need, in addition to sn into two Schoenflies balls, call them xn, yn. If Smale's work, the work of Kervaire and Milnor on one applies our easy lemma to some arbitrarily surgery. But there, things stop; when one moves 1 1 chosen En - c sn- , one can see that, if one splits to n = 4, the issue is not yet settled. The smooth sn open along iBn - 1 ' then what one gets is xn # yn. 4-dimensional Schoenflies problem (i.e., the ques­ A second application of the same easy lemma, this tion whether t, 4 is smoothly standard) is an open time in dimension n, yields the diffeomorphism mystery, to this day. (5) xn#yn =En . This is a good time to take another look at our 3 Now comes the big step. Like in (3), we introduce seemingly innocent Whitehead manifold Wh . As the following, perfectly legitimate object everybody knows, there are no knots in dimension four, and, starting from this fact, it is quite easy (6) zn = xn #Yn #Xn#yn # ... to see that there is a diffeomorphism Using the formulae (4), (5), as well as the asso­ (10) Wh3 x (0, 1) = R 4 ciativity of the composition law #, we can express the zn above in two different ways, namely (or Wh 3 x R = R4 , if you like). By R4 we will mean 4 zn xn#(Yn#Xn)#(Yn#Xn)# ... here the standard, well-known, smooth R . While for n f= 4 it is known (via work of Stallings) Xn - {a boundary pointp E oXn}, that Rn has no DIFF structures other than the one and everybody knows, exactly for n = 4 there are also zn (Xn # yn) # (Xn # yn) # . .. exotic R 4 's (actually loads of them). Starting from the fact that in (1) every Tn is En - {a boundary point q E oEn} . included in the interior of the next Tn +1 , one can The upshot is that rewrite the Wh3 x (0, 1) from (10) as follows

(7) ~n - {p E o~n} = En - {q E oEn} 3 DIFF ' (11) wh x (o, 1) = cr x (o, 1ll u cL x (o, 1ll

0 where ~n means a general n-Schoenflies ball. This u(T 3 X (0, 1)) U .. . is what is called an infinite swindle. Via the stan­ dard one-point compactification, which replaces Now one can add a piece of boundary at the let us say the" ... " in (6) by" ... u {oo} ", one also infinity of the open manifold above and replace it gets, from (7), the by

0 0 0 (8) t,n = En. TOP (12) (T1x(O, 1]) u (T2x (O, 1))u(T3 x (O, 1)) u ... , But several questions may pop up at this point. where the first (0, 1) becomes (0, 1]. So, what kind Firstly, how come the argument above works, of an object is this (12)? Well, it is a smooth while the following deceptively similar "proof" non-compact manifold with non-empty boundary, that 1 = 0 is humbug?: call it M4 , which is such that

(9) 1- 1+ 1- 1+1 - . .. = (1-1)+(1-1) = ... = 0 (13) intM4 = R 4 , oM4 = S1 x D2. = 1 + (-1 + 1) + (-1 + 1) + ... = 1. It actually turns out that our M 4 is one of the Well, you may have noticed that in my little ex­ so-called "Casson handles" (CHs). These CHs com­ position above, I went to some length in stressing prise a whole class of 4-manifolds, discovered by that the zn in (6) is a mathematically well-defined in the mid-1970s. His motivation object. This is certainly not the case for the left­ was to circumvent the failure of the "Whitney hand side of (9). Equation (6) is a special case of process" in dimension four. In higher dimensions, (3), for which we have been very careful to specify where it works very well, the Whitney process was

MAY 2007 NOTICES OF THE AMS 621 AMERICAN MATHEMATICAL SOCIETY an essential ingredient for Smale's h-cobordism theorem. Casson's own construction of the CHs is an intricate, tricky, infinite business, much too June 2007 marks complex to be described here: it is like a very high-powered version of the construction of Wh3 . Casson constructed a whole Cantor set's worth of the end of the CHs. In a certain sense, the moduli space for the CHs is a Cantor set, our M 4 being a very precise AMS Bookstore point of it, something like a 0.1111 ... The various CHs all share the properties of our M 4 listed above, 10th Anniversary including of course (13). Why should we care? Well, the main step in 's proof of the Celebration 4-dimensional TOP Poincare Conjecture was to show that all CHs are topologically standard, in the sense that, for any CH one has

(14) CH = D2 x D2 • TOP So our M 4 form (12) actually turns out to be homeomorphic to D2 x D2 • See whether you manage to prove this little fact with bare hands, without playing Freedman's full symphony of infinite processes! It turns out that, if one could replace, in full generality, TOP by DIFFin (14), then there would be no exotic R 4 's. But, since exotic R 4 's do exist there 0 must also be CHs not diffeomorphic to D2 x D2 • Take advantage One can even explicitly name some. For more than one reason, the mysteriously of BIG discounts deep chasm between DIFF and TOP in dimension four, and exactly there, is an important issue. One and special offers of the attractions of the 4-dimensional smooth Schoenflies problem is that it touches on this before it all ends! issue. Further Reading [1) Travaux de Thurston sur les Surfaces, Seminaire Orsay, Asterisque, 66-67, second edition, 1991. [2) R. KIRBY, The topology of 4-manifolds, Springer Lecture Notes 1374 (1989). (3) M. H. FREEDMAN and F. QUINN, Topology of 4-manifolds, Princeton University Press (1990). Visit the AMS Bookstore www.ams.org/bookstore/ tenthanniversary for the grand finale sale.

622 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Book Review Project Origami: Activities for Exploring Mathematics Reviewed by Helena Verrill

T HOWAS Huu

experience difficulty, how to help them, and a dis­ Project Origami: Activities for Exploring cussion of pedagogy. There are plenty of diagrams Mathematics and handouts, which can also be downloaded Thomas Hull from the publisher's website. Most teachers will A K P,eters, Ltd., 2006 not have the luxury of being able to teach a whole $30.00, 272 pages, ISBN 1568812588 course devoted to the mathematics of origami, but an appendix lists which activities might fit Origami often starts with folding a boat or a hat. into which courses. However, it is inconvenient Traditionally one starts with a single square of that the book has no index. paper, which is folded to create a finished model, The book gives a broad overview of mathe­ without the use of scissors or glue. The end result matics applied to origami, presented at a level could equally well be a bird or a hexagon; in ei­ appropriate for undergraduates, with references ther case the process of folding involves a latent to more advanced works. The activities are roughly understanding of simple geometrical concepts. It ordered according to the following sequence of is not surprising that origami is quite popular topics: with teachers and students in elementary schools. (1) Constructibility, Is it possible to use origami in the higher level (2) Polyhedra, mathematics classroom? (3) Flat foldable crease patterns. An affirmative answer is given in Thomas Hull's Constructibility is covered by the first eight book Project Origami: Activities for Exploring activities. This concerns problems such as deter­ Mathematics. Based on Hull's extensive expe­ mining what points can be constructed as the rience of combining origami and mathematics intersection of two creases, and what shapes can teaching over the last fifteen years, it aims to be folded, when only a certain set of folding help the teacher bring origami into the math­ procedures are allowed. The starting point is the ematics classroom, at the high school, college, activity of folding an equilateral triangle from a and university level. Although there are quite square, followed by successively more and more a number of books covering the mathematics complex constructions. For example, activity 6 of origami, most are more elementary, or not shows a method of combining folding and pencil oriented for classroom use. Hull's book is very marks to plot points on a singular cubic curve. much intended for classroom use; it is a book of Most of these activities would fit nicely into a lesson plans, aimed at the teacher. Twenty-two course on Euclidean geometry in the plane, along classroom "activities" are presented. Each comes with straight-edge and compass constructions. with detailed descriptions, including expected Both origami constructions and straight-edge and ·time taken for folding, where to expect students to compass constructions allow students to learn about theorems and proof; different materials and Helena ,Verrill is professor of mathematics at Louisiana methods are used but can be applied to the same State University. Her email address is verrill@ problems, such as "produce an equilateral trian­ math. l su. edu. gle". Part of the attraction of activity 5, on the

MAY 2007 NOTICES OF THE AMS 623 simple origami construction of trisection of an might be better for a math club activity. Although angle, is that we know this cannot be done with activity 10 on business card modulars does give a straight-edge and compass. way of producing triangular-faced polyhedra, the Many of these activities could be developed results are not very esthetic or stable in general. much further and would be ideal starting points It's great to have polyhedra to hand when teach­ for undergraduate projects. A detailed discussion ing about symmetries, and the cover dodecahedron of Galois theory, the usual tool for dealing with model is very nice. Many students do not know geometric constructibility, is beyond the scope of what a dodecahedron is, and being introduced to the book, though some indications are given at one by folding may be an effective way of interest­ the end of activity 6. An important issue to be ing a student in these objects. One would not want addressed before attempting to prove what can to spend too much time on folding at the expense and cannot be constructed in origami, is to deter­ of explaining the mathematics of symmetries, so mine what exactly are the rules of folding. These it might be necessary to just start students on are much harder to pin down than the rules of the folding, giving them instructions for how to straight-edge and compass constructions. This is complete the model at home. Although there are discussed in the pedagogy sections of activities 5 alternative methods of producing a dodecahedron, and 6. Hull gives a more extensive discussion on Hull's edge unit has the advantage that it can also his Web pages [Hull), where he describes folding be used to create buckyballs and tori. The activities rules known as the "Huzita origami axioms". More covered using this model introduce several con­ recently, a construction called the "multifold" has cepts from graph theory, including Hamiltonian been investigated by Robert Lang and others, who cycles and Euler's theorem. Here origami is used find that including this fold allows even more as a means to illustrate ideas from other areas of possible constructions [Chow and Fan). mathematics, which it does nicely, but the origami Activity 8, with enough material for four class­ has no more relation to the mathematical concepts es, introduces "Haga's origamics". Kazuo Haga than knitting, for example. has developed a series of educational origami ac­ Flat foldability and crease patterns are the tivities, aimed at high school students in Japan, central theme of the remaining activities. If you and has written two books on the topic, currently fold an origami model, which is "flat" when com­ available only in Japanese. These activities also pleted, i.e., it all lies in one plane, as much as involve Euclidean geometry but have a different possible allowing for the paper thickness, and flavor from the first few activities in the book then open it out again to a flat square of paper, and are less comparable with straight-edge and you'll have a pattern of creases. The main question compass constructions. A typical problem asks of activities 15 to 22 is whether a given set of the student to investigate what kinds of polygons crease lines can be obtained in this way. Whereas can be formed when all four corners of a square constructibility questions are concerned with what of paper are folded to a single point on the paper. can be folded using a certain precise set of rules, The aim of Haga's origamics is to develop scien­ for these questions the method of folding is largely tific reasoning skills, and these problems could be irrelevant. Activity 16 leads the student through used in courses on an introduction to proof. some theorems about necessary conditions for a Activity 2, on an approximate method of di­ crease pattern to be flat-foldable, whereas activity viding strips into N equal pieces using a limiting 17 covers impossible crease patterns. An omission process, involves little Euclidean geometry and from this sequence of activities is the map folding would fit better into the beginning of a calculus problem of finding how many ways a map can course. be folded up, only folding along the given grid Polyhedra are covered in activities 9 to 14. pattern of crease lines. These are usually constructed by linking together Whereas the "constructibility" and "polyhedra" many identically folded units, as in the model activities could be used as peripheral activities in shown on the book's cover. This method of con­ courses on Euclidean geometry, , or struction, modular origami, is an excellent way to combinatorics, in the foldability activities, origami produce polyhedra. Many modular origami poly­ is central. These activities tend to use more ad­ hedra are sturdy, elegant, and often easier to make vanced mathematics than most of the rest of the than gluing together cardboard polygons. book, such as matrices and Gaussian curvature. However, in spite of the polyhedron on the front They also perhaps take the student closest to ac­ cover, polyhedra are not a central theme. The over­ tual origami research, which generally seems to be all emphasis is on the mathematics of folding, not carried out by computer scientists and on how to fold any specific model. Those looking rather than mathematicians. for instructions on how to fold all platonic solids Drawbacks of folding in a mathematics class. would be better off with one of the many books This is not an elementary origami book; some devoted to origami polyhedra. Of the models that familiarity with folding is assumed. There is no are given, some, such as the five intersecting tetra­ uniform convention for lines used to indicate hedra, look spectacular, but would be infeasible mountain or valley folds. For inexperienced fold­ to carry out to completion in a single class; they ers, this could cause problems. A teacher wanting

624 NOTICES OF THE AMS VOLUME 54, NUMBER 5 to use origami models in the mathematics class­ room would probably want to have at least one more-elementary book for getting some experi­ ence with folding. Hull gives suggestions in the introduction, and the book also has an extensive bibliography. Folding a successful model usually needs pre­ cise folds; making corrections can be difficult and leaves unwanted creases, weakening the structure. Students unfamiliar with origami probably won't be aware of these problems until too late and may find it frustrating and off-putting to end up with a crumpled piece of paper rather than the beautiful model they expected. In any case, folding complete models is often too time-consuming to fit into a class, especially since there will likely be some students requiring one-on-one help with folding. Whether or not these issues are a prob­ lem may depend on how much models are used in the classes; most activities are not essentially about building models, but about understanding the mathematics of flat-folding origami. However, the finished models are probably the motivation Put Your Math for many students' interest in folding, and stu­ dents who have not been exposed to origami may Intelligence to Work be unmotivated in a class on the mathematics of origami that does not cover making some models. When you join NSA, you join a highly talented Conclusions. There are various reasons for us­ group of Mathematicians who deduce structure ing this book in a class; one is the concept "origami where it is not apparent, find patterns in is fun, let's use it to make mathematics fun!" For some students, particularly mathematics majors, seemingly random sets, and create order out "fun" may be motivation enough to study math­ of chaos. They apply Number Theory, Group ematics; however, such students probably don't Theory, Finite Field Theory, Linear Algebra, need to be lured into mathematics by origami. , , For others, usefulness and applications are more Combinatorics, and more to a world of motivating. Origami does have real world applica­ challenges. They exchange ideas and work tions, from paper cup designs to airbag folding [Cipra], which would be important to mention in a with some of the finest minds and most course devoted solely to the mathematics of origa­ powerful computers in the country. And you mi. Probably this book will appeal most to those can too, when you put your math intelligence teaching mathematics to liberal arts students. Per­ to work at NSA. haps some students otherwise uninterested in mathematics can be drawn into active participa­ For more information and to apply tion in mathematics via the medium of origami. online, visit our Web site. Whatever the reason for including origami in a mathematics class, anyone wishing to do so will find many invaluable ideas in this book and will probably discover that there are more possibilities than first imagined.

References [Cipra] B. A. CIPRA, In the Fold: Origami Meets Math­ ematics, SIAM News, Volume 34, Number 8, October 2001. [Chow and Fan] T. CHOW and C. K. FAN, The power of multifolds: folding the algebraic closure of the rationals, www.NSA.gov/Careers in Proceedings of the Fourth International Conference on WHERE INTELLIGENCE GOES TO WORK Origami in Science, Mathematics, and Education (40SME)

(to appear). U.S. citizenship is required. NSA is an equal opportunity employer. All applicants for [Hull] T. HULL, http: I lwww. merrimack. edul -thull I employment are considered without regard to race, color, religion, sex, national origin, age, marital status, disability, sexual orientation, or status as a parent. OrigamiMath.html.

MAY 2007 NOTICES OF THE AMS 625 If Euclid Had Been Japanese Bill Casselman

Starting with the two points (0, 0) and (1, 0), apply­ One has to be careful. If P lies on .f! then the fold ing the standard operations with straight edge and line constructed is the line through Q perpendicu­ compass, one can obtain any point with coordinates lar to -1!. But if P doesn't lie on .f! then this operation in atower of quadratic extensions of Q. There is an is not always possible, and when it is in fact possible analogous result about origami constructions. it will usually not be unique. Why is this? 1n terms of 1n origami construction, one starts with the con­ algebra, we are looking for a line y = mx + b such figuration of the three lines x = 0, y = 0, x + y = 1 that and applies certain basic origami operations I'll de­ scribe in a moment.ln this case, the points that one YQ = mxQ + b YP- mxp- obtains are those with coordinates in a tower of qua­ (XP',O)=(Xp,yp)-2 ( +m b) [-m,1] dratic and cubic extensions. One half of the proof 1 2 follows closely the argument for the classical case. leading to a quadratic equation for m, which may The interesting part is the explicit construction of have two, one, or zero solutions. If P = (0, 1), for roots. example, then we get the equations The basic object in origami is a line, and one constructs it by folding along it. Mathematical­ m 2 + m(2xQ) + (2YQ- 1) = 0, b = YQ- mxQ ly, folding amounts to an orthogonal reflection from which we see that through the line. The simplest principles of origami construction are that (1) points are constructed m = -XQ ±~X~ + 1 - 2yQ. by intersecting two lines, and, conversely, (2) any This has two real roots as long as two points determine a fold line through them. But there are more interesting ways to "construct" YQ < (x~ + 1)/2, lines. Amore complicated but still practical origami or, equivalently, (xQ, YQ) lies outside the parabola operationisthat(3)given two points P and Q and y=(x2 +1)/2. a line ./!, one folds along a line through Q, taking P to a point P' on .f!.ln this way P, Q, and.f!giverise to a newline.

~ ~ ~ ~ ' ' ~ ~ xeP-~ Q .. P' Fold lines are tangents to the parabola.

Folding P to .f! along a line through Q. Geometrically, finding the line y = mx + b that we are looking for amounts to finding a line through Bill Casselman is professor of mathematics at the Uni­ Q tangent to this parabola, which is both the enve­ versity of British Columbia and graphics editor of the lope of all these lines and the parabola with focus P Notices. His email address is cass@math. ubc. ca. and directrix .f!.

626 NOTICES OF THE AMS VOLUME 54, NUMBER 5 which we may do without loss of generality, and plot the image of various points Q with respect to the various O"p•. The red horizontal lines are those at dis­ tance d(P, Q) from -1!. The image must lie between them, since reflections are isometries. ' '

' ' ' ' ' Q

The parabola is the envelope of the fold lines.

Since every point exterior to this parabola lies on two tangents, we can make more precise the con­ struction stated somewhat imprecisely earlier: Let ' P be a point and -1! a line not containing P. If Q is ' a poini not inside the parabola with focus P and ' directrix-/!, suppose m to be a line through Q tan­ gent to that parabola. Folding along m, taking P to a point on-/!, is an allowable origami operation that, in effect, constructs m. It now becomes plausible that origami can cal­ culate square roots, establishing that it is at least as potent as straight edge and compass. Another allowable origami construction is more characteristic of origami, and more capable than ' any available by means of straight edge and com­ pass. (4) Suppose P and Q to be distinct points, The images of various fold maps. -1! and m two lines with P not on -1!, Q not on m. The new operation folds P onto -/!, Q onto m, The formula for the bisector of the segment be­ in effect constructing the fold line, when this is tween(O, 1)and(t,O)is possible. As before, we must answer some questions: y = tx + ( 1 - t 2 ) I 2 When is this operation possible? To what extent and that for O"(r,o) takes (x, y) to is it unique?To answer, we first look at all possible operations folding Ponto -1!. This is simple, because (x,y) - 2 (Y- tx -t ; 1~2:(1- t2)) [ -t, 1]. if P' is a point on-/!, then the axis of reflection taking P to P' must be the perpendicular bisector of P P'. As y - ± oo this tends asymptotically to the line Let that reflection be O"p• . If we are given a further at height y - 1. The image is therefore intersect­ point Q and line m, we are reduced to asking: Does ed by iJ,ny line m with slope f= 0. As for lines with one of the reflections O"p• take Q to m ? If so, slope = 0, the image spans the entire closed range how many such reflections are there? Another, ly l ::; d(P, Q) except (as the first picture suggests) equivalent, way to pose this question: let C be the when Q lies on they-axis. In all cases where minter­ image of Q with respect to the transformations O"p• sects -1! there exists at least one point on m to which as P' varies along -1! . Does C intersect m ? In how Q is mapped by some O"p• . On the other hand, if m many points? is parallel to -1! then we must assume that d(-1!, m) ::; The ·following figures suggest what happens. In d(P, Q) in order for the construction to be possi­ them, we fix P = (0, 1) and -1! equal to the x-axis, ble, and if d(P, Q) = d(-1!, m) thenmmustbe on the

MAY 2007 NOTICES OF THE AMS 627 side of P opposite to Q. In all cases, there are on­ ly a finite number of possibilities. It turns out that finding these explicitly amounts to solving a cubic equation. So, given our starting configuration and the rules so far given for constructing points and lines, what more can we construct? Q •Bymethod(l),fromthethreelinesx = O,y = 0, x + y = 1 we can construct the three points (0, 0), (1, 0), (0, 1). This configuration of points and lines guarantees that every line constructed has at least two (constructed) points on it, and also that there exists at least one point not on it. • We already know how to construct perpendicu­ lars. Using this construction twice, we see that given a point P not on -8 we can construct the line parallel p to-8throughP. 2. Construct two uniformly spaced horizontal • Givenaline-8andapointP, wecanconstructthe lines. reflection of Pin -8 -i.e., find an allowable fold along a line intersecting the perpendicular to -8 through P. • With these procedures in hand, we can now do anything straight edge and compass can do. • More interesting, we can trisect angles. Since cos 38 = 4cos3 e- 3 cos e trisecting an angle is equivalent to solving a cubic equation. Activity 5 of Hull's book (see review of Project Origami on previous pages) explains how to construct Z/2 (thus doubling the cube), and Ac­ tivity 6 how to construct a root of an arbitrary cubic equationx3 +ax+b = 0.

References [1] DAVID Cox, Galois Theory, John Wiley and Sons, 2004. Chapter 10 is concerned with geometric 3. Fold P and Q to -8 and m. constructions, and §10.3 is concerned with origami.

4. The line from the corner to its reflection Trisection: 1. Start with an angle e. trisects the angle.

628 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Book Review Letters to a Young Mathematician letters to a young L_____mathemuici:~n ______; Reviewed by Emma Carberry

Letters to a Young Mathematician the life of the book, and the issues of relevance to Ian Stewart her change significantly. Stewart handles this by Perseus Books Group, 2006 avoiding in-depth discussions of practical matters, $22.95, 224 pages, ISBN 0465082319 instead keeping his explicit advice witty and brief. The result is an enjoyable collection of essays "So," my grandmother would say, fixing me with whose chief mentoring value lies in the insight they the expectant look of the schoolteacher she once provide into the life of a mathematician. Some of was, "What is it that you do, exactly?". Often this the issues discussed in the latter part of the book, enquiry is interpreted on a purely practical level: such as teaching and the tenure process, do not what are the problems that mathematicians solve, lend themselves well to such treatment, and and why are these important? The outcomes of I feel that the book would have been stronger had our research are what justify its funding. But what it ended earlier in Meg's life. However, it contains my grandmother really wanted to understand is some beautiful pieces of mathematical exposition the more human question of what it is like to do and will be a valuable resource for those wishing mathematics. We may feel that we have some idea to know more about the world of mathematics and of what is involved in building a house or perform­ of mathematicians. ing hand surgery, despite a lack of experience Stewart characterises Letters as his attempt to with these tasks. However few nonmathematicians update parts of Hardy's A Mathematician's Apol­ would assert an understanding of the human ex­ ogy. Hardy's view on the public's perception of perience of mathematical research. mathematics is rosy indeed. "There are now few For students considering embarking upon a studies more generally recognised, for good rea­ mathematical career, the questions of whether sons or bad, as profitable and praiseworthy." "The they find mathematics to be a worthwhile en­ mass of mathematical truth is obvious and impos­ deavor and what being a mathematician is like are ing; its practical applications .. .obtrude themselves of fundamental importance. The main strength on the dullest imagination. The public does not of Ian Stewart's book Letters To a Young Math­ need to be convinced that there is something in ematician is the way in which it addresses these mathematics." Sixty-six years after Hardy penned questions in an entertaining and accessible man­ those words, their veracity is no longer so clear. ner. It is the first scientific entry in Basic Books' There is by no means universal public awareness Art of Mentoring series, whose volumes are each of the role that mathematics plays in daily life, structured in homage to the collection of Rainer and Stewart begins by undertaking the task that Maria Rilke's letters to the young poet Franz Xaver Hardy once deemed unnecessary. Stewart has Kappus. Stewart's book takes the form of letters the expositor's flair for invoking visual images to to his fictitious niece Meg as she progresses from make his point, and I enjoyed his rueful wish for high school through to an assistant professor of red stickers emblazoned with the words "Math mathematics. It provides an intriguing glimpse into inside" to appear on everything that uses math­ the intellectual life of her mathematical uncle: a ematics, to increase awareness of the many roles personal window into the process of jousting with that the subject plays. In his eyes, mathematics is mathematical problems and how being a math­ indeed ubiquitous; soon everything from airplane ematician has affected the way in which he views tickets to the vegetables in the local grocery store the world, Meg ages around fifteen years through is wearing a red sticker. The spacing of birds perching on phone lines prompts a discussion of Emma Carberry is professor of mathematics at the Univer­ crystal lattices, a passing dog becomes the subject . sity of Sydney, Australia. Her email address is car berry@ of gait analysis with applications to robotics and maths.usyd.edu.au. orthopaedic rehabilitation, and he tells how his

MAY 2007 NOTICES OF THE AMS 629 understanding of the rainbow enhances its physi­ beautiful account of this process with a story of cal beauty with an intellectual one. his own that conveys a strong sense of the inner It is natural for high school students to assume world into which Meg is entering. that the type of thinking involved in higher math­ Towards the end of the book Stewart touches ematics is similar in nature, if not in scope, to that on subjects of relevance to young postdocs and they have encountered in mathematics classes. It faculty: teaching, giving talks at conferences, the is also far from the truth, and to describe to Meg pleasures and perils of collaboration, the mores of the type of thinking in which mathematicians the mathematics community, and tenure. These are engage, Stewart first dispels this notion. "Calling matters that will not be of immediate relevance to [these classes] 'mathematics' debases the most readers of this volume, and so the advice is of mathematical thought; it's a bit like using the perfunctory; these letters serve as an indication of term 'composing' to describe routine exercises the kind of things a young mathematician needs in playing musical scales." An indication of what to think about. I feel that the inclusion of these the currency of mathematical thought might be chapters detracts somewhat from the impact of the comes later, in letters written whilst Meg is an book. They offer less for the general reader than undergraduate, on the nature and necessity of do the earlier parts, and for those currently deal­ proof. Unsurprisingly, he does an excellent job of discussing both elementary puzzles and famous ing with these issues, the advice is too simplistic. theorems, distilling the ideas down to their essence Sometimes this is because he focuses on fairly in order to provide an accessible account. A word shallow aspects of the matters under discussion; puzzle (the SHIP-DOCK theorem) is used here to for example his advice on teaching includes a long demonstrate the difference between a proof and description of the things that can go wrong with less rigorous reasoning; a chessboard puzzle the equipment in a lecture theatre, but very little introduces the idea (unfortunately not always about how to teach in a way that maximises stu­ part of the high-school curriculum) that not all dent learning. At other times he directly addresses mathematics problems can be solved; and there more complex issues, but his treatment is too shal­ is a discussion of the impossibility of trisecting low. The most helpful advice is often that which the angle that is eminently readable, if somewhat enables the recipient to avoid or navigate difficul­ lacking in detail. Gauss's logarithmic integral ap­ ties, and Stewart several times glosses over situ­ proximation for the number of primes less than a ations that offer exactly this opportunity for the specified number is invoked to demonstrate that passing on of wisdom. For example, he gives a rosy proofs really are necessary. Gauss's approximation description of the current landscape for women exceeds the correct value for all numbers that have in mathematics, and assures Meg that "The idea ever been tested, yet as Littlewood has shown, the that math is not a suitable subject for women is approximation and the correct value swap places stone-cold dead". Meg could easily conclude from infinitely often. The aesthetics of proof is also his discussion that gender issues in mathematics discussed; he characterises a proof as a story, and are a thing of the past. Unfortunately this is not invokes Erdos's notion of God's Book of exquisite yet true. Good advice is certainly encouraging, but proofs. Indeed, there are a number of forays into not because it pretends that complications may not the philosophy of mathematics, as Stewart ponders exist, but rather because it helps one to see how whether the mathematics of aliens would agree to deal with them. with ours (he is not convinced) and how exactly to Letters to a Young Mathematician succeeds well define mathematics and mathematicians. As with in opening a door into the world of mathematics the rest of the book, the over arching goal of keep­ and enticing the reader inside. I would recom­ ing the material reasonably undemanding does mend it to those curious about mathematics and make the level of discussion somewhat superficial, but the book is filled with references for those life as a mathematician, particularly high school whose appetite has been sufficiently whetted that students and undergraduates considering further they are hungry for more. mathematical study. Readers of the Notices may Central to the question of what it is like to be well find it enjoyable reading, or wish to pass it on a mathematician is the process of tussling with a to their students. Reading it is rather as I imagine problem. Stewart provides readers with a glimpse a conversation with Uncle !an-the-mathematician into the inner sanctum of mathematical creativ­ could be; one does not so much feel the benefit of ity; those cherished moments when the confusion a ream of practical advice, but rather of exposure clears and all is simple and elegant. These mo­ to the inner realm of mathematics, and the enlight­ ments do not appear by magic of course, and my enment that that provides. It goes a good way to favourite part of the book is Stewart's elucidation answering the question of what it is that we do; of this process. He invokes Poincare's three stages: I suspect that my grandmother would have been "preparation", "incubation followed by illumina~ satisfied with Stewart's response to her question. tion", and "verification", and follows up Poincare's

630 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Furstenberg and Smale Receive 2006-2007WolfPrize

On January 15, 2007, the Wolf Foundation an­ nounced that the 2006-2007 Wolf Prize in Math­ ematics will be awarded to HARRY FURSTENBERG and STEPHEN ]. SMALE. Furstenberg, emeritus professor of mathematics at the Hebrew University of Jerusa­ lem, is honored "for his profound contributions to , probability, topological dynamics, analysis on symmetric spaces and homogenous flows." Smale, emeritus professor at the Univer­ sity of California, Berkeley, and a professor at the Toyota Technological Institute at the University of Chicago, is honored "for groundbreaking con­ tributions that have played a fundamental role in shaping differential topology, dynamical systems, , and other subjects in mathematics." The US$100,000 prize will be pre­ Harry Furstenberg Stephen J. Smale sented by the president of the State of Israel in a special ceremony at the Knesset (parliament) in flow on surfaces of constant negative curvature is Jerusalem on May 13, 2007. uniquely ergodic has become a major part of the dynamical theory of Lie group actions. In his study Harry Furstenberg of processes on homogenous spaces, he Harry Furstenberg is one of the great masters of introduced stationary methods whose study led probability theory, ergodic theory, and topological him to define what is now called the Furstenberg dynamics. Among his contributions: the applica­ Boundary of a group. His analysis of the asymp­ tion of ergodic-theoretic ideas to number theory totic behavior of random walks on groups has had and combinatorics and the application of proba­ a lasting influence on subsequent work in this area, bilistic ideas to the theory of Lie groups and their including the study of lattices in Lie groups and discrete subgroups. cocycles of group actions. In probability theory he was a pioneer in study­ In ergodic theory Furstenberg developed the ing products of random matrices and showing how fundamental concept of dynamical embedding. their limiting behavior was intimately tied to deep This led him to spectacular applications in combi­ structure theorems in Lie groups. This result has natorics, including a new proof of the Szemeredi had a major influence on all subsequent work in Theorem on arithmetic progressions and far­ this area, which has emerged as a major branch not reaching generalizations thereof. only in probability but also in Born in Germany in 1935, Furstenberg received and other fields. his Ph.D. from Princeton University in 1958 under In topological dynamics Furstenberg's proof the direction of . Since 1965 of the structure theorem for minimal distal flows Furstenberg has been a professor of mathematics introduced radically new techniques and revolu­ at the Hebrew University. He is a recipient of the tionized the field. His theorem that the horocycle Israel Prize and is a member of the Israel Academy

MAY 2007 NOTICES OF THE AMS 631 of Sciences and Humanities and of the U.S. National joined the faculty at the University of California, Academy of Sciences. Berkeley, in 1960 and retired in 1994. He was a Distinguished Professor at the City University of Stephen J. Smale Hong Kong from 1995 to 2001, when he joined the Stephen ]. Smale contributed greatly, in the late Toyota Technological Institute. Smale received the 19 5Os and early 1960s, to the development of Fields Medal in 1966 and the National Medal of differential topology, a field then in its infancy. Science in 1996. His other honors include the AMS His results on immersions of spheres in Euclidean Veblen Prize (1965) and election to the National spaces still intrigue mathematicians, as witnessed Academy of Sciences. by recent films and pictures on his so-called "ever­ -Based on Wolf Foundation news releases sion" of the sphere. His proof of the Poincare Conjecture for dimensions bigger or equal to 5 is one of the great mathematical achievements of the twentieth century. His h-cobordism theorem has become probably the most basic tool in dif­ ferential geometry. During the 1960s Smale reshaped the view of the world of dynamical systems. His theory of hyperbolic systems remains one of the main developments on the subject after Poincare, and the mathematical foundations of the so-called "chaos-theory" are his work as well. In the early 1960s Smale's work dramatically changed the study of the topology and analysis of infinite­ dimensional manifolds. This change was achieved through his infinite-dimensional version of Morse's critical point theory (known today as "Palais-Smale Theory") and his infinite-dimensional version of Sard's theorem. In the 1970s Smale's attention turned to me­ chanics and economics, to which he applied his ideas on topology and dynamics. For instance, his notion of "amended potential" in mechanics plays a key role in current developments in stability and bifurcation of relative equilibria. In economics Smale applied an abstract theory of optimiza­ tion for several functions, which he developed, to provide conditions for the existence of Pareto optima and to characterize this set of optima as a submanifold of diffeomorphic states to the set of Pareto equilibria. He also proved the existence of general equilibria under very weak assumptions and contributed to the development of algorithms for the computation of such equilibria. It is this last activity that led Smale in the early 1980s to the longest segment of his career, his work on the and computa­ tional mathematics. Against mainstream research on scientific computation, which focused on im­ mediate solutions to concrete problems, Smale developed a theory of continuous computation and (akin to that developed by computer scientists for discrete computations) and designed and analyzed algorithms for a number of specific problems. Some of these analyses constitute mod­ els for the use of deep mathematics in ·the study of numerical algorithms. Born in 1930 in the United States, Smale re­ ceived his Ph.D. from the in 1957 under the direction of . Smale

632 NOTICES OF THE AMS VOLUME 54, NUMBER 5 2007 AwardforanExemplary Program or Achievement in a Mathematics Department

The Award for an Exemplary Program or Achieve­ of students. The department has also invested ment in a Mathematics Department was established considerable efforts to reach out to underrepre­ by the AMS Council in 2004 and was given for the sented minorities, so that 15%-22% of its recent first time in 2006. The purpose is to recognize a graduates are minority students. Undergraduate department that has distinguished itself by under­ REU [Research Experiences for Undergraduates] taking an unusual or particularly effective program participation has been an important component of value to the mathematics community, internally of its program, and this has been broadened to or in relation to the rest of society. Departments of include the new Applied Mathematics Laboratory mathematical sciences in North America that offer where students participate in actual physical ex­ at least a bachelor's degree in the mathematical periments involving robotics and fluid flow. sciences are eligible. The award carries a cash prize The graduate mathematics program has also of US$1,200 and is given annually. seen considerable growth, going from about 112 The award is presented by the AMS Council students in 2000 to over 200 students today. The acting on the recommendation of a selection com­ department has restructured its graduate program mittee. For the 2007 award the members of the into one where the students learn in a research selection committee were: Sheldon Axler, Joel V. group environment early in their studies. Ph.D. Brawley, James H. Curry, Karl Knight (chair), and students take special seminar courses after their Donal B. O'Shea. first year to streamline them into research proj­ The previous recipient of the award is Harvey ects. This seminar series, which has been expanded Mudd College (2006). to include participation in long programs at IP AM, The recipient of the 2007 Award for an Exem­ has students reading current literature and pre­ plary Program or Achievement in a Mathematics senting their work. The department also has a new Department is the MATHEMATICS DEPARTMENT AT THE graduate internship program which allows Ph.D. UNIVERSITY OF CALIFORNIA, . students to work for one summer with a faculty member from another discipline or with someone Citation in industry. The department's postdoctoral pro­ The American Mathematical Society (AMS) pres­ gram is one of the largest in the country, with over ents its second annual Award for an Exemplary thirty postdocs last academic year. Program or Achievement in a Mathematics Depart­ UCLA is home to the Institute for Pure and Ap­ ment to the University of California, Los Angeles plied Mathematics, which was conceived of and (UCLA). The Mathematics Department at UCLA was founded by faculty from the department and is awarded a VIGRE [Vertical Integration of Research currently run by Mark Green, with help by Stanley and Education, a program of the National Science Osher, both long-time department members. The Foundation] grant in 2000 which was renewed department has been influential in creating and in 2005. The department has created a compre­ developing programs as well as training REU stu­ hensive vision for its undergraduate, graduate, dents through the Research in Industrial Projects and postdoc training programs which involves for Students (RIPS) Program. Interactions with the important interactions with the Institute for Pure IP AM have also led to several important initiatives, and Applied Mathematics (IP AM) at UCLA. Through such as the NIH [National Institutes of Health] these unusually large training programs, UCLA has Center for Computational Biology, participation in become one of the largest pipelines to mathemati­ the California NanoSystems Institute, and a leader­ cal careers in the United States. ship role in the new Institute for Digital Research The undergraduate mathematics program at and Education, all of which directly involve many UCLA has seen tremendous growth in the past students and postdocs in the department. ten years, increasing mathematics degrees by 81% The mathematics community is fortunate to from 1996 to 2005. Part of this increase is due to have UCLA present such an outstanding example the department's pioneering broad-based major of an exemplary program in a mathematics de­ which has enough options to draw a diverse group partment.

MAY 2007 NOTICES OF THE AMS 633 Mathematics Programs That Make a Difference

In 2005 the AMS Committee on the Profession The AMS commends the program co-directors (CoProf) began a project to bring recognition to Sylvia Bozeman and Rhonda Hughes for their suc­ outstanding programs that successfully address cess in improving the diversity of the profession the issue of underrepresented groups in math­ of mathematics in the United States. ematics. The project, called Mathematics Programs Description of Program That Make a Difference, each year singles out two Begun in 1998, the EDGE Program is founded on outstanding programs that: (1) aim to bring more the belief that students who have excelled in early individuals from underrepresented minority back­ mathematics courses but may have had limited grounds into some portion of the pipeline, begin­ exposure to advanced mathematics can still earn ning at the undergraduate level and leading to an . Moreover, exposure to graduate-level advanced degree in mathematics, or retain them in mathematics and the culture of graduate school, the pipeline; (2) have achieved documentable suc­ along with a rich support network and positive cess in doing so; and (3) are replicable models. feedback, will significantly enhance a student's Two Programs That Make a Difference were ability to obtain a Ph.D. This philosophy is contrary designated in 2006: the graduate program at the to the popular view that the well-prepared, fast­ University of Iowa and the Summer Institute in thinking graduate student with high GRE scores is Mathematics for Undergraduates/Research Experi­ the most likely to succeed in a graduate program. ence for Undergraduates at Universidad de Puerto The experiences of EDGE students show the ef­ Rico, Humacao. fectiveness of the EDGE philosophy. For 2007 CoProf has identified two Programs The long-range goals of the program are to That Make a Difference: the ENHANCING DIVERSITY increase the presence of women, with a special IN GRADUATE EDUCATION (EDGE) program and the focus on women of color, in the upper ranks of MATHEMATICAL THEORETICAL BIOLOGY INSTITUTE (MTBI). mathematical scientists and to create models for The 2007 CoProf Subcolllnittee on Programs mathematics programs that allow people from all That Make a Difference consisted of backgrounds and cultures to thrive, advance, and Gordon, Chawne Kimber, David Manderscheid, contribute to the profession. and Ivelisse Rubio. The cornerstone of the EDGE program, which for What follows are CoProf's citations and descrip­ the first five years alternated between Bryn Mawr tions of the two programs. The descriptions were College and Spelman College, is the four-week prepared by the Notices based on information sup­ Summer Program, which has both academic and plied by the programs. social components. The academic program con­ sists of two 4-week courses, one in abstract Enhancing Diversity in Graduate Education and linear algebra and one in analysis; problem Citation sessions conducted by three graduate student Be it resolved that the American Mathematical mentors; a minicourse and guest lectures on cur­ Society and its Colllnittee on the Profession rec­ rent areas of mathematical research; T£X sessions; ognize the Enhancing Diversity in Graduate Educa­ and presentations by participants. The program tion Program for its significant efforts to increase provides intense exposure to material and mastery the presence of women, with a special focus on through problem solving by a combination of indi­ women of color, in the upper ranks of mathemati­ vidual effort and teamwork. The social aspects of cal scientists. the program are designed to build a community of The Enhancing Diversity in Graduate Educa­ scholars in a collegial and supportive environment. tion (EDGE) program is a collaborative effort of The activities include a diversity seminar, panel Bryn Mawr College and Spelman College. Since its discussions about graduate school, weekly dinners, inception in 1998, over one hundred women from and reunions of the previous participants. diverse racial and educational backgrounds have In 1998 there were eight student participants. participated in EDGE. Of these women, over 90 Over the years, as funding has allowed, that num­ percent are either actively pursuing or have already ber has grown to fourteen. The selection criteria completed graduate degrees in mathematics. include acceptance into a graduate program in the

634 NOTICES OF THE AMS VOLUME 54, NUMBER 5 mathematical sciences (including statistics and Institute for its significant efforts to encourage , but not computer science), underrepresented minorities to continue in the an expressed desire to obtain a degree, study of mathematics. interest in being part of a network of women schol­ In the last ten years the Mathematical Theo­ ars, academic and leadership potential identified retical Biology Institute (MBTI) has mentored and by faculty recommendation letters, and the need supported over 300 students with more than half for exposure to graduate-level mathematics or the of these students entering graduate school. MTBI's graduate school culture. EDGE participants have all efforts have significantly increased the number been stars in their undergraduate institutions. of U.S. doctoral degrees awarded to members of The second major component of EDGE is the underrepresented minority groups. Follow-Up Mentoring Program. The codirectors and The AMS commends the director of MTBI, Carlos local coordinator arrange for a faculty mentor at Castilla-Chavez, for his high level of commitment each student's graduate institution. In addition, and his successful efforts to improve the diversity they maintain contact with students during the of the profession of mathematics in the United year and provide a small research allowance for States. books and professional travel. Each student is Description of Program invited to return to EDGE the following summer The Mathematical and Theoretical Biology Insti­ for a reunion, and an electronic bulletin board tute (MTBI) and the Institute for Strengthening allows alumnae to share triumphs and challenges Understanding of Mathematics and Science (SUMS, throughout their graduate school years. founded by the late Joaquin Bustoz Jr.) merged in One hundred five outstanding women were ac­ 2004. These two programs have received a total of cepted into the EDGE Program from 1998 to 2006. three presidential mentoring awards. In 2003 the Summer Program became portable, Since 1996, MTBI/SUMS has offered sequential moving to Pomona College with Local Coordinator research experiences for undergraduate and gradu­ Ami Radunskaya, in 2004 back to Spelman with ate students. Most of the participants have been Coordinator Yewande Olubummo, in 2005 to North members of underrepresented minority groups or Carolina A&T State University with Coordinators women. The program has run in conjunction with Janice Oldham and Patricia Shelton, and in 2006 (1996-2003), Los Alamos Na­ to New College of Florida with Coordinator Eirini tional Laboratory (2003-2005), and Arizona State Poimenidou. As of 2005, data show that EDGE University (2004-present). MTBI/SUMS experiences participants are from diverse racial (49% underrep­ are driven by the applications of mathematics and resented minorities) and educational (44% liberal statistics to questions in the biological and social arts) backgrounds. In the first eight years of EDGE sciences. The program has been particularly suc­ (1998-2005), 90 EDGE students entered graduate cessful in providing continuous research mentor­ programs. Approximately 93% of them either have ship training for students who want to work at the earned a graduate degree or are persisting in their interface of applied mathematics and theoretical graduate programs. To date, seven EDGE partici­ and computational biology. pants (7.7%) and three other EDGE graduate men­ Since 1996, MTBI/SUMS alumni have coauthored tors have earned the doctorate in mathematics, 111 technical reports. Some of these reports have and several other participants expect to complete been published or became the basis of the research degree requirements by the end of 2007. Among efforts of MTBI/SUMS-associated faculty, postdocs, the seven doctoral recipients, four are white and graduate students, undergraduate students, visi­ three are African-American. Of the total group, tors, and faculty. MTBI/SUMS has mentored and by 2006, 29% had earned the master's degree and supported 2 77 undergraduate students and 31 discontinued their education, most often to accept graduate students. A large percentage of its under­ employment, and 57% were continuing in a gradu­ graduates have participated multiple times, and 14 ate degree program. of its graduate alumni had participated previously The program has been supported primarily by in MTBI/SUMS as undergraduate students. the AndrewW. Mellon Foundation and the National Over its first ten years of existence, MTBI/SUMS Science Foundation. The EDGE Program is a project sent 130 students from underrepresented minor­ sponsored by the mathematics departments of ity groups to graduate school, and a total of 169 Bryn Mawr and Spelman Colleges with co-directors students overall. Furthermore, 52 percent of those Sylvia Bozeman, Spelman College, and Rhonda were females, including 65 from minority groups Hughes, Bryn Mawr College. and 33 international students, primarily from de­ veloping nations. MTBI/SUMS has been supported Mathematical Theoretical Biology Institute through grants provided by the National Security Citation Agency, the National Science Foundation, the Sloan Be it resolved that the American Mathematical Foundation, Los Alamos National Laboratory, and Society and its Committee on the Profession the offices of the provosts of Cornell University recognize the Mathematical Theoretical Biology and Arizona State University.

MAY 2007 NOTICES OF THE AMS 635 MTBI/SUMS's efforts have significantly in­ creased the number of U.S. Ph.D.'s awarded to About the cover members of underrepresented minority groups. In 2005 Ph.D.'s in the mathematical sciences were Oswald Veblen and colleagues awarded to ten MTBI/SUMS alumni, seven of whom This month's cover accompanies the ar­ are members of underrepresented minorities. ticle by Steve Batterson about the American These seven Ph.D.'s account for one-quarter of all mathematician Oswald Veblen, who had Ph.D.'s awarded that year to U.S. citizens who are a large hand in establishing the School of members of underrepresented minorities (2005 Mathematics at the Institute for Advanced Annual Survey, Notices, February 2006). Of the Study in the 1930s. Einstein (second from ten total MTBI/SUMS alumni Ph.D.'s, seven took left) and Veblen (fourth from left) were the postdoctoral positions and one became an as­ first members of that school. The others sistant professor at the University of Puerto Rico, in the group photograph, which was taken Mayaguez, campus. In 2005 approximately fifteen sometime in the 1940s, are the topologist women who are members of underrepresented ]. W. Alexander (at left, looking away from the minorities received Ph.D.'s in mathematics, and camera, for reasons known only to himself); this group includes five MTBI/SUMS alumnae. The (third from left), the second number of alumni who received their Ph.D.'s in director of the Institute; and Marston Morse. 2006 is about the same as in 2005. As of today, The location is what was then Einstein's office twenty-four MTBI alumni have been awarded in Fuld Hall, subsequently occupied by Arne Ph.D.'s in the mathematical sciences. Beurling and . MTBI/SUMS alumni have attended, or currently In 1939 all of the men in the photograph are attending, universities across the United States, had played an important role in bringing England, Colombia, and Mexico. MTBI/SUMS se­ about the resignation of the first director, quential summer programs have helped establish Abraham Flexner, amidst faculty frustra­ large underrepresented minority communities tion with Flexner's management. Some of at Cornell University, the University of Iowa, and the issues involved were important, such Arizona State University (ASU). MTBI/SUMS alumni as the Institute's relations with Princeton have established a community of minority scholars University-among other things, the anti­ at ASU and facilitated additional recruitment into Semitic attitude of some of the university's the graduate program. Currently there are thirty­ faculty (as told by Alexander in a letter to one minority group members in the ASU graduate Veblen dated September 1934). The story is program, including twenty-six Latinos and five recounted in detail in Chapter 9 of Batterson's African Americans. The ASU mathematics gradu­ book Pursuit of Genius. ate program has the nation's largest population of The photographs are from the archive of students who are minority group members. the Institute. We wish to thank Marcia Tucker MTBI/SUMS summer research programs are librarian of the Schools of Historical Studie~ run like NSF-sponsored workshops. New students and Social Science, for her generous help. take three and a half weeks of intense training in dynamical systems (broadly understood to -Bill Casselman, Graphics Editor include stochastic processes) and modeling in the ([email protected]) biological and social sciences. They also become familiar with computational software packages. Students set the research agenda each summer. Participants form groups of three to four students around a project of their own choice at the end of the initial training period. Each group is assigned a faculty advisor and provided with appropriate graduate student support. Between 20 percent and 33 percent of the undergraduates participate in two summers and often return as graduate student participants or mentors. These summer workshops have produced an average of ten technical reports per year and have instigated the research of a large percentage of its participants (undergradu­ ate, graduate and postdoctoral students, faculty, and visitors).

636 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Jumping Ship: Topology Board Resigns Allyn jackson

"Why should one spend one's life maintaining a together large numbers of journals- even a top-class journal- for Elsevier?" This rhetorical of thousand, in Elsevier's case-and sell them to question, posed by Martin Bridson of Imperial Col­ institutions for a single price. Around this time, lege London, expresses one strong current of feel­ the perception deepened within some segments ing within the mathematical community. Bridson of the mathematical community that Elsevier was one of nine editors of the journal Topology, overcharges for its journals and exploits the published by Elsevier, who resigned en masse in work of mathematicians. While the Topology edi­ the summer of 2006, effective at the end of the tors identified price as their main concern, some year. The following January those editors, together mathematicians may have turned against Elsevier with three new ones, reconstituted themselves as for other reasons (see for example http: I I cage. the editorial board of a new publication, Journal of ugent.bel-npgl elsevierl). Topology, which is owned by the nonprofit London The Topology editors found that an increasing Mathematical Society (LMS) and which will be type­ number of mathematicians refused to submit pa­ set, printed, and distributed by Oxford University pers to the journal or referee for it. "We've been Press. Despite the resignation, Elsevier has made concerned for years about the price and tried to clear its intention to keep Topology going with a bring it down on different occasions, in different new editorial board. ways," explained of the , one of the editors who resigned. "Finally First Mass Resignation in Math we decided we had to do something. We did not Boards of journals in other fields have jumped ship want to preside over a decline in ·the quality of the in recent years, but this appears to be the first time journal." The editors of Topology received a surge such a resignation has happened in mathemat­ of supportive emails right after they announced ics. And it happened with one of the field's top their intent to resign, and a further batch of en­ journals: Topology, founded by J. H. C. Whitehead couraging messages followed the launch of the new in the late 1950s, has an illustrious history and journal of Topology. carried some of the best work in that branch of mathematics in the twentieth century. Whitehead, Prices and Alternatives who was a professor at the University of Oxford, of the University of California, Berke­ personally knew the controversial publishing mag­ ley, has a soft spot for Topology, having published nate Robert Maxwell, an Oxford resident. Because his 1965 Ph.D. thesis there. "I'm sorry to see [Topol­ of this friendship, Topology started its life as a ogy] die," he said. "But it had to happen." In fact journal at Maxwell's company, Pergamon Press, Elsevier is not ready to let the journal die. In a letter which, according to Bridson, seems to have been that appeared in the December 2006 issue of the generally viewed as a "benevolent" supporter of European Mathematical Society Newsletter, Elsevi­ the journal. Over the years Topology has retained er's journals publisher Robert Ross wrote: "We are its close association with Oxford, and many of its committed to the long term future of [Topology] editors have been on the faculty there. After buy­ and its archive and to build upon its impressive ing out Pergamon, Elsevier took over Topology in heritage." Some believe that, after the resignation 1994. of the previous board and the supportive response Not long afterward, the Topology board became from the mathematical community, Elsevier would concerned about the journal's cost. In 2004 the have trouble recruiting new editors. Bridson finds board negotiated a deal with Elsevier that cut the it "unthinkable that any reputable mathematician institutional subscription price in half. But this would join a replacement board." deal had no practical effect because of the rise of The current of opposition to Elsevier that one "bundling", a system whereby publishers package . finds among mathematicians has been stoked over the years by activists like Kirby, who, after watch­ Allyn jackson is senior writer and deputy editor of the ing academic libraries strain under the pressure Notices. Her email address is axj@ams. org. of rising journal prices, has taken up low-cost

MAY 2007 NOTICES OF THE AMS 637 publishing as a personal cause. Kirby helped to that Topology "has never been more available" and launch Geometry and Topology, which began in that over 4,000 institutions around the world have 1996 as a free electronic journal and now charges either print or online access to the journal. "Our a subscription fee that is quite low compared to average price increase is amongst the lowest in the other mathematics journals (the 2007 price is industry," he wrote. "We have worked with libraries US$360 for paper and US$240 for electronic-only to develop flexible purchasing options that offer access). One motivation for establishing G&Twas significant discounts." to give Topology some serious, low-cost competi­ Bundling and consortium deals have not si­ tion and to provide a high-quality alternative for lenced Elsevier's critics, however, who maintain geometers and topologists who were concerned that such arrangements often take decisions about about journal prices. G&T, together with its spin­ subscriptions out of the hands of librarians and off journal Algebraic and , users of journals and put them into the hands of published 5,000 pages in 2006, an output that ac­ administrators higher up in the academic chain cording to Kirby constitutes "a big chunk of topol­ of command. For mathematics, this is "a deeply ogy, some of the best of topology." Kirby's activism insidious trend", Bridson said. "The intellectual has gone yet further. In 2004 he banded together integrity of journal publishing is all too likely to with several colleagues-including Joan Birman get lost in this scale of operation." He believes of Columbia University, Colin Rourke of Warwick that mathematics journals should be owned by University, and Ronald Stern of the University of the mathematical community, and his preferred California, Irvine-to start a nonprofit publishing model is ownership by learned societies. In this enterprise, Mathematical Sciences Publishers, with model, commercial publishers could be used by Paulo Ney de Souza as production manager. MSP the journal owners to produce and distribute jour­ now puts out six journals (five in mathematics and nals, and the owners could shop around to various one in engineering). publishers to get the best deal. What does Topology cost? The 2006 institu­ tional subscription price was US$1,665. One might Effects ofthe Resignation compare that to the US$ 5 70 that the LMS will "The resignation of the Topology editorial board charge for journal of Topology when it starts up highlights the real problem of scholarly publish­ in 2008. However, yearly subscription prices do ing today, which is exorbitant journal prices, and not provide adequate comparisons. One reason is highlighting that is certainly positive," remarked that journals differ in the number of pages they AMS executive director John H. Ewing. "Whether or produce in a year and in the amount of material not creating a new, less expensive journal is also that appears on a page. Journal price comparison positive isn't so clear. If the older, established jour­ surveys try to take these differences into account. One well-known survey was prepared by Ulf nal continues to exist, libraries are now faced with Rehmann at the UniversiUit Bielefeld (see http: I I the problem of subscribing to yet another journal, with no resources to do so. In the short run, that www.math.uni-bie1efe1d.del~rehmanniBIBI AMSIPub 1 i she r. htm1) using 2003 data gathered makes things worse for libraries." In fact, not only by the AMS about journal subscription prices was journal of Topology created to compete with and numbers of pages (http: www. ams. o rgl Topology, so was Geometry and Topology. Thus members hi pljourna1-survey. htm1 ). Rehmann's today there are three journals where there was only survey identifies 55 journals (out of a total of 2 74) one before. Even if libraries wanted to subscribe to that cost US$1.00 per page or more and gives a only the two less-expensive journals, they might price of US$0.99 per page for Topology (more not be able to opt out of Topology if it is offered recent AMS figures show Topology's price rose in a bundle. to US$1.18 per page by 2005). The survey singles The resignation of the Topology board could out Annals of Mathematics as a low-cost, high­ have a big impact if it were to set off a wave of quality journal, at US$0.12 per page. One might get resignations of boards of other commercially a ballpark estimate of what the per-page cost of the published journals. Indeed, rumors have circulated journal ofTopologywill be by noting that the four about other restive editorial boards. Nevertheless, LMS journals in the survey range from US$0.40 to it is unclear if a wave of resignations is in the off­ US$0.51 per page. The four AMS research journals ing. Hitchin noted that, although he is an editor included in the survey range from US$0.19 to for (which is published US$0.29 per page. by Springer and weighs in at US$1.09 per page in Price comparisons based on yearly subscrip­ Rehmann's survey), he does not feel the same kind tion cost have become less meaningful in recent of pressure over that journal as he did as an edi­ years, because bundling and consortium deals have . tor of Topology. One reason is that Mathematische blurred the prices of individual journals. In his let­ Annalen is a general journal that serves a wide ter to the EMS Newsletter, Ross referred to bundling segment of the mathematical community, whereas and consortium arrangements when he asserted Topology serves a more focused, close-knit group,

638 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Letter from Elsevier journals Publisher in electronic distribution, archiving and The following letter to the editor by Elsevier journals administration. The cost per article downloaded has publisher Robert Ross appeared in the December declined to an average of approximately US$2 per 2006 issue of the European Mathematical Society article. Newsletter. Elsevier has invested US$160 million in digitizing and maintaining the digital archive of our entire jour­ This August 2006, Elsevier received the resignation nal program. This facilitates and assures of the Editors of Topology. We regret this decision electronic access and distribution of the research by the Editors as we have appreciated the opportu­ record, allowing instant access throughout the world nity to work with them to publish one of the math or wherever and whenever the is available. community's most historically significant journals. A report by CSFB, quoting a case study at the Uni­ We are committed to the long term future of the versity of California, confirmed.that Elsevier provided journal and its archive and to build upon its impres­ much better value than a simple comparison of list sive heritage. prices would suggest. I More broadly, library statis­ Though we have attempted to address their con­ tics from organizations such as the UK's usu2 have cerns, it has become clear to us that the Editors are begun to show increased access to journal literature no longer interested in working with a commercial and falling unit prices. publisher. We have made a series of proposals to the We want to assure authors-including those with Editors of Topology and we will build on these going papers currently under review with Topology-that forward. the journal will continue. Indeed, subscribers to the At a time when publishers have been seeking to 2007 volume will receive the 2008 volume at no fur­ offer better value and to meet the needs of universi­ ther cost. This offer will apply whether they subscribe ties in consortia collectively purchasing digital access through a paper subscription or one of the electronic to diverse holdings, many scientists have continued options or packages our customers more commonly to be focused on price per page as an indicator of choose. value. We look forward to engaging the mathematics Whereas some in the mathematics community comm_unity to identify how we can work most ef­ might feel Topology has become unaffordable, it has fectively to serve and meet their needs. Pricing is an never been more available. Over 4,000 institutions issue under continuous discussion here, as it is at throughout the world have either print or on-line ac­ all publishers. We again regret the decision of the cess to this journal. During 2006, 27,000 downloads Topology Editors but do appreciate their concerns. have been recorded on this journal alone. Because the Elsevier is working hard to inform and work more majority of our subscribers purchase this journal in closely with all of our journal editors and we want to a larger set of journals, most are paying a fraction publicly thank those who continue to provide a very of the institutional subscription price. At the same necessary service to authors, the community, and to time, the personal subscription price has been held our publishing program. at US$99. Elsevier has taken steps to moderate price in­ I credit Suisse First Boston (CSFB), Equity Research, STM creases. Our average price increase is amongst the publishing sector review, 29 September 2004. lowest in the industry. We have worked with librar­ 2 USU annual library statistics 2004: http: I /www. l boro. ies to develop flexible purchasing options that offer ac.uk/departments/ls/lisu/pages/publications/ significant discounts. We have also made major al s04. html. where public campaigns against expensive journals At the time of this writing, there were just have had a strong influence. under forty signatories. This relatively low number In 2005 a group of mathematicians met at the might simply be due to few mathematicians having Banff International Research Station for a confer­ heard about the Banff Protocol. But there might be another explanation. While most mathematicians ence about the impact that increasing journal would agree in principle that journal prices ought prices is having on the field. They crafted a state­ to come down, the reality of the struggle for jobs, ment they called the "Banff Protocol" that reads: tenure, grants, and advancement in the field means "We agree neither to submit to, referee for, nor that the primary concern of most mathematicians participate in the operation of any journal that is to get their papers published in the best journals charges an excessively high per page subscription they can. Journal price, if it enters the picture at fee, as compared to the average of the 25 highest all, is secondary. Until the balance tips the other impact journals in pure mathematics." In 2004 way and large numbers of mathematicians start abandoning expensive journals, the status quo of the average price per page of those journals was journal cost is here to stay. US$0.59. The list of people who have signed on to the Banff Protocol may be found on the Web page http://members.cox.net/banffprotocol/.

MAY 2007 NOTICES OF THE AMS 639 2007 ]PBM Communications Award

The 2007 Communications Award of the Joint Massachusetts Institute of Technology. He has Policy Board for Mathematics (JPBM) was presented received awards for both his teaching and his re­ at the 113th Annual Meeting of the AMS in New search, including MIT's highest teaching prize, the Orleans in January 2007. E. M. Baker Award for Excellence in Undergraduate The JPBM Communications Award is presented Teaching, and a Presidential Young Investigator annually to reward and encourage journalists Award from the NSF. Strogatz joined the Cornell and other communicators who, on a sustained University faculty in 1994. He is a member of basis, bring accurate mathematical information to the Society for Industrial and Applied Mathemat­ nonmathematical audiences. The award carries a ics, and the Society for Mathematical Biology. cash prize of US$1,000. Strogatz has broad interests in applied math­ Previous recipients of the JPBM Communica­ ematics. At the start of his career, he focused on tions Award are: James Gleick (1988), questions arising in mathematical biology, includ­ Hugh Whitemore (1990), Ivars Peter­ ing the geometry of supercoiled DNA, the dynamics son (1991), Joel Schneider (1993), Mar­ tin Gardner (1994), Gina Kalata (1996), of the human sleep-wake cycle, the topology of Philip]. Davis (1997), Constance Reid three-dimensional chemical waves, and the col­ (1998), Ian Stewart (1999), John Lynch lective behavior of biological oscillators, such as and Simon Singh (special award, 1999), swarms of synchronously flashing fireflies. In the Sylvia Nasar (2000), Keith J. Devlin 1990s his work turned toward nonlinear dynam­ (2001), Claire and Helaman Fergu­ ics and chaos applied to physics and engineering. son (2002), Robert Osserman (2003), Several of these projects were concerned with lru:ge Barry Cipra (2005), and Roger Penrose systems of coupled oscillators, such as arrays of (2006). and superconducting Josephson junctions. The 2007 JPBM Communications In each case the research involved close collabora­ Award was presented to STEVEN H. tions with experimentalists. Currently, with his stu­ Steven H. Strogatz STROGATZ. The text that follows pres­ dents, he has been exploring a variety of complex ents the selection committee's ci­ networks in both the natural and social sciences, tation, a brief biographical sketch, using ideas from graph theory, statistical physics, and the recipient's response upon receiving the and nonlinear dynamics. award. Response Citation I'm thrilled and humbled to receive the JPBM Com­ The 2007 JPBM Communications Award is given to munications Award. Thank you for this wonderful Steven H. Strogatz for his work in the mathematical honor. dynamics of synchrony, including phenomena as My dad was a quiet man, not prone to giving diverse as human sleep, Josephson junctions, and advice or speechifying, so when he did express fireflies. Professor Strogatz not only carried out seminal research in this area, but he also reached himself in this way, it was memorable. He once told out to a wide audience to explain the exciting ideas me that a truly lucky person is one who could feel behind this research in numerous outlets-news­ excited about going to work each day. papers, magazines, and radio. In this sense, he is Teaching mathematics is what I love. There's a model for all research mathematicians. so much to be delighted by: the logic and power of the subject, its colorful history, its stunning Biographical Sketch results-but what inspires me most is its intercon­ After receiving his bachelor's degree in mathemat­ nectedness. Not just the links within mathematics ics from Princeton in 1980, Strogatz spent two itself, but also its connections to everything in the years as a Marshall Scholar at Cambridge Univer­ world around us, from the beating of our hearts to sity. He did his doctoral work in applied math­ the aggravating density waves of rush hour traffic. ematics at Harvard University and then stayed for I've tried to convey the pervasiveness of mathemat­ three years as a National Science Foundation (NSF) ics in my communications with the public and feel postdoctoral fellow. From 1989 to 1994, Strogatz very grateful to be recognized by the JPBM for taught in the Department of Mathematics at the these efforts.

640 NOTICES OF THE AMS VOLUME 54, NUMBER 5 MAAPrizes Presented inNewOrleans

At the Joint Mathematics Meetings in New Orleans Haimo Awards for Teaching in January 2007, the Mathematical Association of The Deborah and Franklin Tepper Haimo Awards America presented several prizes. for Distinguished College or University Teaching were established in 1991. These awards honor col­ Gung and Hu Award for Distinguished lege or university teachers who have been widely Service recognized as extraordinarily successful and whose The Yueh-Gin Gung and Dr. Charles Y. Hu Award teaching effectiveness has been shown to have had for Distinguished Service to Mathematics is the influence beyond their own institutions. most prestigious award made by the MAA. It hon­ The 2007 Haimo Awards were presented to ors distinguished contributions to mathematics jENNIFER QUINN, MICHAEL STARBIRD, and GILBERT and mathematical education, in one particular STRANG. aspect or many, and in a short period or over a The award citation for jennifer Quinn states: career. "Jennifer Quinn has a contagious enthusiasm LEE LORCH of York University in Canada re­ that draws students to mathematics. The joy she ceived the 2007 Gung and Hu Award for "for his takes in all things mathematical is reflected in life-long contributions to mathematics, his con­ her classes, her presentations, her publications, tinued dedication to inclusiveness, equity, and her videos, and her on-line materials. Her class human rights for mathematicians, and especially assignments often include nonstandard activi­ his profound influence on the lives of minority and ties, such as creating time line entries for historic women mathematicians who have benefited from math events, or acting out scenes from the book his efforts." A remarkable teacher and mentor, Proofs and Refutations." In her thirteen years at Oc­ Lorch inspired many to enter mathematical careers cidental College, she developed new courses with who otherwise might not have considered such a innovative teaching methods, including a history path. "He has been throughout his career a vocal of mathematics course that led to a mathematics advocate and energetic worker for human rights "game show" called "The Number Years", which and educational opportunities," the award citation was a huge hit at the Joint Mathematics Meetings states. "His interventions, especially in the 1950s, in 2000. Quinn is a co-editor of the magazine Math led to changes in the policies and practices of the Horizons and co-author with Arthur Benjamin of AMS and the MAA that ensured that all mathemati­ the book Proofs That Really Count: The Art of Com­ cians could participate in the official events of these binatorial Proof, which won the MAA's Beckenbach organizations." Lorch was summoned before the Book Prize. Today Quinn is the executive director House Committee on Un-American Activities and of the Association for Women in Mathematics and cited for contempt for refusing to say whether he teaches occasionally at the University of Puget had ever been a member of the Communist Party. Sound and Pacific Lutheran University in Wash­ He was dismissed from positions in the southern ington state. United States for his political activities; later, two "Michael Starbird's goal is to help his students institutions that had dismissed him awarded him unleash the creativity within them," states the honorary degrees. In his written response to the award citation for Starbird. "He doesn't just prize, Lorch raised pointed questions about the teach them mathematics. He teaches them how to devastation that hit New Orleans as a result of discover and appreciate mathematics for them­ Hurricane Katrina and the continued suffering of selves." In his thirty-two years of teaching at the the city's inhabitants. "Even the AMS homepage University of Texas at Austin, Starbird has had tells us only of Tulane- not of the several afflicted a positive impact on thousands of students. His [historically black colleges and universities]," he influence has also spread through his teaching wrote. "Perhaps no one in these institutions has videos that appear in the Great Courses series of submitted a report. Maybe they do not feel really the Teaching Company. Starbird has received about part of the mathematical community. Why not? a dozen teaching awards. He wrote a textbook What is being done about it?" During the prize with co-author entitled The Heart session, Lorch was greeted with a standing ovation of Mathematics: An Invitation to Effective Think­ by the meeting participants. ing, which is aimed at nonmathematics majors.

MAY 2007 NOTICES OF THE AMS 641 The book received the 2002 Robert W. Hamilton the Earth" (Mathematics Magazine, Vol. 77, No.4, Book Award and has been adopted at over two June 2004, pp. 171-189) as well as a book, Hesiod's hundred colleges and universities. The prize cita­ Anvil: Falling and Spinning through Heaven and tion concludes that Starbird is "creative, articulate, Earth, published by the MAA. indefatigable, and an eloquent communicator and promoter of mathematics." Beckenbach Book Prize According to the award citation for Gilbert The Beckenbach Book Prize, presented since 1982, Strang, "[h]is approaches to teaching linear algebra is awarded to an author of a distinguished, innova­ and mathematics for engineers have changed the tive book published by the MAA. way we all approach these subjects." Strang has WILLIAM P. BERLINGHOFF and FERNANDO Q. GOUVEA taught for forty-four years at the Massachusetts received the 2007 Beckenbach Book Prize for Math Institute of Technology, where he pioneered a through the Ages: A Gentle History for Teachers new way to teach linear algebra, embodied in and Others (MAA and Oxton House Publishers, his textbook Linear Algebra and Its Applications 2004). This book presents twenty-five short histori­ (1976). "This book sparked a revolution in the way cal sketches of important topics in mathematics. linear algebra was taught and has influenced a The prize citation notes that the book has "the multitude of books that have come out since then," the award citation says. "Rather than utilizing a great advantage of being appealingly readable to a wide audience ranging from secondary school theorem-proof format, the book was written in a conversational tone and included many practical and liberal arts students through the mathematical applications." Every year since 1981 more than community's educators and practitioners." The ci­ 300 students out of the MIT class of 1,000 have tation goes on to say: "The beautiful writing makes taken Strang's course. Strang also rethought the it difficult for a reader to put the book down, and teaching of mathematics for engineers, introduc­ it is inviting to jump from one historical sketch ing a new course at MIT and writing a textbook to another." based on it, Introduction to Applied Mathemat­ ics. MIT's Graduate Student Council recognized this course by giving Strang a teaching award in The Euler Book Prize is given to the author(s) of 2003. Strang is also a prolific researcher and has an outstanding book about mathematics. The supervised twenty Ph.D. and five master's degree prize was given for the first time in 2007 to com­ students. "Gil has a deep love of mathematics and memorate the 300th anniversary of the birth of a profound understanding of how mathematics is . used in the sciences and engineering," the award received the 2007 Euler Prize for citation concludes. "He has applied these qualities his book Prime Obsession: and to reshape the way we teach mathematics." the Greatest Unsolved Problem in Mathematics (Jo­ seph Henry Press, National Academies Press, 2003). "Mathematical sagas don't get much better than The Chauvenet Prize recognizes a member of the this, the story of the , close to MAA who has written an outstanding expository 150 years old and referred to by the author of this article. First awarded in 1925, the prize is named book as 'the great white whale of mathematical re­ for William Chauvenet, who was a professor of search,"' the prize citation says. It continues: "[T]he mathematics at the United States Naval Acad­ book covers the rich history of the problem and emy. conveys to the nonspecialist reader the reasons the The 2007 Chauvenet Prize was awarded to problem is interesting and important, not only in ANDREW ]. SIMOSON for his article "The of mathematics but possibly in physics." Hades" (Mathematics Magazine, Vol. 75, No. 5, December 2002, pp. 335-350). Certificates of Meritorious Service Calling this paper both amusing and learned, the prize citation says: "The central question Each year the MAA presents Certificates of Meritori­ of this paper, 'Is the due to gravity ous Service for service at the national level or for stronger or weaker as we descend into the Earth?,' service to a section of the MAA. Those honored in relates to celestial mechanics. The author answers 2007 are: Florida Section: MARILYN REPsHER of Jackson­ this question for several models of the Earth's ville University; Kansas Section: SISTER Jo ANN FEUJN, structure that have been proposed over the cen­ Benedictine College; Michigan Section: JERROLD W. turies. The article is accessible to undergraduates GROSSMAN, Oakland University; Northeastern Section: who have had multivariable calculus and who are DONNA BEERs, Simmons College; Rocky Mountain Sec­ familiar with Newton's law of gravitational force, tion: JANET HEINE BARNETT, Colorado State University, even for students who are not yet in tune with Pueblo; and Texas Section: STUART ANDERSON, Texas abstract mathematics." Simoson wrote another A&M University, Commerce. article on this topic, "Falling down a hole through -Allyn jackson

642 NOTICES OF THE AMS VOLUME 54, NUMBER 5 A WM Awards Presented inNewOrleans

The Association for Women in Mathematics (AWM) recently, a monograph about Brousseau's work and presented two awards at the Joint Mathematics the nature of didactics. Meetings in New Orleans in January 2007. Schafer Prize Louise Hay Award The Alice T. Schafer Prize for Excellence in Math­ The Louise Hay Award for Contributions to Math­ ematics by an Undergraduate Woman was estab­ ematics Education was established in 1990 to lished in 1990. The prize is named in honor of honor the memory of Louise Hay, who was widely Alice T. Schafer, one of the founders of AWM and recognized for her contributions to mathematical one of its past presidents. logic and for her devotion to students. ANA CARAIANI of Princeton University received The 2007 Hay Award was presented to VIRGINIA the 2007 Schafer Prize. A senior at Princeton, Cara­ McSHANE WARFIELD of the University of Washington. iani is already doing professional-level research. In Warfield's early work on Project SEED, a highly regarded program of mathematical activities for the summers of 2005 and 2006 she participated fourth- through sixth-grade students, led to her in the Research Experiences for Undergraduates becoming the leader of the entry-level mathemat­ program at the at Duluth ics courses in the mathematics department at the and undertook research resulting in a paper "On University of Washington, which she continues to wild semi-groups". The prize citation says that oversee. She significantly revised courses for future the paper "introduces new ideas that exhibit sig­ elementary teachers and has served as a mentor nificant ingenuity." Caraiani has done very well for graduate students. From 1994 to 2001, she was in her Princeton coursework and is noted for her project director for "Preparing Future Faculty", in independence and mathematical sophistication. which, among other things, she arranged for gradu­ The prize citation states: "One professor has said ate students to spend time at local community that her work 'made you think that it was a pro­ or four-year colleges, took them to conferences fessional mathematician who was answering the on educational issues, and arranged conferences problems.' Another professor rates her among the with guest speakers. Warfield was instrumental top undergraduate mathematics majors in fifty in creating Washington Teachers of Teachers of years at Princeton." She won the Putnam Competi­ Mathematics (WAToToM), at which members of tion twice, scoring among the top five competitors departments of mathematics and mathematics in both her freshman and sophomore years. The education from around the state get together for a mathematics department at Princeton awarded her weekend of conversation and idea-sharing. During the Class of 1861 Prize in her sophomore year and the past ten years she has played a leading role in the Andrew H. Brown Prize for outstanding juniors. three major teacher enhancement projects funded "She is expected to become a major mathematical by the National Science Foundation: Creating a Community of Mathematics Learners, Extending figure and a world class research mathematician," the Community of Mathematics Learners, and the prize citation concludes. Graduate Teaching Fellows in K- 12 Education, all Three other senior mathematics majors were of which partner University of Washington faculty recognized by the AWM at the Joint Mathematics and graduate students with in-service teachers of Meetings. TAMARA BRODERICK of Princeton University mathematics. Warfield has also made significant and YAIM CooPER of the Massachusetts Institute of contributions to mathematics education research Technology were named the runners-up for the through her collaboration with Guy Brousseau, Schafer Prize, and ALYSON DEINES of Kansas State which has led to publication of several articles, University was an honorable mention recipient. translation and co-editorship of a book, and, most - From A WM announcements

MAY 2007 N OTICES OF THE AMS 643 Mathematics People

Ni will complete his Ph.D. at Princeton under the Jolmson Receives AAAS direction of . He works on the topology of three manifolds and theory. His thesis combines Mentoring Award techniques from Heegaard Floer homology and foliation RAYMOND L. ]OHNSON of the University of Maryland, College theory to prove that knot Floer homology detects fibered Park, has received the 2006 Mentor Award for Lifetime knots. He has written ten papers on this subject and oth­ Achievement from the American Association for the Ad­ ers. He earned his B.S. in 2001 and his M.S. in 2003 from vancement of Science (AAAS). He has mentored twenty­ Peking University. He won a gold medal at the International three students-twenty-two of them African Americans, Mathematical Olympiad in 1997. eight of whom are women-who have received Ph.D. The runners-up for the AIM Fellowship are Andrew degrees in mathematics. Putman (University of Chicago), Andrew Schultz (Stanford In 1969 Johnson became the first African American to University), and Corinna Ulcigrai (Princeton University). earn a Ph.D. degree in mathematics from Rice University. He then became the first African American professor at -From an AIM announcement the University of Maryland. He has devoted his career to increasing participation in the mathematical sciences by African Americans. He has served on the boards of gov­ National Academy of ernors of the Mathematical Association of America (MAA) and the Institute for Mathematics and its Applications Engineering Elections (IMA). He has helped to organize conferences to promote The National Academy of Engineering (NAE) has an­ more participation in mathematics research by African nounced the election of sixty-four new members and nine Americans and other minorities and was a founding mem­ foreign associates, including four whose work involves the ber of the Conference for African American Researchers mathematical sciences. Their names, institutions, and the in Mathematical Sciences. research for which they were elected follow. The Mentor Award for Lifetime Achievement honors ]OHN ]. DoRNING, , Charlottes­ members of the AAAS who have mentored and guided ville, for the development of advanced computational significant numbers of underrepresented students toward methods for nuclear reactor analysis; SILVIO MICALI, Mas­ Ph.D. degrees in the sciences and who have demonstrated sachusetts Institute of Technology, for contributions to scholarship, activism, and community building on behalf modern cryptography through the development of zero­ of underrepresented groups, including women of all racial knowledge protocols and the theory of pseudo-random­ or ethnic groups; African American, Native American, and ness; EvA TARDOS, Cornell University, for contributions to Hispanic men; and people with disabilities. This award the design and analysis of efficient algorithms for network often recognizes individuals with twenty-five or more problems; and LLOYD N. TREFETHEN, Oxford University, for years of success in mentoring students. The recipient contributions to in numerical analysis receives US$5,000 and a commemorative plaque. and its application to the determination of the onset of turbulence. -From an AAAS announcement -From an NAE announcement Yi Ni Receives AIM Five-Year Fellowship YI NI of Princeton University has been named the recipi­ ent of the 2007 American Institute of Mathematics (AIM) Five-Year Fellowship.

644 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Mathematics Opportunities

One of the most pressing needs for the detection of NSF Postdoctoral Research nuclear weapons and material is the development of algo­ Fellowships rithms for intelligent signal processing. Examples include algorithms that can detect a weak signature in ambient The National Science Foundation (NSF) awards Mathemati­ background noise, algorithms that can be programmed in cal Sciences Postdoctoral Research Fellowships (MSPRF) a chip that is embedded directly in a detector, and algo­ for appropriate research in areas of the mathematical rithms that can process information from multiple sensors sciences, including applications to other disciplines. that are detecting weak signals from nuclear material in Awardees are permitted to choose research environments transit and can combine the information in a meaningful that will have maximal impact on their future scientific way and lead to a conclusion with a very low false alarm development. Awards are made in the form of either rate. Other relevant areas of mathematics research include Research Fellowships or Research Instructorships. The network theory, , uncertainty and risk Research Fellowship option provides full-time support for analysis, pattern recognition, image and signal processing, any eighteen academic-year months in a three-year period and data fusion. in intervals not shorter than three consecutive months. The deadline for full proposals is May 2, 2007. See The Research Instructorship option provides a combina­ http://www.nsf.gov/funding/pgm_summ.jsp?pims_ tion of full-time and half-time support over a period of i d=501056. three academic years, usually one academic year full-time -From an NSF announcement and two academic years half-time. Under both options the award includes six summer months; however, no more than two summer months of support may be received in NSF-CBMS Regional any calendar year. Under both options the stipend support Conferences, 2007 for twenty-four months (eighteen academic-year months plus six summer months) will be provided within a forty­ With funding from the National Science Foundation (NSF), eight-month period. the Conference Board of the Mathematical Sciences (CBMS) The deadline for proposals is October 17, 2007. See will hold two NSF-CBMS Regional Research Conferences http://www.nsf.gov/funding/pgm_summ.jsp?pims_ during the summer of 2007. i d=5301&org=DMS. These conferences are intended to stimulate interest and activity in mathematical research. Each five-day con­ -From an NSF announcement ference features a distinguished lecturer who delivers ten lectures on a topic of important current research in one Domestic Nuclear Detection sharply focused area of the mathematical sciences. The lecturer subsequently prepares an expository monograph Office/NSF Academic Research based on these lectures. Initiative Support for about thirty participants will be provided for each conference. Both established researchers and The Domestic Nuclear Detection Office (DNDO) within the interested newcomers, including postdoctoral fellows and Department of Homeland Security (DHS), in partnership graduate students, are invited to attend. with the National SCience Foundation (NSF), has announced Information about an individual conference may be a joint effort to advance fundamental knowledge in new obtained by contacting the conference organizer. The two technologies for the detection of nuclear threats and to conferences to be held in 2007 are as follows. develop intellectual capacity in fields relevant to long­ Numerical Methods for Nonlinear Elliptic Equations, term advances in nuclear detection capability. Research Roland Glowinski, lecturer. May 21-25, 2007, University will focus on detection systems, individual sensors, or of Iowa. Organizers: Kendall E. Atkinson, email: kendall - other research that is potentially relevant to the detection atki nson@ui owa. edu; Weimin Han, email: whan@math. of nuclear weapons, special nuclear material, radiation ui owa. edu; Laurent 0. Jay, email: na. l j ay@na-net. dispersal devices, and related threats. ornl. gov; Brian E. Moore, email: bemoore@math. uiowa.

MAY 2007 NOTICES OF THE AMS 645 Mathematics Opportunities edu; Suely Oliveira, email: [email protected]; will consist of two sessions of five hours each. Problems and David Stewart, email: dstewart@math. ui owa. edu; will come from the fields of algebra, analysis (real and conference email: cbms_2007@math. ui owa. edu; confer­ complex), and combinatorics. The working language will ence website: http: I /www. math. ui ow a. edu/ events/ be English. CBMS2007 /. The deadline for registration is May 31, 2007. For Finite Morse Index Solutions and Related Topics, details, see the website http: I /www. i me-math. o rg. uk/ E. N. Dancer, lecturer. December 16-20, 2007, University or contact John Jayne, University College London, Gower of Texas, San Antonio. Organizer: Shair Ahmad, email: Street, London WClE 6BT, United Kingdom; telephone: shai r. ahmad@utsa. edu; conference website: http: I I +44-20-7679-7322; email: j [email protected]. math.utsa.edu/~ahmad/cbms/. -john jayne, University College London -From a CBMS announcement Call for Proposals for News from the IMA The 2007-2008 thematic program at the Institute for 2008 NSF-CBMS Regional Mathematics and its Applications (IMA), University of Min­ nesota, is "Mathematics of Molecular and Cellular Biology". Conferences This program provides a setting for mathematicians and To stimulate interest and activity in mathematical re­ scientists to explore recent and nascent breakthroughs in search, the National Science Foundation (NSF) intends to molecular and cellular biology. support up to seven NSF-CBMS Regional Research Confer­ The fall workshops emphasize nucleic acid (DNA and ences in 2008. A panel chosen by the Conference Board of RNA) organization, structure, function, and the interac­ the Mathematical Sciences will make the selections from tion between DNA and RNA in the production of proteins among the submitted proposals. and the orchestration of cellular metabolism. A tutorial, Each five-day conference features a distinguished lec­ Mathematics ofNucleic Acids, will be held on September 15, turer who delivers ten lectures on a topic of important 2007. The two workshops are: Mathematics of DNA Struc­ current research in one sharply focused area of the math­ ture, Function, and Interactions, September 16-21, 2007; ematical sciences. The lecturer subsequently prepares an and RNA in Biology, Bioengineering, and Nanotechnology, expository monograph based on these lectures, which is October 29-November 2, 2007. normally published as a part of a regional conference se­ The winter workshops are devoted to protein structure ries. Depending on the conference topic, the monograph and function. A tutorial, Mathematics of Proteins, will be will be published by the American Mathematical Society, held January 10-11, 2008. The two workshops are: Protein by the Society for Industrial and Applied Mathematics, Folding, January 14-18, 2008; and Organization ofBiologi­ or jointly by the American Statistical Association and the cal Networks, March 3-7, 2008. Institute of Mathematical Statistics. The spring workshops focus on the mathematics of Support is provided for about thirty participants at cellular physiology. A tutorial, Network Dynamics and each conference, and both established researchers and Cell Physiology, will be held April 17-18, 2008. The two interested newcomers, including postdoctoral fellows and workshops are: Design Principles in Biological Systems, graduate students, are invited to attend. April 21-25, 2008; and Quantitative Approaches to Cell The proposal due date is April 20, 2007. For further Motility and Chemotaxis, May 2 7-30, 2008. Limited financial information on submitting a proposal, consult the CBMS support is available for the workshops. Detailed informa­ website, http: I /www. cbmsweb. o rg/NSF /2008_ca 11 . tion about this program can be found at http: I /www. htm, or contact: Conference Board of the Mathematical ima.umn.edu/2007-2008/. Sciences, 1529 Eighteenth Street, NW, Washington, DC The IMA is currently accepting applications for post­ 20036; telephone: 202-293-1170; fax: 202-293-3412; email: doctoral fellows, industrial postdoctoral fellows, New [email protected] or [email protected]. Directions Visiting Professors, and general members in -From a CBMS announcement connection with the 2008-2009 IMA thematic program, "Mathematics and Chemistry". The application deadline International Mathematics for New Directions professorships and postdoctoral fel­ lowships is January 5, 2008; applications for general Competition for University memberships are accepted at any time. The IMA actively solicits proposals for programs from Students members of the mathematical sciences community. Pos­ The fourteenth International Mathematics Competition sibilities include annual programs (which run from Sep­ (IMC) for University Students will be held August 3-9, tember to June), summer programs (which typically run 2007, at the American University in Blagoevgrad, Bulgaria. for two to four weeks and involve between 60 and 120 Participating universities are invited to send several stu­ participants), and Hot Topics workshops (which typically dents and one teacher; individual students are welcome. last a few days and treat a topic of exceptional contempo­ Students completing their first, second, third, or fourth rary interest and potential impact). Additional information years of university education are eligible. The competition is available at http: I /www/i rna. umn. edu/ so 1 i cit.

646 NoTICEs oF ruE AMS VOLUME 54, NUMBER 5 Mathematics Opportunities

For detailed information about IMA programs, as well participants for the entire program or to attend individual as the online application forms for IMA workshops and workshops. memberships, see http: I lwwwli ma. umn. edu. Optimal Transport Tutorials: March ll-14, 2008. -From an IMA announcement Workshop I: Aspects of Optimal Transport in Geometry and Calculus of Variations: March 31-April 4, 2008. Workshop II: Numerics and Dynamics for Optimal Trans­ News from IPAM port: April 14-18, 2008. The Institute for Pure and Applied Mathematics (IPAM), Workshop III: Transport Systems in Geography, Geosci­ located at the University of California, Los Angeles, holds ences, and Networks: May 5-9, 2008. long- and short-term research programs and workshops Workshop W: Optimal Transport in the Human Body: throughout the academic year for junior and senior Lungs and Blood: May 19-23, 2008. mathematicians and scientists who work in academia, -!PAM announcement the national laboratories, and industry. In the summer IPAM sponsors a program for undergraduate students International Dobrushin (RIPS) and graduate studentslpostdocs (Graduate Sum­ mer School). !PAM's 2007-2008 academic year programs Foundation are listed below. Please go to http: I lwww . i pam. ucla. edu for detailed information and online application and Roland L. Dobrushin (1929-1995) was one of the greatest registration forms. mathematicians of the twentieth century. He is widely !PAM's Science Advisory Board meets in November, known for his achievements in probability theory, infor­ when it considers program proposals. Program proposals mation theory, statistical physics, and mathematical and from the community are encouraged, and instructions on computer linguistics. He was also a brilliant lecturer and how to do this are available at our website. research director particularly well known for his sup­ port of young mathematicians. Dobrushin was one of IPAM is seeking its next director, to begin July 2008. the founders of the Independent University of Moscow. Information about the position and how to apply is avail­ He created a laboratory at the Institute for Information able on our website. Transmission Problems which now bears his name. The Graduate Summer School. Probabilistic Models of Dobrushin Laboratory became known throughout the Cognition: The Mathematics of Mind, July 9-27, 2007. world for its staff of outstanding mathematicians includ­ Interested individuals may register online. ing three Fields Medal laureates. Mathematics of Knowledge and Search Engines. Sep­ Recently the International Dobrushin Foundation tember 10-December 14, 2007. Please apply by August 1. was established by admirers of his talent. The purpose This long program includes the following workshops, which of the foundation, which is funded by a private endow­ are also open for participation. Individuals may apply online ment, is to support science and scientists, in particular for support to be core participants for the entire program mathematics and mathematicians. Toward this goal, the or to attend individual workshops. foundation is launching the International Dobrushin Mathematics of Knowledge and Search Engines Tutori­ Prize, five Dobrushin scholarships for senior students of als: September ll-20, 2007. the Independent University of Moscow, and a Dobrushin Workshop I: Dynamic Searches and Knowledge Building: grant for a professor. October 1-5, 2007. The International Dobrushin Prize is awarded annually Workshops II: Numerical Tools and Fast Algorithms for to outstanding researchers for the totality of their work Massive Data Mining, Search Engines and Applications: in the domains of research interests of Dobrushin, includ­ October 22-26, 2007. ing , statistical physics, probability Workshop III: Social Data Mining and Knowledge Build­ theory, and mathematical and computer linguistics. The ing: November 5-9, 2007. prize committee, formed of independent experts, accepts Workshop W: Search and Knowledge Building for Biologi­ nominations from any researcher or a group of research­ cal Datasets: November 26-30, 2007. ers. Nominations should be sent before May 10, 2007, Winter 2008 Short Programs. Individuals may apply to dobrushi nawards@yahoo. com and should include a online for support to attend. list of significant publications of the candidate, a brief Scientific Computing Applications in Surgical Simulation summary of several of the candidate's most important of Soft Tissues: January 7-ll, 2008. publications, and a list of three experts who have agreed Challenges in Image Analysis in Molecular Microscopy: to endorse the nomination. January 28-February 1, 2008. The prize is awarded each year on June 20, the birth­ Expanders in Pure and Applied Math: February ll-15, day of Dobrushin. The winner receives a diploma and the 2008. equivalent of US$3,000 dollars and is invited to present Graph Cuts and Related Discrete or Continuous Optimi­ a lecture at the Institute for Information Transmission zation Problems: February 25-29, 2008. Problems in Moscow. Optimal Transport. March 10-June 13, 2008. Please -Dobrushin Foundation announcement apply by February 1, 2008. This long program includes the following workshops, which are also open for participa­ tion. Individuals may apply online for support to be core

MAY 2007 NoTICES OF THE AMS 647 R.eference and Book List

The Reference section of the Notices tion and application forms for the www.awm-math.org/travelgrants. is intended to provide the reader AMS scholarships, see http: I jwww. html; telephone 703-934-0163; email: with frequently sought information in ams.org/outreach/mimoscow.html awm@math. umd. edu; or contact As­ an easily accessible manner. New or contact Math in Moscow Program, sociation for Women in Mathematics, information is printed as it becomes Membership and Programs Depart­ 11240 Waples Mill Road, Suite 200, available and is referenced after the ment, American Mathematical Soci­ Fairfax, VA 22030. first printing. As soon as information ety, 201 Charles Street, Providence, Rl May 2, 2007: Proposals for Do­ is updated or otherwise changed, it 02904-2294; email: student-serv@ mestic Nuclear Detection Office/Na­ will be noted in this section. ams. org. tional Science Foundation Academic April 16, 2007: Applications for Research Initiative. See "Mathematics Contacting the Notices Project NExT: New Experiences in Opportunities" in this issue. The preferred method for contacting Teaching. See http: I /archives. May 31, 2007: Registration for the Notices is electronic mail. The math.utk.edu/projnext/. the International Mathematics Com­ editor is the person to whom to send April20, 2007: Proposals for 2008 petition for University Students. See articles and letters for consideration. NSF-CBMS Regional Conferences. See "Mathematics Opportunities" in this Articles include feature articles, me­ "Mathematics Opportunities" in this issue. morial articles, communications, opinion pieces, and book reviews. issue. June 1, 2007: Applications for The editor is also the person to whom May 1, 2007: Applications for National Academies Christine to send news of unusual interest AWM Travel Grants. See http: I I Mirzayan Graduate Fellowships about other people's mathematics research. Where to Find It The managing editor is the person A brief index to information that appears in this and previous issues of the Notices. to whom to send items for "Math­ AMS Bylaws-November 2005, p. 1239 ematics People", "Mathematics Op­ AMS Email Addresses-February 2007, p. 271 portunities", "For Your Information", "Reference and Book List", and "Math­ AMS Ethical Guidelines-June/July 2006, p. 701 ematics Calendar". Requests for AMS Officers 2006 and 2007 (Council, Executive Committee, permissions, as well as all other Publications Committees, Board of Trustees)-May 2007, p. 657 inquiries, go to the managing editor. AMS Officers and Committee Members-October 2006, p. 1076 The electronic-mail addresses are Conference Board of the Mathematical Sciences-September 2006, noti ces@math. ou. edu in the case of p. 911 the editor and noti ces@ams. org in Information for Notices Authors-june/July 2006, p. 696 the case of the managing editor. The Mathematics Research Institutes Contact Information-August 2006, fax numbers are 405-325-7484 for p. 798 the editor and 401-331-3842 for the National Science Board-January 2007, p. 57 managing editor. Postal addresses New Journals for 2004-]une/]uly 2006, p. 697 may be found in the masthead. NRC Board on Mathematical Sciences and Their Applications-March Upc:oming Deadlines 2007, p. 426 April15, 2007: Applications for AMS NRC Mathematical Sciences Education Board-April 2006, p. 546 "Math in Moscow" Scholarships for NSF Mathematical and Physical Sciences Advisory Committee-February fall 2007. See http: I /www. mccme. I 2007, p. 274 ru/mathi nmoscow or contact Math in Program Officers for Federal Funding Agencies-October 2006, Moscow, P.O. Box 524, Wynnewood, p. 1072 (DoD, DoE); December 2006, p. 1369 (NSF) PA 19096; fax +7095-291-65-01; Stipends for Study and Travel-September 2006, p. 913 email: mi m@mccme. ru. For informa-

648 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Reference and Book List for the fall program. See http: I I erhatics in the news) warrant drawing The Equations: Icons ofKnowledge, www7.nationalacademies.orgl readers' attention to older books. Sug­ by Sander Bais. Harvard University pol i cyfe ll ows or contact The Na­ gestions for books to include on the list Press, November 2005. ISBN 0-674- tional Academies Christine Mirzayan may be sent to noti ces-bookl i st@ 01967-9. Science and Technology Policy ams. org. The Essential Turing, edited by Graduate Fellowship Program, 500 *Added to "Book List" since the B. Jack Copeland. Oxford Univer­ Fifth Street, NW, Room 508, Wash­ list's last appearance. sity Press, September 2004. ISBN ington, DC 20001; telephone: 202- 0-198-2 5080-0. (Reviewed November 334-2455; fax: 202-334-1667; email: An Abundance of Katherines, by 2006.) [email protected]. John Green. Dutton Juvenile Books, Euclid in the Rainforest: Discover­ June 5, 2007: Proposals for En­ September 2006. ISBN 0-525-47688-l. ing Universal Truths in Logic and hancing the Mathematical Sciences 's Automatic Comput­ Math, by Joseph Mazur. Press, Workforce in the Twenty-First Cen­ ing Engine: The Master Codebreaker's October 2004. ISBN 0-131-4 7994-6. tury. See http:llwww.nsf.govl Struggle to Build the Modern Com­ Evolutionary Dynamics: Exploring publicationslpub_summ.jsp?ods_ puter, edited by B. Jack Copeland. the Equations ofLife, by Martin Nowak. key=nsf05 595. , June 2005. Belknap Press, September 2006. ISBN June 30, 2007: Nominations for ISBN 0-198-56593-3. 0-674-02338-2. 2007 Fermat Prize. See http: I lwww. Analysis and Probability: Wavelets, The Fabulous Fibonacci Numbers, math.ups-tlse.friFermatl. Signals, by Palle E. T. Jor­ by Alfred S. Posamentier and Ingmar October 1, 2007: Applications for gensen. Springer, September 2006. Lehmann. Prometheus Books, February AWM Travel Grants. See http: I I ISBN 0-387-29519-4. 2007. ISBN 1-591-02475-7. www.awm-math.orgltravelgrants. Arthur Cayley: Mathematician Fearless Symmetry: Exposing the html; telephone 703-934-0163; email: Laureate of the Victorian Age, by Hidden Patterns ofNumbers, by Avner awm@math. umd. edu; or contact As­ Tony Crilly. Johns Hopkins University Ash and Robert Gross. Princeton Uni­ sociation for Women in Mathematics, Press, December 2005. ISBN 0-801- versity Press, May 2006. ISBN 0-691- 11240 Waples Mill Road, Suite 200, 88011-4. 12492-2. (Reviewed January 2007.) Fairfax, VA 22030. The Artist and the Mathematician: From Cosmos to Chaos: The Sci­ October 5, 2007: Full propos­ The Story of , the ence of Unpredictability, by Peter als for NSF IGERT competition. See Genius Mathematician Who Never Coles. Oxford University Press, Au­ http:llwww.nsf.govlpubsl2007l Existed, by Amir D. Aczel. Thunder's gust 2006. ISBN 0-198-56762-6. nsf07540insf07540.htm. Mouth Press, August 2006. ISBN From Zero to Infinity: What Makes October 15, 2007: Preferred dead­ 1-560-2 5931-0. Numbers Interesting, by Constance line for January entrance in junior­ A Beautiful Math: John Nash, Game Reid. Fiftieth anniversary edition, year program at the Smith College Theory, and the Modern Quest for a A K Peters, February 2006. ISBN Center for Women in Mathematics. Code of Nature, by Tom Siegfried. 1-568-81273-6. (Reviewed February See http:llwww.math.smith.edul Joseph Henry Press, October 2006. 2007.) center. ISBN 0-309-10192-l. Godel's Theorem: An Incomplete October 17, 2007: Proposals for The Best of All Possible Worlds: Guide to Its Use and Abuse, by Torkel NSF Postdoctoral Research Fellow­ Mathematics and Destiny, by Ivar Franzen. A K Peters, May 2 00 5. ISBN 1- ships. See "Mathematics Opportuni­ Ekeland. University of Chicago Press, 568-81238-8. (Reviewed March 2007.) ties" in this issue. October 2006. ISBN-13: 978-0-226- Great Feuds in Mathematics: Ten January 5, 2008: Applications for 19994-8. of the Liveliest Disputes Ever, by Hal IMA postdoctoral and New Directions Bourbaki, a Secret Society of Math­ Hellman. Wiley, September 2006. ISBN program. See "Mathematics Opportu­ ematicians, by Maurice Mashaal. AMS, 0-471-64877-9. nities" in this issue. June 2006. ISBN 0-8218-3967-5. How Mathematics Happened, by ''The Cat in Number/and, by Ivar Peter S. Rudman. Prometheus Books, Book List Ekeland. Cricket Books, April 2006. October 2006. ISBN 1-591-02477-3. The Book List highlights books that ISBN-13: 978-0-812-62744-2. How to Cut a Cake: And Other have mathematical themes and are Descartes: A Biography, by Desmond Mathematical Conundrums, by Ian aimed at a broad audience potentially Clarke. Cambridge University Press, Stewart. Oxford University Press, No­ including mathematicians, students, March 2006. ISBN 0-521-82301-3. vember 2006. ISBN 0-199-20590-6. and the general public. When a book Descartes: The Life and Times of a It's about Time: Understanding Ein­ has been reviewed in the Notices, a Genius, by A C. Grayling. Walker&Com­ stein's Relativity, by N. David Mermin. reference is given to the review. Gen­ pany, November 2006. ISBN 0-8027- Princeton University Press, September erally the list will contain only books 1501-X. 2005. ISBN 0-691-12201-6. published within the last two years, Einstein's Heroes: Imagining the John von Neumann: Selected Let­ though exceptions may be made in World through the Language of Math­ ters, edited by Miklos Redei. AMS, No­ cases where current events (e.g., the ematics, by Robyn Arianrhod. Oxford vember 2005. ISBN 0-8218-3776-l. death of a prominent mathematician, University Press, July 2006. ISBN-13: King of Infinite Space: Donald Cox­ coverage of a certain piece of math- 978-0-195-30890-7. eter, the Man Who Saved Geometry,

MAY 2007 NOTICES OF THE AMS 649 Reference and Book List by Siobhan Roberts. Walker & Com­ ''Negative Math: How Mathematics Shadows of Reality: The Fourth pany, September 2006. ISBN 0-802- Rules Can Be Positively Bent, by Al­ Dimension in Relativity, Cubism, and 71499-4. berto A. Martinez. Princeton Univer­ Modern Thought, by Tony Robbin. *Leonhard Euler, by Emil A. Fell­ sity Press, November 2005. ISBN-13: Press, March 2006. mann. Birkhauser, 2007. ISBN-13: 978- 978-0-691-12309-7. ISBN 0-300-11039-1. (Reviewed April 3-7643-7538-6. The Newtonian Moment: Isaac New­ 2007.) *Leonhard Euler, a Man to Be Reck­ ton and the Making ofModern Culture, The Shoelace Book: A Mathematical oned With, by Andreas K. Heyne and by Mordechai Feingold. New York Guide to the Best (and Worst) Ways to Library and Oxford University Alice K. Heyne. Birkhauser, 2007. ISBN- Lace Your Shoes, by Burkard Polster. Press, December 2004. ISBN 0-195- 13: 978-3-7643-8332-9. AMS, June 2006. ISBN 0-8218-3933-0. ''Letters to a Young Mathematician, 17735-5. Not Even Wrong: The Failure of (Reviewed December 2006.) by Ian Stewart. Perseus Books, April Stalking the Riemann Hypothesis: 2006. ISBN-13: 978-0-465-08231-5. and the Continuing The Quest to Find the Hidden Law (Reviewed in this issue.) Challenge to Unify the Laws of Phys­ of Prime Numbers, by Dan Rock­ The Lifebox, the Seashell, and ics, by . Jonathan Cape, the Soul: What Gnarly Computation April 2006. ISBN 0-224-07605-1. more. Pantheon, April 2005. ISBN Taught Me about Ultimate Reality, Once upon Einstein, by Thibault 0-375-42136-X. (Reviewed September the Meaning of Life, and How to Be D'Amour. A K Peters, March 2006. 2006.) Happy, by Rudy Rucker. Thunder's ISBN 1-568-81289-2. Superior Beings: If They Exist, Mouth Press, October 2005. ISBN The Pea and the Sun: A Math­ How Would We Know?: Game-Theo­ 1-560-25722-9. ematical Paradox, by Leonard M. retic Implications of Omnipotence, A Madman Dreams of Turing Ma­ Wapner. A K Peters, April2005. ISBN Omniscience, Immortality, and In­ chines, by Janna Levin. Knopf, August 1-568-81213-2. (Reviewed October comprehensibility, by Steven Brams. 2006. ISBN 1-400-04030-2. 2006.) Springer, second edition, November The Man Who Knew Too Much: Piano Hinged Dissections: Time to 2007. ISBN-13: 978-0-387-48065-7. Alan Turing and the Invention of the Fold!, by Greg Frederickson. A K Peters, Symmetry and the Monster: The Computer, by David Leavitt. Great October 2006. ISBN 1-568-81299-X. Story of One of the Greatest Quests Discoveries series, W. W. Norton, Piero della Francesca: A Mathe­ of Mathematics, by Mark Ronan. Ox­ matician's Art, by J. V. Field. Yale December 2005. ISBN 0-393-05236-2. ford University Press, May 2006. ISBN University Press, August 2005. ISBN (Reviewed November 2006.) 0-192-80722-6. (Reviewed February 0-300-10342-5. (Reviewed March The Math Instinct: Why You're a 2007.) 2007.) Matl1.ematical Genius (along with Lob­ Prince of Mathematics: Carl Fried­ The Three Body Problem, by Cath­ sters, Birds, Cats, and Dogs), by Keith rich Gauss, by M. B. W. Tent. A K erine Shaw. Allison and Busby, March Devlin. Thunder's Mouth Press, March Peters, January 2006. ISBN 1-568- 2005. ISBN 0-749-08347-6. (Reviewed 2005. ISBN 1-56025-672-9. 81261-2. October 2006.) Mathematicalfllustrations:AManual *Project Origami: Activities for Ex­ The Triumph of Numbers: How ofG eometry and PostScript, by Bill Cas­ ploring Mathematics, by Thomas Hull. Counting Shaped Modern Life, by selman. Cambridge University Press, A K Peters, March 2006. ISBN 1-568- I. B. Cohen. W. W. Norton, July 2006. December 2004. ISBN 0-521-54788-1. 81258-2. (Reviewed in this issue.) ISBN-13: 978-0-393-32870-7. (Reviewed January 200 7 .) Pursuit of Genius: Flexner, Einstein, The Trouble with Physics: The Rise Mathematics and Common Sense: and the Early Faculty at the Institute of String Theory, the Fall of a Science, A Case ofCreative Tension, by Philip]. for Advanced Study, by Steve Bat­ and What Comes Next, by . Davis. A K Peters, October 2006. ISBN terson. A K Peters, June 2006. ISBN 1-568-81270-1. Joseph Henry Press, October 2006. 1-568-812 59-0. ISBN 0-309-10192-1. Measuring the World, by Daniel Reality Conditions: Short Math­ Kehlmann. Pantheon, November 2006. Unknown Quantity: A Real and ematical Fiction, by Alex Kasman. Imaginary History ofAlgebra, by John ISBN 0-3 75-42446-6. Mathematical Association of America, Derbyshire. Joseph Henry Press, May More Mathematical Astronomy May 2005. ISBN 0-88385-552-6. (Re­ 2006. ISBN 0-309-09657-X. Morsels, by Jean Meeus. Willmann­ viewed August 2006.) *Useless Arithmetic: Why Environ­ Bell, 2002. ISBN 0-943396-743. The Road to Reality: A Complete Musimathics: The Mathematical Guide to the Laws of the Universe, mental Scientists Can't Predict the Foundations of Music, Volume 1, by by Roger Penrose. Knopf, February Future, by Orrin Pilkey and Linda Gareth Loy. MIT Press, September 2005. ISBN 0-679-45443-8. (Reviewed Pilkey-Jarvis. Columbia University 2006. ISBN 0-262-12282-0. June/ July 2006.) Press, February 2007. ISBN 0-231- Mystic, Geometer, and Intuitionist: The Secret Life of Numbers: 50 13212-3. The Life of L E. ]. Brouwer. Volume Easy Pieces on How Mathematicians Yearning for the Impossible: The 2: Hope and Disillusion, by Dirk van Work and Think, by George G. Szpiro. Surprising Truths of Mathematics, by Dalen. Oxford University Press, Octo­ Joseph Henry Press, March 2006. ISBN . A K Peters, May 2006. ber 2005. ISBN 0-198-51620-7. 0-309-09658-8. ISBN 1-568-81254-X.

65()1 NOTICES OF THE AMS VOLUME 54, NUMBER 5 American Mathematical Society Contributions

Dear Friends and Colleagues,

During 2006 your generous support helped the Society and our profession in many ways. I thank each of you for that support.

The Young Scholars program is in its seventh year, supporting summer workshops for talented high school students-the future of our profession. We are building an endowment, the Epsilon Fund, to support this program far into the future, and we hope to reach our goal of two million dollars over the next few years. Supporting such programs is important for mathematics.

The Centennial Fellowships play a key role for outstanding young mathematicians at the formative stages of their careers, from three to twelve years beyond the degree. These fellowships are funded by contributions from mathematicians throughout the world.

We use contributions to the General Fund to support all of our activities, including survey work, public awareness, and outreach to mathematicians in the developing world.

Your generosity allows the Society to carry out all these programs and shows that mathematicians care deeply about our profession. Thank you for that expression of caring.

john H. Ewing

Thomas S. Fiske Society

The Executive Committee and Board of Trustees have established the Thomas S. Fiske Society to honor those who have made provisions for the AMS in their estate plans. For further information contact the Development Office at 800-321-4AMS, or deve l opment@ams. or g.

Pedro B. Barquero Isidore Fleischer Ralph Mansfield Theda Salkilld Kathleen Baxter Ramesh Gangolli Trevor McMinn Margaret W. Taft Shirley and Gerald Bergum Sidney Glusman B. A. Taylor Shirley Cashwell Rosalind Guaraldq Franklin P. Peterson Steven H. Weintraub Carl Faith Yanguang Li Moshe Rosenfeld Sally Whiteman KyFan JosephS. Mamelak Bequests Received

Henry M. Schaerf

MAY 2007 NOTICES OF THE AMS 651 AMS Contributions

Gifts in Memory and Gifts in Honor The American Mathematical Society welcomes gifts made in memory or honor of members of the mathematical com­ munity or others. Unless directed toward a special fund or program, such gifts are used to support the general mission of the Society.

Gifts were made in memory of the following individuals: Maurice Auslander by Bernice L. Auslander by Carol-Ann Blackwood Paul Halmos by Susan Schwartz Wildstrom by Sigurdur Helgason William Ted Martin by Sigurdur Helgason Vincent 0. McBrien by Thomas W. Hungerford Vincent 0 . McBrien by joseph W. Paciorek Frederick Mosteller by Norton Starr Irving Reiner by Irma M. Reiner Arnold Ross by Charles W. Misne r Frederick Bodo Strauss by Delmar L. Boyer George Thomas by Susan Schwartz Wildstrom Kathryn B. Toll by Eugene Toll Albert Leon Whiteman by Sally Wh iteman

Contributors to the AMS During 2006 '' Donors who have given for three years consecutively. e Donors who have given to the AMS Epsilon Fund, the endowment for the support of young scholars programs. The names of donors who have given $1,000 or more in a single year are affixed to a plaque that is prominently displayed in the Society's headquarters.

PRESIDENT'S •*M. Susan Montgomery with matching gift from •*Gerald E. & Shirley Bergum <''Everett C. Dade •* Cathleen S. Morawetz Google ' Manuel P. Berriozabal ' Robert J. Daverman ASSOCIATES * Jacques Neveu Anonymous (3) ' David S. Berry e*Paul L. Davis (Gifts of $5,000 and above) •* Samuel Murray Rankin III •* John A. Berton • Stephen M. DeBacker * Norton Starr David & Monika Eisenbud- PATRONS * Martin Billik e*Guy M. De Primo To establish the Leonard <*Jean E. de Valpine •* joan S. Birman <''Robert L. Devaney Eisenbud Prize <*Steven H. Weintraub (Gifts of $100 and above) •*Richard L. Bishop e" Charles E. Dickerson Estate of Barbara J. Beechler •* Susan Schwartz Wildstrom •*William W. Adams •*Peter B. Bjorklund <*Loyal Durand * Sigurdur Helgason Genji Yoshino <* Irving Adler Carol·Ann Blackwood E"' Peter L. Duren •*Harry Lucas Jr. ' Rena}. Zieve & Greg * Roy L. Adler '*Paul E. Bland • lincoln K. Durst ' Oehmke Charitable Fund Kuperberg , ,, Alfred Aeppli • Delmar L. Boyer ' John W. Duskin Jr. ' Paul J. Sally Jr. Anonymous (2) ' J Ralph Alexander Jr. Raul Bravo ' Earl F. Ecklund Jr. * Robert & Maria W. Steinberg • Frank W. Anderson •*David M. Bressoud <*Elmer Eisner * Sally Whiteman-Alber! SPONSORS ' Tsuyoshi Ando <*James G. Bridgeman ' Jessie Ann Engle Leon Whiteman Prize H. Robert Andrews • Robert C. Brigham ' Solomon Feferman (Gifts of $500 and above) •*Kenneth I. Appel * Robert L. Bryant '' John C. Fenley ASSOCIATES * Arielle & Jerald Brodkey <*Richard F. Arenstorf <*Robert Bumcrot •''Gerald B. Folland Philanthropic Fund Manfred K. Armbrust ' Robert Calderbank '' Paul Fong (Gifts of $1,000 and above) , Richard A. Askey ' Jonathan W. Aronson ' James J. Callahan • Deborah S. Franzblau • *Richard D. Anderson ' Robert J. Blattner James G. Arthur • Sylvain E. Cappell ' Daniel E. Frohardt Edward D. Baker Karl E. Byleen • Scott E. Axelrod <*Concordia C. Chen Brock Fuller ' Felix E. Browder * Pierre E. Conner Jr. ' Raymond G. & Christine W. •* Stuart Citrin • William R. Fuller •''Nathaniel Chafee ' John B. Conway Ayoub •* Alfred Clark Jr. <*William Fulton ' Richard A. Cleveland * William L. Duren Jr. ' Edward A. Azoff <* Ed Cline <*Richard L. Gantos •*William Craig Clifford J. Earle <*Donald G. Babbitt ' John Coffey John B. Garnett <*john H. Ewing <*Timothy & Laurie Francis- John M. Bachar Jr. •*Daniel I. A. Cohen • Frederick W. Gehring '*George F. Haddix Wright ' Paul M. Bailyn ' Frederick R. Cohen ' Pedro J. Geraldo Cabrera * Carl E. Harrell •* Joseph Kist Joseph A. Ball <*Paul Dana Cole <*Anthony A. Gioia e*Bill Hassinger Jr. •*Robert V. Kohn • Claude W. Bardos • Donald B. Coleman ' Milton Alfred Glass • *John M. Hosack e*Gary ]. Kurowski Theodore J Barth <*Peter S. Constantin '' Andrew M. Gleason * Phyllis & Donald Kahn <*David B. Massey Frances B. Bauer <*Arthur H. Copeland Jr. James G. Glimm Philanthropic Fund <*James W. Maxwell E Patricia E. Bauman •*Louis J. Cote <*Richard P. Goblirsch ' Peter D. Lax <*Richard M. Schoen <*Edward Beckenstein ' Walter Craig J. K. Goldhaber •*George F. Leger <* Joel H. Spencer •* John A. Beekman • Michael G. Crandall • Martin Golubitsky & Barbara <*William James Lewis * Murray Marvin Stokely III <*George Benke <*Philip C. Curtis Jr. Lee Keyfitz

652 N OTICES OF THE AMS V OLUME 54 NUMBER 5 AMS Contributions

e Carolyn S. Gordon e J. ]. Malone e*Dev P. Sinha E Stephanie B. Alexander e*Gebhard Biieckle e Ronald L. & Fan Chung e Giles W. Maloof e* John R. Smart e* Gerald L. Alexanderson e'' S. Elwood Bohn Graham e* JosephS. Mamelak e William F. Smyth e* JohnS. Alin Miklos Bona * Wilfred Martin Greenlee Joseph F. Manogue e Louis Solomon e* Anne H. Allen <''Francis Bonahon e Robert Louis Griess Jr. * Edward Manougian e Aruna & David Spencer e Steven Alpern e William M. Boothby e*Phillip A. Griffith e Stefano Marchiafava e* Stephen E. Spielberg e William G. Altig e Leonard John Borucki e*Elizabeth Grossman & e*Thomas ]. Marlowe Jr. E Paul G. Spirakis e Vrege Jolfai Amirkhanian Wieb Bosma Joshua Boorstein e M. H. Martin e Olaf P. Stackelberg e Jose M. Amores Esteban e*Rupert D. Boswell Jr. e*Victor W. Guillemin ' Wallace S. Martindale III e*lvar Stakgold .-·'Fredric Davis Ancel C. Chaya Boughan , Richard K. Guy e* Jacob R. Matijevic e*Richard P. Stanley e* Joel H. Anderson ' Djamel Bourbia ' Mark Haiman Arthur P. Mattuck e*Ronald J. Stern ' Marlow E. Anderson e'' Aldridge K. Bousfield • Deborah Tepper Haimo e*R. Daniel Mauldin •''Glenn H. Stevens e*Michael T. Anderson e*Ward D. Bouwsma '*Richard M. Hain e* Stephen B. Maurer ' Stephan A. Stolz e*S usan Andim a * Paul ]. Bowron , Alfred W. Hales ' Thomas L. McCoy ' Tina H. Straley e*Peter P. Andre e*Mike Boyle * Mary-Elizabeth Hamstrom ' 0. Carruth McGehee •''Richard W. Sullivan e*George E. Andrews e* JohnS. Bradley e John L. Hank e*T. G. McLaughlin Margaret W. Taft with Philip M. Anselone e*Richard C. Bradley e Carsten Hansen * Brockway & Audrey W. matching gift from Oracle e*Peter H. Anspach ' Steven B. Bradlow ' David Harbater McMillan Corp. ,·•stuart S. Antman , Wray G. Brady Andrew William Harrell '' Robert F. McNaughton Jr. Jean E. Taylor ' Howard Anton * Louis R. Bragg e* Joe T. Harris Jr. Paul Meier e Laurence R. Taylor e''Myla M. Archer e*Kenneth A. Brakke Egbert Harzheim e*Ernest A. Michael Edward C. Thoele ' Vernor Arguedas E* Alberto Branciari e* Adam O'Neill Hausknecht John]. Michels ' John A. Thorpe e*Thomas E. Armstrong E Fred Brauer Mikihiro Hayashi ' David Middleton Eugene Toll Robert L. Arnberg George U. Brauer ' David R. Hayes ' Stephen S. Miller e*Selden Y. Trimble V Esperanza Blancaflor Arugay e Henry G. Bray e*William S. Heck e*Kenneth C. Millett ' Joann Stephanie Turisco e Bernice L. Auslander John C. Breckenridge e*Henry Helson e*Guido Mislin ' Harald Upmeier ' George Austin-Martin ' Michael A. Breen ' John P. Hempel ' Charles W. Misner ' H. N. VanEck ' Roger A. Avelsgaard e*Richard P. Brent e*Gerald A. Heuer Gordon D. Mock Donovan H. VanOsdol e*Sheldon Axler e joseph Edward Brierly '' Gloria C. Hewitt ' Alberto Cezar Moreira e*Wolmer V. Vasconcelos e*George Bachman * Judith E. Broadwin William R. Hintzman * Yasuhiro Morita e*David A. Vogan Jr. , Alexandru loan Badulescu e* Jerald S. Brodkey ' Samuel S. Holland Jr. e*Larry J. Morley e* Justin Clement Walker e*Richard J. Bagby e Farris Wayne Brown e* James E. Householder ' Robert A. Morris John H. Walter ' Charles R. Baker Johnny E. Brown e*W. L. Hoyt Paul S. Muhly e''Evelyn K. Wantland ' Kirby A. Baker * Kenneth S. Brown e*Tiao-Tiao Hsu e* Albert A. Mullin e1'Buck Ware ' Robert S. Baker E*Lawrence G. Brown e* James G. Huard ·• Peter E. Ney * Frank W. Warner III ' Erik jan Balder e*Robert F. Brown e*George W. Hukle * Paul ]. Nikolai •''Seth L. Warner e* John T. Baldwin ' Sharan Inez Brown e*Craig L. Huneke ' e*William E. Warren ' William R. Ballard ' W. Dale Brownawell e*Thomas W. Hungerford e*Nobuo Nobusawa ' David L. Webb e Edward ]. Barbeau Jr. e Robert R. Bruner * Franklin T. Iha ' Sunsook Noh ' Anthony A. Weidner e*Carlo Bardaro Barry W. Brunson ' Godfrey L. Isaacs e F. Alexander Norman ' Joel L. Weiner e'' Julio Edgardo Barety e Giacomo Bruzzo e*I. Martin Isaacs e Paul D. Olson ' Alexander H. Weintraub * Bruce H. Barnes e*Billy F. Bryant ' William H. Jaco e 0 . Timothy O'Meara e Brian Cabell White e*Domingo Barrera-Rosillo ' Rebecca A. Buchanan Bjarni Jonsson * Takashi Ono e*Kathleen B. Whitehead e Britt W. Barrett e*Nicholas P. Buchdahl e*Henry Price Kagey Robert Osserman e*Roger A. & Sylvia Margaret e* jose Barros-Neto e* joseph T. Buckley •''Herbert M. Kamowitz e* Joseph W. Paciorek Wiegand Karl F. Barth ' Richard S. Bucy e''Derek G. Kane e* Jingyal Pak e* John F. Wilkinson e*Philip R. Bartick e* Sebastian loan Buhai e Louis H. Kauffman ' Taxiarchis Papakostas e Susan Gayle Williams e Paolo de Bartolomeis ' Pierre Victor Bulens e* James E. Keisler e Thomas H. Parker ' Izaak Wirszup e* Alexander Barvinok e* Stephen S. Bullock e* John F. Kellaher * Henry J. Passerini ' Christopher Wolfe e* '' Daniel Willis Bump e*Wayne G. Kellner e*Donald S. Passman e*Scott A. Wolpert e*R. Michael Beals e* Almut Burchard e*Harry Kesten e* Alexander Perlin * Jay A. Wood e*William A. Beck e R. B. Burckel * Thomas C. Kipps W. V. Petryshyn e* Alan C. Woods * William H. Beckmann Krzysztof Burdzy e* Allan M. Kirch ' Mollie Pflumm ' Tsu C. Wu e*Robert Beig e John Brian Burghduff ' jane & James Kister e''Mark A. Pinsky e''Tatsuhiko Yagasaki ' Harold Bell e* James E. Burke e*Maria Margaret Kiawe * George Piranian e*Masayuki Yamasaki ' Steven R. Bell e* Donald L. Burkholder Peter Klein * VeraS. Pless Catherine Huafei Yan e''Wolfgang Bell e*Warren T. Burns Jr. e*Roland R. Kneece Jr. • Aleksey Popelyukhin ,•]. Michael Yohe e*Katalin A. Bencsath ' Stefan A. Burr ' George H. Knightly e*Paul H. Rabinowitz ' Sam Wayne Young e*Carlos Benitez e'' Ralph Stevens Butcher * Antoni A. Kosinski James v. Ralston e* Allen D. Ziebur e* Georgia Benkart e*Thomas R. Butts e*Eric J. Kostelich ' George N. Raney e''Paul F. Zweifel ' Irving 0. Bentsen e Robert Lawrence Byrom e*Thomas R. Kowalski e*M. M. Rao Anonymous (70) e*Gary D. Berg ' Rotraut C. Cahill ' Ralph M. Krause , Michael Reid ' Alan E. Berger M. Carme Calderer ' joseph B. Kruskal e''Robert ]. Reynolds ' Steven B. Berger ' William C. Calhoun e* •''Marc A. Rieffel FRIENDS e'' George M. Bergman ' Eric Cane! ' S. T. Kuroda * John Roe (G ifts of less than $100) e*Leonard D. Berkovitz e*Corrado Cardarelli e* B. Lacampagne e*Vijay K. Rohatgi Johan F. Aarnes ·• Salvatore D. Bernardi e* ]on F. Carlson * Jeanne LaDuke ' Hugo Rossi e Ian M. Aberbach <''Christopher Bernhardt e*David W. Carter ' Carl E. Langenhop ' Richard L. Roth e*Ciarence M. Ablow ' Swanhild Bernstein Thomas ]. Carter e* Joseph A. Langsam ' Zalman Rubinstein , Thundiyil Samuel Abraham ' Carlos M. Betancourt Paolo Casali e*Richard G. Larson ' David Ryeburn e*William P. Abrams •*Gerhard Betsch ' James R. Case Richard K. Lashof e* Alexander A. Rylov * lain T. A. C. Adamson ' Edward james Bevan e*Robert G. Cawley e*Walter R. Lawson e* Jeffrey R. Sachs ' Arnold M. Adelberg ' Nicholas]. Bezak e Afton H. Cayford * Alan C. Lazer e*Habib Salehi e* Jeffrey D. Adler e*Meempat Bhaskaran Thomas E. Cecil e*William G. Leavitt ' Samuel Schechter ' Alan C. Adolphson ' Marilyn S. Bickel , * Gulbank D. Chakerian e* John M. Lee ' Michael Schlessinger N. U. Aluned * Klaus D. Bierstedt e Richard A. Champion Jr. e*Robert N. Leggett Jr. ' Bertram M. Schreiber T. M. G. Ahsanullah e*Terrence Paul Bisson ' Bruce Chandler e* Joan R. Leitzel e*Mark Schroder 'e*T. Aikou e*Denis L. Blackmore ' Weita Chang e*H. W. Lenstra e* Jerry D. Schuur •''Michael I. Aissen David E. Blair e*Ruth M. Charney e*William]. LeVeque ·• Stuart A. Seligson •''Ethan ]. Akin •''William D. Blair e*Ronald ]. Chase e'~Norman e* John D. Blanton e*Elliott H. Lieb E. Sexauer ' Stanley A. Alama e*Pak Soong Chee e*Sally Irene Lipsey ' Abdulalim A. Shabazz e*Michael 0 . Albertson e''Steven E. Blasberg ' * Albert T. Lundell e*Freydoon Shahicli e Kenneth S. Alexander ' David Alan Blomberg e*Kwan·Wei Chen e*Russell D. Lyons , Richard ]. Shaker e Roger K. Aiexander e''David S. Bloom * Robert Chew

MAY 2007 N OTICES OF THE AMS 653 AMS Contributions

e*Theodore S. Chihara e*Ogun Dogru e Simon John Fraser e*William D. Hahn Popescu Ionela E Julius B. Chi.ni e''Pierre E. Dolbeault ' Christopher L. Frenzen e DawitHaile e Lynne Kamstra lpina * Choong Yun Cho * Igor Dolgachev e* Peter J. Freyd ' Douglas F. Hale '' Ron Irving e" jal R. Choksi e* james P. Donaly e*Eugene M. Friedman e*Robert Joseph Halk ··· Richard E. Isaac e* Charles C. Chouteau e Robert S. Doran e* Merwyn M. Friedman e*Brian C. Hall '' Noburo Ishii * William E. Christilles e Jay Robert Dorfman e Robert David Friedman e* james E. Hall e*Noboru Ito e* e'' Jim Douglas Jr. e*Jurg M. Frohlich e''David Handel N. M. Ivochkina * Philip T. Church e*Ronald G. Douglas e Bent Fuglede , Zeljko Hanjs V e* Eric Robert Jablow e*R ichard C. Churchill e Martin J. Dowd e* Hisanori Fujita ' Zhifeng Hao '' William Burkiey jacob e Erhan Cinlar e* Alex J. Dragt e Koji Fukuda e Heiko Harborth e*Robert E. Jaffa e*Kevin F. Clancey Ronald Lewis Drake e F. Brock Fuller e john Michael Harby e John Antone Jaksich e*Chester Dodge Clark Alexander N. Dranishnikov e Robert A. Fuller Robert M. Hardt David M. James John Thomas Clark * Arthur A. Drisko e*J effry B. Fuqua e''Beverly Bailey Hargraves e''Melvin F. Janowitz e* Robert A. Clark e* Bruce K. Driver e*Steven Allen Gabriel Dorothee D. Haroske e Heera La! Janwa e* jack D. Clayton * Thomas L. Drucker e*Andr ei Gabrielov Steven Guy Harris e* Herbert Jarszick e Matthew Clegg ' Krishan La! Duggal Marvin C. Gaer e*Fred F. Harrop •''Trevor M. Jarvis e*Phillp A. Cobb e*Steve N. Dulaney e*Michael E. Gage George Hassiotis e*George A. Jennings e* Alan Cobham , Stephen C. Dumars e Luis Gallardo e*Akio Hattori e*Charles H. Jepsen e James A. Cochran e*William Dart Dunbar Jr. e Jean H. Gallier ' Ralph H. Hautau e David Jerison e*John C. Cock * Kanat Durgun e Luis E. Galup ' Trevor 0. Hawkes e* Eugene C. Johnsen e* James Wesley Cogdell e*Timothy R. Eaton Tord H. Ganelius •''Melvin L. Heard * Trygve Johnsen e*Amy Cohen e*Patrick Barry Eberlein e* Joseph M. Gani e William R. Hearst III e*D. Randolph Johnson e Donald L. Cohn Dorothea M. Eberz-Wagner * Howard Garland e''Leo Hellerman Dale Martin Johnson e*George Cole e* Allan L. Edmonds * Lioyd A. Gavin e Simon Hellerstein e David Copeland Johnson e A. Jolm Coleman e H. P. Edmundson e Dale E. Geer e" judson Hendelman e*Donald G. Johnson e*Vincent E. Coli Jr. e'' C. Henry Edwards e Jean Raymond Genet e*Francis McVey Henderson e* Guy johnson Jr. * Daniel Comenetz ' Paul J. Eenigenburg e Ross Geoghegan e Christopher M. Herald e'' Harold H. Johnson e*Frank F. Connor Isaac Efrat '*John C. Georgiou e'' james B. Herreshoff e* Jon L. Johnson e Bruce P. Conrad e*William I. Eggers ' Hillel H. Gershenson e Patricia Hersh e Kenneth David Johnson ' Kenneth L. Cooke ' Kurt Merrell Ehlers e*Murray Gerstenhaber e*Michael J. Heumos e*Theodore D. johnson * Thomas A. Cootz ' Leon Ehrenpreis e Margaret P. Gessaman e Yasunari Higuchi David W. Jonah * Heinz 0 . Cordes •''Stanley Mamour Einstein- e Abraham Gill e''Hugh M. Hilden e*William B. Jones e*H. Cornet Matthews e*Tepper L. Gill e*Peter J. Hilton e Kristin Halla J6nsd6ttir Steve A. Corning e Alexander S. Elder e*Maurice Eugene Gilmore e David M. Himmelblau e*Virginia V. jory * Thomas Carney Corrigan •''David Eliezer ' Andrew Michael Girard •''Nancy Hingston e* Joaquim]. A. Judice ' Chris Cosner e'' Joanne Elliott e* Samuel Gitler e John J. Hirschfelder e Winfried Just e*Malcolm A. Coulter e''Steven P. Ellis e*Earl C. Gladue e F. E. P. Hirzebruch e* james H. justice Carl C. Cowen ··· Richard S. Elman e*Raoul F. Gloden e''Peter David Hislop e* jeffry N. Kahn ' Robert H. Cowen e"Paul W. Eloe e*Kazimierz A. Goebel e*Chungwu Ho e Peter ]. Kahn ' Marion Lonnie Cox e John F. Elter e Valentina Gogovska e*Arthur M. Hobbs e* Yuichir6 Kakihara , Petru Teodor Craciunas e''Thomas J. Emerson e Edray Herber Goins e''Harry Hochstadt e Agnes M. ·Kalemaris jessica Marguerite Craig e Koji Endo e* Robert Gold Melvin Hochster Diana Kalish e*Stephen H. Crandall e'' Hans P. Engler e*Samuel Goldberg e* Scott H. Hochwald e N.J. Kalton ' Henry Crapo '' Philip G. Engstrom e* Seth I. Goldberg e*J onathan P. E. Hodgson e" Richard A. Kanner e*Thomas M. Creese e Kaspar Daniel Ernst e* William Mark Goldman e*Helmut H. W. Hofer e Tsuneo Kanno ' Ernest S. Croot III e Linda L. Eroh e*Daniel A. Goldston e David A. Hoffman e* Stanley Kaplan e*Donald L. Curlovic e'~ Kumar Eswaran Michael Golomb e* Michael E. Hoffman e*Wilfred Kaplan ' William David Curry II e''Leonard Evens e*J ose Luis Gomez Pardo Detlev W. Hoffmann e*Ioannis Karatzas e''Herbert J. Curtis e W. Norrie Everitt e* Kenneth R. Goodear l e*Tom H0holdt e* Martin Lewis Karel Raul E. Curto , Edward R. Fadel! e* Roe W. Goodman * Charles S. Holmes ,,, Julian R. Karelitz Jim M. Cushing e'' Carl Faith ' Steven Gore e* Philip John Holmes Victor]. Katz e*David Scott Cyphers e*Edson de Faria e* john A. Goth Richard B. Holmes e* Arthur Kaufman Marius Dadarlat e* Bruno Farina e*Yasuhiro Goto e Roger H. Homer e Kiran S. Kedlaya ' H. Garth Dales e William M. Farmer e* Clau de Goutier e Jennifer L. Hopkins e* Anthony ]. Keeping Stephen Michael Dalton Amassa C. Fauntleroy e*Kevin A. Grasse e'' Jean MacGregor Horn Arnold Peter Keet ' David B. Damiano e Ruth G. Favro e Jack E. Graver Jim E. Haste e Niovi Kehayopulu e'' James N. Damon * George F. Feeman e Alan S. Graves e*Fredric T. Howard e Bernhard M. Keller e*Martin P. Dana e''Mark E. Feighn e*Larry K. Graves * Henry C. Howard ,*Edward L. Keller Donald A. Darling e Michael Bertrand Fein ' john W. Gray e Everett W. Howe e David C. Kelly e''Boris A. Datskovsky e Paul Feit james A. Green * ]. S. Hsia e John F. Kennison * Donald M. Davis e Arnold D. Feldman Vladimir A. Greenberg e Wu-Hsiung Huang e*Haidar G. Khajah e* Martin D. Davis e''Norman Feldman e* Curtis Greene James A. Huckaba e* Efim Khalimsky Thomas Charles Davis Dominique Fellah e* Peter H. Greene e*Arc hibald Perrin Hudgins e Dima Khavinson e'' Anthony T. Dean e Helaman Ferguson e* Frederick P. Greenleaf e*Denise Huet t*Kazuo Kido e*Daniel G. Degiorgi e Jose Humberto Ferreira Rosa Stanley ]. Greif e Anne Hughes * Rudger W. Kieboom e Ronald W. DeGray e*lan M. Ferris Alain A. Grigis e* Mark E. Huibregtse ,*Peter C. Kiessler e* Percy Alec Deift e James Allen Fill ' Helmut Groemer e* Birge K. Huisgen- Michael K- H. Kiessling e*Dominick Del Casale '' David V. Finch e* Leonard Gross Zimmermann Steven]. Kifowit e* Aristide Deleanu e Leib Finkelstein e*Edward H. Grossman •·'James E. Humphreys * Jun Kigami * Franklin D. Demana e Gene Fiorini e*Robert Andrew Grossman E Bruce Hunt Cbul Kim e* Ralph E. DeMarr ' William T. Fishback e* Edward A. Grove t*Karen C. Hunt Wilfred M. Kincaid e* Jocben Denzler e''Benji N. Fisher <''Rosalind]. Guaraldo e''Paul M. Hunt e* Donald R. King ' John E. Derwent e Marjorie Fitting-Gifford e* Estber E. Guerin * Beryl E. Hunte e Walter W. Kirchherr e* Charles A. Desoer ef• Uri Fixman e juan Mateos Guilarte e* Marcel Hupperich e Alexander Kirillov Jr. '' Dennis DeTurck e*Bernard A. Fleishman e Craig R. Guilbault e james F. Hurley e Alexander A. Kirillov Sr. e* Fred I. Diamond •''Richard J. Fleming * Wynne Alexander Guy e* Michael G. Hurley e* Ellen E. Kirkman e* Charles R. Diminnie e Wendell H. Fleming e Kwang Chul Ha e''Edward L. Hutton e'' Jan Kisyilski , David Dirninnie e Russell Alan Foltz-Smith e Seymour Haber e* ] ang C. Hwang E Phyllis M. Kittel e*Gerald P. Dinneen e*Juli e A. Fondurulia e Petros Hadjicostas ,*Tom Ilmanen e*Peter H. Kleban ' Peter G. Dodds e Robert A. Fontenot e Esfandiar Haghverdi e Ettore Infante E* Bruce A. Kleiner Zane Dodson ' John Michael Fox e*Peter Hagis Jr. e'' Hiroshi Inoue e*Juli a Knight E Boro Doering e* Hans-Bjorn Foxby Susan G. Hahn e Arnold ]. Insel e* Marvin I. Knopp

654 NOTICES OF THE AMS V OLUME 54, NUMBER 5 AMS Contributions

' Donald E. Knuth ' Tsai-Sheng Liu •*Michael J. Miller ' Brad G. Osgood , Amado Reyes Donald I. Knutson Xiaoyan Liu Nicholas S. Miller •*Steve G. Oslon * Fazlollah Reza t*Tsuyoshi Kobayashi •*George W. Lofquist •* Russell G. Miller •* James M. Osterburg , Lisa M. Rezac * Richard M. Koch Terry M. Lohrenz •*Thomas Len Miller ' William Oswald ' Charles W. Rezk <'Yoshiharu Kohayakawa <''Walter L. Lok Victor Saul Miller ,., Javier Otal •*Tong-Shieng Rbai •* Kurt Siegfried Kolbig ' Charles ]. Lombardo •*William Da\~d Miller Carol B. Overdeep •*Martin G. Ribe ' R. ]. Kolesar John M. Long ., Milton A. Mintz •*Michio Ozeki •*Stephen ]. Ricci e~'Yasuo Komori <*William C. Lordan <*Norman D. Mirsky •*Istvan Ozsvath •''Richard S. Rich ' Pasaad Kongtaln •* Michael P. Loss •*Michal Misiurewicz •*Richard G. Pace Horst P. Richter Heinz J. Konig •*Tsu-Ming Lu ' Stephen Ames Mitchell ' Victor P. Palamodov ' John H. Rickert ' M. Koras ' Thomas G. Lucas •*Theodore Mitchell Fotios C. Paliogiannis •*Benjamin Rickman ' Nicholas ' Eric E. Lund ' Lothrop Mitten thai ' Bruce P. Palka •*Eleanor G. Rieffel <'' jerald]. Kovacic ' Norman Y. Luther ' Hisao Mizumoto •''Diethard Ernst Pallaschke , Carl R. Riehm ' Istvan Kovacs •*Kirill C. H. Mackenzie ' Ismail ]. Mohamed ' Nikolaos S. Papageorgiou •*Ronald Edgar Rietz ' Leonid V. Kovalev •* James Joseph Madden ' ]. Donald Monk •''Kyoo-Hong Park ' Robert D. Rigdon ' Erwin Kozakiewicz •'' Adolf Mader <''Peter L. Montgomery ' Elwood G. Parker Gregory Riggs Irwin Kra ' Kazem Mahdavi •*Barbara B. Moore e* George D. Parker , Timothy Rupert Riley e David P. Kraines * Mehran Mahdavi * Hal G. Moore •'' Alberto Parmeggiani •'' Jose Rio Allan M. Krall e Franz Maier •*Richard A. Moore <''Walter R. Parry •*Thomas W. Rishel •* ]urg Kramer john C. Mairhuber * W. Keith Moore •*David A. Pask ' Roberto R. Rivera •*Raymond F. Kramer Jr. e*Peter Malcolmson •*frank Morgan •* John]. Pastor Joel L. Roberts , Herbert C. Kranzer ' Frederik Malfait ' John W. Morgan <'' Leonid Pastur ,., Joseph B. Roberts e·'Bernd Krauskopf ,,, Joseph Malkevitch George W. Morgenthaler ' Donald A. Patterson •''Lois J, Roberts •*Gary R. Krumpholz ' Dennis R. Maim , Joseph R. Morris " Walter M. Patterson III ' E. Arthur Robinson Jr. ' Arthur H. Kruse ' Salvador Malo ' L. E. Morris E'' Charles M. Patton ' Stanley A. Robinson * Yoshiki Kurata •*David M. Malon * John A. Morrison <''Sandra 0. Paur , Tom Roby •*Robert P. Kurshan ' Pauline Mann-Nachbar •*jo seph G. Moser ' Krzysztof Pawalowski , David W. Roeder e*Ruishi Kuwahara ' Haria Mantellini ' Ronald G. Mosier •''Robert G. Payton ' Hartley Rogers Jr. •''Nosup Kwak e Hrant Babken Marandjian * Pierre Marie Moussa ' Erik Kjaer Pedersen •*David E. Rohrlich ' jean Pierre Lafon ' Peter D. March ' Allan Muir •''Lambertus A. Peletier •''Judith Roitman jeffrey C. Lagarias •* Margaret 0. Marchand ' Francesco Munari ' Richard P. Pembroke •'' josephine Jardin Romero •*Daniel Y. Lam ' Eugene A. Margerum •*Marvin G. Mundt , John w. Pennisten •*Guillermo Romero Melendez •* John Patrick Lambert '' David E. Marker ' James R. Munkres " Paul M. Pepper •*William L. Root * William A. Lampe <'' Charles Michel Marie ' David C. Munton •" juan C. Per a! ' Mario Rosati Clifton A. Lando •'"Murray Angus Marshall •*Grattan P. Murphy , Robert V. Perlis ' Nicholas J. Rose •*Peter S. Land weber •''Mario Umberto Martelli ' R. Bradford Murphy ··· William L. Perry * Robert A. Rosenbaum ' William E. Lang ' Donald A. Martin •* Jan Mycielski Charles Samuel Peskin ,.Jon athan M. Rosenberg •*David C. Lantz ' Jeremy Leander Martin ' Donald E. Myers •* justin R. Peters , '· Gerald Rosenfeld <''Arnold Lapidus ' Nathaniel F. G. Martin •*Takasi Nagahara ' Bent E. Petersen , John Rosenknop •* Michel L. Lapidus ' Senisho Philip Mashike •* Alexander Nagel ' Norbert Peyerimhoff , Kenneth A. Ross •* Peter A. Lappan Jr. Bernard Maskit e* Kuniaki Nakamitsu ' Anthony V. Phillips ' Sharon Cutler Ross * Margaret M. LaSalle Robert M. Mason •* Kazumi Nakano •*Ilya Piatetski-Shapiro Adrian S. Roth * George Laush , Attila Mate M. Zuhair Nashed ' Jonathan Pila , ' Mitchell J, Rothstein <*Lorraine D. Lavallee ' John Norman Mather ef'J ames B. Nation * Anand Pillay •*Christel Rotthaus e* Albert F. Lawrence '*Frank H. Mathis Claudio Francisco Nebbia e* Steven Pincus •*Virginia G. Rovnyak •* john W. Lawrence ' Ronald M. Mathsen •*J oseph Neggers ' Thane Earl Plambeck , Ranjan Roy •*H. Blaine Lawson Jr. Anastasios Matzavinos •*Csaba Nemethi •* John William Poduska Sr. ' James Samuel Royer e Jimmie D. Lawson ' Donald E. Maurer e"' Walter D. Neumann * john C. Polking •*Melvin Glenn Royer <''Robert F. Lax •'' John C. Mayer ' Siegfried F. Neustadter •*Robert]. Pollack * Wimberly C. Royster * Robert K. Lazarsfeld ' Raymond A. Mayer Jr. •*Lee P. Neuwirth •* Harriet S. Pollatsek ' Daniel Ruberman Ian]. Leary t*Rafe Mazzeo ' Eugene D. Nichols ' Florian Pop •* Robert ]. Rubin Dong Hoon Lee * Michael ]. McAsey ' Liviu I. Nicolaescu ' Richard C. Potter Joachim H. Rubinstein , Ko tik K. Lee ' Jon McCammond ' Heinrich Niederhausen ' Mark Adam Pavich • Ernst A. Ruh ]. Larry Lehman Dennis ]. McCaughan Seiki Nishikawa •*Robert T. Powers , Daniel P. Runyan * Gerald M. Leibowitz ' Gregory L. McColm ' Togo Nishiura * Narahari Umanath Prabhu •*Stanley J. Russell •''Manoel Jose M.S. Lemos Robert M. McConnel •*Ricardo H. Nochetto ' Martin E. Price ' Bernard Russo <''james I. Lepowsky ' Robert A. McCoy •*Scott R. Nollet •*David S. Protas •* Leon W. Rutland Jr. •*Steven C. Leth ' Marjorie Frost McCracken ' Virginia A. Noonburg ' Loki Der Quaeler ' Dorothy S. Rutledge Howard A. Levine Michael M. McCrea * John W. Norris ' Adbeel N. Quinones t*Cihan K. Saclioglu * Robert ]. Levit ' Leon R. McCulloh ' Tom Eldon Norwood ' Eric Todd Quinto •* Anthony Sacremento ' Michael David Levy <'' William D. Mcintosh ' Sergei Yakovlevich Novikov £'' Baker Rader t*Takashi Sakai ' Douglas Lewan •''Thomas G. McKay , Olav Kristian Nygaard , David E. Radford •*Hector N. Salas •*Andrew M. Lewis ' Robert W. McKelvey •*Duane Q. Nykamp •* Louis B. Rail , Luis C. Salinas ' D . ] . Lewis <'' Lionel W. McKenzie ' James E. Nymann Dinakar Ramakrishnan ' Scott Anthony Sallberg •*George M. Lewis ' James P. McKeon e··~ serge Ochanine •*Melapalayam S. Ramanujan •*Laurent Saloff-Coste ' *H. L. Lewis <''Robert C. McOwen * Mitsuyuki Ochiai '' R. Michael Range ' Daniel Saltz •''Roger T. Lewis •'' John C. Meakin •* Andrew M. Odlyzko •'' Salvatore Rao , Jose Luis Sanchez Palacio •* Frederick W. Leysieffer •'' David Meier •* Andrew P. Ogg ' Louise Arakelian Raphael £·' Robert W. Sanders ' james E. L'heureux John E. Meier '' Yong-Geun Oh ' Wayne Mark Raskind ' Jeffrey D. Sandstrom Congming Li · Anders Melin ' jack E. Ohm ' Dwijendra K. Ray-Chaudhuri * Angel San Miguel ' Richard Allan Libby ,,, Jose M. R. Mendez-Perez ' Mayumi Ohmiya ' Marko Razpet ' Ulderico Santarelli e Zvie Liberman ' Edward P. Merkes •*Michael L. O'Leary <''Maxwell 0 . Reade ' ]. M. Sanz-Serna * Stephen Lichtenbaum David M. Merriell Loren D. Olson * Douglas C. Reber •*Donald E. Sarason Eduardo Lima de Sa Bruce E. Meserve ' Frank W. ]. Olver •*Don Redmond ,*Hiroki Sat o t''Shen Lin t*Siavash Meshkat ' John Arthur Oman * Christopher L. Reedy * Stanley A. Sawyer ' Gregory L. Linder Arne Meurman •*Philip]. O'Neil •*David E. Reese <* Richard C. Scalzo , .• John E. Lindgren •* Jean-Pierre G. Meyer Barrett O'Neill * Ernestine Reeves-Hicks •*Paul T. Schaefer •''Peter A. Linnell William A. Michael * Yoshitsugu Oono •*Eugenio Regazzini , Juan Jorge Schaffer •*Miriam A. Lipschutz-Yevick , .• Marvin V. Mielke ' Seth F. Oppenheimer e*Irma M. Reiner •*Doris W. Schattschneider •*William G. Lister •* Michael H. Millar ' Edward T. Ordman ' john H. Reinoehl * Gideon Schechtman ' John B. Little ' Ellen Rarnmelkamp Miller Peter P. Orlik * Robert B. Reise! ' Markus Schmidmeier <''Chiu-Chu Melissa Liu ' Jack M. Miller * Kent Orr ··· Richard S. Rempel ' Harvey]. Schmidt Jr. t*Ming Chit Liu * Kenneth S. Miller •*Mason S. Osborne ' Michael Bela Revesz ' Wolfgang M. Schmidt

MAY 2007 NOTICES OF THE AMS 655 AMS Contributions

Norman L. Schryer <*Wilbur L. Smith • Krystyn Styczen • Juan Luis Valle • Andrew P. Whitman * John Schue * Joel A. Smoller <*Kelly John Suman Johannes A. Van Casteren Kenneth G. Whyburn * George W. Schueller E Alan David Sokal • Andrei A. Suslin Aernout C. D. Van Enter G. Kenneth Williams •*Paul E. Schupp * Emilio del Solar-Petit • Myron M. Sussman E Werner Varnhorn Lawrence R. Williams • Charles Freund Schwartz • Bruce Michael Solomon Ellen E. Swanson •*Michael Voichick e* Charles K. Williamson * Gerald W. Schwarz David Robert Solomon •*William J. Sweeney •*Dan-Virgil Voiculescu • Susan Williamson • Willi Schwarz Kamal Nasir Soltanov Roman Sznajcler e*Paul A. Vojta • Paul A. Willis •*Michael]. Schwietzer • Sung Yell Song • Earl J. Taft * Hans W. Volkmer e* Alan Stephen Wilmshurst • Warner Henry Harvey Scott III • *Linda R. Sons e*Kazuaki Taira • Diane Voss • Richard Wilson • George F. Seelinger John]. Sopka e*Lajos F. Takacs • David H. Wagner * Robert Lee Wilson •*Howard A. Seid • Ralf]. Spatzier e*Hisao Tanaka •*Philip D. Wagreich • George Washington Whnbush •*George Seifert •*Birgit Speh Jun-Ichi Tanaka •* Jonathan M. Wahl • John W. Wingate Kent Seinfeld • *Dennis Spellman * Yoshihiro Tanaka • Sebastian Walcher * Eric J. Wingler •*George B. Seligman • David H. Spring • Daniel Louis Tancreto e*David B. Wales •''F. Wintrobe e Francesco Serra Cassano • Richard H. Squire • Folkert M. Tangerman • Homer F. Walker • Bettina Wiskott • Robert E. Seydel Jr. Aravamuthan Srinivasan • Peter Tannenbaum e*William Wallace • Louis Witten • Patrick Shanahan * Ram P. Srivastav • James]. Tattersall • Nolan R. Wallach * James J. Woeppel e*Priti Shankar * ]. T. Stafford • Nicholas F. Taussig •*John Thomas Walsh e*Stephen D. Wolthusen •*Gerald Peter Shannon • Friedemann W. Stallmann * Michael D. Taylor • Hans Ulrich Walther • Sung-Sik Woo •*Henry Sharp Jr. • William L. Stamey • Zachariah C. Teitler Roger P. Ware • Japheth L. M. Wood • John C. Shepherdson e*Paul H. Stanford • Jean-Marc Terrier •*Bette L. Warren •*John W. Wood • Richard B. Sher <''Lee James Stanley •*Paul M. Terwilliger Lawrence C. Washington e*George V. Woodrow III e*Kenichi Shiraiwa e*Walter C. Stanley * Andrew]. Terzuoli e*Robert H. Wasserman • Christopher T. Woodward •*William Ivan Shorter * Charles S. Stanton •*Francisco Javier Thayer e*Michiaki Watanabe Yasuko Yamazaki • Vladimir Shpilrain • Edmund Beauclerc Staples III • ''Lawrence E. Thomas e*Sh6ji Watanabe •*Michael Yanowitch •*Steven E. Shreve * Christopher W. Stark • Edward G. Thurber • Shuji Watanabe e''Fawzi M. Yaqub • Warren E. Shreve •* e'' James G. Timourian e*Toshihiro Watanabe e*Suresh Yegnashankaran • Stanley R. Shubsda Jr. • Sherman K. Stein • Frederick C. Tinsley William C. Waterhouse • Peter Yff • Alexander S. Shumovsky •* Charles !. Steinborn Wladyslawa Toczycki e*David S. Watkins Ki-Jo Yoo <*Stuart J. Sidney David L. Stenson * Daniel B. ]. Tomiuk <*Mark E. Watkins • Donald F. Young • Franc;ois Sigrist •*Ellen M. Stenson Nicolo Goodrich Torre * Edward C. Waymire • Radu Zaharopol e* Allan]. Silberger • Kenneth Stephenson Craig A. Tracy e*Cary H. Webb e*Nobuo Zama e*Daniel S. Silver • Peter ]. Sternberg •*Charles R. Traina • Hans F. Weinberger * Jean-Claude Zambrini • Patrick]. Sime • T. Christine Stevens • William R. Transue • Gideon L. Weinstein • Hermann Zapf • Giorgio Simeoli • John R. Stock Marc Troyanov •*Michael I. Weinstein e*Franc;ois Zara • lloyd D. Simons • Paul K. Stockmeyer • Long-Yi Tsai • Walter Weinstein * Michel M. Zarka • Charles C. Sims Gunter H. Stolz •*Ralph P. Tucci Guido L. Weiss e*Thomas Zaslavsky e*Iakov G. Sinai •*Lawrence D. Stone • Edward C. Turner <''MichaelS. Weiss Steven M. Zelditch •''David B. Singmaster * Emil J. Straube •* Johan Tysk • lloyd R. Welch David E. Zitarelli e*Hardiv H. Situmeang •*Walter A. Strauss e*Tomio Umeda • Jon A. Wellner • Steven M. Zucker e*Walter S. Sizer * Robert S. Strichartz • Yasushi Unai Unai •*David M. Wells Anonymous (276) e*Thomas Skill e* Gerhard 0. Strohmer • Michael Ungs * John C. Wenger • Michael Slattery • Daniel W. Stubbs •*John A. W. Upton •*Henry C. Wente Eric V. Slud • Garrett James Stuck Alejandro Uribe •* Aric ]. Werly Wayne Stewart Smith Eric A. Sturley • Colleen A. Vachuska • John Wermer

656 N OTICES OF THE AMS VOLUME 54 N UMBER 5 From the AMS Secretary

Officers of the Sodety 2006 and 2007 Updates

Except for the members at large of Members at Large Publications Committees the Council, the month and year of All terms are for three years and Bulletin Editorial Committee the first term and the end of the pres­ expire on January 31 following the Susan]. Friedlander 7/05-1/09 ent term are given. For members at year given. large of the Council, the last year of Colloquium Editorial Committee 2006 Paul]. Sally Jr. 2/05-1/08 the present term is listed. James W. Cannon Sylvain E. Cappell Journal oftheAMS Editorial Council Beverly E. ]. Diamond Committee 2/04-1/0 7 President Mark Goresky Robert K. Lazarsfeld 2/07- 1/10 James G. Arthur 2/05-1/07 Alejandro Uribe Mathematical Reviews Editorial James G. Glimm 1/07-1/09 2007 Committee President elect Sara Billey Jonathan I. Hall 2/06-1/09 James G. Glimm 2/06-1/07 Carolyn S. Gordon Sheldon H. Katz Mathematical Surveys and Immediate Past President Monographs Editorial Committee James G. Arthur 2/07-1/08 Michael F. Singer Catherine H. Yan ]. Tobias Stafford 2/05-1/08 Vice Presidents 2008 Mathematics ofComputation Ha'im Brezis 2/05-1/08 Editorial Committee Robert L. Bryant 2/07-1/10 William M. Goldman Craig L. Huneke Chi-Wang Shu 2/02-1/08 Ruth M. Charney 2/06-1/09 Proceedings Editorial Committee Vaughan F. R. Jones 2/04-1/07 Judy Anita Kennedy Ken Ono Ronald Fintushel 2/06-1/10 Secretary Judy L. Walker Transactions and Memoirs Editorial Robert ]. Daverman 2/99-1/09 2009 Committee Associate Secretaries Robert L. Devaney Robert Guralnick 2/05-1/09 Susan]. Friedlander 2/96-1/08 Frank S. Quinn MichelL. Lapidus 2/02-1/08 Board of Trustees Katherine St. John Matthew Miller 2/05-1/09 Marjorie Senechal James G. Arthur (ex officio) 2/05-1/07 Lesley M. Sibner 2/93-1/09 Francis Edward Su John B. Conway 2/01-1/11 Treasurer John M. Franks (ex officio) 2/99-1/09 John M. Franks 2/99-1/09 Members of Executive Eric M. Friedlander 2/00-1/10 Associate Treasurer Committee James G. Glimm (ex officio) 2/07-1/09 Donald E. McClure 2/03-1/09 Members of the Council, as provided Linda Keen 2/99-1/09 for in Article 7, Section 4 (last sen­ Donald E. McClure (ex officio) 2/03- tence), of the Bylaws of the Society. 1/09 Jean E. Taylor 2/03-1/08 Sylvan Cappell 2/06-1/10 Carol S. Wood 2/02-1/11 Ruth M. Charney 2/07-1/11 Walter L. Craig 2/03-1/07 Robert Guralnick 2/05-1/09 Paul ]. Sally Jr. 2/04-1/08

MAY 2007 NOTICES OF THE AMS 657 AMERICAN MATHEMATICAL SOCIETY

PUBLISHER

Applications and nominations are invited for the position of Publisher of the American Mathematical Society.

The publisher oversees all editorial and acquisitions activities of the Society's publishing program for journals and books (but not Mathematical Reviews), which includes coordinating the activities of eight editorial boards for journals, nine editorial boards for books, three acquisitions editors, and several governing bodies related to AMS publishing. The publisher is the primary representative of the AMS publication program to the scholarly and publishing community and should normally hold a doctorate in the mathematical sciences with a record of research.

Responsibilities of the publisher include:

• Direction of the Acquisitions Department (for books) • Leadership in setting scientific and editorial standards for books and journals • Development and implementation of long-range plans for the AMS book and journal programs • Budgetary planning and control for all scientific aspects of AMS books and journals • Representing publications to the Council, the Board of Trustees, the Committee on Publications, and the Editorial Boards Committee. (The publisher serves on the two latter committees, ex officio.)

The publisher reports directly to the executive director, but works closely with the associate executive director for publishing, who oversees all non-scientific aspects of publishing (production, printing, distribution, promotion, and marketing).

The publishing departments are in the AMS headquarters, which are located in Providence, Rhode Island, not far from the Brown University campus. A printing and warehouse facility is nearby in Pawtucket, Rhode Island. The Rhode Island offices of the AMS employ approximately 150 staff. The appointment will be for three to five years, with possible renewal, and will commence in 2008. The starting date and length of term are negotiable. The publisher position is full time, but applications are welcome from individuals taking leaves of absence from another position. Salary is negotiable and will be commensurate with experience.

Nominations and applications should be sent on or before August 1, 2007, to:

Dr. John Ewing, Executive Director American Mathematical Society 201 Charles Street Providence, RI 02904 [email protected]

Applications should include a curriculum vitae, information on editorial and administrative experience, and the names and addresses of at least three references. The American Mathematical Society is an Equal Opportunity Employer. AMERICAN MATHEMATICAL SOCIETY

MATHEMATICAL REVIEWS EXECUTIVE EDITOR

Applications and nominations are invited for the position of Executive Editor of Mathematical Reviews (MR).

The executive editor heads the MR Division of the AMS and is responsible for all phases of its operations. The major activity of the division is the creation of the MR database along with its derived products, MathSciNet and the paper journal. The executive editor is the primary repre­ sentative of Mathematical Reviews to the mathematics community and should normally hold a doctorate in the mathematical sciences with a record of research.

Responsibilities of the executive editor include:

• Direction of all staff in the Ann Arbor office of the American Mathematical Society • Leadership in setting scientific and editorial standards for MR • Development and implementation of long-range plans for the MR Division • Budgetary planning and control for the MR Division • Relations with reviewers and authors

The managing editor and administrative staff assist the executive editor in carrying out day-to-day tasks and non-editorial aspects of MR production. The MR Editorial Committee provides high-level scientific advice about the editorial standards of MR. The executive editor reports to the executive director.

The MR Division is located in Ann Arbor, Michigan, near the campus of the University of Michigan, and editors enjoy many faculty privileges at the university. MR employs approximately 70 personnel, including associate editors, copyeditors, bibliographic specialists, information technology staff, and clerical support.

The appointment will be for three to five years, with possible renewal, and will commence in the summer of 2008. The starting date and length of term are negotiable. The executive editor position is full time, but applications are welcome from individuals taking leaves of absence from another position. Salary is negotiable and will be commensurate with experience.

Nominations and applications should be sent on or before August 1, 2007, to:

Dr. John Ewing, Executive Director American Mathematical Society 201 Charles Street Providence, Rl 02904 [email protected]

Applications should include a curriculum vitae, information on editorial and administrative experience, and the names and addresses of at least three references. The American Mathematical Society is an Equal Opportunity Employer. Mathematics Calendar

The most comprehensive and up-to-date Mathematics Calendar information is available on the AMS website at http://www.ams.org/mathcal/.

May 2007 '' 7-11 Random and Dynamic Graphs and Networks, UCLA, Los Angeles, California. '' 4-5 3 1st SIAM Southeastern-Atlantic Section Meeting, The Uni­ versity of Memphis, Memphis, Tennessee. Scientific Overview: This workshop will bring together experts with Description: The conference will feature plenary talks, special diverse backgrounds to discuss current challenges in modeling and sessions, contributed lectures and a student paper session. Plenary analyzing networked structures, with a specific focus on dynamics talks will be given by Pete Casazza (University of Missouri), Erica *of* networks (i.e., how do real-world networks such as the World­ Flapan (Pomona College), Yuriko Renardy (Virginia Tech) and Wide-Web evolve over time) and dynamics ''over* networks (i.e., for Michael Shearer (North Carolina State University). networks that carry some form of traffic, what is its dynamic and Deadlines/Information: Visit: http : I /www. msci. memphis. edu/ how does it interact with the network). This workshop is part of siam- seas/. !PAM's long program on Random Shapes. Application/Registration: An application/registration form is * 4-6 Workshop on Financial Engineering for Actuarial Mathe­ available at http: I /www. ipam. ucla. edu/programs/rsws3/. The matics, University of Michigan, Ann Arbor, Michigan. application is for people requesting financial support to attend the Program: The workshop is aimed at researchers working at workshop. If you don't intend to do this, you may simply register. the interface of finance and insurance, including probabilists, Applications will be accepted up until March 26, 2007. For the financial mathematicians, and actuaries. Conference themes include fullest consideration we urge you to apply as early as possible. insurance pricing and hedging; risk measures; event and mortality Successful applicants will be notified as soon as funding decisions risk in insurance markets; insurance contract design; are made. We have funding to support the attendance of recent methods in actuarial applications. Registration is free, but space is Ph.D.'s, graduate students, and researchers in the early stages of limited. their career. Encouraging the careers of women and minority math­ lnvit

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

660 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Mathematics Calendar

Siena, Italy. ,., 26-28 Mathematical Theories of Abstraction, Substitution and Aim: The aim is to gather people working in the many research Naming in Computer Science, ICMS, , Scotland. streams originating from Girard's main achievements of the recent Information: http: I /www. icms. org. uk/workshops/maththeoCS/ years. Themes: For each of the four main themes- Linear Logic (specif­ ically, Proof Nets and Geometry of Interaction), Ludics, Implicit '' 30-June 1 Short Course on Sparse Representations and High Complexity and Operator Algebras- there will be in-depth lectures Dimensional Geometry (in conjunction with the AMS 2007 Von (3 to 4 hours), with emphasis on the state of the art and prospects Neumann Symposium), UCLA, Los Angeles, California. for future development. There will also be some time for 30- minute Aim: This associated short course will be held five weeks prior contributed papers and for discussion of general perspectives and to the 2007 Von Neumann Symposium. The principle aim is to philosophical foundations. provide a firm foundation in sparse approximation to members Organizers: V. M. Abrusci (Roma), C. Faggian ( Paris), S. Martini of the scientific community new to the topic - before the Von (Bologna), S. Ronchi Della Rocca (Torino), A. Ursini (Siena). Neumann Symposium. Each speaker will give three informal 50 Speakers: Patrick Baillot, Pierre-Louis Curien, Alberto Dal Lago, minute tutorials on his or her area of expertise, conveying his or Claudia Faggian, Jean-Yves Girard, Paul Andre Mellies Michele her unique perspective. Speakers will be in residence for the entire Pagani, Laurent Regnier, Kazushige Terui. three day meeting and will be available to participants for in depth Information: http: I /www. unisi . it/eventi/LOGIC/. discussions. Speakers: Anna Gilbert (University of Michigan), Justin Romberg '' 1 8-20 International Conference on Special Functions & their (Georgia Institute of Technology), Jared Tanner (University of Utah), Applications (7th Annual Conference of SSFA, India), St. Thomas Joel Tropp (University of Michigan), Roman Vershynin (UC Davis), College, Arunapuram P.O., Pala, Kottayam, Pin 686574, Kerala, Jing Zou (University of Maryland). India. Application/Registration: An application/ registration form is Information: http: I lwww . ssfa. gq. nul conf. htm. available at http: I lwww. ipam. ucla. edulprogramslvn2007 1. The application is for people requesting financial support to attend the '' 21-2 5 Image Processing for Random Shapes: Applications to workshop. If you don't intend to do this, you may simply register. Mapping, Geophysics and , UCLA, Los Angeles, We urge you to apply as early as possible. Applications received California. by April 18, 2007, will receive fullest consideration. Successful Scientific Overview: Random shapes occur in many physical and applicants will be notified as soon as funding decisions are made. biological models and applications, and image processing of these We have funding to support the attendance of recent Ph.D.'s, random shapes is very challenging. This workshop will bring graduate students, and researchers in the early stages of their together experts in image processing, mathematics, biology and career; however, mathematicians and scientists at all levels who medicine, and physical sciences. Topics discussed in the workshop are interested in this area are encouraged to apply for funding. include, but are not limited to: brain imaging and measures Encouraging the careers of women and minority mathematicians of complexity; random fields in brain science; complexity in and scientists is an important component of IPAM's mission and cortex or brain morphology; shrinking and wrinkling of anatomical we welcome their applications. structures; filaments and large scale structures in the cosmos; Information: email: sbeggsl!lipam. ucla. edu. random fields; geometry and the Gaussian free field; distribution of dark matter; 3D image processing and graphics for complex june 2007 surfaces in the geophysical sciences; computational geometry for complex sets and surfaces. This workshop is part of !PAM's long '' 2 00 7 International Conference on Learning, Johannesburg, South program on Random Shapes. Africa. Application/Registration: An application/ registration form is Information: http: I /www . LearningConference. com. available at http: I lwww. ipam. ucla. edulprogramslrsws41. The application is for people requesting financial support to attend the '' 1 0-1 51 PM LogicConference2007,SchoolofMathematics,Institute workshop. If you don't intend to do this, you may simply register at for Studies in and Mathematics (IPM), Tehran, any time. Applications will be accepted up until April 9, 2007. For Iran. the fullest consideration we urge you to apply as early as possible. Goal: The goal of the conference is to give an introduction to Successful applicants will be notified as soon as funding decisions current research topics in various aspects of are made. We have funding to support the attendance of recent as well as presentation of the latest applications into mathematics Ph.D.'s, graduate students, and researchers in the early stages of and computer science. their career. Encouraging the careers of women and minority math­ Call for Papers: Participants who wish to contribute a talk should ematicians and scientists is an important component of !PAM's submit their abstracts to logic20071!lipm. ir no later than April mission and we welcome their applications. 30, 2007. Local expenses will be paid for those whose abstracts are Information: email: sbeggsl!lipam. ucla. edu. accepted. Deadlines: Deadline for submission of abstract of contributed '' 26-27 Vlora Conference on Algebra, , and Cryp­ talks: April30, 2007. Deadline for registration: May 21, 2007. tography, Vlora, Albania. Contact: School of Mathematics, IPM, Niavaran Square, Niavaran Description: The main goal is to bring together researchers in a Street, P.O. Box: 19395-5746, Tehran, Iran; tel: (+98) 21 22290928; relaxing atmosphere in order to foster cooperation. fax: (+98) 21 22290648; email: logic20071!lipm. ir. For more in­ Topic: This year's topic of the conference will be algebra, coding formation, please see the following link: http: I lwww. ipm. ac. ir I theory, and cryptography. logic2007 1. Information: http: I /www. albmath. org/vlconf/. Organizers: T. Shaska, shaskal!loakland. edu; Edgar Martinez Moro, '' 11-16 QTRF4- Quantum Theory: Reconsiderations and Founda­ [email protected]. tions 4, International Centre for Mathematical Modelling in Physics, Registration: There is no registration fee. However, you must Engineering and Cognitive Sciences at Vaxjo University, Sweden. register in advance in order to attend the conference. No exceptions Description: This conference is devoted to quantum foundations, will be made for anyone. We will arrange to pick you up from the especially clarification of fundamental questions (physical, mathe­ airport, to help with hotels, etc. matical and even philosophical). This conference is devoted to the Information: http: I /www. albmath. org/vlconf/. 80 years of the Copenhagen Interpretation.

MAY 2007 NOTICES OF THE AMS 661 Mathematics Calendar

Invited speakers: V. Belavkin, University of Nottingham, UK, I. '' 26-30 ICDS International Conference on Dynamical Systems Bengtsson, Stockholm University, Sweden, E. Binz, University of 2007, Abant Yzzet Baysal University, Bolu, Turkey. Mannheim, Germany, Yu. Bogdanov, , Rus­ Aim: The aim of this conference is to present new research sia, T. H. Boyer, City College of the CUNY, USA, G. Cassinelli activities (theoretical and practical) on geometrical aspects of Universita di Genova, Italy, G. D'Ariano, University of Pavia, Italy, W. dynamic systems and the recent developments to other scientists. De Baere, University of Ghent, Belgium, H. D. Doebner, University High Quality Research Submissions are invited for technical papers of Clausthal, Germany, W. Freudenberg, University of Cottbus, describing original unpublished results of theoretical, empirical, Germany, B. Hellsing, Chalmers Goteborg, Sweden, K. Hess, Uni­ conceptual, and experimental work. versity of Illinois, USA, R. Hudson, University of Nottinghem, Confirmed Speakers: E. Pawlusevusz, Bialsytock, Poland; G. Naber, UK, G. Jaeger, Boston University, USA, A. Yu. Khrennikov, Vaxjo , USA; G. Khadekar, Nagbur, India; G. Todorova, Ten­ University, Sweden, P. Lahti, University of Turku Finland, J-A. nessee, USA; R. Santilli, Florida, USA; Robert Devaney, Boston, USA; Larsson, Linkoping University, Sweden, N. Landsman, University of T. Dereli, Ystanbul, Turkey; T. Onder Ankara, Turkey; Giuseppe Nijmegen, The , M. Manko, Lebedev Physical Institute, Quartieri Ytaly; Alptekin Erkollar, Klagenfurt, Austria; Erik Trell, Moscow, Russia, V. Manko, Lebedev Physical Institute, Moscow, Linkoping, Sweden; RaulFalc6n Gafarnian, Sevilla, Spain; Juan Nunes Russia, Th. Nieuwenhuizen, Amsterdam University, The Nether­ Valdes, Sevilla, Spain; Allaberen Ashrallyev, Ystanbul, Turkey. lands, Yu. Ozhygov, Moscow State University, Russia, ] .G. Pereira, Deadlines: The last day for registration is 30 April 2007. The last Universidade Estadual Paulista, Brasil, D. Petz, University of Buda­ day for submitting papers is 30 May 2007. pest, , A. Plotnitsky, Purdue University, USA, P. Rocchi, IBM, Contact: Yzzet Baysal Kampiisii, Fen Edebiyat Fakiiltesi, Matematik Rome, Italy, M. Scully, University of Texas-Austin, USA, G.L. Sewell, Boliirnii, 14280 Bolu, Turkey; tel: +90-374-2541000; fax: +90- Queen Mary, University of London, UK, S. Stenholm, KTH - Royal 374-2534642;http: I /www. dynamicalsystems . ibu. edu. tr; email: Institute of Technology, Sweden, W. von Waldenfels, University of dynamicalsytems©ibu . edu.tr. Heidelberg, Germany. Deadline: Abstract and application form has to be sent before '' 30-July 1 Advanced Numerical Techniques in Applied Dynamical April 15th, 2007. Systems, Centre de recherches mathematiques, Universite de Information: http: I /www. vxu. se/ msi/aktuellt/konferens/ Montreal, Montreal, Canada. index.xml. Lectures: Lectures will be aimed in the first instance at graduate students and postdoctoral fellows who have a need for advanced '' 1 5-1 7 Algebraic Methods in , Chalmers numerical techniques in the study of applied dynamical systems University of Technology and Gothenburg University, Gothenburg, that arise in their research. It is expected that there will also be Sweden. experienced researchers in dynamical systems and its applications Organizers: Volodymyr Mazorchuk, Lyudmila Turowska. who will be interested in the course. The lectures will be followed Information: http: I /www . math. chalmers. se/-turowska/ by work sessions, where the lecturers and additional experts will be workshop/index.html. present to assist the participants with basic numerical experiments and with more advanced numerical computations that arise in the '' 1 6- 1 9 XVI Congreso Nacional de Matematicas, Medellin, Colom­ research of the participants. Assistance and advice will be available bia. on several levels, ranging from technical assistance with software Information: http://www. scm .org. co/index.php?hoja= and issues related to numerical algorithms, to theoretical issues Congreso2007/ . related to mathematical models under consideration. Format: The format of the course will consist of invited lectures '' 18- 22 Cherednik Algebras, , Edinburgh, by experts in applied dynamical systems who are known for their Scotland. contributions to the theory and the software implementation of Information: http: I /www . icms . org. uk/ workshops/cheralg/ . advanced numerical techniques, and who are also recognized as excellent lecturers. The lecturers will be strongly encouraged to '' 19-22 Thirteenth Annual Conference of African American Re­ include illustrative computer demonstrations in their presentation. searchers in the Mathematical Sciences (CAARMS13), Northeast­ Organizers: E. Doedel (Concordia), H. Osinga (Bristol). ern University and University of Massachusetts, Boston, Massachu­ Information: email: paradis@crm. umontreal. ca. setts. Organizers: Donald R. King ( d.king@neu. edu), Alfred G. Noel July 2007 (anoel©math. umb . edu), William A. Massey (wmassey©princeton. edu). '' 2-6 Advanced Algorithms and Numerical Software for the Bifurcation Analysis of Dynamical Systems, Centre de recherches '' 2 5- 28 The 2007 World Congress in Computer Science, Compu­ mathematiques, Universite de Montreal, Montreal, Canada. ter Engineering, and Applied Computing: WORLDCOMP'07, Las Description: There is an increased need for advanced computa­ Vegas, Nevada. tional and visualization tools in the study of dynamical systems Information: A link to each conference's URL can be found at: that arise in important physical applications, and that are modeled http: //www.worldacademyofscience .org/worldcomp07/ . by complicated or large-scale systems of equations. The computa­ tional tools must provide capabilities that go well beyond simple *25- 29 Number Theory and Computability, ICMS, Edinburgh, simulation: for example, they identify and classify important bi­ Scotland. furcations, and they determine critical manifolds in Information: http : I /www . i cms. org. uk/ workshops/numtheocomp/ or in parameter space that separate regions of fundamentally different dynamics. Such tools must also be applicable to con­ servative systems and to systems possessing certain symmetries. '' 25- 29 Summer School on Algebra and Combinatorics, CELC, State-of-the-art graphical tools are typically required for the rep­ University of Lisbon, Portugal. resentation and the interpretation of such computational results. Desuiption: The "Summer School on Algebra and Combinatorics" This workshop will bring together computational scientists and ap­ will consist of courses of five one-hour lectures, supplemented plied mathematicians who develop such advanced computational by seminars and/ or posters and informal talks. The school is and visualization tools, or who have a strong need for such tools mostly addressed to graduate students or researchers interested in their investigations. The emphasis will be on the numerical in algebraic aspects of combinatorics. analysis of discrete and continuous dynamical systems that model Information: http: I /alg- comb . cii. fc . ul. pt/. important physical phenomena. Specific topics include algorithms

662 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Mathematics Calendar

and software for the computation and visualization of bifurcations, and scientists is an important component of IPAM's mission and invariant manifolds, and traveling wave phenomena, in nonlinear we welcome their applications. ordinary differential equations, delay and functional differential equations, and in certain classes of partial differential equation, '' 16-20 Optimal Transportation, and Applications to Geophysics especially nonlinear parabolic systems. and Geometry, University of Edinburgh, Edinburgh, Scotland. Organizers: E. Doedel (Concordia), H. Osinga (Bristol). Organizer: Interational Centre for Mathematical Sciences. Information: http: I /www. crm. math. ca/ Dynamics2007 / . Information: http: I /www. icms . org. uk/ workshops/ optransgeo/

'' 2-6 9th Conference on Orthogonal Polynomials Special Func­ tions and Applications (OPSFA9), International Center for Math­ '' 18-August 3 X Diffiety School-School in Geometry of Partial ematical Meetings, Marseille, France. Differential Equations, Santo Stefano del Sole (AV), Italy. Scopes: They are traditionally rather wide and include selected Aim: The aim of the School is to introduce undergraduate and Ph.D. topics on: Special Functions, OP and q-Extensions, q-Difference Equa­ students in Mathematics and Physics as well as post-doctoral re­ tions, OPRL and OPUC, Real and Complex Jacobi Matrices, Jacobi searchers in a recently emerged area of Mathematics and Theoretical Matrices and Integrability, Hermite-Pacte Approximants, Contin­ Physics: Secondary Calculus. ued Fractions, Determinate and Indeterminate Moment Problems, What is a Diffiety: A diffiety is a new geometrical object that Riemann-Hilbert Approach to OP, Matrix OP. properly formalizes the concept of the solution space of a given Organizers: Jacek Gilewicz (Toulon, France), Roland Triay (Mar­ system of (nonlinear) PDEs, much as an algebraic variety does seille, France), Galliano Valent (Paris, France). with respect to solutions of a given system of algebraic equations. lnformation:http: I /www. cpt. univ-mrs. frrcosmo /WEB_OPSFA9/ Secondary Calculus is a natural diffiety analogue of the standard OPSFA 9 . h tml. Calculus on smooth manifolds, and as such leads to a very rich general theory of nonlinear PDEs. Moreover, it appears to be the ,., 8- 21 37th Probability Summer School, Saint-Flour, France. unique natural language for quantum physics, just as the standard Speakers:]. Buzzi, F. den Hollander,]. Mattingly. Calculus is the natural language for . Information: http: I /math. univ -bpclermont. fr /stflour/. Scientific Director: Alexandre M. Vinogradov; email: school07ru

MAY 2007 NOTICES OF THE AMS 663 Mathematics Calendar

glish version log in to http: I /altenua. uctea. edu. cor claxvii/ Description: On the occasion of the 60th birthday of Valentin ingles index. htm. Lychagin. Focus: Geometric and algebraic methods of investigations of PDEs * 23-August 31 Cemracs'07: Summer Mathematical Research Cen­ for compatibility, solvability and integrability with applications to ter on Scientific Computing and Its Applications, CIRM, Luminy, mathematical physics. France. Organizers: Boris Kruglikov, Per Jakobsen. Description: This year's Cemracs is devoted to pre- and post­ Deadline: June 1, 2007. processing aspects of numerical simulation, including mesh gener­ Information: http : I /www .math. u i t. no/pdes2007 / index .html. ation and adaptation, error estimation, scientific visualisation, etc. It will consist of two consecutive events: A one-week summer school '' 1 3-1 7 Groups, rings, Lie and Hopf algebras. II, Bonne Bay Marine (July 23-July 30) and a five-week research center (July 30-August Station of Memorial University of Newfoundland, Norris Point, NL, 31) Canada, AOK 3VO. Topics: Cemracs'07 research center will host several research Description: Survey lectures and research talks on the topics in projects on the topics of applied mathematics and scientific the title of the workshop. computation, sponsored by industrial or public research funds. Lectures by invitation: Antonio Giambruno (University of Palermo), One or two young researchers (Ph.D. students) are invited to work Sergei Ivanov (University of Illinois - Urbana-Champaign), Eric on each project under the supervision and help of senior scientists. Jespers (University of Brussels), Erhard Neher (University of Ottawa), The projects will start during the summer school and last until the Dmitri Nikshych (University of New Hampshire), Sudarshan Sehgal end of August. (University of Alberta), Vera Serganova (University of California - Information: email: pppebay©sandia.gov. Berkeley). Information: email: aac©math. mun. ca. * 2 5- 2 7 Symbolic-Numeric Computation 2007 (SNC 2007), London, Canada. '' 1 5- 18 1nternational Conference on Integral Geometry, Harmonic Description: The goal of the present workshop is to support the Analysis and Representation Theory in honor of Sigurdur interaction and integration of symbolic and numeric computing. Helgason on the occasion of his 80th birthday, University of SNC 2007 is affiliated with the 2007 International Symposium Iceland, Reykjavik, Iceland. on Symbolic and Algebraic Computation (ISSAC 2007). Co-located Topics: Harmonic analysis on symmetric spaces, The Radon trans­ with this workshop will be PASCO 2007, the 2007 International form and generalizations, the on symmetric spaces, Workshop on Parallel Symbolic Computation. SNC and PASCO will representation theory, complex and algebraic methods in harmonic be held immediately prior to the ISSAC 2007 meeting, both at the analysis and representation theory. University of Western Ontario, Canada. ISSAC 2007 will be held Speakers: Jean-Phillipe Anker, Erik van den Ban, Leticia Barchini, nearby in Waterloo, Canada. Jacques Faraut, Mogens Flensted-Jensen, Fulton Gonzales, Simon Information: http: I /www . orcca. on. ca/co nferences/sn c2007 / . Gindikin, Sigurdur Helgason, Toshiyuki Kobayashi, Adam Koranyi, Bent Orsted, Gestur Olafsson, Eric Opdam, Toshio Oshima, Angela * 2 9-Aug u st 4 Reconnect Conference 2007, DyDAn Center, Rutgers Pasquale, Todd Quinto, Francois Rouviere, Henrik Schlichtkrull, University, Piscataway, New Jersey. Robert Stanton, Gudlaugur Thorbergsson, , Nolan Description: Reconnecting teaching faculty to the mathematical Wallach, Joseph Wolf. sciences enterprise and exposing researchers in government and Deadline: Registration: June 30, 2007. industry to relevant current research. The DyDAn Summer Recon­ Information: http: I /www . raunvis . hi. is/ He l gason/ ; email: nect Conferences will expose faculty teaching undergraduates to Hel gasonConferen ce©raunvis.hi.is. the role of the mathematical sciences in homeland security by introducing them to a current research topic that will be relevant '' 27-29 Automata 2007: 13th International Workshop on Cellular for classroom presentation. The conferences also offer researchers Automata, The Fields Institute for Research in Mathematical in government or industry the opportunity to learn about recent Science, Toronto, Canada. techniques in emerging application areas relevant to homeland se­ Organizers: AnnaT. Lawniczak (University of Guelph) and Henryk curity. These workshops offer the opportunity for junior faculty as Fuks (Brock University). well as mid-level and senior faculty, and government and industry Program: The workshop will revolve around all important theo­ professionals to explore research questions in a new area of the retical and applied aspects of cellular automata (CA) and discrete mathematical sciences. Anyone may apply. will be given complex systems. to faculty whose primary job is undergraduate teaching and those Invited Speakers: RamonAlonso-Sanz (Polytechnic Univ. of Madrid, working at government labs. Two-year college faculty are welcome Spain), Stefania Bandini (Univ. of Milano-Bicocca, Italy), Franco to apply. Faculty from groups under-represented in mathematics Bagnoli (Univ. of Florence Italy), Bastien Chopard (Univ. of Geneva, are also encouraged to apply. Switzerland), Andreas Deutsch (Tech. Univ. Dresden, Germany), Topic in Law Enforcement and Homeland Security. Samira El Yacoubi (Univ. of Perpignan, France), Nazim Fates (LORIA Principal Speaker: WilliamM. Pottenger, Rutgers University, billp© Univ. Nancy 1, France), Katsunobu Imai ( Univ., Japan), dimacs.rutger s . edu. Raymond Kapral (Univ. of Toronto, Canada), Petr Kurka (Charles Conference Organizer: Fred S. Roberts, Rutgers University Univ. Prague, Czech Republic), Maria Elena Umaga (UAEM, Mexico), ( froberts©dimacs . rutgers . edu). Pietro Lio (Univ. of Cambridge, UK), Danuta Makowiec (Univ. of Information: Contact Coordinator: reconnect©dimacs . rutgers. Gdansk, Poland), Armin Mikler (Univ. of North Texas, USA), Angelo edu, or telephone at (732) 445-4304; or visit the website: http: I I B. Mingarelli (Carleton Univ., Canada), Kenichi Morita (Hiroshima dydan.rutgers.edu/reconnect/. Univ., Japan), HenningS. Mortveit (Virginia Tech., USA), Pedro P.B. de Oliveira (Univ. Presbiteriana Mackenzie, Brasil), Marcus Pivato August 2007 (Trent Univ., Canada), Raul Rechtman S (UNAM, Mexico), Peter Sloat (Univ. of Amsterdam, The Netherlands), Gianluca Tempesti (Univ. '' 6- 1 0 Mathematical Virology, Edinburgh, Scotland. of York, UK), Burton Voorhees (Athabasca Univ., Canada), Gabriel Information: http : I /www . icms . org. uk/wor kshops/mathvir/ . A. Wainer (Carleton Univ., Canada) Jain Yuan (Tsinghuan Univ., Beijing, China). '' 1 2- 1 7 Geometry and Algebra of PDEs, University of Tromso, Deadlines: May 30, 2007: Reduced fee registration ends June Norway. 1, 2007: Regular fee registration starts July 15, 2007: Abstract

664 N OTICES OF THE AMS VOLUME 54, N UMBER 5 Mathematics Calendar

submission deadline August 27, 2007: On site registration with '' 1 0-11 journees Jean-Yves Girard, Conference in Honour of his higher fee. 60th Birthday, Institut Hemi Poincare, Paris, France. Information: http: I /www . fields. utoronto. ca/programs/ Program: Through our choice of invited speakers, we hope to scientific/07-08/automata07/. illustrate the wide range of scientific interests of Jean-Yves Girard over thirty-five years, from the complexity of proofs to quantum '' 27-31 LMS workshop on motives, quadratic forms and algebraic mechanics, from system F to the geometry of computation, from groups, Queen's University, Belfast, United Kingdom. denotational semantics to Von Neumann algebras. Description: The workshop is planned to bring together experts, Organizers: Michele Abrusci (Roma III), Pierre-Louis Curien (CNRS young and old, on various aspects of research in Chow motives, -Paris 7, chair), Martin Hyland (Cambridge), Giuseppe Longo (ENS, quadratic forms and algebraic groups, as well as graduate students, Paris), Mitsu Okada (Keio Univ., Tokyo), Phil Scott (Univ. of Ottawa), postdocs and others who wish to learn about the subject areas. Jacqueline Vauzeilles (Paris 13, co-chair). Courses: N. Karpenko (Jussieu), Motives and quadratic forms; B. Speakers: Patrick Dehornoy, Gerard Huet, Herman Jervell Yves Totaro (Cambridge), Quadrics and Birational Geometry; C. Weibel Lafont, Olivier Laurent, Thierry Paul Peter Selinger, Glynn Winskel. (Rutgers), Motivic Cohomology. Information: http: I /www-lipn. univ-paris13. fr/jyg60/. Speakers and Invited Participants: A. Bak (Bielefeld), E. Bayer­ '' 1 0-December 1 4 Mathematics of Knowledge and Search Engines Fluckiger (Lausanne), R. de J eu (Durham), D. Hoffmann (Nottingham), (Long Program), UCLA, Los Angeles, California. M. Karoubi (Paris 7), B. Koeck (Southampton), D. Lewis (Dublin), Scientific Overview: This long program at IPAM will be devoted J.-P. Tignol (Louvain-la Neuve), N. Vavilov (St. Petersburg), A. Vishik to new mathematics and methodologies of knowledge engines: the (Moscow), A. Wadsworth (UC San Diego), K. Zainoulline (Muenchen). mathematical procedures used to extract knowledge from large Sponsor: London Math. Society. databases. While this includes topics related to search engines it is Information: http: I I queensworkshop. googlepages. com/. mainly devoted to the more general problem of finding features in a database or using defined features to search within a database. September 2007 Activities: There will be an active program of research activities, seminars and workshops throughout the period and core partici­ '' 3-7 Conference in Numerical Analysis 2007 (NumAn 2007): pants will be in residence at IPAM continuously for these fourteen Recent Approaches To Numerical Analysis: Theory, Methods weeks. The program will open with tutorials, and will be punctuated and Applications, Kalamata, Greece. by four major workshops and a culminating workshop at UCLA's Overview: NumAn provides an opportunity to learn of new devel­ Lake Arrowhead Conference Center. Several distinguished senior opments and to present original research results in all areas of researchers will be in residence for the entire period. Between Numerical Analysis such as Theory, Methods and Applications. the workshops there will be a program of activities involving the Call for Papers: Further information is available at: http: I /www. long-term and short-term participants, as well as visitors. elsevier.comlwpslfindljournaldescription.cws_home/ Application: Please apply online to request financial support to 505613lauthorinstructions/.To submit a paper, send an email attend and participate for extended periods up to the entire length to:[email protected]. of the program. Applications for individual workshops are separate Important Dates: Submission of abstracts: March 31, 2007. Notifi­ and will be posted on individual workshop home pages. For the cation of acceptance: April 30, 2007. Full versions of the accepted fullest consideration we urge you to apply as early as possible presentations to be included in the Proceedings of the Conference: but no later than August 1, 2007. Successful applicants will be May 31, 2007. notified as soon as funding decisions are made. We have funding Financial Support: Some financial support will be available from to support the attendance of recent Ph.D.'s, graduate students, the conference, to cover expenses of graduate students and post­ and researchers in the early stages of their career. Encouraging the doctoral fellows. Applications should be sent electronically to the careers of women and minority mathematicians and scientists is conference e-mail address: numan2007@math. upatras. gr. an important component of IPAM's mission and we welcome their Information: Contact: email: numan2007@math. upatras. gr; http: applications. //www.math.upatras.gr/numan20071. Information: The application and more information is available at http:llwww.ipam.ucla.edulprogramslse2007l. '' 3-7The Riemann-Hilbert Problem and Toeplitz Operators, Heriot Watt University, Edinburgh, Scotland. '' 11-14 Fall 2007 Workshop for Young Researchers in Mathemat­ Organizer: International Centre for Mathematical Sciences. ical Biology (WYRMB), Columbus, Ohio. Information: http: I lwww . icms. org. uklworkshopsltoepopl. Information: http: I lwww. mbi. ohio-state . edulpostdocworkshopl fwyrmb2007.html '' 4-8 Potential Theory and , Albae, . * 1 7-21 Manifolds with nonnegative , American Dedicated: To the memory of Aurel Cornea. Institute of Mathematics, Palo Alto, California. Organizers: "Simian Stoilow" Institute of Mathematics of the Description: This workshop, sponsored by AIM and the NSF, will Romanian Academy, University of Alba Julia and University of be devoted to manifolds with nonnegative and positive curvature. Pitesti. Significant progress has been made recently following a program Main speakers: H. Aikawa (Hokkaido) [to be confirmed], D. Bakry suggested by K. Grove that one should study this condition under (Toulouse), A. Bendikov (Wroclav), H. -P. Blatt (Eichstaett),]. Bliedtner the presence of a large isometry group. The workshop will consider (Frankfurt), K. El Mabrouk (Monastir), S.-L. Eriksson (Tampere), D. various special cases, with significant time spent working on Feyel (Evry), B. Fuglede (Copenhagen), M. Fukushima (Osaka), problems. S. Gardiner (Dublin), ]. Glover (Gainesville), K. GowriSankaran Organizers: Kristopher Tapp and Wolfgang Ziller. (Montreal), W. Hansen (Bielefeld), N. Jacob (Swensa), K. Janssen Deadline: May 15, 2007. (Duesseldorf), K. Kuwae (Kumamoto), A. de La Pradelle (Paris), Information: http: I I aimath. org/ ARCC/workshops/ ]. Lukes (Prague), T. Lyons (Oxford)[to be confirmed], Z.-M. Ma nnsectcurvature.html. (Beijing), H. Maagli (Tunis), I. Netuka (Prague), M. Rao (Gainesville), M. Roeckner (Bielefeld), W. Stannat (Darmstadt), J. Vesely (Prague) * 1 7-21 SECURECOMM 2007: Third International Conference on [to be confirmed]. Security and Privacy for Communication Networks, Nice, France. Information: http: I lwww. imar. rorpuricelconferences/afis­ Information: For information regarding the conference, please albac. pdf; email: lucian. beznea@imar. ro. visit: http: I lwww. securecomm. orgl.

MAY 2007 NOTICES OF THE AMS 665 Mathematics Calendar

* 20-22 Finsler Geometry (Mathematics and Physics), Institut de Aims and Topics: Keeping the tradition of the AIMS conference Recherche Mathematique Avancee, , France. series, the conference covers all major areas of analysis and Organizers: Athanase Papadopoulos and Vladimir Turaev. dynamics, with emphases on theory, methods, application, modeling Invited speakers: Yves Benoist (ENS Paris), Christian Duval (Mar­ and computations. seille), Patrick Foulon (Strasbourg), Hubert Goenner (Goettingen), Format: There will be plenary talks; 30-minute special session Vladimir Matveev (Jena), Gabriel Paternain (Cambridge), Jean­ talks; 20-minute contributed talks; and poster sessions. If you are Baptiste Pomet (Sophia-Antipolis), Hans Bert Rademacher (Leipzig), interested in proposing a special session, please contact Shouchuan Marc Troyanov (Lausanne), Constantin Vernicos (Neuchatel), Ab­ Hu at General@aimSciences. org. delghani Zeghib (ENS Lyon). Organizing Committee: Jianzhong Su (Chair; AIMS2008@uta. edu), Information: email: papadopoulos@math. u-strasbg. fr. Jianping Zhu (Co-chair), Tuncay Aktosun, Gaik Ambartsoumian, Alain Bensoussan, Hristo V. Kojouharov, Cecelia Levings, Yue Liu, '' 24-29 18th Congress of Unione Matematica ltaliana, Bari, Italy. Peter Moore, Hua Shan. Information: email: segreteria@congressoumi2007. it; http : I I Scientific Committee: Shouchuan Hu (Chair) Jerry Bona, Alberto www.congressoumi2007.it. Bressan, Adrian Constantin, Amadeu Delshams, Hiroshi Matano, Alain Miranville, Wei-Ming Ni, N.S. Papageorgiou, Jianzhong Su, November 2007 jianping Zhu. Coordinator: Xin Lu; lux@uncw. edu. Deadline: February 29, 2008: For both submission of abstracts and * 1-December 31 Program on Bose-Einstein Condensation and early registration. Quanle ized Vortices in Superfluidity and Superconductivity, In­ Information: http: I laimsciences . orgl AIMS-Conferencel2008l stitute for Mathematical Sciences, Singapore, Singapore. index.htm. Topics: Review the most recent and advanced development in the research on Bose-Einstein condensation and quantized vortices in superfluidity and superconductivity, from experiment to theory, simulation and application; present the recently developed mathe­ matical theories, including modeling, analysis and computational techniques, that are relevant to BEC and quantized vortices; discuss and compare different recently proposed scientific models for BEC, especially for BEC at finite temperatures, and fermion condensation; identify critical scientific issues in the understanding of BEC and quantized vortices and the difficulties that are common to both disciplines; accelerate the interaction of applied and computational mathematics with physics and materials science, and promote this highly interdisciplinary research that has emerging applications; develop and foster international in a new era of scientific research. Organizing Committee: Weizhu Bao (National University of Singa­ pore), Fanghua Lin (Courant Institute, New York University). Information: For enquiries on scientific aspects of the program, please email Weizhu Bao at bao_weizhu@nus. edu. sg.

January 2008 * 14-1 8 The uniform bounded ness conjecture in arithmetic dy­ namics, American Institute of Mathematics, Palo Alto, California. Description: This workshop, sponsored by AIM and the NSF, will be devoted to arithmetic properties of preperiodic points for morphisms on projective space. The hope is to create new approaches to the study of arithmetic properties of periodic and preperiodic points for (quadratic) polynomials, for one-dimensional rational maps, and for projective morphisms of higher dimension. A specific goal of the workshop is to develop tools and a for proving the first (highly) nontrivial case of the uniform boundedness conjecture in dynamics, namely for quadratic polynomials in one variable over Q. Organizers: Matthew Baker, Robert Benedetto, Liang-Chung Hsia, and j oseph H. Silverman. Dead line: September l, 2007. Information: See http: I laimath. orgl ARCC/workshops/ ari thdynamics . html.

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. May 2008 '' 18-21 The 7th AIMS International Conference on Dynamical Syste ms, Differential Equations and Applications, University of Texas at Arlington, Arlington, Texas.

666 NOTICES OF THE AMS VOLUME 54, NUMBER 5 New Publications Offered by the AMS

M. Takakura, and H. Ogi, Unbounded conditional expectations for Analysis 0*-algebras; M. Joia, A Radon-Nikodym theorem for completely multi-positive linear maps and its applications; A. Kokk, Commutativity criteria for Gelfand-Mazur algebras; G. Lassner, ------Topological Algebras Topological algebras and quantum cosmology [the Abstract]; CONTEMPORARY A. L. Khlass and M. Oudadess, Representation of extensions, MATHEMATICS ------and Applications of 1[, endowed with a discrete absolute value; M. Leinert, Another proof of the Shirali-Ford theorem; A. Mallios, On Topological Anastasios Mallios and Marina algebra spaces; A. Mallios and A. Oukhouya, On combinatorially Algebras and Haralampidou, University of regular topological algebras; A. Najrni, Topological algebras Applications Athens, Greece, Editors with continuous characters; G. F. Nassopoulos, Spectral Anastasios Mallios decomposition and in commutative locally C*-algebras; Marina ~~~t~~mpidou The Fifth International Conference on L. Oubbi, Locally A-convex algebras revisited; M. Oudadess, On @ Topological Algebras and Applications different versions of Vidav-Palmer theorem; A. Oukhouya, On ------was held in Athens, Greece, from June combinatorially regular Frechet algebra; 0. Panova, Description AmancanMi!lhlll!la!~ISac•aiY 27th to July 1st of 2005. The main topic of closed maximal one-sided ideals in several classes of real of the conference was general theory of Gelfand-Mazur algebras; A. Y. Pirkovskii, Strictly flat cyclic topological algebras and its various applications, with emphasis Frechet modules and approximate identities; A. Y. Pirkovskii on the "non-normed" case. In addition to the study of the and Y. V. Selivanov, Homologically trivial Frechet algebras; internal structure of non-normed, and even non-locally convex C. P. Podara, On strictly flat Frechet modules; N. V. Rao, topological algebras, there are applications to other branches of T. V. Tonev, and E. T. Toneva, Uniform algebra isomorphisms mathematics, such as differential geometry of smooth manifolds, and peripheral spectra; C. Trapani, Bounded and strongly and mathematical physics, such as quantum relativity and bounded elements in Banach quasi *-algebras; Y. Tsertos, On dual quantum cosmology. of unbounded operators coordinate systems; W. D. Zelazko, Operator algebras on locally and related non-normed topological algebras are intensively convex spaces. studied here. Other topics presented in this volume are topological Contemporary Mathematics, Volume 427 homological algebra, topological algebraic geometry, sheaf theory andK -theory. May 2007, 442 pages, Softcover, ISBN-10: 0-8218-3868-7, ISBN- Contents: Z. Abdelali and M. Chidarni, Topologisation 13: 978-0-8218-3868-6, 2000 Mathematics Subject Classification: et multiplication dans certaines algebres; M. Abel, On 46H05, 46Hl0, 46H20, 46J05, 46K05, 46Kl0, 46Ml8, 47L60, Serre-Swan-Mallios theorem; M. Abel, Topological algebras 16D40, 17A40, All AMS members US$95, List US$119, Order code with idempotently pseudoconvex von Neumann homology; CONM/ 427 M. Amyari and M. S. Moslehian, Hyers-Ulam-Rassias stability of derivations on Hilbert C* -Modules; J. Arhippainen, On extensions of Stone-Weierstrass Theorem; H. Arizmendi, A. Carrillo, and L. Palacios, On Q1-algebras; F. Bagarello, Some results on the algebraic approach to quantum dynamics; S. J. Bhatt, Topological Differential Equations algebras and differential structures in C* -algebras; S. J. Bhatt, A. Inoue, and H. Ogi, On C* -spectrality of locally convex *-algebras in C* -algebras; D. G. Birbas, Ptak function, positive elements and the positive cone of a unital LC ''-algebra; J. Bonet, CONTEMPORARY Control Methods Topologizable operators on locally convex spaces; A. J. C. Martin MATHEMATICS and M. Haralampidou, On locally convex H *-triple systems; in PDE-Dynamical M. Chahboun, Harmonic functional calculus in m-p-complete Control Methods Systems A- p-normed algebras; R. Choukri, A concept of finiteness in in POE-Dynamical topological algebras; T. Chryssakis, Square roots of strongly Systems Fabio Ancona, University Fabio Ancona positive elements in lmc algebras; A. Kinani, Harmonic functions 1renaloslecka of Bologna, Italy, Irena Wolteruttmon operating on contractions in m-convex algebras; A. Kinani, RobertoTrlggkml Lasiecka, University of Virginia, Edi!OfS M. A. Nejjari, and M. Oudadess, Some characterizations using Charlottesville, VA, Walter cone notions in m-convex algebras; M. Fragoulopoulou, A. Inoue, Littman, University of Minnesota, and K.-D. Kiirsten, On the completion of a C*-normed algebra Amencan Matl'lemanca l Saclnty under a locally convex algebra topology; R. I. Hadjigeorgiou, On Minneapolis, MN, and Roberto Silov's idempotent theorem; M. Haralampidou, On generalized Triggiani, University of Ambrose algebras; A. Y. Helemskii, Tensor products in quantum Virginia, Charlottesville, VA, functional analysis: The non-matricial approach; A. Inoue, Editors

MAY 2007 NOTICES OF THE AMS 667 New Publications Offered by the AMS

While rooted in controlled PDE systems, this 2005 AM S-IMS-SIAM Nonlinear Equations Summer Research Conference sought to reach out to a rather distinct, yet scientifically related, research community in and mathematics interested in PDE-based dynamical systems. M.S. BirmanandN. N. Uraltseva, Indeed, this community is also involved in the study of St. Petersburg State University, dynamical properties and asymptotic long-time behavior (in particular, stability) of PDE-mixed problems. It was the editors' Russia, Editors conviction that the time had become ripe and the circumstances propitious for these two mathematical communities-that of This volume is devoted to the memory PDE control and optimization theorists and that of dynamical of the famous Saint Petersburg specialists-to come together in order to share recent advances mathematician Olga Aleksandrovna and breakthroughs in their respective disciplines. This conviction Ladyzhenskaya. For many years she was further buttressed by recent discoveries that certain ran the Saint Petersburg Seminar energy methods, initially devised for control-theoretic a-priori on mathematical physics, which became a basis for the estimates, once combined with dynamical systems techniques, scientific school she created. The ten articles in the volume, yield wholly new asymptotic results on well-established, written by students and colleagues of 0. A. Ladyzhenskaya, nonlinear PDE systems, particularly hyperbolic and Petrowski-type are mainly devoted to boundary value problems for partial PDEs. differential equations and to spectral problems for differential operators. These expectations are now particularly well reflected in the contributions to this volume, which involve nonlinear parabolic, as Contents: A. Arkhipova, Quasireverse Holder inequalities well as hyperbolic, equations and their ; aero-elasticity, in parabolic metric and their applications; M. Sh. Birman elastic systems; Euler-Korteweg models; thin-film equations; and N. D. Filonov, Weyl asymptotics of the spectrum of the Schrodinger equations; beam equations, etc. In addition, the Maxwell operator with non-smooth coefficients in Lipschitz static topics of Helmholtz and Morrey potentials are also domains; A. M. Budylin and V. S. Buslaev, Semiclassical prominently featured. A special component of the present volume pseudodifferential operators with discontinuous symbols and focuses on hyperbolic conservation laws, to take advantage of their applications to the problems of statistical physics; recent theoretical advances with significant implications also L. D. Faddeev, What is complete integrability in quantum on applied problems. In all these areas, the reader will find mechanics; N. Ivochkina, Geometric evolution equations state-of-the-art accounts as stimulating starting points for further preserving convexity; B. A. Plamenevskii, On spectral properties research. of elliptic problems in domains with cylindrical ends; N. Kikuchi and G. Seregin, Weak solutions to the Cauchy problem for the Contents: F. Ancona and A. Marson, Asymptotic stabilization Navier-Stokes equations satisfying the local energy inequality; of systems of conservation laws by controls acting at a V. A. Solonnikov, Schauder estimates for the evolution of the single boundary point; G. Auchrnuty, Variational principles generalized Stokes problem; T. A. Suslina, Homogenization of a for finite-dimensional initial value problems; G. Avalos and periodic parabolic Cauchy problem; N. N. Uraltseva, Boundary P. Cokeley, Boundary and localized null controllability of struc­ estimates for solutions of elliptic and parabolic equations with turally damped elastic systems; A. V. Balakrishnan, Nonlinear discontinuous nonlinearities. aeroelastic theory: Continuum models; S. Benzoni-Gavage, R. Danchin, S. Descombes, and D. jamet, Stability issues in American Mathematical Society Translations-Series 2 the Euler-Korteweg model; A. Bressan and W. Shen, Optimality (Advances in the Mathematical Sciences), Volume 220 conditions for solutions to hyperbolic balance laws; I. Chueshov May 2007, 244 pages, Hardcover, ISBN-10: 0-8218-4209-9, and I. Lasiecka, Long-time dynamics of a sernilinear wave ISBN-13: 978-0-8218-4209-6,2000 Mathematics Subject Classifica­ equation with nonlinear interior/boundary and sources tion: 35-06, All AMS members US$87, List US$109, Order code of critical exponents; R. M. Colombo and M. Garavello, On the TRANS2/220 p-system at a junction; A. V. Fursikov, Analyticity of stable invariant manifolds of 1D-sernilinear parabolic equations; G. Hegarty and S. Taylor, Boundary feedback stabilization of nonlinear beam models; V. Isakov, Increased stability in the Geometry and Topology continuation for the Helmholtz equation with variable coefficient; J. R. King, Microscale sensitivity in moving-boundary problems for the thin-film equation; W. Littman and S. Taylor, The heat and Schrodinger equations: Boundary control with one shot; The Ricci Flow: J. Serrin, A remark on the Morrey potential; G. Todorova and B. Yordanov, Nonlinear dissipative wave equations Techniques and with potential; R. Triggiani and X. Xu, Pointwise Carleman estimates, global uniqueness, observability, and stabilization Applications for Schrodinger equations on Riemannian manifolds at the Part 1: Geometric Aspects H 1 (Q)-level. Contemporary Mathematics, Volume 426 Bennett Chow, University of California, San Diego, April2007, 404 pages, Softcover, ISBN-10: 0-8218-3766-4, ISBN-13: CA, and East China Normal 978-0-8218-3766-5, 2000 Mathematics Subject Classification: 35- XX, 49-XX, 93-XX, All AMS members US$87, List US$109, Order University, Shanghai, People's code CONM/426 Republic of China, Sun-Chin Chu, National Chung Cheng University, Chia-Yi, Taiwan,

668 NOTICES OF THE AMS VOLUME 54, NUMBER 5 New Publications Offered by the AMS

David Glickenstein, University of Arizona, Tucson, for which ]. Milnor was awarded the Fields Medal in AZ, Christine Guenther, Pacific University, Forest 1962. Grove, OR, James Isenberg, University of Oregon, Contents: Exotic spheres: Introduction: How these papers came to Eugene, OR, Tom Ivey, College of Charleston, SC, be written; On manifolds homeomorphic to the ?-sphere; On the Dan Knopf, University of Texas, Austin, TX, Peng Lu, relationship between differentiable manifolds and combinatorial manifolds; Sommes de varietes differentiables et structures University of Oregon, Eugene, OR, Feng Luo, Rutgers differentiables des spheres; Differentiable structures on spheres; University, Piscataway, N], and Lei Ni, University of A procedure for killing groups of differentiable California, San Diego, CA manifolds; Differentiable manifolds which are homotopy spheres; with M. A. Kervaire, Groups of homotopy spheres: I; Differential This book gives a presentation of topics in Hamilton's Ricci flow topology; Expository lectures: Introduction; with J. R. Munkres, for graduate students and mathematicians interested in working Lectures on differential topology (Notes by]. R. Munkres); Lectures in the subject. The authors have aimed at presenting technical on differentiable structures; Smooth manifolds with boundary; material in a clear and detailed manner. In this volume, geometric Relations with algebraic topology: Introduction; with R. Bott, On aspects of the theory have been emphasized. The book presents the parallelizability of the spheres; Some consequences of a the theory of Ricci solitons, Kahler-Ricci flow, compactness theorem of Bott; On the Whitehead homomorphism]; with M.A. theorems, Perelman's monotonicity and no local Kervaire, Bernoulli numbers, homotopy groups and a theorem of collapsing, Perelman's reduced distance function and applications Rohlin; Cobordism: Introduction; On the cobordism ring n *; On to ancient solutions, and a primer of 3-manifold topology. Various the cobordism ring n * and a complex analogue, part I; Travaux technical aspects of Ricci flow have been explained in a clear and de Milnor sur le cobordisme; A survey of cobordism theory; A detailed manner. The authors have tried to make some advanced survey of cobordism (Erratum); Spin structures on manifolds; material accessible to graduate students and nonexperts. The book Remarks concerning spin manifolds; On the Stiefel-Whitney gives a rigorous introduction to Perelman's work and explains numbers of complex manifolds and of spin manifolds; A technical aspects of Ricci flow useful for singularity analysis. concluding amusement: ; Bibliography; Throughout, there are appropriate references so that the reader Index. may further pursue the statements and proofs of the various Collected Works, Volume 19 results. Contents: Ricci solitons; Kahler-Ricci flow and Kahler-Ricci June 2007, 329 pages, Hardcover, ISBN-10: 0-8218-4230-7, ISBN- solitons; The compactness theorem for Ricci flow; Proof of the 13: 978-0-8218-4230-0, LC 2006048014,2000 Mathematics Subject compactness theorem; Energy, monotonicity, and breathers; En­ Classification: 01A75, 57-06,All AMS members US$55,ListUS$69, tropy and no local collapsing; The reduced distance; Applications Order code CWORKS/ 19.3 of the reduced distance; Basic topology of 3-manifolds; Basic Ricci flow theory; Other apsects of Ricci flow and related flows; Glossary; Bibliography; Index. Mathematical Surveys and Monographs, Volume 135 Logic and Foundations May 2007, 536 pages, Hardcover, ISBN-10: 0-8218-3946-2, ISBN- 13: 978-0-8218-3946-1, 2000 Mathematics Subject Classification: 53C44, 53C25, 58]35, 35K55, 35K05, All AMS members US$87, ------CONTEMPORARY Advances in Logic List US$1 09, Order code SURV / 13 5 MATHEMATICS ------Su Gao and Steve Jackson, Advances University of North Texas, Collected Papers of in Logic Denton, TX, and Yi Zhang, Sun Yat-Sen University, Guangzhou, SuGao Steve Jackson Vi Zhong People's Republic of China, Differential Topology Editors Editors

John Milnor, Stony Brook ------The articles in this book are based on University, NY, Editor Amenc.Inllathemat.caiSocoety talks given at the North Texas Logic Conference in October of 2004. The main The field of differential topology goal of the editors was to collect articles representing diverse fields underwent a dramatic development within logic that would both contain significant new results and be period between 1955 and 1965. This accessible to readers with a general background in logic. Included collection of articles written by one of in the book is a problem list, jointly compiled by the speakers, the creators of this field contains not only original papers but that reflects some of the most important questions in various also previously unpublished expository lectures. It includes areas of logic. This book should be useful to graduate students commentary by the author, filling in some of the historical and researchers alike across the spectrum of mathematical context, and outlining subsequent developments. It includes a rich logic. bibliography of newer and older papers, providing a wider and Contents: j. R. Steel, A stationary-tower-free proof of the derived deeper understanding of the subject. It also outlines the actual model theorem; I. Farah, A proof of the Li -absoluteness theorem; state of the art and provides an index that will allow the reader to S. Bold and B. Lowe, A simple inductive measure analysis browse easily through the book. for cardinals under the of ; S. Lempp and Of particular interest are the articles related to the existence T. Slaman, The complexity of the index sets of N0-cateogrical of exotic differentiable structures on spheres, the achievement theories and of Ehrenfeucht theories; W. Calvert, S. S. Goncharov,

MAY 2007 NOTICES OF THE AMS 669 New Publications Offered by the AMS

and J. F. Knight, Computable structures of Scott rank wfK D. G. C. McKeon, Using the renormalization group; J. Palmer, in familiar classes; R. Solomon, Thin classes of separating Short distance behavior of scaling functions for the planar ising sets; A. Blass, Voting rules for infinite sets and boolean model; I. Todorov, Constructing conformal field theory models; algebras; B. Kastermans, Very mad families; C. M. Boykin and S. Weinzierl, The art of computing loop ; J. Zinn-Justin, S. Jackson, Borel boundedness and the rounding property; The transition temperature of the weakly interacting Bose S. Gao, A. W. Miller, and W. A. R. Weiss, Steinhaus sets and gas. Jackson sets; S. Gao, S. Jackson, and Y. Zhang, A problem Fields Institute Communications, Volume 50 list. Contemporary Mathematics, Volume 425 March 2007, 404 pages, Hardcover, ISBN-10: 0-8218-4273-0, ISBN- 13: 978-0-8218-4273-7, LC 2006048035,2000 Mathematics Subject April2007, 150 pages, Softcover, ISBN-10: 0-8218-3819-9, ISBN-1 3: Classification: 81Tl5, 81Tl6, 76F30, 60K35, 30C99, 60]65, 82B41, 978-0-8218-3819-8, LC 2006047984, 2000 Mathematics Subject 82B43, All AMS members US$95, List US$119, Order code FIC/ 50 Classification: 03C52, 03D25, 03D35, 03D80, 03£05,03£15,03£55, 03£60, 05C12, 52C20, All AMS members US$39, ListUS$49, Order code CONM/ 425 The Conceptual Mathematical Physics Foundations of Quantum Mechanics Leonard Eisenbud

Universality and This book provides a clear and logical path to understanding what FIELDS INST! rt TE Renormalization COMMIJNlCAl IONS quantum mechanics is about. It will From Stochastic Evolution be accessible to undergraduates with Universality and to Renormalization of minimal mathematical preparation: Renormalization all that is required is an open From Stochastic Evolution to Quantum Fields Renormalization of Quantum Fields mind, a little algebra, and a first course in undergraduate

Ilia Binder physics. Dirk Kreimer Ilia Binder, University of Toronto, Editors ON, Canada, and Dirk Kreimer, Quantum mechanics is arguably the most successful physical f:) theory. It makes predictions of incredible accuracy. It provides the Institut des Hautes Etudes structure underlying all of our electronic technology, and much Scientifiques, Bures-sur-Yvette, of our mastery over materials. But compared with Newtonian France, Editors mechanics, or even relativity, its teachings seem obscure-they have no counterpart in everyday experience, and they sometimes This book covers a wide range of phenomena in the contradict our simplest notions of how the world works. A natural sciences dominated by notions of universality and full understanding of the theory requires prior mastery of very renormalization. The contributions in this volume are equally advanced mathematics. This book aims at a different goal: to teach broad in their approach to these phenomena, offering the the reader, step by step, how the theory came to be and what, mathematical as well as the perspective of the applied sciences. fundamentally, itis about. They explore renormalization theory in and statistical physics, and its connections to modern mathematics Most students learn physics by learning techniques and as well as physics on scales from the microscopic to the formulas. This is especially true in a field like quantum macroscopic. mechanics, whose content often contradicts our common sense, and where it's tempting to retreat into mathematical This item will also be of interest to those working in probability formalism. This book goes behind the formalism to explain and applications. in direct language the conceptual content and foundations of Titles in this series are co-published with the Fields Institute quantum mechanics: the experiments that forced to for Research in Mathematical Sciences (Toronto, Ontario, construct such a strange theory, and the essential elements of its Canada). strangeness. Contents: S. Arnone, T. R. Morris, and 0. ]. Rosten, Manifestly Contents: The failure of classical theory; Consequences of gauge invariant exact renormalization groups; R. 0. Bauer, a mistrust of theory; Properties of electrons, photons; The SLE(8/3) and Brownian excursions in annuli; V. Beffara, Cardy's De Broglie relations; An analysis of electron ; formula on the triangular lattice, the easy way; K. Ebrahirni-Fard Heisenberg's principle of indeterminancy; Interpretations of the and L Guo, Rota-Baxter algebras in renormalization of Heisenberg principle; Dynamical properties of microsystems; perturbative quantum field theory; J. A. Gracey, Practicalities of Determinism and state; Statistical determinism; Probability renormalizing quantum field theories; S. Hollands, Quantum field amplitudes; The superposition principle; Summary and comment; theory in curved spacetime; A. R. Its, B.-Q. ]in, and v. E. Korepin, Index. Entropy of XY spin chain and block Toeplitz determinants; AMS Chelsea Publishing N.-G. Kang, On the quantitative boundary behavior of SLE; M. J. Kozdron and G. F. Lawler, The configurational measure May 2007, 148 pages, Hardcover, ISBN-10: 0-8218-4179-3, ISBN-13: on mutually avoiding SLE paths; D. Kreimer, Dyson-Schwinger 978-0-8218-4179-2, LC 2006052989, 2000 Mathematics Subject equations: From to number theory; G. F. Lawler and Classification: 81-01, All AMS members US$26, List US$29, Order J. R. Lind, Two-sided SLE813 and the infinite self-avoiding polygon; code CHEL/360.H

670 NOTICES OF THE AMS VOLUME 54, NUMBER 5 New AMS-Distributed Publications

Number Theory David A. Vogan, ]r., Isolated Unitary Representations: Isolated unitary representations; Bibliography; Wen-Ching Winnie Li, Ramanujan Graphs and Ramanujan Hypergraphs: Ramanujan graphs and Ramanujan hypergraphs; Ramanujan graphs and connections with number theory; Ramanujan hypergraphs; Automorphic Forms Bibliography. and Applications lAS/Park City Mathematics Series, Volume 12 Peter Sarnak, Princeton May 2007,427 pages, Hardcover, ISBN-10: 0-8218-2873-8, ISBN-13: University, N], and Freydoon 978-0-8218-2873-1, LC 2006048036, 2000 Mathematics Subject Shahidi, Purdue University, West Classification: ll-06, l1Fl2, l1F66, l1F70, 11F72, l1G40, l1M36, Lafayette, IN, Editors 11 T60, 22E46, 81QSO, All AMS members US$60, ListUS$75, Order code PCMS/12 The theory of automorphic forms has seen dramatic developments in recent years. In particular, important instances of Langlands functoriality have been established. This volume presents three weeks of lectures from New AMS-Distributed the lAS/Park City Mathematics Institute Summer School on automorphic forms and their applications. It addresses some of the general aspects of automorphic forms, as well as certain recent Publications advances in the field. The book starts with the lectures of Borel on the basic theory of automorphic forms, which lay the foundation for the lectures by Cogdell and Shahidi on converse theorems and Applications the Langlands-Shahidi method, as well as those by Clozel and Li on the Ramanujan and graphs. The analytic theory of GL(2)-forms and L·functions are the subject of Michel's lectures, while Terras covers arithmetic . Coding Theorems The volume also includes a chapter by Vogan on isolated unitary representations, which is related to the lectures by of Classical and Clozel. Quantum Information This volume is recommended for independent study or an advanced topics course. It is suitable for graduate students Theory and researchers interested in automorphic forms and number K. R. Parthasarathy, Indian theory. Statistical Institute, New Delhi, Titles in this series are co-published with the Institute for India Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National The aim of this little book is to convey Council of Teachers of Mathematics (NCTM) receive a 20% discount three principal developments in the from list price. evolution of modern information theory: Shannon's initiation of Contents: Introduction; Armand Borel, Automorphic Forms on a revolution in 1948 by his interpretation of Boltzmann entropy Reductive Groups: Automorphic forms on reductive groups; as a measure of information yielded by an elementary statistical Bibliography; L. Clozel, Spectral Theory of Automorphic Forms: experiment and basic coding theorems on storing messages Spectral theory of automorphic forms; Mostly SL(2); The spectral and transmitting them through noisy communication channels decomposition of L 2 ( G( Q) \ G (A)): Arthur's conjectures; Known in an optimal manner; the influence of ergodic theory in the bounds for the cuspidal spectrum and the Burger-Sarnak method; enlargement of the scope of Shannon's theorems through the Applications: Control of the spectrum; All reductive adelic works of McMillan, Feinstein, Wolfowitz, Breiman and others and groups are tame; Bibliography; ]ames W Cogdell, L-functions and its impact on the appearance of the Kolmogorov-Sinai invariant Converse Theorems for GLn: L-functions and converse theorems for elementary dynamical systems; and finally, the more recent for GLn; Fourier expansions and multiplicity one; Eulerian work of Schumacher, Holevo, Winter, and others on the role integrals for GLn; Local L-functions; Global L-functions; Converse of in the quantum avatar of the basic theorems; Converse theorems and functoriality; Bibliography; coding theorems when messages are encoded as quantum states, Philippe Michel, Analytic Number Theory and Families of transmitted through noisy quantum channels, and retrieved by Automorphic L·functions: Analytic number theory and families generalized measurements. of automorphic L-functions; Analytic properties of individual A publication of Hindustan Book Agency. Distributed on an L-functions; A review of classical automorphic forms; Large sieve exclusive basis by the AMS in North America. Online bookstore inequalities; The subconvexity problem; Some applications of rights worldwide. subconvexity; Bibliography; Freydoon Shahidi, Langlands-Shahidi Method: Langlands-Shahidi Method; Basic concepts; Eisenstein Contents: Entropy of elementary information sources; series and L-functions; Functional equations and multiplicativity; Stationary information sources; Communication in the Holomorphy and boundedness; Applications; Bibliography; Au­ presence of noise; Quantum coding theorems; Bibliography; drey Terras, Arithmetical Quantum Chaos: Arithmetical quantum Index. chaos; Finite models; Three symmetric spaces; Bibliography;

MAY 2007 NOTICES OF THE AMS 671 New AlMS-Distributed Publications

Hindusta n Book Agency 14F22, 14F05, 14B12, 16K50, 14G99,53C21, 53C20,58]60, 58]05, February 2007, 168 pages, Hardcover, ISBN-10: 81-85931-75-5, 35]60, 60G50, 60]65, 60] 55, 28A80, Individual member US$112, ListUS$124, OrdercodeAST/ 307 ISBN-13: 978-81-85931-75-3, 2000 Mathematics Subject Classifica­ tion: 94A24, 68P30, All AMS members US$27, List US$34, Order codeHIN/33 Curvature Problems Curvature Problems Claus Gerhardt, Ruprecht- ctaus Gerhardt Karls-Universitdt, Heidelberg, Geometry and Topology Germany Applying analytic methods to geometric problems has proved to be extremely Seminaire Bourbaki fruitful in the last decades. Among the new techniques, with the help of Volume 2004/2005 which many problems have been solved, Exposes 938-951 ii'··-·-··- curvature flows and a priori estimates for fully non-linear elliptic partial differential As in the preceding volumes of this equations are especially important. seminar, one finds here fourteen survey The use of curvature flows started with the groundbreaking lectures on topics of current interest: paper of Hamilton in which he considered the Ricci flow three on algebraic geometry, two on which is driven by the Ricci curvature. Huisken then studied differential geometry, one about the the mean curvature flow. These fundamental papers created Poincare conjecture, one on dynamic a new analytical tool for solving problems in geometry and systems, one on number theory, one physics. on the fundamental lemma, one about In the present book we consider curvature problems in Riemannian the Andre-Oort conjecture, one about quadratic forms, one on and Lorentzian geometry which have in common that either the algebraic topology, one on mathematical physics, and one on extrinsic curvature of closed hypersurfaces is prescribed or that probabilities. curvature flows driven by the extrinsic curvature are studied This item will also be of interest to those working in algebra and and used to obtain some insight into the nature of possible algebraic geometry, differential equations, mathematical physics, singularities. probability, and number theory. This book is supposed to be an advanced textbook for graduate A publication of the Societe Mathematique de France, Marseilles students and researchers interested in geometry and general (SMF), distributed by the AMS in the U.S., Canada, and relativity. Mexico. Orders from other countries should be sent to the A publication of International Press. Distributed worldwide by the SMF. Members of the SMF receive a 30% discount from American Mathematical Society. list. Contents: Foundations; Curvature flows in semi-Riemannian Contents: Novembre 2004: 0. Biquard, Metriques Kii.hleriennes manifolds; Hypersurfaces of prescribed curvature in Riemannian courbure scalaire constante : unicite, stabilite; X. Buff, La a manifolds; Hypersurfaces of prescribed curvature in Lorentzian mesure d'equilibre d'un endomorphisme de IP'k(

672 N OTICES OF THE AMS VOLUME 54, NUMBER 5 New AMS-Distributed Publications

proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves; Number Theory B. Mazur, W. Stein, and J. Tate, Computation of p-adic heights and log convergence; R. Pollack and T. Weston, Kida's formula and congruences; P. Schneider and J. Teitelbaum, Banach-Hecke A Collection of algebras and p-adic Galois representations; A. Scholl, Higher DOCUMENTA MATHEMATICA fields of norms and (¢,[)-modules; J. Silverman, Divisibility JOURNAl-Dl':ft Manuscripts Written sequences and powers of algebraic integers; R. Taylor, On OoonliNDf.:T 1996 the meromorphic continuation of degree two L-functions; in Honour of J. Tilouine, Siegel varieties and p-adic Siegel modular forms; john H. Coates on J. Wintenberger, On p-adic geometric representations of ACOI.l..r.cTI0/1 OI'MANUSCKII'ffl WRITT&NINIIO:

!.I~KO,S.LICHTF;NIIAUM , Ivan B. Fesenko, University ISBN-13: 978-3-936609-28-8, 2000 Mathematics Subject Classifica­ B. PERRIN-HIOU, P. SCUN>:(Dt;R of Nottingham, England, tion: 11-XX, All AMS members US$94, List US$118, Order code Stephen Lichtenbaum, Brown DOCMATH/ 4 University, Providence, RI, Bernadette Perrin-Riou, Universite Paris-Sud, Orsay, France, and Peter Schneider, UniversiUit Munster, Germany, Editors

This volume is dedicated to Professor John H. Coates, an outstanding contributor to number theory, both through his pioneering and fundamental mathematical works and through the magnificent mathematical school he has established. It contains 24 articles written by 38 authors on a wide range of topics in the cutting edge of research in number theory, algebraic geometry and analysis: zeta functions and L-functions, automorphic and modularity issues, Galois representations, arithmetic of elliptic curves, Iwasawa theory, noncommutative Iwasawa theory, and p-adic analysis. This volume will be of interest to researchers and students in these and neighboring fields. A publication of the Documenta Mathematica. The AMS distributes this series, beginning with volume 3, in the United States, Canada, and Mexico. Contents: K. Ardakov and K. Brown, Ring-theoretic properties of Iwasawa algebras: A survey; G. Banaszak, W. Gajda, and P. Krason, On the image of l-actic Galois representations for Abelian varieties of type I and II; S. Bocherer and A. Panchishkin, Admissible p-adic measures attached to triple products of elliptic cusp forms; D. Burns and M. Flach, On the equivariant Tamagawa number conjecture for Tate motives, part II; D. Burns and 0. Venjakob, On the leading terms of Zeta isomorphisms and p-adic L-functions in non-commutative Iwasawa theory; K. Buzzard and F. Calegari, The 2-adic eigencurve is proper; L. Clozel and E. Ullmo, Equidistribution Adelique des Tares et equidistribution des points CM; R. Coleman and K. McMurdy, Fake CM and the stable model of Xo(Np3 ) ; D. Delbourgo, Lambda-adic Euler characteristics of elliptic curves; E. de Shalit, Coleman integration versus Schneider integration on semistable curves; R. Greenberg, On the structure of certain Galois cohomology groups; M. Harris, J.-S. Li, and C. Skinner, p-adic L-functions for unitary Shimura varieties I: Construction of the Eisenstein measure; H. Hida, Anticyclotomic main conjectures; F. Jarvis, Optimal levels for modular mod 2 representations over totally real fields; K. Kato, Universal norms of p-units in some non-commutative Galois extensions; S. Kobayashi, An elementary

MAY 2007 NOTICES OF THE AMS 673 Classified Advertisements Positions available, items for sale, services available, and more

CALIFORNIA ILLINOIS send cover letter, CV, research statement and three letters of reference to: UNIVERSITY OF CALIFORNIA lOS UNIVERSITY OF CHICAGO Search Committee Stevanovich Center for Financial The Stevanovich Center for Financial ANGElES Mathematics Institute for Pure and Applied Mathematics c/ o Terri Rossi Mathematics Applications are invited for the following 5734 S. University Avenue Chicago IL 60637 position: Research Associate (Instruc­ The Institute for Pure and Applied Math­ The University of Chicago is an Affirma­ ematics (IP AM) at UCLA is seeking its next tor)/Lecturer Outstanding candidates in the areas of Financial Mathematics/ tive Action/Equal Opportunity Employer. director, to begin the position in July 2008. 000036 Candidates with imagination, energy, and Statistics//Economics are welcome. The Research Associate (Instruc­ experience are encouraged to apply. It is tor)/Lecturer will have his/her appoint­ necessary that IPAM's director possess ment in the Department of Statistics or OHIO sufficient scientific distinction to be of­ Mathematics, and will be associated with fered a tenured faculty position at UCLA. the new Stevanovich Center for Financial THE OHIO STATE UNIVERSITY Further information about IPAM and its Mathematics. The initial appointment is College of Mathematical and Physical programs is available at www. i pam. ucla. for one year, with possible renewal for Sciences edu. Candidates are asked to send a CV an additional year. The successful candi­ Department of Mathematics and cover letter to di rectorsearch@ date, who must have a Ph.D. by the date i pam. ucla. edu. For fullest consider­ The Department of Mathematics in the of the appointment, will be expected to College of Mathematical and Physical ation, applications should be received by teach one course per academic year, and June 1, 2007; however, applications will be Sciences at The Ohio State University is otherwise at liberty to do research. The will have available several Arnold Ross considered until the position is filled. For position provides the opportunity for a Assistant Professorships. These term po­ a detailed job description, go to http: I I new Ph.D. to focus on his/ her research. sitions are primarily teaching positions, www.ipam.ucla.edu/jobopenings/ The Stevanovich Center for Financial renewable annually for up to a total of director. html. UCLA is an Equal Oppor­ Mathematics is jointly operated by the De­ three years. Candidates are expected to tunityI Affirmative Action Employer. partments of Mathematics, Statistics, and present evidence of excellence in teaching 000035 Economics. Interested applicants should and research. Further information on the

Suggested uses for classified advertising are positions available, books or June 28, 2007; October 2007 issue-July 26, 2007; November 2007 issue-August lecture notes for sale, books being sought, exchange or rental of houses, 28, 2007; December 2007 issue-October 1, 2007. and typing services. U.S. laws prohibit discrimination in employment on the basis of color, age, The 2007 rate is $110 per inch or fraction thereof on a single column(one­ sex, race, religion, or national origin. "Positions Available" advertisements inch minimum), calculated from top of headline. Any fractional text of 1/2 from institutions outside the U.S. cannot be published unless they are inch or more will be charged at the next inch rate. No discounts for multiple accompanied by a statement that the institution does not discriminate on ads or the same ad in consecutive issues. For an additional $10 charge, these grounds whether or not it is subject to U.S.laws. Details and specific announcements can be placed anonymously. Correspondence will be wording may be found on page 1373 (vol. 44). forwarded. Situations wanted advertisements from involuntarily unemployed math­ Advertisements in the "Positions Available" classified section will be set ematicians are accepted under certain conditions for free publication. Call with a minimum one-line headline, consisting of the institution name above toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or 401-45 5-4084 body copy, unless additional headline copy is specified by the advertiser. worldwide for further information. Headlines will be centered in boldface at no extra charge. Ads will appear Submission: Promotions Department, AMS, P.O. Box 6248, Providence, in the language in which they are submitted. Rhode Island 02940; or via fax: 401-331-3842; or send email to cl­ There are no member discounts for classified ads. Dictation over the a5 5 ad 5 @am 5 . or g. 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: June/ July 2007 billed upon publication. issue-April27, 2007; August 2007 issue-May 29, 2007; September 2007 issue-

674 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Classified Advertisements department can be found at http: I jwww. In addition, the institute periodically funds standalone source for calculus review. math.ohio-state.edu. participation in international conferences. Visit http: I jwww. magi math. com to All candidates should apply online at It is also possible to raise grant money for download a free sample in PDF format http://www.math.ohio-state.edu/ research through various government and and to obtain ordering information. Order applications/ and have at least three industry sources. Faculty enjoy several one today and recommend it to your letters of recommendation sent to: personal benefits including on-campus students! Advisory Committee housing with free high- LAN con­ 000038 Department of Mathematics nection, free or subsidized medical care, The Ohio State University and easy access to schools on campus for 231 W. 18th Avenue children. Further information is available Columbus, OH 43210 at:http://www.math.iitb.ac.in/.Ap­ If you cannot apply online, please send plications including a curriculum vitae, a vitae, research statement, and teaching list of publications, a statement describing statement to the above address. Please current and planned research, a state­ direct inquiries to facul tysearch@math. ment outlining teaching experience, and ohi a-state. edu. at least three letters of recommendations should be sent to: Head, Department of To build a diverse workforce, Ohio State Mathematics, liT Bombay, Powai, Mumbai encourages applications from minorities, 400076, India. Applications can also be veterans, women, and individuals with dis­ sent by email to: head. math@i i tb. ac. in abilities. Flexible work options available. or by fax to (+91-22) 2572 3480. EEO/ AA Employer. 000033 000032

TENNESSEE MEXICO UNIVERSIDAD NACIONALAUTONOMA TENNESSEE STATE UNIVERSITY DE MEXICO Assistant/ Associate Professor Institute for Research in of Mathematics Applied Mathematics and Systems Teach undergraduate and graduate math­ ematics courses; Ph.D . in mathematics, or The Department of Mathematical and mathematical statistics at the time of ap­ Numerical Methods of the Institute for pointment; minimum of five years' teach­ Research in Applied Mathematics and ing experience at a regionally accredited Systems of the Universidad Nacional Aut­ four-year college or university required noma de Mexico in the campus of Mexico for appointment to the rank of associ­ City invites applicants for tenure-track ate professor. Please visit our website at and postdoctoral positions for research­ http: I jwww. jobs. tnstate. edu for full ers in mathematical physics. description and online application. TSU is Candidates working in direct and in­ an EO/ AA/M/F employer. verse problems in spectral and scattering 000034 theories and in quantum computation and information are of particular interest. Applicants are required to have a Ph.D. and for tenure-track position at least two INDIA years of postdoctoral experience. To apply: Send a brief cover letter, a INDIAN INSTITUTEOFTECHNOLOGY curriculum vita, a statement describing BOMBAY your current and planned research and Department of Mathematics three letters of recommendations to: Dr. Rafael del Rio or Dr. Ricardo Weder, Applications are invited for visiting and IIMAS-UNAM, Departamento de Metodos permanent faculty positions at all levels. Matematicos y Numericos, Apdo. Postal Applicants should have a Ph.D. and an 20-726 Admon. No. 20, Delegacion Alvaro excellent academic record. Outstanding Obregon 01000 Mexico, D.F. candidates in all areas of Mathematical Emails: [email protected] Sciences are encouraged to apply. Current and weder@servi dor. unam. mx. departmental interests include Algebra, Algebraic Geometry, Algebraic Topology, For information about the institute Combinatorics, Differential Geometry, please visit our electronic page http: I I Functional Analysis, Harmonic Analysis, www. i i mas. unam. mx. Number Theory, Numerical Analysis, Par­ 000037 tial Differential Equations, Probability and Statistics. The Department of Mathemat­ ics and liT Bombay offer an environment BOOKS AVAILABLE conducive to research. Teaching duties are about 5 hours a week and consist of CALCULUSFORTHEFORGETFUL at most two courses per semester at the by Wojciech Kosek undergraduate (B.Tech.), postgraduate (M.Sc.), or doctoral (Ph.D.) levels. A sub­ A new short calculus review book, 160 stantial "seed grant" of up toRs. 5,00,000 pages, 6x9 in., can be used as a supple­ is available for each new faculty member. ment to any calculus textbook or as a

MAY 2007 NOTICES OF THE AMS 675 Meetings & Conferences oftheAMS

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

Special Sessions Zacatecas, Mexico Applied Category Theory: Graph-Operad Logic, Zbig­ Universidad Aut6noma de Zacatecas niew Oziewicz, UNAM, and Hanna Makaruk, Los Alamos National Laboratory. May 23-26,2007 Applied Mathematics in Petroleum Industry Problems, Wednesday - Saturday Jorge Velasco-Hernandez, Instituto Mexicano del Petroleo, and William Fitzgibbon, University of Houston. Meet ing #1 028 Convexity, Luis Montejano, UNAM, and Paul Goodey, Seventh ]oint International Meeting of the AMS and the University of Oklahoma. Sociedad Matematica Mexicana. Differential Geometry, Rafael Herrera Guzman, CIMAT, Associate secretary: Matthew Miller Guanajuato, and Haydee Herrera, Rutgers University. Announcement issue of Notices: April 2007 Functional and Harmonic Analysis (in honor of Mischa Program first available on AMS website: Not applicable Cotlar), Salvador Perez Esteva, UNAM, Cuernavaca, and Program issue of electronic Notices: Not applicable Josefina Alvarez, New Mexico State University. Issue of Abstracts: Not applicable Holomorphic Dynamics in the Riemann Sphere (in mem­ ory of Adrien Douady), Monica Moreno Rocha, CIMAT, Deadlines Guanajuato, and Araceli Medina-Bonifant, University of For organizers: Expired Rhode Island. For abstracts: April 20, 2007 Low-Dimensional Topology, Mario Eudave-Munoz, UNAM, and Jennifer Schultens, University of California Invited Addresses Davis. Monica Clapp, Universidad Nacional Aut6noma de Mathematical Physics, Jaime Cruz Sampedro, Universi­ Mexico, Multiple solutions to an elliptic equation with criti­ dad Autonoma del Estado de Hidalgo, and Daniel Tataru, cal nonlinearity. University of California Berkeley. Edward L. Green, Virginia Polytechnic Institute & State University, Koszul A lgebras: A very special class of Metric Differential Geometry, Catherine Searle, UNAM, algebras. Cuernavaca, and Gerard Walschap, University of Okla­ Kiran S. Kedlaya, Massachusetts Institute of Technol­ homa. ogy, A differential approach to computing zeta functions Nonlinear Boundary Value Problems, Monica Clapp, over finite fields. UNAM, and Alfonso Castro, Harvey Mudd College. John Lott, University of Michigan, Ann Arbor, Curvature Nonlinear Waves, Petr Zhevandrov, Universidad Mi­ of metric spaces. choacana de San Nicolas de Hidalgo, and Arturo Vargas Gelasio Salazar, Universidad Aut6noma de San Luis and Gustavo Cruz, UNAM. Potosi, Crossing numbers of graphs in surfaces. Operator Theory and Complex Analysis, Enrique Petr Zhevandrov, Universidad Michoacana de San Ramirez de Arellano and Nikolai Vasilevski, CINVESTAV, Nicolas de Hidalgo, Title to be announced. and Raul Curto, University of Iowa, Iowa City.

676 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings & Conferences

Representation Theory (in honor ofEdward Green's 60th Control and Optimization of Non-linear PDE Systems, birthday), Christof Geiss, UNAM, and Gordana Todorov, Irena Lasiecka, University of Virginia, and jan Sokolowski, Northeastern University. Systems Research institute. Rings and Modules, Maria jose Arroyo, UAM-Iztapalapa, Dynamical Systems, Steven Hurder, University of and Sergio R. Lopez-Permouth, Ohio University. Illinois at Chicago, Michal Misiurewicz, Indiana Univer­ sity-Purdue University Indianapolis, and Pawel Walczak, University of Lodz. , Poland Dynamics, Control and Optimization of Finite Dimen­ sional Systems: Theory and Applications to Biomedicine, University of Warsaw Urszula Forys, Warsaw University, Urszula Ledzewicz, Southern Illinois University, and Heinz Schaettler, Wash­ July 31 -August 3, 2007 ington University. Tuesday - Friday Ergodic Theory and Topological Dynamics, Dan Ru­ dolph, Colorado State University, and Mariusz Lemanc­ Meeting #1 029 zyk, Nicholas Copernicus University. First ]oint International Meeting between the AMS and the Extremal and Probabilistic Combinatorics, joel Spencer, Polish Mathematical Society New York University-Courant Institute, and Michal Karon­ Associate secretary: Susan]. Friedlander ski and Andrzej Rucinski, Adam Mickiewicz University. Announcement issue of Notices: To be announced Function Spaces, Theory of Operators and Geometry Program first available on AMS website: Not applicable of Banach Spaces, Henryk Hudzik, Adam Mickiewicz Program issue of electronic Notices: Not applicable University, Anna Kaminska, University of Memphis, and Issue of Abstracts: Not applicable Mieczyslaw Mastylo. Geometric Applications of Homotopy Theory, Yuli B. Deadlines Rudyak, University of Florida, Boguslaw Hajduk, War­ For organizers: Expired saw University, jaroslaw Kedra, University of Aberdeen, For consideration of contributed papers in Special Ses­ and Aleksy Tralle, The College of Economics & Comp sions: To be announced Science. For abstracts: To be announced Geometric Function Theory, Michael Dorff, Brigham Young University, Piotr Liczberski, University of Lodz, Invited Addresses Maria Nowak, Biblioteka Instytutu Matematyki, and Ted , Rutgers University, Title to be an­ Suffridge, University of Kentucky. nounced. , Mladen Bestvina, University Tomasz J. Luczak, Adam Mickiewicz University, Title of Utah, Tadeusz Januszkiewicz, Ohio State University, to be announced. and jacek Swiatkowski, University of Wroclaw. , Massachusetts Institute of Technol­ Geometric Topology, jerzy Dydak, University of Ten­ ogy, Title to be announced. nessee, Slawomir Nowak, and Stanislaw Spiez, University Ludomir Newelski, University of Wroclaw, Title to be of Warsaw. announced. Invariants of Links and 3-manifolds, Mieczyslaw Dab­ Madhu Sudan, Massachusetts Institute of Technology, kowski, University of Texas at Dallas, jozef H. Przytycki, Title to be announced. George Washington University, Adam S. Siroka, State Anna Zdunik, Warsaw University, Title to be an­ University of New York at Buffalo, and Pawel Traczyk, nounced. Warsaw University. Issues in Reforming Mathematics Education, Jeremy Special Sessions Kilpatrick, University of Georgia, and Zbigniew Semadeni, Arithmetic Algebraic Geometry, Grzegorz Banaszak, University of Warsaw. Adam Mickiewicz University, , North­ Mathematics of Large Quantum Systems, Michael Loss, western University, Wojciech Gajda, Adam Mickiewicz Georgia Institute of Technology, jan Philip Solovej, Uni­ University, Piotr Krason, Szczecin University, and Wi­ versity of Copenhagen, and jan Derezinski, University eslawa Nizio. of Warsaw. Complex Analysis, Zeljko Cuckovic, University of To­ Noncom mutative Geometry and Quantum Groups, Paul ledo, Zbigniew Blocki, Jagiellonian University, and Marek Baum, Pennsylvania State University, and Ulrich Kraehmer Ptak, University of Agriculture. and Tomasz Maszczyk. , Robert Devaney, Boston Univer­ Partial Differential Equations of Evolution Type, Susan sity, Jane N. Hawkins, University of North Carolina, and j. Friedlander, University of Illinois at Chicago, and Grze­ Janina Kotus, Warsaw University of Technology. gorz A. Karch, University of Wroclaw. Complexity of Multivariate Problems, joseph F. Traub, Quantum Information Theory, Robert Alicki, University Columbia University, Grzegoorz W. Wasilkowski, Univer­ of Gdansk, and Mary Beth Ruskai, Tufts University. sity of Kentucky, and Henryk Wozniakowski, Columbia Topological Fixed Point Theory and Related Top­ University. ics, jerzy Jezierski, University of Agriculture,

MAY 2007 NOTICES OF THE AMS 677 Meetings & Conferences

Wojdech Kryszewski, Nicholas Copernicus University, evening. A conference dinner-party will close the meeting and Peter Wong, Bates College. on Friday evening. Please watch the conference website Topology of Manifolds and Transformation Groups, Sla­ for more information. womill' Kwasik, Tulane University, Krzysztof Pawalowski, Adam Mickiewicz University, and Dariusz Wilczynski, Travel Utah State University. By Air: The most convenient airport is Warsaw. It takes ap­ proximately 30-45 minutes to get from the airport to the Conf erence Site and Web Site conference site and hotels by bus or cab. Polish Airlines The meeting will be held from Tuesday, July 31, through LOT is the only carrier flying non-stop from the U.S. (New Friday, August 3, at the campus of the University of War­ York and Chicago) to Poland. U.S. citizens do not require saw, located in downtown Warsaw, close to the historical a visa to enter Poland. districts of the city. All U.S. citizens traveling by air must have a valid Most of the information in this announcement is taken passport to re-enter the United States after traveling out from the website maintained by the local organizers. of the country. See these pages maintained by the Bureau Watch http: I /www. ams. ptm. org. pl I for additional program details and links to sites for hotels, tours, and of Consular Affairs, U.S. Department of State for specific much other local information. information about travel to Poland at http:/ /travel. state.gov/travel/tips/regional/regional_ll74. Accommodations html. Participants should make their own arrangements for ac­ commodations. Please contact hotels directly for current room rates. Please watch the conference website for more Chicago, Illinois information on accommodations as well as for housing to be offered in university guest rooms. DePaul University Hotels within walking distance (< 15 min) from the conference site include the following: October 5-6, 2007 Friday - Saturday Hotel Sofitel-Victoria ('"'"""') ht tp://www.orbis.pl/?s=123,0,1161&h=27&j=2 Meeting #1 030 Hotel Bristol ('"'"""') Central Section ht tp://www.royalmeridienbristolhotel.com/ Associate secretary: Susan J. Friedlander Le_Royal_Meridien_Bristol_Home.html Announcement issue of Notices: August 2007 Hotel Harenda ('"') Program first available on AMS website: August 16, 2007 ht tp://www.hotelharenda.eom.pl/i .php?i=ll Program issue of electronic Notices: October 2007 Hotel Mazowiecki (") Issue of Abstracts: Volume 28, Issue 3 http://www.hotelbelwederski .pl/mazowiecki/ eng/index.html Deadlines Hotel Gromada ('""') For organizers: Expired http://www.hotele.gromada.pl/hotele_en. For consideration of contributed papers in Special Ses­ php?id_hotel=17 sions: June 19, 2007 Old Town Apartments For abstracts: August 7, 2007 http://www.warsawshotel.com/warsaw-hotels. htm Invited Addresses Resildence Diana Martin Golubitsky, University of Houston, Title to be http://www.residencediana.com/ announced. Resiidence St. Andrew's Palace http://www.residence.com.pl/en_index.php Matthew J. Gursky, University of Notre Dame, Title to Royal Route Residence be announced. http://www.warsaw-apartments.net/ Alex Iosevich, University of Missouri, Title to be an­ nounced. Registration and Meeting Information David E. Radford, University of Illinois at Chicago, Title Sessions will be held at the University of Warsaw. See the to be announced. conference website to register on line. There will be a conference registration fee. Special Sessions Algebraic Coding Theory (in honor of Harold N. Ward's Social Events and Tours retirement)(Code: SS 19A), Jay A. Wood, Western Michigan All participants are invited to the Opening Ceremonies University. that will take place at 3:00p.m. on Tuesday, July 31, and Algebraic Combinatorics: Association Schemes andRe­ the Welcome Reception the same evening. Cultural and lated Topics (Code: SS 1A), Sung Y. Song, Iowa State Uni­ social events will be planned for Wednesday and Thursday versity, and Paul Terwilliger, University of Wisconsin.

678 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings & Conferences

Algebraic Geometry (Code: SS SA), Lawrence Man Hou Ein and Anatoly S. libgober, University of Illinois New Brunswick, New at Chicago. Analysis and CR geometry (Code: SS 12A), Song-Ying Jersey li, University of California Irvine, and Stephen S-T Yau, University of Illinois at Chicago. Rutgers University-New Brunswick, Busch Applied Harmonic Analysis (Code: SS 13A), Jonathan Campus Cohen and Ahmed I. Zayed, DePaul University. Automorphic Forms: Representation Theory of p-adic October 6-7, 2007 and Adelic Groups (Code: SS 8A), Mahdi Asgari and Anan­ Saturday - Sunday tharam Raghuram, Oklahoma State University. Differential Geometry and its Applications (Code: SS Meeting #1 031 17A), Jianguo Cao, University of Notre Dame. Eastern Section Ergodic Theory and Symbolic Dynamical Systems (Code: Associate secretary: Lesley M. Sibner SS 7A), Ayse A. Sahin and Ilie D. Ugarcovici, DePaul Announcement issue of Notices: August 2007 University. Program first available on AMS website: August 16, 2007 Extremal and Probabilistic Combinatorics (Code: SS 3A), Program issue of electronic Notices: October 2007 Jozsef Balogh, University of Illinois at Urbana-Champaign, Issue of Abstracts: Volume 28, Issue 3 and Dhruv Mubayi, University of Illinois at Chicago. Geometric Combinatorics (Code: SS 15A), Caroline J. Deadlines Klivans, University of Chicago, and Kathryn Nyman, For organizers: Expired Loyola University Chicago. For consideration of contributed papers in Special Ses­ Graph Theory (Code: SS 20A), Hemanshu Kaul and sions: June 19, 2007 Michael J. Pelsmajer, Illinois Institute of Technology. For abstracts: August 7, 2007 Hop{ Algebras and Related Areas (Code: SS 2A), Yev­ Invited Addresses genia Kashina and Leonid Krop, DePaul University, M. Susan Montgomery, University of Southern California, Satyan L. Devadoss, Williams College, Title to be an­ and David E. Radford, University of Illinois at Chicago. nounced. Mathematical Modeling and Numerical Methods (Code: Tara S. Holm, University of Connecticut, Title to be SS 16A), Atife Caglar, University of Wisconsin-Green announced. Bay. Sir Roger Penrose, University of Oxford, Title to be an­ nounced (Einstein Public Lecture in Mathematics). Nonlinear Conservation Laws and Related Problems Scott Sheffield, Courant Institute and Institute for (Code: SS llA), Cleopatra Christoforou and Gui-Qiang Advanced Study, Title to be announced. Chen, Northwestern University. Mu-Tao Wang, Columbia University, Title to be an­ Numerical and Symbolic Techniques in Algebraic Ge­ nounced. ometry and Its Applications (Code: SS 18A), GianMario Besana, DePaul University, Jan Verschelde, University of Special Sessions Illinois at Chicago, and Zhonggang Zeng, Northeastern Commutative Algebra (Code: SS 4A), Jooyoun Hong, Illinois University. University of California Riverside, and Wolmer V. Vascon­ Sequence Spaces and Transformations (Code: SS lOA), celos, Rutgers University. Constantine Georgakis, DePaul University, and Martin Geometric Analysis of Complex Laplacians (Code: SS Buntinas, Loyola University of Chicago. 8A), Siqi Fu, Rutgers University, Camden, Xiaojun Huang, Singular Integrals and Related Problems (Code: SS 14A), Rutgers University, New Brunswick, and Howard J. Jaco­ Laura , Florida International University, and A. M. bowitz, Rutgers University, Camden. Stokolos, DePaul University. Mathematical and Physical Problems in the Foundations Smooth Dynamical Systems (Code: SS 6A), Marian Gidea, of Quantum Mechanics (in honor of Shelly Goldstein's 60th Northeastern Illinois University, and Ilie D. Ugarcovici, birthday) (Code: SS 3A), Roderich Tumulka and Detlef DePaul University. Diirr, Miinchen University, and Nino Zanghi, University The Euler and Navier-Stokes Equations (Code: SS 4A), of Genova. Alexey Cheskidov, University of Michigan, and Susan J. Partial Differential Equations in Mathematical Physics Friedlander and Roman Shvydkoy, University of Illinois (in honor of Shelly Goldstein's 60th birthday) (Code: SS 2A), at Chicago. Sagun Chanillo, Michael K.-H. Kiessling, and Avy Soffer, from Mathematical and Numerical Rutgers University. Viewpoints (Code: SS 9A), Gabriel Koch, University of Chi­ Partial Differential Equations of Mathematical Physics, I cago, Catalin Constantin Turc, Caltech and University of (dedicated to the memory of Tom Branson) (Code: SS 7A), North Carolina at Charlotte, and Nicolae Tarfulea, Purdue Sagun Chanillo, Michael K.-H. Kiessling, and Avy Soffer, University Calumet. Rutgers University.

MAY 2007 NOTICES OF THE AMS 679 Meetings & Conferences

Probability and Combinatorics (Code: SS 1A), Jeffry N. Recent Developments in 2-D Turbulence (Code: SS 3A), Kahn and Van Ha Vu, Rutgers University. MichaelS. Jolly, Indiana University, and Greg Eyink, Johns of the Continuum (Code: SS SA), Simon R. Hopkins University. Thomas, Rutgers University. Topics in Mathematical Physics (Code: SS 4A), Rafal To ric Varieties (Code: SS 6A), Milena S. Hering, Institute Komendarczyk, University of Pennsylvania, and Robert for Mathematics and Its Applications, and Diane Macla­ Michal Owczarek, Los Alamos National Laboratory. gan, Rutgers University. Variational Problems in Condensed Matter(Code: SS SA), lia Bronsard, McMaster University, and Tiziana Giorgi, Albuquerque, New New Mexico State University. Mexico Murfreesboro, University of New Mexico Tennessee October 1 3-14, 2007 Middle Tennessee State University Saturday - Sunday November 3-4, 2007 Meet ing #1 032 Saturday - Sunday Western Section Meeting #1 033 Associate secretary: Michel L. Lapidus Announcement issue of Notices: August 2007 Southeastern Section Program first available on AMS website: August 30, 2007 Associate secretary: Matthew Miller Program issue of electronic Notices: October 2007 Announcement issue of Notices: September 2007 Issue of Abstracts: Volume 28, Issue 4 Program first available on AMS website: September 20, 2007 Deadlines Program issue of electronic Notices: November 2007 For organizers: Expired Issue of Abstracts: Volume 28, Issue 4 For consideration of contributed papers in Special Ses­ Deadlines sions: June 26, 2007 For organizers: Expired For abstracts: August 21, 2007 For consideration of contributed papers in Special Ses­ Invited Addresses sions: July 17, 2007 For abstracts: September 11, 2007 Emmanuel Candes, California Institute of Technology, Title to be announced. Invited Addresses Alexander Polischuk, Univerity of Oregon, Title to be Sergey Gavrilets, University of Tennessee, Mathemati· announced. cal models of speciation. Eric Raines, University of California Davis, Title to be Daniel K. Nakano, University of Georgia, Title to be announced. announced. William E. Stein, University of California San Diego, D. Savage, North Carolina State University, Title SAGE: Software for Algebra and Geometry Experimenta­ to be announced. tion. Sergei Tabachnikov, Pennsylvania State University, Ubiquitous billiards. Special Sessions A ffine Algebraic Geometry (Code: SS 2A), David Robert Special Sessions Finston, New Mexico State University. Advances in A lgorithmic Methods for Algebraic Struc­ Computational Applications of Algebraic Topology tures (Code: SS 3A), James B. Hart, Middle Tennessee State (Code: SS 6A), Ross Staffeldt, New Mexico State Univer· University. sity. Applied Partial Differential Equations (Code: SS 4A), Yuri Computational Methods in Harmonic Analysis and Sig­ A. Melnikov, Middle Tennessee State University, and Alain nal Processing (Code: SS 1A), Emmanuel Candes, California ]. Kassab, University of Central Florida. Institute of Technology, and Joseph D. Lakey, New Mexico Billiards and Related Topics (Code: SS 6A), Sergei State University. Tabachnikov, Pennsylvania State University, and Richard Harmonic Analysis Applied to Partial Differential Equa­ Schwartz, Brown University. tions (Code: SS 7A ), justin Homer, University of California Combinatorial Methods in Continuum Theory (dedicated Berkeley, Changxing Miao, Institute of Applied Physics to ]o Heath, Auburn University, on the occasion of her re­ and Computational Mathematics, and Jiaong Wu, Okla­ tirement) (Code: SS 8A), Judy Anita Kennedy, University homa State University. of Delaware and Lamar University, Krystyna M. Kuper-

680 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings & Conferences berg, Auburn University, and Van C. Nail, University of Matthew J. Visser, Victoria University of Wellington, Richmond. Title to be announced. Differential Equations and Dynamical Systems (Code: SS lA), Wenzhang Huang and Jia Li, University of Alabama, AMS Special Sessions Huntsville, and Zachariah Sinkala, Middle Tennessee State Computability Theory, Rodney G. Downey and Noam University. Greenberg, Victoria University of Wellington. Graph Theory (Code: SS 2A), Rong Luo, Don Nelson, Dynamical Systems: Probabilistic and Semigroup Meth­ Chris Stephens, and Xiaoya Zha, Middle Tennessee State ods, Arno Berger, University of Canterbury, Rua Murray, University. University of Waikato, and Matthew ]. Nicol, University Mathematical Modeling in Biological Systems (Code: SS of Houston. 9A), Zachariah Sinkala, Middle Tennessee State University, Hopf Algebras and Quantum Groups, M. Susan Mont­ and Wenzhang Huang and Jia Li, University of Alabama gomery, University of Southern California, and Yinhuo at Huntsville. Zhang, Victoria University of Wellington. Mathematical Tools for Survival Analysis and Medical Infinite-Dimensional Groups and Their Actions, Chris­ Data Analysis (Code: SS 7A), Curtis Church, Middle Ten­ topher Atkin, Victoria University of Wellington, Greg nessee State University, Chang Yu, Vanderbilt University, Hjorth, University of California Los Angeles/University and Ping Zhang, Middle Tennessee State University. of Melbourne, Alica Miller, University of Louisville, and Splines and Wavelets with Applications (Code: SS SA), Vladimir Pestov, University of Ottawa. Don Hong, Middle Tennessee State University, and Qing­ Matroids, Graphs, and Complexity, Dillon Mayhew, tang Jiang, University of Missouri-St. Louis. Victoria University of Wellington, and James G. Oxley, Louisiana State University. New Trends in Spectral Analysis and Partial Differential Equations, Boris P. Belinskiy, University of Tennessee, Wellington, New Chattanooga, Anjan Biswas, Delaware State University, and Boris Pavlov, University of Auckland. Zealand , David B. Gauld, University of Auckland, and Scott E. Morrison, University of California Victoria University of Wellington Berkeley. December 12-1 5, 2007 Special Functions and Orthogonal Polynomials, Shaun Cooper, Massey University, Diego Dominici, SUNY New Wednesday - Saturday Paltz, and Sven Ole Warnaar, University of Melbourne. Meeting #1 034 First joint International Meeting between the AMS and the New Zealand Mathematical Society (NZMS). San Diego, California Associate secretary: Matthew Miller San Diego Convention Center Announcement issue of Notices: June 2007 Program first available on AMS website: Not applicable January 6-9, 2008, Sunday- Wednesday Program issue of electronic Notices: Not applicable Issue of Abstracts: Not applicable Meeting #1 035 joint Mathematics Meetings, including the 114th Annual Deadlines Meeting of the AMS, 91 st Annual Meeting of the Mathemati­ For organizers: Expired cal Association of America (MAA), annual meetings of the For consideration of contributed papers in Special Ses­ Association for Women in Mathematics (A WM) and the sions: To be announced National Association of Mathematicians (NAM), and the For abstracts: To be announced winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and AMS Invited Addresses Applied Mathematics (SIAM). Marston Conder, University of Auckland, Title to be Associate secretary: Michel L. Lapidus announced. Announcement issue of Notices: October 2007 Rodney G. Downey, Victoria University of Wellington, Program first available on AMS website: November l, Title to be announced. 2007 Michael H. Freedman, Microsoft Research/University Program issue of electronic Notices: January 2008 of California Santa Barbara, Title to be announced. Issue of Abstracts: Volume 29, Issue l Gaven J. Martin, Massey University, Title to be an­ nounced. Deadlines Assaf Naor, Microsoft Research/Courant Institute, Title For organizers: Expired to be announced. For consideration of contributed papers in Special Ses­ Theodore A. Slaman, University of California Berkeley, sions: July 26, 2007 Title to be announced. For abstracts: September 20, 2007

MAY 2007 NOTICES OF THE AMS 681 Mee1tings & Conferences New York, New York Bloomington, Indiana Courant Institute of New York University Indiana University March 1 5-1 6, 2008 April4-6, 2008 Saturday - Sunday Friday - Sunday

Meeting #1 036 Meeting #1 038 Eastern Section Central Section Associate secretary: Lesley M. Sibner Associate secretary: Susan J. Friedlander Announcement issue of Notices: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be an- Program first available on AMS website: To be an- nounced nounced Program issue of electronic Notices: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Issue of Abstracts: To be announced

Deadlines Deadlines For organizers: August 15, 2007 For organizers: September 4, 2007 For consideration of contributed papers in Special Ses­ For consideration of contributed papers in Special Ses­ sions: To be announced sions: To be announced For abstracts: To be announced For abstracts: To be announced

Special Sessions Special Sessions L-Functions and Automorphic Forms (Code: SS lA), Alina Combinatorial and Geometric Aspects of Commutative Bucur, Institute for Advanced Study, Ashay Venkatesh, Algebra (Code: SS lA), Juan Migliore, University of Notre Courant Institute of Mathematical Sciences, Stephen D. Dame, and Uwe Nagel, University of Kentucky. Miller, Rutgers University, and Steven]. Miller, Brown University. Claremont, B«:tton Rouge, California L<)uisiana Claremont McKenna College Louisiana State University, Baton Rouge May 3-4, 2008 Saturday - Sunday March 28-30, 2008 Meeting #1 039 Friday - Sunday Western Section Meeting #1 037 Associate secretary: Michel L. Lapidus Southeastern.Section Announcement issue of Notices: To be announced Associate secretary: Matthew Miller Program first available on AMS website: To be an- Announcement issue of Notices: To be announced nounced Program first available on AMS website: To be an- Program issue of electronic Notices: To be announced nounced Issue of Abstracts: To be announced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: To be announced For organizers: October 4, 2007 Deadlines For consideration of contributed papers in Special Ses­ For organizers: August 28, 2007 sions: To be announced For abstracts: To be announced For consideration of contributed papers in Special Ses­ sions: To be announced Special Sessions For abstracts: To be announced Dynamical Systems and Differential Equations (Code: SS lA), Adolfo Rumbos, Pomona College, Mario Martelli, Claremont McKenna College, and Alfonso Castro, Harvey Mudd College.

682 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings & Conferences Rio de Janeiro, Brazil Middletown, Instituto Nacional de Matematica Pura e Connecticut Aplicada (IMPA) Wesleyan University june 4-7,2008 October 11-12, 2008 Wednesday - Saturday Saturday - Sunday Meeting #1 040 Meeting #1 042 First ]oint International Meeting between the AMS and the Eastern Section Sociedade Brasileira de Matematica. Associate secretary: Lesley M. Sibner Associate secretary: Lesley M. Sibner Announcement issue of Notices: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be an- Program first available on AMS website: To be an- nounced nounced Program issue of electronic Notices: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Issue of Abstracts: To be announced Deadlines Deadlines For organizers: March 11, 2008 For consideration of contributed papers in Special Ses­ For organizers: To be announced 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 Kalamazoo, Michigan Vancouver, Canada Western Michigan University University of British Columbia and the October 1 7-19, 2008 Friday - Sunday Pacific Institute of Mathematical Sciences (PIMS) Meeting #1 043 Central Section October 4-5,2008 Associate secretary: Susan]. Friedlander Saturday - Sunday Announcement issue of Notices: To be announced Program first available on AMS website: To be an- Meeting #1 041 nounced Program issue of electronic Notices: To be announced Western Section Issue of Abstracts: To be announced Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be an- For organizers: March 17, 2008 nounced For consideration of contributed papers in Special Ses­ Program issue of electronic Notices: To be announced sions: To be announced Issue of Abstracts: To be announced For abstracts: To be announced

Deadlines For organizers: March 9, 2008 Huntsville, Alabama For consideration of contributed papers in Special Ses­ University of Alabama, Huntsville sions: To be announced For abstracts: To be announced October24-26,2008 Friday - Sunday

Meeting #1 044 Southeastern Section Associate secretary: Matthew Miller Announcement issue of Notices: To be announced

MAY 2007 NOTICES OF THE AMS 683 Meetings & Conferences

Program first available on AMS website: To be an­ nounced Washington, District Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced of Columbia Deadlines Marriott Wardman Park Hotel and Omni For organizers: March 24, 2008 For consideration of contributed papers in Special Ses­ Shoreham Hotel sions: To be announced January 7-10,2009 For abstracts: To be announced Wednesday - Saturday ]oint Mathematics Meetings, including the 115th Annual Meeting of the AMS, 92nd Annual Meeting of the Math­ Sl1anghai, People's ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (A WM) and Republic of China the National Association of Mathematicians (NAM), and the Fudan University winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Deoember 1 7-21,2008 Applied Mathematics (SIAM). Wednesday - Sunday Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2008 Meeting #1 045 Program first available on AMS website: November 1, First ]oint Interntional Meeting Between the AMS and the 2008 Shanghai Mathematical Society Program issue of electronic Notices: January 2009 Associate secretary: Susan]. Friedlander Issue of Abstracts: Volume 30, Issue 1 Announcement issue of Notices: To be announced Program first available on AMS website: To be an- Deadlines nounced For organizers: April1, 2008 Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Ses­ Issue of Abstracts: To be announced sions: To be announced For abstracts: To be announced Deadlines For organizers: To be announced For consideration of contributed papers in Special Ses­ sions: To be announced Urbana, Illinois For abstracts: To be announced University of Illinois at Urbana-Champaign Invited Addresses March 27-29,2009 L.. Craig Evans, University of California Berkeley, Title Friday - Sunday to be announced. Southeastern Section Zhi-Ming Ma, Chinese Academy of Sciences, Title to be Associate secretary: Susan]. Friedlander announced. Announcement issue of Notices: To be announced , Stanford University, Title to be an­ Program first available on AMS website: To be an- nounced. nounced Richard Taylor, Harvard University, Title to be an­ nounced. Program issue of electronic Notices: To be announced Xiaoping Yuan, Fudan University, Title to be an­ Issue of Abstracts: To be announced nounced. Deadlines Weiping Zhang, Chern Institute, Title to be an­ nounced. For organizers: August 29, 2008 For consideration of contributed papers in Special Ses­ sions: To be announced For abstracts: To be announced

684 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings & Conferences Raleigh, North Issue of Abstracts: To be announced Deadlines For organizers: March 30, 2009 Carolina For consideration of contributed papers in Special Ses­ North Carolina State University sions: To be announced For abstracts: To be announced April4-5, 2009 Saturday - Sunday Southeastern Section San Francisco, Associate secretary: Matthew Miller Announcement issue of Notices: To be announced Program first available on AMS website: To be an- California nounced Moscone Center West and the San Fran­ Program issue of electronic Notices: To be announced cisco Marriott Issue of Abstracts: To be announced January 6-9,2010 Deadlines Wednesday - Saturday For organizers: September 4, 2008 ]oint Mathematics Meetings, including the 116th Annual For consideration of contributed papers in Special Ses­ Meeting of the AMS, 93rd Annual Meeting of the Math­ sions: To be announced ematical Association of America (MAA), annual meetings For abstracts: To be announced of the Association for Women in Mathematics (A W'M) and the National Association ofMat hematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), San Francisco, with sessions contributed by the Society of Industrial and Applied Mathematics (SIAM). California Associate secretary: Matthew Miller Announcement issue of Notices: October 2009 San Francisco State University Program first available on AMS website: November 1, 2009 April25-26, 2009 Program issue of electronic Notices: January 2010 Saturday - Sunday Issue of Abstracts: Volume 31, Issue 1 Western Section Associate secretary: Michel L. Lapidus Deadlines Announcement issue of Notices: To be announced For organizers: Aprill, 2009 Program h.rst available on AMS website: To be an- For consideration of contributed papers in Special Ses­ nounced sions: To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced Deadlines Lexington, Kentucky For organizers: September 25, 2008 For consideration of contributed papers in Special Ses­ University of Kentucky sions: To be announced For abstracts: To be announced March 27-29,2010 Saturday - Monday Southeastern Section Boca Raton, Florida Associate secretary: Matthew Miller Announcement issue of Notices: To be announced Florida Atlantic University Program first available on AMS website: To be an- nounced October 30- November 1, 2009 Program issue of electronic Notices: To be announced Friday - Sunday Issue of Abstracts: To be announced Southeastern Section Associate secretary: Matthew Miller Deadlines Announcement issue of Notices: To be announced For organizers: August 28, 2009 Program first available on AMS website: To be an- For consideration of contributed papers in Special Ses­ nounced sions: To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced

MAY 2007 NOTICES OF THE AMS 685 Meetings & Conferences

Deadlines New Orleans, For organizers: April1, 2011 For consideration of contributed papers in Special Ses­ Louisiana sions: To be announced For abstracts: To be announced New Orleans Marriott and Sheraton New Orleans Hotel San Diego, California January 5-8,2011 San Diego Convention Center and San Wednesday - Saturday Diego Marriott Hotel and Marina ]oint Mathematics Meetings, including the 117th Annual Meeting of the AMS, 94th Annual Meeting of the Math­ january 9-12, 2013, Wednesday- Saturday ematical Association of America, annual meetings of the ]oint Mathematics Meetings, including the 119th Annual Association for Women in Mathematics (A WM) and the Meeting of the AMS, 96th Annual Meeting of the Math­ National Association of Mathematicians (NAM), and the ematical Association of America, annual meetings of the winter meeting of the Association for Symbolic Logic (ASL), Association for Women in Mathematics (A WM) and the with sessions contributed by the Society for Industrial and National Association of Mathematicians (NAM), and the Applied Mathematics (SIAM). winter meeting of the Association for Symbolic Logic (ASL), Associate secretary: Susan ]. Friedlander with sessions contributed by the Society for Industrial and Announcement issue of Notices: October 2010 Applied Mathematics (SIAM). Program first available on AMS website: November 1, Associate secretary: Lesley M. Sibner 2010 Announcement issue of Notices: To be announced Program issue of electronic Notices: January 2011 Program first available on AMS website: To be an- Issue of Abstracts: Volume 32, Issue 1 nounced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: To be announced For organizers: April1, 2010 Deadlines For consideration of contributed papers in Special Ses­ For organizers: April1, 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 B

686 NOTICES OF THE AMS VOLUME 54, NUMBER 5 Meetings and Conferences of the AMS

Associate Secretaries of the AMS 249), Chicago, IL 60607-7045; e-mail: susan@math. nwu. edu; Western Section: MichelL. Lapidus, Department of Math­ telephone: 312-996-3041. ematics, University of California, Sproul Hall, Riverside, CA Eastern Section: Lesley-M. Sibner, Department of Math­ 92521-0135; e-mail: [email protected]; telephone: ematics, Polytechnic University, Brooklyn, NY 11201-2990; 951-827-5910. e-mail: lsi bner@duke. poly. edu; telephone: 718-260-3505. Central Section: Susan-}. Friedlander, Department of Math­ Southeastern Section: Matthew Miller, Department of Math­ ematics, University of Illinois at Chicago, 851 S. Morgan (M/ C ematics, University of South Carolina, Columbia, SC 29208-0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690.

The Meetings and Conferences section of the Notices March 27-29 Urbana, Illinois p. 684 gives information on all AMS meetings and conferences April4-5 Raleigh, North Carolina p.685 approved by press time for this issue. Please refer to the page April 25-26 San Franciso, California p.685 numbers cited in the table of contents on this page for more Oct. 30-Nov. 1 Boca Raton, Florida p. 581 detailed information on each event. Invited Speakers and Special Sessions are listed as soon as they are approved by the 2010 cognizant program committee; the codes listed are needed January 6-9 San Franciso, California p.685 for electronic abstract submission. For some meetings the Annual Meeting list may be incomplete. Information in this issue may be March 27-29 Lexington, Kentucky p.685 dated. Up-to-date meeting and conference information can be found at www. ams. org/meeti ngs/. 2011 January 5-8 New Orleans, Louisiana p.686 Meetings: Annual Meeting 2007 2012 May 23-26 Zacatecas, Mexico p.676 January 4-7 Boston, Massachusetts p.686 July 31-August 3 Warsaw, Poland p. 677 Annual Meeting October 5-6 Chicago, Illinois p.678 2013 October 6-7 New Brunswick, New Jersey p.679 January 9-12 San Diego, California p.686 October 13-14 Albuquerque, New Mexico p.680 Annual Meeting November 3-4 Murfreesboro, Tennessee p.680 2014 December 12-15 Wellington, New Zealand p.681 January 15-18 Baltimore, Maryland p.686 2008 Annual Meeting January 6-9 San Diego, California p.681 Annual Meeting Important Information Regarding AMS Meetings March 22-23 New York, New York p.682 Potential organizers, speakers, and hosts should refer to March 28-30 Baton Rouge, Louisiana p.682 page 78 in the the January 2007 issue of the Notices for gen­ April4-6 Bloomington, Indiana p.682 eral information regarding participation in AMS meetings and May 3-4 Claremont, California p.682 conferences. June 4-7 Rio de Janeiro, Brazil p.683 October 4-5 Vancouver, Canada p.683 Abstracts October 11-12 Middletown, Connecticut p.683 Speakers should submit abstracts on the easy-to-use interac­ October 17-19 Kalamazoo, Michigan p.683 tive Web form. No knowledge of lb'JEX is necessary to submit October 24-26 Huntsville, Alabama p.683 an electronic form, although those who use IbTEX may submit abstracts with such coding, and all math displays and simi­ December 17-21 Shanghai, People's larily coded material (such as accent marks in text) must Republic of China p. 684 be typeset in lb'JEX. Visit http://www.ams.org/cgi-bin/ abstracts/abstract. pl . Questions about abstracts maybe 2009 sent to abs-i nfo@ams. org. Close attention should be paid to January 7-10 Washington, DC p.684 specified deadlines in this issue. Unfortunately, late abstracts Annual Meeting cannot be accommodated.

Conferences: (see http: I /www. ams. org/meeti ngs/ for the most up-to-date information on these conferences.) June 16-July 6, 2007: Joint Summer Research Conferences, Snowbird, Utah. July 8-July 12, 2007: von Neumann Symposium on Sparse Representation and High-Dimensional Geometry, Snowbird, Utah.

MAY 2007 NOTICES OF THE AMS 687 AMBRIDGE

Outstanding titles in MATHEMATICS from Cambridge .

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More Math Put nam and Beyond Mathematics and the Int o LaTeX, Razvan Gelca, Texas Tech University and Titu Aesthetic 4th Edition Andreescu, University ofTexas at Dallas Nathalie Sinclair, Michigan State University; George Gratzer, This book serves as a study guide for the David Plmm, University of Alberta, Canada; University of Manitoba, Winnipeg, MB, Putnam Mathematics Competition and as a and William Higginson, Queen's University, Canada transition from high school problem-solving Ontario, Canada to university level mathematical research. There are several Latex guides, but this one Using numerous problems reflecting the 2006. 288 pp., 78 illus. (CMS Books in wins hands down for the elegance of its spirit of the Putnam exam, the book helps Mathematics) Hardcover undergraduates gain proficiency across the approach and breadth of coverage. ISBN 987-0-387-30526-4 .- $59.9S • Amazon.com, Best of 2000, Editor's range of subjects comprising the core Choice mathematics curriculum. 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The second part is devoted to 2007. Approx. 300 p. 44 illus. (Stochastic financial modeling by the benchmark Geometry and Topology in Modelling and Applied Probability. Volume approach. Hamiltonian Dynamics and 57) Hardcover 2006. 700 p. 199 ill us. (Springer Finance) ISBN 978-0-387-306797-7 .- $59.95 Marco Pettlni, Osservat. Astrofisico Arcetri, Hardcover Firenze, Italy ISBN 978-3-540-26212-1 .- $89.95 Random Fields and Geometry This book explores the foundations of RJ. Adler, Israel institute ofTechnology, hamiltonian dynamical systems and Compact Lie Groups Haifa, Israel and Jonathan Taylor, Stanford University, California statistical mechanics, in particular phase Mark R. Sepanski, Baylor University, Texas transition, from the point of view of This monograph is devoted to a completely Assuming no prior knowledge of Lie groups, geometry and topology. 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