American Mathematical Society

Notable textbooks from the AMS Graduate and undergraduate level publications suitable fo r use as textbooks and supplementary course reading.

A Companion to Analysis A Second First and First Second Course in Analysis T. W. Korner, University of Cambridge, • '"l!.li43#i.JitJij England Basic Set Theory "This is a remarkable book. It provides deep A. Shen, Independent University of and invaluable insight in to many parts of Moscow, Moscow, Russia, and analysis, presented by an accompli shed N. K. Vereshchagin, Moscow State analyst. Korner covers all of the important Lomonosov University, Moscow, R ussia aspects of an advanced calculus course along with a discussion of other interesting topics." " ... the book is perfectly tailored to general -Professor Paul Sally, University of Chicago relativity .. . There is also a fair number of good exercises." Graduate Studies in Mathematics, Volume 62; 2004; 590 pages; Hardcover; ISBN 0-8218-3447-9; -Professor Roman Smirnov, Dalhousie University List US$79; All AJviS members US$63; Student Mathematical Library, Volume 17; 2002; Order code GSM/62 116 pages; Sofi:cover; ISBN 0-82 18-2731-6; Li st US$22; All AlviS members US$18; Order code STML/17 • lili!.JUM4-1"4'1 Set Theory and Metric Spaces • ij;#i.Jit-iii Irving Kaplansky, Mathematical Analysis Sciences R esearch Institute, Berkeley, CA Second Edition "Kaplansky has a well -deserved reputation Elliott H . Lleb M ichael Loss Elliott H. Lieb, Princeton University, for his expository talents. The selection of Princeton, NJ, and Michael Loss, Georgia topics is excellent." I nstitute of Technology, Atlanta, GA -Professor Lance Small, UC San Diego AMS Chelsea Publishing, 1972; "I liked the book very much. The topics chosen 140 pages; Hardcover; ISBN 0-8 218-2694-8; were su ited toward concepts that I wanted List US$29; All AMS members US$26; students to master." Order code CHEL/298.H -Professor Gary Sampson, Auburn University Graduate Studies in M athematics, Volume 14; 2001 ; 346 pages; Hardcove r; ISBN 0-8 218-2783-9; List US$41; All AMS members US$33; • IMWt.iiJ Order code GSM/14.R Partial Differential Equations Lawrence C. Evans, University of California, Berkeley, CA • lili!.Ji*3£WM'i "This excellent textbook is meant as an introduction to mathematical analysis of The Knot Book partial differential equations ... [This book is An Elementary Introduction to the recommended] as the first textbook for Mathematical Theory of Knots anyone who wants to learn the theory of Colin C. Adams, Williams College, partial differential equations." Williamstown, MA - European Mathematical Society NeJJJsletter Graduate Studies in M athematics, Volume 19; "I thought the book was very well suited for an 1998; 662 pages; Hardcover; ISBN 0-8218-0772-2; undergraduate knot theory/topology course. List US$79; All AMS members US$63; · The exposition was very clear." O rder code GSM/19 - Professor Jennifer Taback, BoJVdoin College 2004; 307 pages; Sofi:cover; ISBN 0-8218-3678-1; • For more information about these titles visit www.ams.org/bookstore List US$29; 1\11 AMS members US$23; Order code KNOT ·.....~-~ ~------~----­To order an exam copy visit www.ams.org/bookstore-exam-copy 3/06 N Editors INTERNATIONAL Morris Weisfeld Editor-in-Chief Enrico Arbarello Joseph Bernstein MATHEMATICS Manjul Bhargava Thomas Bloom Enrico Bombieri RESEARCH PAPERS Richard E. Borcherds Alexei Borodin Jean Bourgain http://www.hindawi.com/journals/imrp/ Marc Burger James W. Cogdell IMRP provides very fast publication of lengthy research articles of high current interest in Tobias Colding all areas of mathematics. All articles are fully refereed and are judged by their contribution Brian Conrad to the advancement of the state of the science of mathematics. Issues are published as Corrado De Concini frequently as necessary. Each issue will contain only one article. IMRP is expected to publish Percy Deift 400± pages in 2006. Robbert Dijkgraaf Nils Dencker Articles of at least 50 pages are welcome and all articles are refereed and judged for S. K. Donaldson correctness, interest, originality, depth, and applicability. Submissions are made by e-mail to Yakov Eliashberg [email protected]. An abstract for each article should be included. A copy may also Edward Frenkel be sent to an editor. Only an acknowledgment from the editorial office officially establishes Denis Gaitsgory the date of receipt. Submissions not prepared using TeX should be typed or printed on one Timothy Gowers side of the page, be double-spaced (including references), and have ample margins. Check Emmanuel Hebey contact information for submission by fax or post. When articles are accepted, production Dennis Hejhal begins immediately; authors should be available to assist the editorial staff. Helmut Hofer Please note that it is a condition of submission of a paper that the author(s) permit editing Yasutaka Ihara the paper for language correctness, readability, and compliance with the IMRP mathematical Michio Jimbo style. Kurt Johansson Masaki Kashiwara Illustrations can be in color. There are no page charges. Each author shall receive 100 Kiran S. Kedlaya complimentary reprints with covers. Carlos Kenig SergiuKlainerman Recently Published Articles Toshiyuki Kobayashi • Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Maxim Kontsevich Critical Growth, N. Ghoussoub and Frederic Robert Igor Krichever Shigeo Kusuoka • Topological Properties of Hamiltonian Circle Actions, Gilles Lebeau Dusa Mcduff and Susan Tolman Joachim Lohkamp • Almost-Commuting Variety, 'D-Modules, and Ch erednik Algebras, John Lott Wee Liang Gan and Victor Ginzburg Nikolai Makarov Yu. I. Marrin • Weak Identity Arrows in Higher Categories, Joachim Kock Barry Mazur • Isomonodromic Deformation of Resonant Rational Connections, Fabien Morel M. Bertola and M. Y. Mo Bjorn Poonen Eric Opdam eriodic Algebra-Geometric Coefficients, • On the Spectrum of Jacobi Operators w ith Quasip Michael Rapoport Vladimir Batchenko and Fritz Gesztesy Igor Rodnianski Peter Sarnak Subscription Information Freydoon Shahidi Subscription E-mail: [email protected] Carlos Simpson Stanislav Smirnov e only, $214.50 Institutional subscription rates for 2006 (400± pages) are $195.00 for onlin Michael Struwe for print only, and $234.00 for both online and print editions. Subscribers to the online G.Tian only edition or to the print and online editions of the journal in 2006 will get perpetual John Toth access to volumes 2005- 2006 (i.e., permanent online access to volumes 2005- 2006 even if the Takeshi Tsuji subscription is canceled in the future). Dan-Virgil Voiculescu All orders should be accompanied by payment and sent to Hindawi Publishing Corporation, Andrei Zelevinsky 410 Park Avenue, 15th Floor, #287 pmb, New York, NY 10022, USA. Maciej Zworski

I--IJ N Q7\WJ Hindawi Publishing Corporation, 410 Park Avenue, 15th Floor, #287 pmb, New York, NY 10022, USA; II A Fax: l -866-446-3294 (USA, toll-free); URL: http://www.hindawi com; E-mail: [email protected] NEW [6 FORTHCOMING from Birkhiiuser

Basic Algebra Number Theory Differential Geometry and ANTH 'ON~ W. KNAPP, State University of New York, Stony An Introduction via the Distribution of Analysis on CR Manifolds Brook, NY - Primes SORIN DRAGOMIR, Universita della Basilicata, Romano, This text presents a unifying approach to algebra for BENJAMIN FINE, Fairfield University, Fairfield, CT; Potenza, Italy; GIUSEPPE TOMASSINI, Scuola Normale students who are planning a career in which advanced ~ GERHARD ROSENBERGER, Universittit Dortmund, Superiore, Pisa, Italy mathematics will be used professionally. Key concepts, Germany The study of CR manifolds lies at the intersection of tools, and results are systematically developed. Most I This book provides an introduction and overview of importantly, the student achieves a global perspective of three main mathematical disciplines: partial differential number theory based on the distribution and properties equations, complex analysis in several complex the subject by seeing its vital role in analysis, topology, of primes. This unique approach provides both a firm geometry, and number theory. variables, and differential geometry. While the POE and background in the standard material as well as an complex analytic aspects have been intensely studied in Three prominent themes recur: the analogy between overview of the whole discipline. All the essential topics the last fifty years, much effort has been recently made integers and polynomials in one variable over a field, the are covered: fundamental theorem of arithmetic, theory to understand the differential geometric side of the interplay between linear algebra and group theory, and of congruences, quadratic reciprocity, arithmetic subject. This monograph provides a unified presentation the relationship between number theory and geometry. functions, and the distribution of primes. Analytic of several differential geometric aspects in the theory More than 400 problems at the ends of chapters develop number theory and algebraic number theory both of CR manifolds and tangential Cauchy-Riemann related topics and point to additional applications. A90- receive a solid introductory treatment. equations. page section at the end of the book gives detailed hints The book's user-friendly style, historical context, and for most of the problems, complete solutions for many. wide range of exercises ranging from simple to quite 2006/584 PP./HARDCOVER ISBN 0-8176-4388-5/$109.00 Basic Algebra may be used for self-study as well as in a difficult (with solutions and hints provided for se lect PROGRESS IN MATHEMATICS, VOLUME 246 classroom, beginning at the advanced undergraduate, ones) make it ideal for self study as well as classroom first-year graduate level. After a standard one-variable use. calculus course, the reader should know how to read Complex Variables with and write a proof, and be comfortable with some linear 2006/APPROX. 320 PP., 11 ILLUS./HARDCOV ER algebra; some prior exposure to groups is helpful but ISBN 0-8176-4472-5/$49.95 (TENT.) Applications not necessary. Complete details regarding Basic S. PONNUSAMY, Indian Institute of Technology, Madras, Chennai, India; HERB SILVERMAN, College of Charleston, SC Algeb~a, including a table of contents and preface, are An Invitation to Quantum available on the Birkhauser website: Complex numbers can be viewed in several ways: as an http://www.springer.com/0-8176-3248-4. Cohomology element in a field, as a point in the plane, and as a two­ Kontsevich's Formula for Rational Plane dimensional vector. Exploited properly, each 2006/APPROX. 450 PP., 40 ILLUS./HARDCOVER perspective provides crucial insight into the ISBN 0-8176-3248-4/$69.95 (TENT.) Curves CORNERSTONES JOACHIM KOCK, UniversitatAutiinoma de Barcelona, interrelations between the complex number system and Barcelona, Spain; ISRAEL VAINSENCHER, Universidade its parent, the real number system. In this work, the Federal de Minas Gerais, Pampu/ha- Bela Horizonte, Brazil Number Theory and Its authors explore these relationships by adopting both This elementary introduction to stable maps and generalization and specialization methods to move from Mathematical Structures quantum cohomology presents the problem of counting real variables to complex variables, and vice versa, while rational plane curves mostly from the viewpoint of simultaneously examining their analytic and geometric A Problem-Solving Approach characteristics. TITU ANDREESCU, The University of Texas at Dallas, enumerative geometry. The preliminary material begins Richardson, TX; DORIN AND RICA, "Babes-Bolyai" University, with an introduction to stable pointed curves and The engaging exposition is replete with discussions, C/uj-Napoca, Romania culminates with a proof of the associativity of the remarks, questions, and exercises, motivating not only One of the oldest, liveliest branches of mathematics, quantum product. Kontsevich's formula is initially understanding on the part of the reader, but also number theory is noted for its theoretical depth and established in the framework of classical enumerative developing the tools needed to think critically about applications to other fields, including representation geometry, then as a statement about reconstruction for mathematical problems. The material includes theory, physics, and cryptography. The forefront of Gromov-Witten invariants, and finally, using generating numerous examples and app lications relevant to number theory is replete with sophisticated and famous functions, as a special case of the associativity of the engineering students, along with some techniques to open problems; at its foundation, however, are basic, quantum product. evaluate various types of integrals. elementary ideas that can stimulate and challenge The book is ideal for self-study, as a text for a mini­ 2006/496 PP., 1091LLUS./HARDCOVER beginning students. course in quantum cohomology, or as a special topics ISBN 0-8176-4457-1/$59.95 This introductory textbook takes a problem-solving app­ text in a standard course in intersection theory. The roach to number theory, situating each concept within book will prove equally useful to graduate students in the framework of an example or a problem for readers the classroom setting as to researchers, geometers, to solve. The book will appeal to senior high school and and physicists working in the field. undergraduate students and instructors. It is a clear, ac­ 2006/ 176 PP., 30 ILLUS./HARDCOVER cessible introduction to the subject and a source of fas­ ISBN 0-8176-4456-3/$49.95 cinating problems and puzzles for readers at all levels. PROGRESS IN MATHEMATICS, VOLUME 249

2006/A PPROX. 300 PP., 150 ILUUS./HARDCOVER ISBN 0-8176-3245-X/$49.95 (TEN T.)

CALL: 1-800-777-4643 • FAX: {201) 348-4505 E-MAIL: [email protected] • www.birkhauser.com Please mention promotion #012252 when ordering. Prices are va lid in the Americas only Birkhiiuser and are subject to change without notice. For price and ordering information outside the Americas, please contact Birkhauser Verlag AG by E-mail: [email protected] Boston · Basel · Berlin 012252x Notices March 2006 Feature Articles 318 Computing over the Reals: Foundations for Scientific Computing Mark Braverman and Stephen Cook The authors explain the "bit-model" of computability and complexity of real functions and subsets of real n-space and argue that this is a good way to formalize problems of scientific computation.

330 Find Me a Hash Susan Landau Hash functions are easy-to-compute compression functions that are used as compact repesenta­ tions, or digital fingerprints, of data and to provide message integrity. Some hash functions in current use have been shown to be vulnerable. The author argues that their replacements should be based on a mathematical theory, which has yet to be created.

334 Mushroom Billiards Mason Porter and Steven Lansel Mushroom billiards are examples of systems with mixed regular-chaotic dynamics whose relatively simple geometry makes their precise analysis possible. The authors analyze and illustrate a number of these problems. Communications Commentary

333 The Einstein Public Lecture in 31 7 Opinion: The Media and Mathematics 2006 Mathematics Look at Each Other 338 WHAT IS ... a Graph Minor? Philip]. Davis BojanMohar 340 Proof-A Movie Review 346 Sullivan Receives 2004 National Reviewed by Daniel Ullman Medal of Science 343 The Curious Incident of the Dog ir' 34 7 Donaldson and Narasimhan the Night-time-A Book Review Receive 2006 King Faisal Prize Reviewed by Helmer Aslaksen 348 Moment of Proof jonathan David Farley 3 51 Impact Factor and How It Relates to Quality of Journals ,Vitali !l!ilman 353 Programs that Make a Difference Notices Departments of the American Mathematkal Society Mathematics People ...... 358 Hirachi Receives Bergman Prize, Mathematics Wunderkind Wins Awarded TWAS EDITOR: Andy Magid Siemens-Westinghouse Competition, Raman Prize, Polterovich and Tsai Awarded AYJdre Aisenstadt Prize, ASSOCIATE EDITORS: Susanne C. Brenner, Bill Casselman (G raphics Editor ), Montalbcin Awarded ASL Sacks Prize, Pillichshammer Wins Robert]. Daverman, Nathaniel Dean, Rick Durrett, Information-Based Complexity Young Researcher Award, Otto Susan Friedlander, Robion Kirby, Steven G. Krantz, Scholarships Awarded, Professor Elliott H. Ueb, Mark Saul, Karen E. Smith, Audrey Receives Leibniz Prize, Rhodes Terras, Usa Traynor of the Year Awards Announced, Szpiro a Finalist for Descartes SENIOR WRITER and DEPUTY EDITOR: Prize, Authors Receive Chauvenet Prize for Notices Article. Allyn Jackson MANAGING EDITOR: Sandra Frost Mathematics Opportunities ...... 363 Program Director Positions at CONTRIBUTING WRITER: Elaine Kehoe AMS Congressional Fellowship, NSF, Summer Program for Women Undergraduates, PRODUCTION ASSISTANT: Muriel Toupin International Mathematics Competition for University Students, PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Murphy, Lori Nero, Arlene O'Sean, Karen Call for Nominations for Information-Based Complexity Prize, Ouellette, Donna Salter, Deborah Smith, Peter Sykes Project NExT: New Experiences in Teaching. ADVERTISING SALES: Anne Newcomb For Your Information ...... 366 SUBSCRIPTION INFORMATION: Subscription prices Mathematics Awareness Month , April 2006. for Volume 53 (2006) are US$430 list; US$344 institu­ tional member; US$258 individual member. (The sub­ Inside the AMS ...... 367 scription price for members is included in the annual dues.) A late c harge of 10% of the subscription price AMS Sponsors NExT Fellows, Correction, Deaths of AMS will be imposed upon orders received from nonmem­ Members. bers after January 1 of the subscription year. Add for postage: Surface delivery outside the United States Reference and Book list ...... 368 and !ndia-US$20; in !ndia-US$40; expedited delivery to destinations in North America- US$35; elsewhere­ US$87. Subscriptions and orders for AMS publications Mathematics Calendar ...... 374 should be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. New Publications Offered by the AMS ...... 381 · All orders must be prepaid. ADVERTISING: Notices publishes situations wanted Classified Advertisements ...... 388 and classified advertising, and display advertising for publishers and academic or scientific organizations...... 389 Advertising material or questions may be faxed to Meetings & Conferences of the AMS ...... 401-331-3842 (indicate "Notices advertising" on fax cover sheet). Meetings and Conferences Table of Contents ...... 399 SUBMISSIONS: Articles and letters may be sent to the editor by email at noti ces@math . ou. edu, by fax at 405-325-5765, or by postal mail at Department of Mathematics, 601 Elm, PHSC 423, University of Okla­ homa, Norman, OK 73019-0001. Email is preferred. Correspondence with the managing editor may be sent to noti ces@ams. or g. For more information, see the section "Reference and Book List". NOTICES ON THE AMS WEBSITE: Most of this publi­ cation is available electronically through the AMS web­ From the site, the Society's resource for delivering electronic products and services. Use the URL http: I /www . ams. AMS Secretar~ org/noti ces/ to access the Notices on the website.

!Notices of the American Mathematical Society (ISSN 0002- Call for Nominations for AMS Exemplary Program Prize .... . 372 9920) is published monthly except bimonthly in june/ July by the American Mathematical Sociery at 201 Charles Street, Prov­ idence, RJ 02904-2294 USA, GSTNo.12189 2046 RT****. Pe­ Call for Nominations for Leroy P. Steele Prizes ...... f .... .373 riodicals postage paid at Providence, RJ, and additional mail­ ing offices. POSTMASTER: Send address change notices to NoticesoftheAmerican Mathematical Society, P.O. Box 6248, Providence, RJ 02940-6248 USA.] Publication here of the So­ ciery's street address and the other information in brackets above is a technical requirement of the U.S. Postal Service. Te l: 401-455-4000, email: noti ces@ams. org. © Copyright 2006 by the American Mathematical Society. All rights reserved. Printed in the United States of America.The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durabiliry. 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

whom the subject is poison? Does it lie with the nature of The Media and Mathematics mathematics, which at its higher reaches is a difficult sub­ Look at Each Other ject? With the nature of the media? With the journalists who write about mathematics? With the mathematicians The word media here will mean not only the newspapers themselves? It would appear that all of the above are op­ and television, but also novels, stories, plays, movies, mu­ erative. seums; in short, all modes of popular communication of A number of programs intended to cure the situation mathematics. are in place. The professional mathematical societies and The complaints of mathematicians. Briefly: mathe­ a number of individual writers work hard to call attention matics gets very little media coverage of recent develop­ to recent accomplishments in the field to a variety of clien­ ments. It would like much more. And when mathematics teles. Some institutes such as MSRI have a rotating jour­ is treated, the media frequently get it wrong. Technical de­ nalist in residence. The United States Air Force has given tails are mostly omitted, and the media's "take" is irksome. a grant to the University of Southern California to teach The media's point of view. Gina Kalata, a widely-read screenwriting to scientists in an attempt to produce movies science writer for the New York Times, wrote me and television shows that show scientists in more sym­ Newspapers are not there to educate or to teach pathetic ways. How all these professional projects diffuse people about the mathematics that underlies into the popular media and public awareness is a matter search engines unless there is something you of conjecture. can say about that mathematics that makes it What kind of reportage would I like to see? In contra­ new and compelling. The fact that the mathe­ diction to what Kalata has said, why is it not the duty of a matics is there is not enough. With most things newspaper to educate? I would suggest that newspapers we use-a car, an iPod, a DVD, most of us don't run articles that give a semblance of understanding of the really care how it works. degree to which mathematics underlies today's world-yes, Sara Robinson, a science writer who majored in math­ a semblance. We are living in an age that is mathematized and ematics and was Journalist in Residence at MSRl (Mathe­ is increasingly so. I would think that at the very least it should matical Sciences Research Institute in Berkeley) recalled that be possible for a newspaper to educate us to the fact that as a fledgling reporter, she was told by a senior reporter mathematics is formatting a good portion of today's life and that the goal of science reportage was to give the reader to point out where this is occurring. It need not give readers merely "an illusion of understanding of the technical sub­ an understanding of the technical mathematics; that is too ject matter". She also reported a statement made by Rob much to expect. But I would hope that clever writers might Finer, a former editor of the New York Times, that point out how mathematics is altering our lifestyles and do Mathematics has no emotional impact. What it in a manner that would not lead Garfield the Cat to say "ho physicists do challenges peoples' notion of ori­ hum". Is it too much to hope that qualitynewspapers might gins and creations. Mathematics doesn't change in the future run such articles as any fundamental beliefs or what it means to be "Professors Smith and Jones show how eigenvalues human. help in search engine strategy. President asks science ad­ Mathematical fiction. In fiction one finds the popular visor what Egg Values are." stereotype of the mathematician, as brilliant, somewhat "Numerical algorithms of aero-hydro-elastodynamics mad, socially inept or reclusive, obsessive, living in the used in the design of the Swiss yacht that won the Amer­ clouds, given to the arcane and/or the fantastic. The math­ ica's Cup." ematicians or mathematics depicted come wrapped in the "Mathematical wavelets aid in gallstone treatments at following sensational themes: magic, codes, espionage, Massachusetts General Hospital." the devil, ghosts, secret messages, other worlds, futurism, "The Surgeon General urges medical schools to require madness, autism, apocalyptism, mysticism, the occult, ob­ probability theory for admissions." sessions, prizes, dystopias, evil mathematical productions "The Attorney General urges law schools to require and cults, machines that turn into sorcerers' apprentices, probability theory for admissions." alternate time concepts. Apparently, there is a steady mar­ "Yogi Berra praises Markoffian applications to baseball ket for this kind of literature, and mathematicians them­ strategy." selves are writers, producers, and readers of it. Mathematics I hope that such items have already been run and that is often regarded by the average person as a kind of magic, as I read my morning paper with my eggs and coffee, I have and this view fits right in with the fictional themes. It may simply missed them. be that, with its abhorrence of mathematics, the general - Philip]. Davis public ignores the mathematics totally and simply goes for Brown University a whopping good story. [email protected] Where does the difficulty of communication lie, and what can be done? Does it lie with mathematics educa­ This is an abridged version of a longer article to be published tion in the lower grades, often taught by teachers for in a Festschrift for Christine Keitel of the Free University, Berlin.

MARCH 2006 NOTICES OF THE AMS 317 Computing over the Reals: Foundations for Scientific Computing Mark Braverman and Stephen Cook

Introduction computation model. Alan Turing defined the no­ The problems of scientific computing often arise tion of a single computable real number in his from the study of continuous processes, and ques­ landmark 1936 paper [Tur36]: a real number is tions of computability and complexity over the computable if its decimal expansion can be com­ reals are of central importance in laying the foun­ puted in the discrete sense (i.e., output by some Tur­ dations for the subject. The first step is defining ing machine). But he did not go on to define the a suitable computational model for functions over notion of computable real function. the reals. There are now two main approaches to model­ Computability and complexity over discrete ing computation with real number inputs. The first spaces have been very well studied since the 1930s. approach, which we call the bit-model and which Different approaches have been proved to yield is the subject of this paper, reflects the fact that equivalent definitions of computability and nearly computers can store only finite approximations equivalent definitions of complexity. From the tra­ to real numbers. Roughly speaking, a real function dition of formal logic we have the notions of re­ f is computable in the pit model if there is an al­ cursiveness and Turing machine. From computa­ gorithm which, given a good rational approxima­ tional complexity we have Turing machine variants tion to x, finds a good rational approximation to and abstract Random Access Machines (RAMs). All f(x). of these converge to define the same well-accepted The second approach is the algebraic approach, notion of computability. The Church-Turing thesis which abstracts away the messiness of finite ap­ asserts that this formal notion of computability is proximations and assumes that real numbers can broad enough, at least in the discrete setting, to in­ be represented exactly and each arithmetic oper­ clude all functions that could reasonably be con­ ation can be performed exactly in one step. The strued to be computable. complexity of a computation is usually taken to be In the continuous setting, where the objects are the number of arithmetic operations (for example, numbers in JR. , computability and complexity have additions and multiplications) performed. The al­ received less attention, and there is no one accepted gebraic approach applies naturally to arbitrary rings and fields, although for modeling scientific Mark Braverman is a Ph.D. student in the Department of computation the underlying structure Computer Science, University of Toronto. His email address is usually JR. ismbraverm@cs. toronto. edu. Partially supported by an or C. Algebraic complexity theory goes back to the NSERC Postgraduate Scholarship. 1950s (see [BM75, BCS97] for surveys). For scientific Stephen Cook is Distinguished University Professor in the computing the most influential Department of Computer Science, University of Toronto. model in the algebraic setting is due to Blum, Shub, His email address is sacook@cs. toronto. edu. Partially and Smale (BSS) [BSS89]. The model and its theory supported by an NSERC Discovery Grant. are thoroughly developed in the book [BCSS98]

318 NOTICES OF THE AMS VOLUME 53, NUMBER 3 7

6 5

-1 -0.5 0.5 1.5 2

Figure 1. The Koch snowflake and the graph of the equation y = ex . (see also the article [Blum04] in the Notices for an formulating a good general definition in the BSS exposition). In the BSS model, the computer has reg­ model along these lines is not straightforward: see isters which can hold arbitrary elements of the un­ [Sma97] for an informal treatment and [BrvOS] for derlying ring R. Computer programs perform exact a discussion and a possible formal model. arithmetic (+, -, ·, and "'"" in the case R is a field) For uncomputability results, BSS theory con­ and can branch on conditions based on exact com­ centrates on set decidability rather than function parisons between ring elements(=, and also < ,in computation. A set C s; JE. n is decidable if some the case of an ordered field). Newton's method, for BSS computer program halts on each input x E JE. n example, can be nicely presented in the BSS model and outputs either 1 or 0, depending on whether as a program (which may not halt) for finding an x E C. Theorem 1 in [BCSS98] states that if C s; JE. n approximate zero of a rational function, when is decided by a BSS program over lE. then C is a R = lE. . A nice feature of the BSS model is its gen­ countable disjoint union of semi-algebraic sets. A erality: the underlying ring R is arbitrary, and the number of sets are proved undecidable as corol­ resulting computability theory can be studied for laries, including the Mandelbrot set and all non­ each R. In particular, when R = Z 2, the model is degenerate Julia sets (Figure 2). However again it equivalent to the standard bit computer model in is hard to interpret these undecidability results in the discrete setting. terms of practical computing, because simple sub­ One of the big successes of discrete com­ sets of lE. 2 which can be easily "drawn", such as the putability theory is the ability to prove uncom­ Koch snowflake and the graph of y =ex (Figure 1) putability results. The solution of Hilbert's lOth are undecidable in this sense [Brt03]. problem [Mat93] is a good example. The theorem In the bit model there is a nice definition of de­ states that there is no procedure (e.g., no Turing cidability (bit-computability) for bounded subsets machine) which always correctly determines of JE. n. For the case of JE. 2 , the idea is that the set is whether a given Diophantine equation has a solu­ bit-computable if some computer program can tion. The result is convincing because of general ac­ draw it on a computer screen. The program should ceptance of the Church-Turing thesis. be able to zoom in at any point in the set and draw A weakness of the BSS approach as a model of a neighborhood of that point with arbitrarily fine scientific computing is that uncomputability re­ detail. Such programs can be easily written for sults do not correspond to computing practice in simple sets such as the Koch snowflake and the the case R = JR . Since intermediate register values graph of the equation y = ex, and more sophisti­ of a computation are rational functions of the in­ cated programs can be written for many Julia sets puts, it is not hard to see that simple transcendental (as will be seen below). A Google se.arch on the functions such as ex are not explicitly computable World Wide Web turns up programs that apparently by a BSS machine. In the bit model these functions do the same for the Mandelbrot set. However no are computable because the underlying philosophy one knows how accurate these programs really is that the inputs and outputs to the computer are are. The bit-computability of the Mandelbrot set is rational approximations to the actual real numbers an open question, although we will see later that they represent. The definition of computability in it holds subject to a major conjecture. Because of the BSS model might be modified to allow the pro­ the Church-Turing thesis, a proof of bit­ gram to approximate the exact output values, so uncomputability of the Mandelbrot set would carry that functions like ex become computable. However some force: any program for any computing device

MARCH 2006 NOTICES OF THE AMS 319 purporting to draw it must draw it wrong, at least Examples of"Easy" and "Hard" Problems at some level of detail. "Easy" Problems In the rest of the paper we concentrate on the We start with examples of problems over the reals bit model, because we believe that this is the most that should be "easy" according to any reasonable accurate abstraction of how computers are used to model. The everyday operations, such as addition, solve problems over the reals. The bit model is not subtraction, and multiplication widely appreciated in the mathematical community, should be consid­ ered easy. More generally, any of the operations perhaps because its principal references are not that written to appeal to a wide audience. In contrast can be found on a common calculator can be re­ the BSS model has received widespread attention, garded as "easy", at least in some reasonable re­ partly because its presentation [BSS89], and espe­ gion. Such functions include, among others, sinx, cially the excellent reference [BCSS98], not only ex, JX, andlogx. present the model but provide a rigorous treatment Functions with singularities, such as x 7 y and of matters of interest in scientific computation. The tanx, are easily computable on any closed region authors deserve credit not just for presenting an which excludes the singularities. The computa­ elegant model, but also for arousing interest in tional problem usually gets harder as we approach foundational issues for numerical analysis and in­ the singularity point. For example, computing tan x spiring considerable research. Undoubtedly the becomes increasingly difficult as x tends to~. be­ concept of abstracting away round-off error in cause the expression becomes increasingly sensi­ computations over the reals poses important nat­ tive in x. ural questions. One example is linear program­ Some basic numerical problems that are known ming: Although polynomial-time algorithms are to have efficient solutions should also be relatively known for solving linear programming problems "easy" in the model. This includes inverting a well with rational inputs, these algorithms assume that conditioned matrix A . A matrix is well conditioned the problem description size (in bits) is available in this setting if A - 1 does not vary too sharply as an input parameter. This is not the usual frame­ under small perturbations of A . work for either the bit model or the BSS model. The Simple problems that arise naturally in the dis­ commonly-used simplex algorithm can be neatly crete setting should usually remain simple when formalized in the BSS model, but it requires expo­ passing to the continuous setting. This includes nential time in the worst case. It would be very nice problems such as sorting a list of real numbers and to find a polynomial-time BSS algorithm for linear finding lengths of shortest paths in a graph with programming. Such an algorithm would be called real edge lengths. strongly polynomial-time in the field of linear pro­ When one considers subsets of JR 2 , a set should gramming, and its existence is an important open be considered "easy" if we can draw it quickly with question. an arbitrarily high precision. Examples include sim­ Here is an outline of the rest of the paper. The ple "paintbrush" shapes, such as the disc B((O, 0), 2) next section gives examples of easy and hard com­ in JR 2 , as well as simple fractal sets, such as the Koch putational problems over the reals. The following snowflake (Figure 1). section motivates and defines the bit model both To summarize, the model should classify a prob­ for computing real functions and subsets of JR n. lem as "easy", if there is an efficient algorithm to Computational complexity issues are discussed. solve it in some practical sense. This algorithm After that we consider the computability and com­ may be inspired by a discrete algorithm, a numer­ plexity of the Mandelbrot and Julia sets as a par­ ical-analytic technique, or both. ticular application of the definitions. Simple pro­ grams are available which seem to compute the "Hard" Problems Mandelbrot set, but they may draw pieces which Naturally, the "hard" problems are the ones for should not be there. The bit-computability of the which no efficient algorithm exists. For example, Mandelbrot set is open, but it is implied by a major it is hard to compute an inverse A - 1 of a poorly conjecture. Many Julia sets are not only computable, conditioned matrix A . Note that even simple nu­ but are efficiently computable (in polynomial time). merical problems, such as division (x - 1) 7 (y - 1), On the other hand some Julia sets are uncom­ become increasingly difficult in the poorly condi­ putable in a fundamental sense. Finally, the last sec­ tioned case. It becomes increasingly hard to eval­ tion discusses a fundamental question related to uate the latter expression as x and y approach 1. the Church-Turing thesis: are there physical sys­ Many problems that appear to be computation­ tems that can compute functions which are un­ ally hard arise while trying to model processes in computable in the standard computer model? nature. A famous example is theN-body problem, Some of the material presented here is given in which simulates the motion of planets. An even more detail in [BrvOS]. See [Ko91] and [WeiOO] for harder example is solving the Navier-Stokes equa­ general references on bit-computability models. tions used in simulations for fluid mechanics. We

320 NOTICES OF THE AMS VOLUME 53, NUMBER 3 a c

b d

Figure 2. a: the Mandelbrot set; b·d: Julia sets with parameter values of i, -1.57, and -0.15 + 0.7i, respectively. will return to questions of hardness in physical sys­ complete for a class called #P, which is believed to tems in the last section. subsume NP. Other problems that should be hard are the nat­ Quasi-fractal Examples: The Mandelbrot and ural extensions of very difficult discrete problems. Julia Sets Consider, for example, the Travelling Salesman The Mandelbrot and the Julia sets are common ex­ Problem (TSP). In this problem we are given a graph amples of computer-drawn sets. Beautiful high­ G = (V, E) with costs c(e) associated with the edges e E E. Our goal is to minimize the cost of a Hamil­ resolution images have become available to us with tonian cycle (a cycle which goes through each ver­ the rapid development of computers. Amazingly, tex exactly once) in G. This problem is widely be­ these extremely complex images arise from very lieved not to have an efficient solution in the simple discrete iterated processes on the complex discrete case. In fact it is NP-complete in this case plane C ([GJ79]), and having a polynomial time algorithm For a point c E C , define fc(z) = z 2 +c. c is said for it would imply that P = NP, which is believed to be in the Mandelbrot set M if the iterated se­ to be unlikely. There is no reason to think that it quence c, fc(c), fdfc(c)), ... does not diverge to oo. should be any easier in the continuous setting While (we believe) very precise images of M can be (where the costs c(e) need not be integers) than in generated on a computer, proving that these im­ the discrete case. ages approximate M would probably involve solv­ The hardness of numerical problems may sig­ ing some difficult open questions about it. nificantly vary with the domain of application. Con­ sider for example the problem of computing the The family of Julia sets is parameterized by functions. In the simple case of quadratic functions integral I(x) = J; f(t)dt. While it is very easy to com­ pilte I(x) from f(x) in the case f is a polynomial, fc(z) = z 2 + c, the Fatou set Kc is the set of points integrating a general polynomial time computable x, such that the sequence x, fc(X), fc(fc(x)), ... does Lipschitz function is as hard as counting the num­ not diverge to oo. The Julia set ] c is defined as the ber of the diffe;:ent shortest routes in the Travel­ boundary of Kc . While many Julia sets, such as the ling Salesman Problem. The latter problem is ones in Figure 2, are quite easy to draw, there are

MARCH 2006 NOTICES OF THE AMS 321 explicit sets of which we simply cannot produce dyadic rationals to express approximations to x and useful pictures. ex, where the dyadic rationals form the set As we see, a priori it is not clear whether these m sets should be "easy" or "hard". This gives rise to ][]) = { 2n I mE Z, n E N}. a series of questions: • Is the Mandelbrot set computable? We can then take a dyadic q = fs E ][]) such that • Which Julia sets are computable? lx- q l < 2- m, and evaluate Pn(q) within an error • Can the Mandelbrot set and its zoom-ins be of 2- (n+2l using finite precision dyadic arithmetic. drawn quickly on a computer? This gives us an approximation y E ][]) of Pn(q) • Which Julia sets and their zoom-ins can be drawn such that ly- Pn(q) l < 2- (n+2l. In our example, an quickly on a computer? assumption lx- q l < 2- (n+4l guarantees that These questions are meaningless unless we agree IPn(X)- Pn(q)l < 2- (n+2l, and we can take upon a model of computation for this setting. In m = n + 4. To summarize, we have the following sections we develop such a model, based on "drawability" on a computer. lex- yl :::; lex - Pn(X)I + IPn(X)- Pn(q)l+ IPn(q)- yl < r(n+ll + 2- (n+2l + r(n+2l = 2- n, The Bit Model and y is the answer we want. The running time of Bit Computability for Functions the computation is dominated by the time it takes The motivation behind the bit model of computa­ to compute our approximation to Pn(q). Note that tion is idealizing the process of scientific com­ this entire computation is done over the dyadic ra­ puting. Consider, for example, the simple task of tionals and can be implemented on a computer in computing the function x .... ex for an x in the in­ time polynomial inn. terval [-1, 1]. The most natural solution appears To find the answer we only needed to know x to be by taking the Taylor series expansion around within an error of 2- (n+4l. This is especially im­ 0: 00 k portant when one tries to compose several com­ X 1 (1) e = 'XL -k,. putations. For example, to compute eex- within an k=O . error 2-n on the appropriate interval, it suffices to Obviously in a practical computation we will only know x within an error of 2- (n+S)_ be able to add up a finite number of terms from While evaluating the function ex is hardly a chal­ (1). How many terms should we consider? By tak­ lenge for scientific computing, the process de­ ing more terms, we can improve the precision of scribed above illustrates the main ideas in the bit the computation. On the other hand, we pay the model of computation. Below are the main points price of increasing the running time as we consider that we have seen in this example and that char­ more terms. The answer is that we should take just acterize the bit model for computing functions. Sup­ enough terms to satisfy our precision needs. pose we are trying to compute f : [a, b] ~ IR n. De­ Depending on the application, we might need the note the program computing f by Mr- value of ex within different degrees of precision. • The goal of Mr(x, n) is to compute f(x) within the Suppose we are trying to compute it with a preci­ error of 2- n; sion of 2- n. That is, on an input x we need to out­ • Mr computes a precision parameter m, it needs put a y such that lex- yl < 2- n. It suffices to take to know x within an error of 2- m to proceed with n + 1 terms from (1) to achieve this level of ap­ the computation; proximation. Indeed, assuming n :2:: 4, for any • Mf receives a dyadic q = fs such that X E [-1, 1], lx-ql < 2- m; • Mf computes a dyadic y such that 00 00 lf(x) - Yl < 2- n; ex - In ~kl = I I ~kl < I ~1 kl < I k=O k! k=n+l k! - k=n+l k! - • the running time of Mr(x, n) is the time it takes to compute m plus the time it takes to compute ~ ..!:_ < ~ _1_ = 2-(n+l) y from q (both of which have firlite representa­ L k' L 2k+l k=n+l · k=n+l tions by bits). Note that the entire computation of Mr is per­ In fact, a smaller number of terms suffices to formed only with finitely presented dyadic num­ achieve the desired precision. We take a portion of bers. There is a nice way to present the computa­ the series that yields error 2- (n+l) < 2- n, to allow tion using oracle terminology. An oracle for a real room for computation (round-off) errors in the number x is a function

322 NOTICES OF THE AMS VOLUME 53, NUMBER 3 limitation would be that the time it takes Mr to read would contradict the solution to Hilbert's lOth the output of

MARcH 2006 NOTICES OF THE AMS 323 It is convenient to take E = 4 · 2-n-our computa­ between the computability notions for these ob­ tion precision in this case. jects- through plots of a function's graph. Suppose now that we are trying to construct P Theorem 3. Let D c JR k be a closed and bounded as a union of pixels of radius 2- n centered at grid computable points (2 - (n+k) · z)k. The basic decision we have to domain, and let f : D ~ lR be a contin­ uous function. Then is computable make is whether to draw each particular pixel or f as a function if and only if the = not, so that the union P would satisfy the condi­ graph fr {(x,f(x)): xED} is computable as a set. tions above. To ensure that P covers S, we include all the pixels that intersect with S. To satisfy the Example: Consider the set S = {(x, y): x, y second condition, we exclude all the pixels that are E [0, 1], x3 + y3 = 1}. It is the graph of the function 2- n-far from S. If none of these conditions holds, f(x) = 3-Jl=X3 on [0, 1], which we have seen to be we are in the "gray" area, and either including or computable. By Theorem 3, S is a bit-computable excluding a pixel is fine. In other words, we should set. This is despite the fact (pointed out in [Blum04]) compute a function from the family that by the cubic case of Fermat's Last Theorem the only rational points inS are (0, 1) and (1, 0). if B(d, 2- n) n S =f 0 A more detailed discussion (2) f(d, n) = { if B(d, 2 · 2- n) n S = 0 of bit computabil­ ~ ity for sets can be found in [BW99, WeiOO, BrvOS]. 0 or 1 otherwise Computational Complexity in the Bit Model Since the basic object in the discussion above is a for every point d E (2 - (n+k) · z)k. Here f is com­ Turing Machine, the computational complexity for puted in the classical discrete sense. the bit model follows naturally from the standard Example: A "simple" set, such as a point, line seg­ notions of computational complexity in the discrete ment, circle, ellipse, etc. is computable if and only setting. Basically, the time cost of a computation if all of its parameters are computable numbers. is the number of bit operations required. For example, a circle is computable if and only if For example, the time complexity T rr(n) for com­ the coordinates of its center and its radius are puting the number rr is the number of bit opera­ computable. tions required to compute a 2-n-approximation of rr. The time complexity Tex (n) of computing the ex­ The way we have arrived at the definition of bit­ ponential function x ~ ex on [-1, 1] is the number computability might suggest that it is specifically of bit operations it takes to compute a tailored to computer-graphics needs and is not 2- n-approximation of ex given an x E [-1, 1]. This mathematically robust. This is not the case, as will running time is assessed at the worst possible ad­ be seen in the following theorem. Recall that the missible x. We have seen that Te x(n) is bounded by Hausdorff distance between bounded subsets of JR k a polynomial inn, and it is not hard to see that the is defined as same holds for T rr(n). dH(S, T) = inf{d: S c B(T, d) and T c B(S, d)}. This computational complexity notion can be used to assess the hardness of the different nu­ We have the following. merical-analytic problems arising in scientific com­ Theorem 2. LetS c JRk be a bounded set. Then the puting. For example, the dependence of matrix in­ following are equivalent. version hardness on the condition number of the matrix fits nicely into this setting. 1. A function from the family (2) is computable. Another example is a result by Schonhage [Sch82, 2. There is a program that for any E > 0 gives an Sch87] showing how the fundamental theorem of E -approximation of S in the Hausdorff metric. algebra can be implemented by a polynomial time 3. The distance function ds(x) = inf{lx - yl: algorithm in the bit model. More precisely, he has y E S} is bit-computable. shown how to solve the following problem in time 1. and 3. remain equivalent even if S is not bounded. O((n 3 + n2s)log3(ns)): Given any polynomial P(z) = anzn + · · · + ao with aJ E C Example: The finite approximations Ki of the Koch and IPI = .Z::v lavl:::; 1, and given any snowflake are polygons that are obviously com­ s E N, find ap­ proximate linear factors LJ(Z) = UJZ + VJ putable. The convergence Ki ~ K is uniform in the (1::;j::;n) such that IP-LlLz···Lnl<2 Hausdorff metric. So K can be approximated in the - s holds. Hausdorff metric with any desired precision. Thus Some of the the Koch snowflake is bit-computable. early work regarding the computa­ tional complexity of operators such as taking de­ The last characterization of set bit-computabil­ rivatives and integration was done in [KF82]. A ity in Theorem 2 connects the computability of sets more detailed exposition of the results can be and functions. There is another natural connection found in [Ko9 1];

324 NOTICES OF THE AMS VOLUME 53, NUMBER 3 The complexity of computing a set is the time How Hard Are the Mandelbrot and julia T(n) it takes to decide one pixel. More formally, it Sets? is the time required to compute a function from First let us consider questions of computability of the family (2). Thus a set is polynomial-time com­ the Mandelbrot set M. Despite the relatively sim­ putable if it takes time polynomial in n to decide ple process defining M, the set itself is extremely one pixel of resolution 2 -n. complex and exhibits different kinds of behaviors To see why this is the "right" definition, suppose as we zoom into it. In Figure 3 we see some of the we dre trying to draw a setS on a computer screen variety of images arising in M. pixel resolution. A which has a 1000 x 1000 The most common algorithm used to compute 2-n-zoomed in picture of S has 0(22n) pixels of M is presented on Figure 4. The idea is to fix some size 2- n, and it would take time O(T(n) · 22n) to number T - the number of steps we are willing to compute. This quantity is exponential inn, even if iterate. Then for every grid point c iterate T(n) is bounded by a polynomial. But we are draw­ fc(z) = z 2 + c on c for at most T steps. If the orbit ingS on a finite-resolution display, and we will need escapes B(O, 2), we know that c f/. M. Otherwise, we to draw only 1000 · 1000 = 106 pixels. Deciding say that c E M. This is equivalent to taking the in­ these pixels would require 0(106 · T(n)) = O(T(n)) polynomial map steps. This running time is polynomial in n if and verse image of B(O, 2) under the fT (c)= fc o fc o · · · o fc(c). In Figure 4 (right) a few only if T(n) is polynomial. Hence T(n) reflects the "true" cost of zooming in when drawing S. T times Beyond the Continuous Case of these inverse images and their convergence to As we have seen earlier, the bit model notion of Mare shown. computability is very intuitive for sets and for con­ One problem with this algorithm is that its analy­ tinuous functions. However, by Theorem 1 it com­ sis should take into account roundoff error in­ pletely excludes even the simplest discontinuous volved in the computation z - z 2 +c. But there functions. For example the step function can be de­ are other problems as well. For example, we take fined by an arbitrary grid point c to be the representative 0, ifxlimit behavior of the orbit of c. is computable if we can plot its graph. By Theorem E.g., in the main cardioid, the orbit of c converges 3, this definition extends the standard bit­ to an attracting point. These components are called computability definition in the continuous case. Ob­ hyperbolic components because they index the hy­ viously it makes Sex computable whenever ex is perbolic Julia sets that will be discussed b elow. computable. Douady and Hubbard have shown that Conjecture

MARCH 2006 NOTICES OF THE AMS 325 Figure 3. A variety of different images arising in zoom-ins of the Mandelbrot set (in black).

4 implies the density of hyperbolic components in computing l c for each value of c. What we can M: hope for is a machine computing J c when given ac­ cess to the parameter c with an arbitrarily high pre­ Conjecture 5. Hyperbolic components are dense cision. The existence of such a machine and the inM. amount of time the computation takes depend on Hertling [Hert05] has shown that Conjecture 5 the properties of the particular Julia set. An ex­ implies the computability of M. There is a possi­ cellent exposition on the properties of Julia sets can bility that M is computable even without this con­ be found in [MilOO]. jecture holding, but it is hard to imagine such a con­ Computationally, the "easiest" case is t:4at of th.e struction without a much deeper understanding of hyperbolic Julia sets. These are the sets for which the structure of M. Moreover, even if M is com­ the orbit of the point 0 either diverges to oo or con­ putable, questions surrounding its computational verges to an attracting orbit. Equivalently, tl;lese are complexity remain wide open. As we will see, the the sets for which there is a smooth metric J.1 on a situation is much clearer for most Julia sets. neighborhood N of l c such that the m~p fc !is A Julia set]r is defined for every rational func­ strictly expanding on N in J.l. Hence, points escape tion r from the Riemann sphere into itself. Here we the neighborhood of fc exponentially fast. That is, restrict our attention to Julia sets corresponding if dUe. x) > 2- n, then the orbit of x will escape to quadratic polynomials of the form fc(z) = z2 +c. some fixed neighborhood of l c in O(n) steps. This Recall that the Fatou set Kc is the set of points x gives us the divergence speed estimate we lacked such that the sequence x, fc(x), fc(fc(x)), . . . does in the computation of the Mandelbrot set M and not diverge to oo. The julia set ] c = oKc is the shows that in this case l c is computable in poly­ boundary of the Fatou set. nomial time. The set M can be viewed as the set of For every parameter value c there is a different parameters c for which f c is connected. The hy­ set J c. so in total there are uncountably many Julia perbolic Julia sets· correspond to the values of c sets, and we cannot hope to have a machine which are either outside M or in the hyperbolic

326 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Input c Z+-C, i<---0

yes

Figure 4. The naive algorithm for computing M, and some of its outputs. components inside M. It can be shown that if Con­ The Church-Turing thesis (CT), in its common jecture 5 holds, all the points in the interior of M interpretation, is the belief that the Turing ma­ correspond to hyperbolic Julia sets as well. None chine, which is computationally equivalent to the of the points in oM correspond to hyperbolic Julia idealized common computer, is the most general sets. model of computation. That is, if a function can be We have just seen that the most "common" Julia computed using any physical device, then it can be sets are computable relatively efficiently. These computed by a Turing machine. are the Julia sets that are usually drawn, such as Negative results in computability theory depend the ones on Figure 2. This raises the question of on the Church-Turing thesis to give them strong whether all Julia sets are computable so efficiently. practical meaning. For example, by Hilbert's lOth The answer to this question is negative. In fact, it Problem, Diophantine equations cannot be gener­ has been shown in [BY04] that there are some val­ ally solved by a Turing Machine. This implies that ues of c for which Jc cannot be computed (even with this problem cannot be solved on a standard com­ oracle access to c). The construction is based on puter, which is computationally equivalent to the Julia sets with Siegel disks. A parameter c which Turing Machine. We need the CT to assert that the "fools" all the machines attempting to compute f c problem of solving these equations cannot be is constructed, via a diagonalization argument sim­ solved on any physical device and thus is truly ilar to the one used in other noncomputability re­ hard. sults. Thus, while "most" Julia sets are relatively When we discuss the computational complexity easy to draw, there are some whose pictures we of problems, we are not only interested in whether might never see. a function can be computed or not, but also in the time it would take to compute a function. The Ex­ tended Church-Turing thesis (ECT) states that any Computational Hardness of Physical physical system is roughly as efficient as a Turing Systems and the Church-Turing Thesis machine. That is, if it computed a function fin time In the previous sections we have developed tools T(n), then f can be computed by a Turing Machine which allow us to discuss the complexity of com­ in time T(n)c for some constant c. putational problems in the continuous setting. As In recent years, the ECT has been questioned, in in the discrete case, true hardness of problems de­ i particular by advancements in the theory of quan­ J pends on the belief that all physical computational tum computation. In principle, if a quantum com­ devices have roughly the same computational puter could be implemented, it would allow us to power. In this section we present a connection be­ factor an integer N in time polynomial in log N tween tractability of physical systems in the bit [Shor97]. This would probably violate the ECT, model, and the possibility of having computing since factoring is believed to require superpoly­ devices more powerful than the standard com­ nomial time on a classical computer. On the other puter. This provides further motivation for ex­ hand there is no apparent way in which quantum ploring the computability and computational com­ computation would violate the CT. plexity of physical problems in the bit model. The Let fbe some uncomputable function. Suppose discussi@n is based in part on [Yao02] . that we had a physical system A and two feasible

MARCH 2006 NOTICES OF THE AMS 327 X f(x) f

Figure 5. Computing fusing a "hard" physical device A (left); a fair coin cannot be considered a "hard" device.

translators 1> and l/1, such that 1> translates an numerical simulations a computer can perform input x to f into a state cj>(x) of A. The evolution are bit simulations, hardness of some robust sys­ of A on uo = cj>(x) should yield a state O"T such that tem A for the bit model implies a positive answer l/J(O"; : = f(x) (Figure 5). This means that at least in for(*). On the other hand, proving that all com­ prinl ) le we should be able to build a physical de­ putationally hard systems are inherently highly vice which would allow us to compute an uncom­ unstable would yield a negative answer to this putable function. To compute f on an input x, we question. translate x into a state uo = cj>(x) of A. We then Note that even if the answer to(*) is affirma­ allow A to evolve from state uo to ur-this is the tive and CT does not hold, and there exists some part of the computation that cannot be simulated physical device A that can compute an uncom­ by a computer. We translate O"T to obtain the out­ putable function f, it does not imply that this de­ put f(x) = l/J(O"T). vice could serve some "useful" purpose. That is, it To make this scheme practical, we should require might be able to compute some meaningless func­ A to be robust around uo = cj>(x), at least in some tion f, but none of the "interesting" undecidable probabilistic sense. That is, for a small random problems, such as the Halting Problem or solv­ pertvrbation uo + E of uo, ljJ(O"T) should be equal ability of Diophantine equations. to f(. ) with high probability. It is apparent from this discussion that such an Acknowledgments A should be hard to simulate numerically, for oth­ The authors are grateful to the following people for erwise fwould be computable via a numerical sim­ helpful comments on a preliminary version of this ulation of A. On the other hand, "hardness to sim­ paper: Eric Allender, Lenore Blum, Bill Casselman, ulate" does not immediately imply "computational Peter Hertling, Ken Jackson, Charles Rackoff, hardness". Consider for example a fair coin. It is Michael Shub, Klaus Weihrauch, and Michael Yam­ virtually impossible to simulate a coin toss nu­ polsky. We would like to thank Philipp Hertel for merically due to the extreme sensitivity of the . supplying a program that was helpful in produc­ process to small changes in initial conditions. De­ ing the images. spite its unpredictability, a fair coin toss cannot be used to compute "hard" functions because it lacks References robustness. In fact, due to noise, for any initial [Blum04] L. BLUM, Computing over the reals: Where Tur­ conditions that put the coin far enough from the ing meets Newton, Notices A mer. Math. Soc. 51 (2004), 1024-1034. ground, we know the probability distribution of the [BSS89] L. BLUM, M. SHUB, and S. SMALE, On a theory of com­ outcome: 50% "heads" and 50% "tails". Another ex­ putation and complexity over the real numbers: NP­ ample where "theoretical hardness" of the wave completeness, recursive functions and universal ma­ equation does not immediately imply a violation of chines. Bull. Amer. Math. Soc. 21 (1989), 1-46. the CT is presented in [WZ02]. [BCSS98] L. BLUM, F. CUCKER, M. SHUB, and S. SMALE, Com­ This leads to a question that is closely related plexity and Real Computation, Springer, New York, to the CT: 1998. [BM75] A. BORODIN and I. MUNRO, The Computational Com­ ( *) Is there a robust physical system plexity of Algebraic and Numeric Problems, Elsevier, that is hard to simulate numerically? New York, 1975. [Brt03] V. BRATIKA, The emperor's new recursiveness: The This is a question that can be formulated in the epigraph of the exponential function in two models framework of bit-computability. Since the only of computability, Words, Languages & Combinatorics

328 NOTICES OF THE AMS VOLUME 53, NUMBER 3 III, (Masami Ito and Teruo Imaoka, eds.), pp. 63-72, World Scientific Publishing, Singapore, 2003. ICWLC 2000, Kyoto, Japan, March 14-18, 2000. [BW99] V. BRAITKA and K. WEIHRAUCH, Computability of sub­ Korea Institute for Advanced sets of euclidean space I: Closed and compact subsets, Study Theoretical Computer Science, 219 (1999), 65-93. (BY04] M. BRAVERMAN and M. YAMPOLSKY, Non-computable julia sets, ]. Amer. Math. Soc., to appear. School of Mathematics [Brv05] M. BRAVERMAN, On the complexity of real functions. Available from http://www.arxiv.org/abs/ Postdoctoral Research cs. CC/0502066+. Also in Proc. of46th IEEE FOCS, pp. Fellowships 155-164, 2005. [BCS97] P. BURGISSER, M. CLAUSEN, and M. A. SHOKROLLAHI, Al­ The School of Mathematics at the Korea gebraic Complexity Theory, Springer, New York, 1997. Institute for Advanced Study (KIAS) invites (G]79] M. R. GAREY and D. S. ]OHNSON, Computers and In­ applicants for positions at the level of A Guide to the Theory of NP-Complete­ tractability: postdoctoral research fellows in the mainstream ness, W. H. Freeman and Company, 1979. KIAS, [Grz55] A. GRZEGORCZYK, Computable functionals, Fund. areas of pure and applied mathematics. Math. 42 (1955), 168-202. incepted in 1996, is committed to the excellence [Hert05] P. HERTLING, Is the Mandelbrot set computable? of research in basic sciences (mathematics, Math. Logic Quart. 51(1) (2005), 5-18. theoretical physics, and computational sciences) [KF82] K. Ko and H. FRJEDMAN, Computational complexity through high-quality research programs and a of real functions, Theor. Camp. Sci. 20(3) (1982), strong faculty body consisting of distinguished 323-352. scientists and visiting scholars. [Ko91] K. Ko, Complexity Theory of Real Functions, Birkhauser, Boston, 1991. Applicants are expected to have [Lac55] D. LACOMBE, Extension de la notion de fonction demonstrated exceptional research potential, recursive aux fonctions d'une ou plusieurs variables contributions beyond or reelles, C. R. Acad. Sci. Paris, 240 (1955), 2478-2480; including major 241 (1955), 13-14, 151-153. through the doctoral dissertation. The annual (Mat93] Y. MATIYASEVICH, Hilbert's Tenth Problem, MIT salary ranges approximately from US$2 5,000 to Press, Cambridge, London, 1993. US$36,000 and, in addition, research funds of [Maz63] S. MAzUR, Computable Analysis, Rozprawy Matem­ US$6, 700-US$9,500 are available per year. The atyczne, Vol. 33, Warsaw, 1963. initial appointment for the position is for two [MilOO]]. MILNOR, Dynamics in One Complex Variable-In­ years and is renewable for up to two additional troductory Lectures, second edition, Vieweg, 2000. years, depending on research performance and (PR89] M. B. POUR-ELand]. I. RICHARDS, Computability in the needs of the research program at KIAS. Analysis and Physics, Perspectives in Mathematical Applications will be reviewed twice a year, Logic, Springer, Berlin, 1989. [Sch82] A. SCHONHAGE, The fundamental theorem of alge­ June 30 and December 31, and selected appli­ bra in terms of computational complexity, Technical cants will be notified in a month after the report, Math. Institut der. Univ. Tiibingen, 1982. review. In exceptional cases, applications can be [Sch87] __ , Equation solving in terms of computa­ reviewed other times based on the availability tional complexity, Proceedings of the International of positions. The starting date of the appoint­ Congress of Mathematicians, 1986, (A. Gleason, ed.), ment is negotiable. Applications must include Amer. Math. Soc., 1987. a complete vitae with a list of publications, a [Shor97] P. SHOR, Polynomial-time algorithms for prime research plan, and three letters of recommen­ factorization and discrete logarithms on a quantum dation, and should be mailed to: computer, SIAM]. Comput. 26 (1997), 1484-1509. [Sma9 7] S. SMALE, Complexity theory and numerical analy­ sis, Acta Numerica 6 (1997), 523-5 51. [Tur36] A. M. TURJNG, On Computable Numbers, with an Korea Institute for Advanced Study (KIAS) Application to the Entscheidungsproblem, Proc. Lon­ 207-43, Cheongryangri-dong, Dongdaemun-ku, don Math Soc., (1936), 230-265. Seoul l30-722, Korea [WeiOO] K. WEIHRAUCH, Computable Analysis, Springer, Tel: 82-2-958-3 74 7 Berlin, 2000. Fax: 82-2-958-3770 (WZ02] K. WEIHRAUCH and N. ZHONG, Is wave propagation E-mail: [email protected] computable or can wave computers beat the Turing machine?, Proc. London Math Soc. 85(3) (2002), 312- 33 2. [Yao02] A. YAo, Classical physics and the Church-Turing thesis,]. of ACM 50 (2003), 100- 105. Available from http://www.eccc.uni - t rier.de/eccc- reports/ 2002/TR0 2- 062/.

MARCH 2006 NOTICES OF THE AMS 329 Find Me a Hash Susan Landau

e're accustomed to hearing about authentication, and as one-time passwords; they are the unreasonable effectiveness of employed in many Internet protocols including W mathematics, delightful-and unex­ SSL/TLS, the transport-layer protocol that enables pected-applications of theory to secure Web transactions, IPsec, and SSH. the real world. In the world of the In­ Because hash functions "shrink" data, collisions ternet, we've seen it in the use of number theory between hashes are inevitable. There are three fun­ in public-key cryptography (the Diffie-Hellman sys­ damental properties that a cryptographic hash tem, the RSA algorithm, elliptic curve cryptosys­ should satisfy: pre-image resistance (sometimes tems), in the utilization of graph theory in net­ called non-invertibility): it should be computation­ work design. In the world of Internet data security, ally infeasible to find an input which hashes to a currently we face the opposite situation: a problem specified output, second pre-image resistance: it in search of mathematical theory. The problem is should be computationally infeasible to find a sec­ hash functions. ond input that hashes to the same output as a A hash function is an easy-to-compute com­ specified input, and collision resistance: it should pression function that takes a variable-length input be computationally infeasible to find two different and converts it to a fixed-length output. The hashes inputs that hash to the same output. In 1979 Merkle in which we are interested, called cryptographic [10, pp. 12-13] and Gideon Yuval [12] indepen­ hash functions, are "one-way", which is to say, they dently observed that because of the "birthday" should be easy to compute and "hard", or compu­ paradox-the well-known result that in a group of tationally expensive, to invert1 . Hash functions are twenty-three people, the probability that two peo­ used as a compact representation of a longer piece ple share the same birthday is slightly more than of data-a digital fingerprint-and to provide mes­ half-on average one needs to search only square sage integrity. The way hashes are used to provide root of the search space to find a collision. Thus integrity is that the hash value of a particular piece hash functions of n bits are at best n/2 bits secure of data, ho, is computed at an initial time to. When against collision attacks. the data needs to be used later at time t1, the hash, A common method for hash function design is h1, is recomputed. If the two hashes are equal, iteration: use a small set of operations combined then the data has not been altered. Ralph Merkle, in what is called a round. This way the algorithm a co-inventor of public-key cryptography, calls is put together using a small set of instructions, an hashes the "duct tape" of cryptography. Among important issue for implementations (the same de­ other things, hashes are used to ascertain soft­ sign paradigm is also true for block ciphers). Merkle ware integrity, in digital signatures, in message and Damgard independently showed how to take Susan Landau is at Sun Microsystems. Her email address a collision-resistant compression function and it­ is susan. l andau@sun. com. erate it to produce a collision-resistant hash 1 Hash functions may be "keyed" or "unkeyed" depending on the application; we will ignore that distinction here, Editor's Note: The month of April is Mathe­ although we note that keyed hash functions include matics Awareness Month. The 2006 theme for Message Authentication Codes, or MACs, whose theory is MAM is Mathematics and Internet Security. well understood.

330 NOTICES OF THE AMS VOLUME 53, NUMBER 3 function [11], [2]. The idea, called the Merkle­ HA VAL, MD4, and RlPEMD) [13]. There had already Damgard construction, goes as follows: Let f be a been a move away from MD 5, but this was the final collision-resistant compression function that maps blow. At the same meeting, Eli Biham and Rafi n + m bits to n bits. Break the input I into blocks Chen [1] showed how to find "near" collisions in x1, ... , xk of length n, padding out the last block SHA-0 and Antoine Joux demonstrated an actual with O's as necessary. Start with a random string collision attack on SHA-0 [5]. SHA-1 still seemed of n bits, called the initialization vector, IV. Let b safe. be the length of I and define an extra block Xk+l In 2005 the situation got worse. Wang, in col­ which holds the right-justified representation of b laboration with Yiqun Lisa Yin and Hongbo Yu, written in binary (assume that b < 2n ); the left part showed a collision attack on SHA-1 that took 269 of Xk+l is filled with O's. Then the hash function h steps (instead of the expected 280) [14]; then Wang, is defined by: h(x) = Hk+l where II denotes con­ in collaboration with Andrew Yao and Frances Yao, catenation and: demonstrated a collision attack on SHA-1 that re­ quired only 263 steps [15]. The good news is that Ho=IV this was not a second pre-image attack and the at­ Hi =f(Hi-lllxi), 1 :s; i :s; k+ 1. tack does not mean that all protocols using SHA-1 for integrity were at risk (for example, the That h is collision-resistant follows from the usage of SHA-1 in the "handshake" of SSL 3.0/TLS observation that any collision for h would imply a protocol is not affected by these attacks). But it did fat some stage i. collision for mean that SHA-1 should be replaced as quickly as The Merkle-Damg<'ud construction underlies possible and should not be used for new applica­ most popular hash functions. A half-dozen years tions. How to proceed was both clear and cloudy. ago, there were several popular cryptographic hash SHA-256, also developed by NSA is waiting in the functions from which to choose, including MDS and wings, and is already a FIPS (as are SHA-384 and SHA-1. MDS, developed by Ron Rivest, is a 128-bit SHA-512). SHA-256 is three to four times slower hash that is a strengthened form of an earlier than SHA-1, but that is not the real problem. Mov­ Rivest hash function, MD4. Because of the birthday ing a new algorithm into the infrastructure, whether attack, MD 5 can only be 64 bits strong; the short SHA-256 or a direct SHA-1 replacement (including length contributed to the National Institute of Stan­ the established RlPEMD-160, SHA-256 truncated dards and Technology (NIST) decision not to cer­ to 160 bits, or a "patched" SHA-1), is not an easy tify MDS as a Federal Information Processing Stan­ task. Although computer manufacturers under­ dard (FIPS). FIPS certification means the system is stand the importance of replacing SHA-1, and approved for sale to the federal government, an im­ SHA-256 is in the next operating systems being portant market for the computer industry. Instead fielded by Microsoft, Sun, and other manufactur­ NIST certified SHA-0, the Secure Hash Algorithm ers, SHA -1 and MD 5 will remain in legacy systems (SHA-0 was originally simply SHA). The algorithm for years to come. And while SHA-256 may share was developed by the National Security Agency some of the structure of SHA-1 and thus be po­ (NSA) and was also based on MD4; it was a 160-bit tentially vulnerable to attack, at 256 bits, the al­ hash that was built to work with Skipjack, the block gorithm is large enough, and strong enough, to cipher with 80-bit key that was part of the Clipper suffice for now. This is the clear step forward. system. Two years later NSA proffered SHA-1, an The cloudy part is what happens next. Ten years algorithm that differed from SHA-0 by a circular ago, there was a need to replace the Data Encryp­ shift operation in the round function, which also tion Standard (DES), the encryption algorithm with became a FIPS. The rapid development of SHA-1 a 56-bit key that had been functioning since the shortly after SHA-0 caused users to shy away from 1970s (see, for example, [6], [7]). Fortunately, since the earlier algorithm-and indeed weaknesses were the 1980s there had been fundamental research into found in SHA-0-but both MDS and SHA-1 have the design of block ciphers, much of it from ideas been widely deployed (MDS was used in many pro­ learned through attacks on DES. The strength of the tocols despite not being a FIPS). And as a result we Advanced Encryption Standard (AES), approved as are in trouble. a FIPS in 2002, is based on that research [8]. It is MDS was already in difficulty in 1993, when clear that we need new hash functions, but hash Bert den Boer and Antoon Bosselaers found prob­ research is not in the same place as block ciphers lems with its compression function [3]; further were a decade ago. Until we really understand the problems were discovered three years later by Hans underpinnings of secure hash functions2 , it does Dobbertin [4]. The situation became a great deal not make sense for NIST to begin a serious com­ worse in 2004. At a cryptography meeting in Santa petition for the next one. DES, a good algorithm for Barbara, California, Xiaoyun Wang, Denggou Feng, Xuejia Lai, and Hongbo Yu received a standing ova­ 2 There are hash functions based on hard mathematical tion for work showing collision attacks on MDS (the problems, making them likely to be secure, but these hash attacks also applied to several other hash functions: functions are inefficient and not used in practice.

MARcH 2006 NOTICES OF THE AMS 331 The Value is Greater than the Sum developing an understanding of the security of block-structured ciphers, provided practice for cryptanalysts. SHA-2 56 is probably the right place to start to do the same for secure hash functions. What is the theory of hash functions? It is not often that mathematicians are asked to develop a theory for duct tape, but there is a clear and pre­ sent need to do so now for cryptographic hash functions.

References [1] E. BIHAM, R. CHEN, A. ]oux, P. CARRBAULT, W. ]ALBY, and C. LEMUET, Collisions in SHA-0 and Reduced SHA-1, Advances in Cryptology-Eurocrypt '05, pp. 36-57, 2005. [2] I. B. DAMGARD, A Design Principle for Hash Functions, Advances in Cryptology-CRYPTO '89, Springer­ Verlag LNCS 455, pp. 416-427. [3] BERT DENBoER and ANTOON BOSSELAERS, Collisions for the Compression Function of MD5, Advances in Cryptol­ ogy, Proceedings Eurocrypt '93, Springer-Verlag LNCS 765, 1994,pp. 293-304. [4] HANs DoBBERTIN, Cryptanalysis of MD5, Rump Session, Eurocrypt 1996. [5] ANTOINE Joux, Collisions for SHA-0, Rump Session, CRYPTO '04, August, 2004. [6] SUSAN LANDAU, Standing the test of time: The Data En­ cryption Standard, Notices of the American Mathe­ matical Society, March 2000, pp. 341-349. [7] __ , Communications security for the twenty-first century: The Advanced Encryption Standard, Notices of the American Mathematical Society, April2000, pp. 450-459. [8] __ , Polynomials in the nation's service: Using algebra to design the Advanced Encryption Standard, American Mathematical Monthly, February 2004, pp. 89-117. [9] ALFRED MENEZES, PAUL VAN OORSCHOT, and SCOTT VAN­ STONE, Handbook ofApplied Cryptography, CRC Press, 2001. [10] RALPH MERKLE, Secrecy, Authentication, and Public Key Systems, UMI Research Press, Ann Arbor, Michigan, 1979. [11] __ , A fast software one-way hash function, jour­ nal of Cryptology 3 (1979), 43-58. [12] GIDEON YuvAL, How to swindle Rabin, Cryptologia 3 (1979), 187-190. (13] XiAOYUN WANG, DENGGUO FENG, XUEJIA LA!, and HONGBO Yu, Collisions for Hash Functions, Rump Session, CRYPTO '04, August, 2004. [14] XiAYUN WANG, YIQUN LISA YIN, and HONGBO Yu, Finding Collisions in the Full SHA-1, Advances in Cryptology­ CRYPTO '05, pp. 17-36. Save 20% (15] X:!AYUNW ANG, ANDREW YAO, and FRANCES YAO, New Col­ Use Code: AMS $47.50; CD-ROM lision Search for SHA-1, Rump Session, CRYPTO '05, ISBN 1-56881-283-3 August 2005.

Call: 781.416.2888 Email: [email protected]

www. a kpeters. com

332 NOTICES OF THE AMS VOLUME 53, NUMBER 3 The American Mathematical Society~ together with the Mathematical Sciences Research Institute~ presents... · The 2006 Einstein Public Lecture in Mathentatics A lecture given by Benoit Mandelbrot

lr~ E NATURE OF ROU.GHNE~S ;;-1 I MATHEMATICS, SCIENCE, AND ART !

I ! ! For Plato, Euclid, Newton, and Einstein, space

1 was smooth. The kind of roughness that is obvi- 1 ous in everyday situations was dismissed as too iI complex for scientific study. It has now become 1 accepted that much of the apparently shapeless roughness in the real world is in fact self-similar, that it can be handledowith the help of fractal "pathologies", and also that fractals predate their use in mathematics. They have been a clearcut and remarkable feature in art since time im­ memorial. I will describe a grand Fractal Ring, which begins in art, progresses through pure math­ ematics and diverse sciences, and returns in the end to art. - Benoit Mandelbrot in his abstract for the lecture

This will be the second public lecture in the series ini­ world. He has thus stimulated enormous public and pro­ tiated by the American Mathematical Society in 2005 fessional interest in the beauty and power of fractals, and to celebrate the one-hundredth anniversary of Einstein's in the general question of "roughness" in nature. "annus mirabilis". The year 1905 marked the publica­ tion by Albert Einstein in Germany of three funda­ Benoit Mandelbrot is the Sterling Professor Emeritus of mental papers that changed the course of twentieth­ Mathematical Science at Yale University, and IBM century physics. Einstein later moved to the United Fellow Emeritus at the IBM Research Center. Among States, where he become a founding member of the his many awards and honors are the Wolf Prize for School of Mathematics at the Institute for Advanced Physics in 1993, the Japan Prize for Science and Tech­ Study in Princeton. nology in 2003, and the first Science for Art Prize of Moet-Hennessy-Louis Vuitton in 1988. He is a mem­ The lecture will take place on Saturday, April29, 2006, ber of the American Academy of Arts and Sciences, the at the AMS Sectional Meeting in San Francisco. It will U.S. National Academy of Sciences, and the Americar be aimed at members of the general public, but will also Philosophical Society, and he is the recipient of be of interest to professional mathematicians. numerous honorary degrees.

Benoit Mandelbrot is the founder of fractal geometry. He is also an inspiring and eloquent public speaker. In his study of disparate physical, mathematical, and social phenomena, Mandelbrot has discovered remarkable order in complicated and seemingly unre­ James G. Arthur, AMS President lated data, thereby changing the way we look at the For more information see www.ams.org/ meetings/eillstein-lect.html Mushroom Billiards Mason A. Porter and Steven Lanse I

rt/2

0 •••••••••••o

-rt/ 2 0 Arc length s of reflection point around boundary 4+3rc/2

Figure 1. (left) Axially symmetric mushroom billiard with a semicircular cap. The red trajectory lies in the integrable region of phase space (the ambient space of positions and velocities), and the blue one lies in the chaotic region. (Right) Billiard map showing discretizations of these trajectories. The vertical lines delineate singular points (corners) between different arcs of the mushroom.

wo 20th century discoveries transformed and dynamical systems a,nd can even be constructed scientists' understanding of nonlinear in experimental settings such as microwave cavi­ phenomena [16). One was Kolmogorov­ ties [12) and atom optics [10) . T Arnold-Moser (KAM) theory, which Geometrically, an orbit of the billiard flow of a demonstrated the stability of regular dy­ confined particle is described by a union of line seg­ namics for small perturbations of Hamiltonian sys­ ments, with the link after a boundary collision de­ tems [4, 5, 11, 15). The other was the theory of sto­ termined according to the rule that the angle of in­ chasticity of dynamical systems (loosely called cidence equals the angle of reflection. Keeping "chaos theory"), which demonstrated the stability track only of the location and incidence angles of' of strongly irregular dynamics under small per­ turbations [3, 18, 20). To gain a better under­ the collisions gives a discrete-time billiard map. standing of complicated nonlinear dynamics, there The best-known examples of chaotic billiards, have been extensive studies of model systems such whose flow is hyperbolic, ergodic, mixing, and as billiards. Bernoulli [17, 19), are the "dispersing" Sinai billiar:d The mathematical idealization of a (classical) (a square table with a circular barrier at its center) billiard, a ubiquitous Hamiltonian system [9, 21), and the Bunimovich stadium (shaped lik.e a rec­ consists of a confined point particle colliding elas­ tangle with tWo "focusing" circular caps).'. Neigh­ tically against the boundaries of a container of boring parallel orbits diverge when the coll~de some shape [17, 19). Uke their real-life namesakes, with dispersing components of a billiard's bound­ these sorts of billiards have long held the interest ary. In chaotic focusing billiards, neighboring par­ of both mathematicians and physicists. They have allel orbits converge at first, but divergence prevails led to numerous advancements in ergodic theory over convergence on average. Divergence and con­ vergence are balanced in integrable billiards such Mason A. Porter is a postdoctoral scholar in the depart­ as circles and ellipses, the position spaces ("con­ ment of physics and Center for the Physics of Information, California Institute of Technology. His email address is figuration spaces") of which are continuously foli­ [email protected]. ated by one or more families of "caustics". A curve Steven Lanse/ is a graduate student in the department of is called a caustic if whenever any link of some tra­ electrical engineering, Stanford University. His email ad­ jectory is tangent"to it, then all other links of the dress is sl ansel @stanford. edu. same trajectory are also tangent to it.

334 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Figure 2. (Left) Billiard map for elliptical mushroom with right:left cap ratio of 1:1. Islands of integrability arise from the presence of elliptical caustics (Right). (The curves at the top and bottom of the rightmost region of phase space, which can also be seen in two other regions, are due to the elliptical caustics.)

FJl}ly chaotic and integrable systems are atypical, relative volume fractions in phase space of initial as most Hamiltonian systems (not just billiards) are conditions l~ading to integrable and chaotic tra­ actually mixed, with islands of integrability situated jectories. Using these mushrooms, Bunimovich has in a "chaotic sea" formed by one or more ergodic also shown that the stationary distribution of non­ components that occupy a subset of positive measure interacting particles in a container can be nonuni­ (volume) of the system's ambient space of positions form [7]. The recent study of two, three, and four and velocities (called its "phase space"). Mixed Hamil­ finite-radius disks in a circular container has shown tonian systems are notorious for being difficult to that this can also happen with interacting particles analyze rigorously, as the approaches developed for [14]. completely integrable and chaotic systems both Although they do not have the KAM island hi­ fail at the boundaries between chaotic and regular erarchies of generic mixed Hamiltonian systems, regions. Computational investigations of mixed mushroom billiards possess "sticky" chaotic tra­ systems are similarly difficult, as small islands are jectories, characterized by long tails in recurrence­ not easy to find numerically. time statistics-they have a power-law distribu­ In 2001, Leonid Bunimovich introduced a class tion rather than an exponential one-that of billiard containers shaped like idealized mush­ experience long periods of almost regular motion rooms [6] (reminiscent of the kind many of us saw [1, 2]. The simple geometry of mushrooms allows in certain 1980s video games), providing a class of the analytical study of such statistics, which are examples with mixed regular-chaotic dynamics known to strongly influence global properties of whose relatively simple geometry makes precise Hamiltonian systems, including correlation decay analysis feasible. This discovery has thus made it and phase-space transport. The long tails arise possible to address some delicate questions about from a zero-measure family of angle-preserving the dynamics of systems with coexisting islands and "marginally unstable periodic orbits", with zero chaotic regions [1, 6, 7]. Lyapunov exponent and real eigenvalues of mod­ The simplest mushroom billiards (see Figure 1), ulus one, that lie inside the chaotic region but consisting of a semicircular cap with a stem of never visit the mushroom stem. some shape attached to the cap's base, provide a Mushrooms with elliptical caps have one chaotic continuous transition between integrable (semi)cir­ component and either zero, one, or two islands cular billiards and ergodic (semi)stadium billiards when the stem is sufficiently long. (Extra KAM is­ as the stem width is increased from zero to the di­ lands arise when the stem is too short, as one can ameter of the circle. There are a large variety of ad­ see especially clearly in Figure 2.) This statement j missible stem shapes, but we'll stick to rectangu­ was proven in [6] and is illustrated by Figures 2-4, lar ones here. which show what happens to the elliptical mush­ Bunimovich proved that for circular mushrooms, room's· billiard map as the rectangular stem is trajectories that remain in the cap are integrable, shifted from left to right. One island of integrability, whereas those that enter the stem are chaotic (ex­ associated with trajectories tangent to the cap's el­ Cept for a set of measure zero). This gives a pre­ liptical caustics, exists if and only if the stem does cise, complete characterization of mushroom bil­ not intersect the edges of the cap (see Figures 2 and liard trajectories. One can take advantage of this 3) and the other, due to trajectories tangent to its knowledge in :;everal ways. Changing the dimen­ hyperbolic caustics, exists if and only if the stem sions of a mushroom billiard controllably alters the does not contain the center of tlie cap's base (see

MARcH 2006 NOTICES OF THE AMS 335 Figure 3. Right: left cap ratio of 1:5. Islands arise from both elliptical and hyperbolic caustics (Right). (The central curves in the right two regions of phase space are due to the hyperbolic caustics.)

Figures 3 and 4) [6]. By contrast, circular mush­ arbitrarily many integrable and chaotic compo­ rooms contain no islands if the stem intersects · nents, each encompassing an arbitrary fraction of the edge of the cap and do not require the extra the phase space volume, using so-called "honey condition concerning the stem length (no KAM is­ mushrooms", which have multiple stems and caps lands arise from the stem being short). The dif­ [6].Three-dimensional generalizations of mush­ ference when the stem intersects the edge of the room billiards have also been examined [8]. Finally, cap arises because elliptical billiards have both el­ the eventual study of quantum mushroom billiards, liptical and hyperbolic families of caustics and whose dynamics satisfy the Schrodinger equation circular billiards have only the elliptical one. As the with homogeneous Dirichlet boundary conditions, stem is shifted to the right, the volume in phase should shed considerable light on the quantization space occupied by the elliptical island shrinks, ul­ of systems with mixed dynamics. timately disappearing when the stem reaches the edge of the cap. The orbits that were tangent to Acknowledgements elliptical caustics now enter the stem, making the We thank the editors for many useful comments top and bottom of phase space chaotic. There is still during the preparation of this paper. Bill Cassel­ an island in the center corresponding to orbits tan­ man provided numerous suggestions and graphi­ gent to hyperbolic caustics that do not enter the cal assistance, The figures were initially created with stem. a graphical user interface (GUI) to simulate bil­ Generalizations of mushroom billiards have also liards on Matlab [13]. This GUI was written as part been studied. One can construct billiards with of a research experience for undergraduates

i j

Figure 4. Right: left cap ratio of 0. That is, the right end of the stem touches the edge of the elliptical cap. Islands arise from hyperbolic caustics (Right).

336 NOTICES OF THE AMS VOLUME 53, NUMBER 3 program funded by a National Science Foundation [20] S. SMALE, Differentiable dynamical systems, Bz.illetin VIGRE grant awarded to the School of Mathemat­ of the American Mathematical Society 73 (196 7), pp. ics at Georgia Tech. MAP also acknowledges sup­ 747-817. port from the Gordon and Betty Moore Foundation [21] D. SzAsz, ed., Hard Ball Systems and the Lorentz Gas, of Mathematical Sciences, Caltech's Center for the Physics of Infor­ vol. 101 of Encyclopaedia through Springer-Verlag, Berlin, Germany, 2000. mation. References [1] E. G. ALTMANN, A. E. MOTTER, and H. KANTz, Stickiness in mushroom billiards, Chaos 15 (2005), No. 033105. [2] __ ,Stickiness in Hamiltonian systems: From sharply divided to hierarchical phase space, Physical Review E, in press (nlin.CD/ 0601008). [3] D. V. ANosov andY. G. SINAl, Some smooth ergodic sys­ tems, Russian Mathematical Surveys 22 (196 7), 103-167. About the Cover [4] V. I. ARNow, Proof of A. N. Kolmogorov's theorem on the preservation of quasiperiodic motions under small Phase Portrait of a Mushroom perturbations of the Hamiltonian, Russian Mathe­ The cover shows a close-up of part of the phase matical Surveys 18 (1963), 9-36. space diagram of one of the "mushrooms" de­ [5] __ , Small divisor problems in classical and celes­ scribed by Mason Porter and Steven Lansel in tial mechanics, Russian Mathematical Surveys 18 this issue. The full diagram is shown below, and (1963), 85-192. what the cover shows of this is at the bottom. [6] L. A. BUNIMOVICH, Mushrooms and other billiards with divided phase space, Chaos 11 (2001), 802-808. -Bill Casselman, Graphics Editor [7] __ , Kinematics, equilibrium, and shape in Hamil­ ([email protected]) tonian systems: The 'LAB' effect, Chaos 13 (2003), 903-912. [8] L. A. BUNIMOVICH and G. DEL MAGNO, Semi-focusing bil­ liards: Hyperbolicity, Communications in Mathemati­ cal Physics 262(1) (2006), 17-32. [9] P. CVITANOVIC, R. ARTUSO, R. MAlNIERI, G. TANNER, and G. VATTAY, Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen, 11 ed., February 2005. http://ChaosBook.org. [10] N. FRIEDMAN, A. KAPLAN, D. CARASSO, and N. DAVIDSON, Observation of chaotic and regular dynamics in atom­ optics billiards, Physical Review Letters 86 (2001), 1518-1521. [11] A. N. KOLMOGOROV, On conservation of conditionally periodic motions under small perturbations of the Hamiltonian, Dokl. Akad. Nauk. SSSR 98 (1954), 527-530. [12] A. KUDROLLI, V. KrDAMBI, and S. SRIDHAR, Experimental studies of chaos and localization in quantum wave func­ tions, Physical Review Letters 75 (1995), 822-825. [13] S. LANSEL and M.A. PORTER, A CUI billiard simulator for Matlab. nlin.CD/ 0405033 (2004). [14] S. LANSEL, M. A. PORTER, and L. A. BUNIMOVICH, One­ particle and few-particle billiards, Chaos, (2006), in press (nlin.CD/ 0508037). [1 5]]. MOSER, On invariant curves of area-preserving map­ I J pings of an annulus, Nachr. Akad. Wiss. Gdttingen Math. Phys. Kl. II (1962), 1-20. [16] A. ScoTT, ed., Encyclopedia ofNonlinear Science, Rout­ ledge, Taylor & Francis Group, New York, NY, 2005. [17] Y. SINAl, WHAT IS ... a billiard, Notices of the American Mathematical Society 51 (2004), 412-413. [18] Y. G. SINAl, On the foundation of the ergodic hy­ .pothesis for a dynamical system of statistical me­ chanics, Dokl. Akad. Nauk. SSSR 153 (1963), 1261-1264. [19] Y. G. SINAl, Dynamical systems with elastic reflections, Russian Mathematical Surveys 2 5 (1970), pp. 13 7-188.

MARCH 2006 NOTICES OF THE AMS 337 W H A T I S a Graph Minor BojanMohar

A minor of a graph G describes a substructure of conjecture that every Kn-minor-free graph is G that is more general than a subgraph. If we take (n- I)-colorable. Hadwiger's Conjecture is still a sub graph of G and then contract some connected wide open and is one of the most challenging prob­ pieces in this subgraph to single points, the re­ lems in graph theory. sulting graph is called a minor of G. Formally, a At the beginning of the 1980s, Neil Robertson graph M is a minor of another graph G if G con­ and Paul Seymour developed the theory of graph tains pairwise disjoint trees Tv 5:::; G, one for each minors in a series of twenty long papers. It took vertex v of M, such that whenever u and v are ad­ twenty-one years [2, 3] to publish this seminal jacent vertices of M, there is an edge in G joining work, which had a tremendous impact not only on Tu and Tv. Closely related to this definition are the various branches of graph theory but also on many operations of deletion and contraction of an edge, other areas, most notably theoretical computer each of which clearly produces a minor. Conversely, science. The ultimate result of Robertson and Sey­ we can obtain any minor M of a graph G, by first mour was the proof of Wagner's Conjecture [3] deleting all edges except those in subgraphs Tv that (finite) graphs are well-quasi-ordered by the (v E V(M)) and those connecting Tu, Tv for graph minor relation, which is equivalent to the fol­ uv E E(M), and secondly contracting the edges in lowing statement: the trees Tv. In every infinite collection of graphs, there are two such that one is a minor of the other. A class Jv1. of graphs is minor-closed if for every G E Jv1., all minors of G are also in Jv1.. Examples of minor-closed classes are the collection of all planar graphs and, more generally, all graphs that Figure 1. The Kuratowski graphs K3,3 and Ks. can be embedded in a fixed surface. Every minor­ closed class Jv1. can be described by specifying the The first important appearance of graph mi­ set of all minor-minimal graphs that are not in Jv1. nors is in the following version of the Kuratowski - these graphs are called the forbidden minors for Theorem: Jvl.. The well-quasi-ordering of graphs with respect A graph G can be embedded in the plane to the minor relation (is planar) if and only if neither the com­ is equivalent to the follow­ ing important result: plete graph Ks nor the complete bipar­ tite graph K3,3 is a minor of G. For every minor-closed family of graphs, In topology, this theorem is usually expressed in the set of forbidden minors is finite. an equivalent form saying that no subgraph of G This is a far-reaching generalization of the Ku­ is homeomorphic to Ks or K3,3. ratowski Theorem with a slight disadvantage that Wagner proved in 1937 thatthe Four Color The­ its proof is nonconstructive-it does not yield a orem is equivalent to the statement that the ver­ procedure for finding the forbidden minors for tices of every graph without Ks minors can be col­ the family Jvl., nor does it yield a constructive ored with four colors so that adjacent vertices bound on the number of forbidden minors. receive different colors. This led Hadwiger to Robertson and Seymour also proved that for Bojan Mohar is professor of mathematics at the Univer­ every fixed graph M, there is an algorithm that has 3 sity of Ljubljana and a Canada Research Chair holder at time complexity O(n ) and that for a given graph Simon Fraser University, Burnaby, Canada. His email ad­ G of order n decides if M is a minor of G. This yields dress is mohar@sfu. ca. another powerful result:

338 NOTICES OF THE AMS VOLUME 53, NUMBER 3 For every minor-closed family :M of genus (so that M itself cannot be embedded inS), graphs there exists a cubic time algo­ and a bounded number of other "path-like" graphs rithm for testing membership in :M­ that are attached to G only along the face bound­ one simply checks if a given graph con­ aries onS. This shows that up to "small" and "well­ tains some forbidden minor for :M. behaved" perturbations, graphs with minor M ex­ cluded are essentially 2-dimensional. This is a For instance, any of the following questions can surprising fact-graph minors have been intro­ be solved in O(n3) time, where n is the order of the duced with the aim of proving and generalizing the input graph G: Kuratowski theorem, and yet, the structure theory (1) Let I be a fixed surface and G a graph. Can G of graph minors cannot be done without studying be embedded in I? graphs embedded in surfaces. 3 (2) Is G linklessly embeddable in JR The theory of graph minors has several power­ 3 (3) Is G knotlessly embeddable in JR ? ful consequences for theoretical computer science, A linkless embedding of a graph is one where most notably in computational complexity and in no two cycles are linked, and a knotless embedding the theory of algorithms. Besides the aforemen­ is one where no cycle is knotted. It is known that tioned consequences of well-quasi-ordering, the the set of forbidden minors for linkless embeddings concept of "tree-like" structure, formally giving consists of precisely seven graphs which can be ob­ rise to the notion of the tree-width of graphs, has tained from the Petersen graph by a sequence of been widely applied in the theory of algorithms. Y~ and ~Y operations (the "Petersen family"). On Extensive research is being carried out today to the other hand, only sporadic examples of forbid­ extend graph minor results to matroids. The goal den minors for knotless embeddings are known in this area is to prove a conjecture of Gian-Carlo (one of them is K7). Rota, claiming that matroids representable over The set of forbidden minors is not known for any finite field are well-quasi-ordered with respect embeddings of graphs in general surfaces. Excep­ to matroid minors. Part of this extensive project tions are the sphere, whose forbidden minors are has already been accomplished by Jim Geelen, Bert Ks and K3,3 by the Kuratowski Theorem, and the Gerards, and Geoff Whittle. projective plane, which has 3 5 forbidden minors. In closing, let us briefly return to edge-deletions Computer searches have shown that there are sev­ and edge-contractions since they are closely re­ eral thousands of forbidden minors for embed­ lated to some other aspects of graph minors. In the dings in the torus, and there is no indication how 1950s, Tutte introduced a two-variable polyno­ to obtain the complete list. Despite these difficul­ mial now known as the Tutte polynomial, defined ties, in some specific cases, like testing embedda­ for ~ graph G recursively as follows. If G has no bility of graphs in a fixed surface, efficient algo­ edges, then T(G; x, y) = 1. Otherwise, let e be an rithms have been devised using different edge of G. If e is a loop, then T(G;x, y) = approaches. yT(G- e;x,y); if e is a cutedge, then T(G;x,y) = Another significant consequence of Robertson x T(G/e;x, y); otherwise T(G; x, y) =T(G- e; x, y)+ and Seymour's work concerns the related structure. T(G 1e; x, y). It can be shown that the Tutte poly­ Wagner proved that graphs that do not contain Ks nomial is well-defined for arbitrary graphs. Its co­ as a minor are precisely those that can be built from efficients are nonnegative integers, and it turns planar graphs and subgraphs of a certain cubic out that T(G; x, y) is essentially the same as the par­ graph on 8 vertices by pasting these together in a tition function of the Potts model, studied in the­ tree-like manner. The paper "Graph Minors XVI" oretical physics in relation to phase transition and shows that a similar structure is obtained if we ex­ critical phenomena. The special case· when clude any fixed graph M as a minor. Every graph y = -1/x is equivalent to the Jones polynomial of that does not contain M as a minor can be built an alternating link via a projection of the link in from building blocks, each of which is a graph that the plane and medial graph construction. can be "nearly embedded" in some surface in which Proofs of graph minor results are usually hard M cannot be embedded. Different building blocks and technical. This explains why most textbooks intersect each other in bounded size sub graphs and cover only the basic facts about minors. An ex­ form a tree-like structure. This is a "rough struc­ ception is [1], which is also a good reference for fur­ ture result" in the sense that none of its ingredi­ ther reading. ents can be eliminated, and in the sense that the structure itself describes a minor-closed family Further Reading such that certain other graphs are excluded from [1] R. DIESTEL, Graph Theory, 3rd Edition, Springer, 2005. this family. [2] N. ROBERTSON and P. D. SEYMOUR, Graph minors. I. Ex- To be slightly more precise, each building block cluding a forest,]. Combin. Theory Ser. B 35 (1983) B is a graph that contains a "small" vertex set A 39-61. such that B - A can be written as the union of a [3] __ , Graph minors. XX. Wagner's conjecture,]. Com­ graph G embedded in some surface S of bounded bin. Theory Ser. B 92 (2004) 325-357.

MARCH 2006 NOTICES OF THE AMS 339 Movie Review

Proof

Reviewed by Daniel Ullman

Proof struggle as the offspring Mira max Films of a famous, brilliant, but Starring Gwyneth Paltrow and Anthony Hopkins unstable mathematician Screenplay by David Auburn and Rebecca Miller father. She has extraordi­ Directed by John Madden nary mathematical ability, allowing her to tackle a re­ David Auburn's brilliant play Proof, which search problem for which her opened on Broadway in 2000, is now a Miramax film limited formal education starring Anthony Hopkins and Gwyneth Paltrow as ought not have prepared her. In this respect, she mathematicians, father and daughter. In recent is something like the title character in the film years, there have been several feature films in­ Good Will Hunting. Both Will and Catherine are un­ volving mathematicians, and Proof is one of the usually brilliant but need psychiatric help, he for best. Hopkins plays Robert, a man who at one time childhood trauma, she for depression (or is it schiz­ was a brilliant young mathematician but is beset ophrenia, like her father?). Both resist treatment. by mental illness. The diagnosis is not specified, Will gets his and, predictably, weeps and is cured. but one can infer that it is schizophrenia. Paltrow Catherine, on the other hand, refuses to submit to plays Catherine, who seems to have inherited her the psychiatric plans that have been arranged for father's brilliance as well as his instability. her. Still, her need for psychiatric help is apparent. It is easy to draw parallels between Robert and But Auburn adds an interesting twist: Catherine John Nash Jr., whose story is now widely known is female. When Catherine asks Hal, the young pro­ even outside of mathematical and economic circles, tege of Robert at the University of Chicago, if he owing to the success of Sylvia Nasar's book A Beau­ knows any female mathematicians, he stammers, and eventually tiful Mind and the Ron Howard film of the same says "There's a woman at Stanford. name. Like Nash, Robert suffers from an illness that I don't remember her name." "Sophie Germain?" tests Catherine, in response. causes him to believe that there are encrypted mes­ "Yeah. I think I've met her at meetings," says Hal, revealing his ignorance. sages being communicated to him via newspapers The absence of women in mathematics is a theme and magazines. Like Nash, Robert attempts to do in the film, but it is also an inconsistency between mathematics during his illness but his efforts re­ the film and reality. Nowadays it would be un­ veal not brilliance but the tragic depths of his men­ thinkable for someone in the field not to know of tal illness. Robert is said to have done revolution­ an~ w?men mathematicians. The era of Sofhie Ger­ ary work in three areas before the age of thirty: mam 1s past. game theory, algebraic geometry, and nonlinear operator theory. That seems close enough to Nash Of these three films, Proofis the one t;hat' most that the reference there is impossible to miss. realistically illustrates the world of mathematics But Proof is more about the daughter Catherine and mathematicians. Matt Damon and Ben Affleck than it is about the father Robert. Auburn must have co-wrote and co-star in Good Will Hunting. It is been fascinated by the story from Nasar's book of clear that they are fascinated by the story of Srini­ vasa Ramanujan. Their John Nash III, who like his father suffered from brilliant yet unschooled character Will, like Ramanujan, emerges from the mental illness. The setting for Proofis Catherine's wrong side of the tracks and clashes culturally with the mathematicians with whom he collabo­ Daniel Ullman is professor of mathematics at George rates. Perhaps Good Will Hunting was envisioned Washington University. His email address is doing for the legend West Side [email protected]. of Ramanujan what

340 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Story did for the legend of Romeo and Juliet. It is 1729-the smallest integer that can be expressed a kind of fictionalization of the Ramanujan story as the sum of two cubes in two different ways­ set among modern street toughs. But it is not clear they similarly explain too much to each other. that Damon and Affleck have much of an under­ Mathematicians, particularly father and daughter, standing of how and why mathematicians do what would have a silent rapport on this. they do. The combinatorial problems on the black­ Auburn and Rebecca Miller, daughter of play­ board that Will solves are not the type to baffle pro­ wright Arthur Miller, wrote the screenplay for the fessional mathematicians. And there are phrases film, which was directed by John Madden (of Shake­ that ring hollow to a mathematician's ear. For ex­ speare in Love fame). Madden should be credited ample, the fictional Fields medalist from MIT de­ with capturing the feeling of the mathematical scribes Ramanujan as having created "some of the world; he consulted effectively with Fields medal­ most exciting math theory ever done". A mathe­ ist Timothy Gowers of Cambridge University in matician wouldn't use that phrasing. This Fields preparation for the film. It is richer and deeper, si­ medalist also uses the phrases "solve a theorem" multaneously both funnier and more serious, than and "prove a problem". These mix-ups broke the either A Beautiful Mind or Good Will Hunting. David spell for me. The feeling of actual mathematicians Auburn has more to say to mathematicians than doing actual mathematics is not present in this film. do Damon, Affleck, or Akiva Goldsman, the screen­ Neither is it in the film A Beautiful Mind. The film writer for A Beautiful Mind. Proofis a multifaceted gives the impression that Nash's deep insight into story about sibling rivalry, about gender ability game theory was motivated by the dilemma of how (did Lawrence Summers catch it?), about mental ill­ to compete for the attention of pretty girls in a bar. ness, about trust in relationships, all set within I can imagine that Nash used the competition for the world of mathematics. It is also a mystery female attention as a way of explaining his work about the authorship of a discovered manuscript. to non-mathematicians. But the notion that he dis­ Great stage plays rarely make blockbuster covered his ideas in a flash of insight upon the sight movies, and this may be another example. The play of a beautiful woman is a romanticized version of won a Pulitzer Prize, a Tony Award, and played for the story suitable for a movie audience. It is not how an astonishing 917 performances on Broadway. mathematicians work. The movie does not show us Not bad for a first effort from Auburn! It seems un­ a mathematical seminar or a mathematical con­ likely that the film will be as successful, even with versation. We see no mention of mathematics as its all-star cast. To me, Proof works slightly better enjoyable or as beautiful (despite the title of the on the live stage, with its stark setting and small film). cast, and with intermission to punctuate the shock­ In Proof, though, the mathematical life is more ing last line of the first act. In particular, part of realistically rendered. Three of the four characters the fascination of the play is the way the action re­ (Robert, Catherine, and Hal) work in mathematics. volves neatly around the one setting, the back We see them read and study and write. Robert de­ porch of Robert's and Catherine's home. The ac­ scribes the pleasure he feels when mathematical tivities in other parts of the house and other parts ideas are flowing. Catherine describes to Hal how of town are implied, cleverly woven into the action she felt when doing mathematics, speaking of "beau­ back on the porch. One of my favorite lines from tiful, elegant proofs, like music". Hal describes his the play is absent from the film: When Catherine fear that his own mathematics research doesn't asks what it means about her mental health that pass muster when compared with Robert's. David she is having a drink with (an image of?) her de­ Auburn did his homework and is really able to ceased father, her father replies sadly and convey how mathematicians work. The script re­ poignantly "It could be a bad sign." The film reveals veals that he knows about John Nash, yes, but also the last name of Robert and Catherine: Llewellyn. about Ramanujan, Germain, Paul Erdos, and Andrew There are far more flashbacks in the film, some even Wiles. His characters seem very much like they be­ momentary. Catherine is 27 and 24 years old in the long in the world of real mathematicians. film where she was 2 5 and 21 years old in the play. There are a couple of breaks from realism in This is presumably to accommodate Paltrow, who Proofwhere characters speak in a way that is for was 32 years old when Proofwas filmed. the benefit of the audience rather than the way The film medium adds a new dimension to any mathematicians would actually talk among them­ stage play. A camera can follow a character in a way selves. When Hal remembers what a Germain prime that an audience cannot. The film Proof gets around is, he speaks to Catherine in a way that would be Robert's and Catherine's house. So we have the patronizing to another mathematician. After giv­ requisite bedroom scene in the film, a scene that ing the definition, he offers: "Like two. Double plus is delicately left to our imagination when we see one is five, another prime." When Robert and the play. We visit the party that is merely heard off­ Catherine recall the fact, made famous by a story stage in the play. The film gets out into Chicago, of Ramanujan and G. H. Hardy, about the number Auburn's home town, as well. There are crowds of

MARCH 2006 NOTICES OF THE AMS 341 people at a funeral and on the campuses of the Uni­ themselves, in being recognized as brilliant, than versity of Chicago and Northwestern University. The they are in advancing mathematics. Will's only in­ play, by contrast, has but four characters in it. Jake terest in mathematics seems to be as a tool to Gyllenhaal, who plays Hal in the film, gives a warm demonstrate intellect. The young John searches and sympathetic portrayal of a young mathemati­ mightily for some big idea that will make others cian, more charming than Hals I've seen on stage. notice him. Catherine accuses Hal of stealing math­ And Hope Davis gives a fine performance in the film ematical results for his own advancement. as the sane, normal, but ultimately annoying Claire, Last, mathematicians are young. Will certainly sister to Catherine. The film adds a couple of minor is young, while the aging Fields medalist seems un­ characters, university mathematicians, with spoken certain whether he can understand Will's work. lines. John does all his work when young, certainly. And Still, on the whole the movie is similar to the play. Hal worries that he too is over the mathematical To say "based on the play by David Auburn" is to hill at age twenty-six. Once mathematicians reach understate the connection. Entire scenes are taken a certain age, Hal in the play suggests that they verbatim from the play. Roughly 80 percent of the "might as well teach high school". In the film, Hal lines in the film are straight from the play. By con­ quips "I'm twenty-six. You know, the downward trast, the film version of A Beautiful Mind is only slope." About the assumption that mathematical loosely based on Nasar's book. ability is the province of the young, Robert in the The real significance of the film is that it brings play says "this is a stereotype that happens to be the story to a wider audience, just as A Beautiful true." Mind was seen in film version much more than it These impressions are of course all stereotypes. was read in book version. (Nasar's book New was a I have not heard any evidence that mental illness York Times best seller, but far fewer Americans read is positively correlated with mathematical talent or books than see movies.) The vast majority of Amer­ mathematical interest. Regarding arrogance, my icans may well have an image of a mathematical ge­ own experience is that mathematicians tend to nius that is shaped primarily by the case studies find mathematics humbling and that they share of Will Hunting, John Nash, and Robert and Cather­ with other mathematicians a certain fellowship. I ine Llewellyn in Proof. I suspect that they will draw suspect that mathematicians may the following conclusions about mathematicians be less social on from these feature films: average than the general population, but probably First, mathematicians are disturbed and need the same can be said for others who work in cere­ psychiatry. Will is emotionally disturbed, John is bral disciplines, where extended solitary concen­ paranoid and schizophrenic, and Catherine suf­ tration is required. I doubt self-promotion is a trait fers from depression (at least). A reasonable in­ that is attributable to mathematicians more than ference is that mathematical talent is itself a psy­ to any other professionals. And while many young chiatric illness, that madness is a natural result of mathematicians accomplish great things, I doubt a mind that can reason mathematically. Or perhaps very much that mathematical ability must neces­ it is the converse, that madness induces a state in sarily wane with age. Much more likely, it is men­ which the ability to reason mathematically is tal energy that wanes with age. These films asso­ heightened. It is never easy to infer causation from ciate mathematicians with brilliance rather than correlation. diligence, which in turn suggests that mathemati­ Second, mathematicians are arrogant and rude. cal work is easy rather than hard. It makes sense Will carries his intellect like a weapon, brandish­ then that a failure to be productive is seen as a loss ing it on psychiatrists, on irritating Harvard stu­ of intellect rather than a loss of drive or stamina. dents, and even on the Fields medalist to demon­ Although Robert is mentally ill and Catherine is strate his superiority. John is portrayed as antisocial, Proof does as much to dispel stereo­ obnoxious, such as when he cuts down a colleague types as it does to reinforce them. It presents math­ by telling him that his ideas have not an ounce of ematicians who seem passionate about their work originality in them. Catherine, too, seems rarely to for its own sake, and in this respect it presents a be nice to anyone but her father. more realistic picture of the mathematicians I know. Third, mathematicians are antisocial. Neither Hal is musical, athletic, energetic, and funny, but John nor Catherine seems to have any friends. Will also vulnerable, in many ways exactly like a regu­ does have friends, but his behavior lands him re­ lar person who is a mathematician. Proof shows us peatedly in jail. Hal actually describes mathemati­ mathematicians, young and old, working together cians as wild party animals, but that characteriza­ with fervor to examine an exciting new manuscript. tion seems to be mostly for laughs, since the One imagines it might be a proof of the Riemann stereotype is opposite. hypothesis or perhaps the twin primes conjecture. Fourth, mathematicians are competitive and self­ Perhaps in the sequel we'll find out which result it promoting. They are more interested in advancing is and get the details of the proof.

342 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Book Review

The Curious Incident of the Dog in the Night-time Reviewed by Helmer Aslaksen

The Curious Incident of the Dog "To the curious incident of the dog in in the Night-time the night-time." Mark Haddon Vintage, 2004 "The dog did nothing in the night-time." Paperback, $12.95 ISBN 1-400-03271-7 "That was the curious incident," re­ marked Sherlock Holmes.

If you have spent the last two years on a planet The Curious Incident is at once gripping, touch­ where this book has not been a runaway bestseller, ing, and funny. That's great, you might say, but why and you have no clue what it is about, then you are is it being reviewed in the Notices of the AMS? Is it in for a treat. because the main character is a mathematical The novel is narrated by a fifteen-year-old boy prodigy? Is it because he suffers from Asperger's who introduces himself as follows: "My name is Syndrome? Or is there another reason? Christopher John Francis Boone. I know all the The portrayal of mathematicians in popular countries of the world and their capital cities and media is not always flattering, to say the least. The every prime number up to 7,507." One night book and movie A Beautiful Mind [6], for example, Christopher discovers that the neighbor's dog has left many mathematicians with mixed feelings. Yes, been murdered with a garden fork Sherlock Holmes it's great to see a movie where the hero is a math­ is one of his heroes, so he decides to find out who ematician and is played by a Hollywood hunk, but does it reinforce a stereotype of mathematicians killed the dog and to write a book about it. In the as the ultimate "mad scientists"? Many friends told process he discovers the truth about the es­ me they understand me much better after seeing trangement of his parents. the movie. I have a strong urge to point out to The Hound of the Baskervilles is Christopher's them that I have not yet been diagnosed with para­ favorite book, but the curious title of Christopher's noid schizophrenia. But I usually play it safe, smile, story comes from the Sherlock Holmes short story and say "Thank you!" "Silver Blaze": The main character of The Curious Incident also "Is there any point to which you would has mental problems. He suffers from Asperger's wish to draw my attention?" Syndrome, or high-functioning autism. The syn­ drome is named after Hans Asperger, a Viennese Helmer Aslaksen is associate professor of mathematics at pediatrician who described the condition in 1944. the National University of Singapore. His email address is He noticed that many people with mild autism [email protected]. excel in mathematics and science. According to

MARCH 2006 NOTICES OF THE AMS 343 James [4]: "Very briefly, the criteria for Asperger's I.am sure that many of include severe impairment in reciprocal social in­ us will be touched by var­ teraction; all-absorbing narrow interests; imposi­ ious parts of Christo­ tion of routines and interests on self and others; pher's story. One of my problems of speech, language, and nonverbal com­ favorite parts is when he munication; and sometimes motor clumsiness. The says (end of Chapter 71): casual observer may notice an aversion to direct "Then, when I've got a eye-contact, peculiarities of expression, difficulty degree in Maths, or in coping in social situations, and an obsession Physics, or Maths and with a particular subject, such as computer science." Physics, I will be able to There are several interesting articles about As­ get a job and earn lots of perger's Syndrome among mathematicians, money and I will be able physicists, and computer scientists. A good start­ to pay someone who can ing point is the paper by James [4). In [1), the au­ look after me and cook thors diagnose the Fields Medalist Richard my meals and wash my Borcherds. The book by Baron-Cohen [2] has a clothes, or I will get a lady to marry me and be my whole chapter about Borcherds. The article by wife and she can look after me so I can have com­ Ochert [3] gives a fascinating picture of the Berke­ pany and not be on my own." I do not know about ley Mathematics Department. the "earn lots of money" part, but fortunately there One might very well ask: "Do we really need an­ are a lot of understanding and supportive spouses other book about a 'crazy' mathematician?" I feel who can accept the sometimes peculiar habits of that there is a fundamental difference between a mathematician. The Curious Incident and A Beautiful Mind. Para­ In another part of the story, Christopher strug­ noid schizophrenia is an extreme condition, and gles to be allowed to take the A level mathematics hardly any mathematicians see themselves or their exam. He attends a "Special Needs" school, and no­ colleagues in the descriptions of John Nash. How­ body from his school has ever taken any A level ever, I believe that many of us will relate to some exam. At first the school does not want him to do aspects of Christopher's behavior. There are for­ it. But his father goes to the headmistress and says tunately few mathematicians Who suffer from full­ (Chapter 71): "Christopher is getting a crap enough . blown autism, but many of us have some of the deal already, don't you think, without you shitting traits to a lesser extent, and some even argue that on him from a great height as well. Jesus, this is a dash of autism is essential for success in math­ the one thing he is really good at." I was so happy ematics. (After reading the book, my wife told me for Christopher that he had an understanding fa­ that Christopher reminded her of me. I don't feel ther, and their falling out at the end was the sad­ bad about that. I am just happy that I have an un­ dest part of the book for me. derstanding and supportive wife who can appre­ When the book was reviewed in the biggest ciate somebody with a high Systemizing Quotient newspaper in Singapore, the reviewer recounted the [2]!) trauma she had felt while taking the A level math­ One thing I admire about the mathematical com­ ematics exam, and how fascinating it was for her munity is our tolerance towards unconventional be­ to read about somebody who desperately wanted havior. What I liked best about A Beautiful Mind was to take the exam. She found herself cheering for the way the mathematical community supported Christopher taking the exam, even though it had Nash. I recently visited a university where a fa­ darkened her own life for years. So why does The Curious Incident deserve to be mous professor is known to have Asperger's Syn­ j drome. I asked some of my friends there how he reviewed in the Notices? In my opinion, it is because was doing. They looked at me in total confusion. the book contains some real mathematics. Mark What did I mean by "How is Professor X doing?" Haddon once said in an interview [7}: "[I)f Y9U enjoy He had recently had dinner at the home of one of math and you write novels, it's very rare th'at you'll them, and there was nothing wrong with him as far get a chance to put your math into a novel. I leapt as they could tell. It was a bit unfortunate that he at the chance." He sure did! There is an ~ppendix had spent most of the dinner under the table read­ that contains Christopher's solution to his favorite ing Encyclopaedia Britannica, but they did not think A level problem: "Prove the following result: 'A tri­ of that as a sign of any problem. So while other peo­ angle with sides that can be written in the form ple might be appalled by some aspects of Christo­ n2 + 1, n2 - 1 and 2n (where n > 1) is right an­ pher's behavior, most mathematicians would feel gled.' Show by means of a counter example, that quite at home with much of it. Borcherds once said the converse is false." Christopher originally wanted in an interview that every department he had ever to put the proof in the main text, but his teacher, visited had at least one person who was clearly more Siobhan, told him to put it in an appendix (Chap­ strange than he. ter 233). "And I ~as going to write out how I

344 NOTICES OF THE AMS VOLUME 53, NUMBER 3 answered the question except Siobhan said it wasn't [3] AYALA OcHERT, The mathematical mind: Madness, very interesting, but I said it was. And she said peo­ genius, and what mathematicians are really like, Cal· ple wouldn't want to read the answers to a maths ifornia Monthly, April 2002, http: I jwww. alumni . question in a book, and said I could put the answer berkeley.edu/Alumni/Cal_Monthly/April_2002/ in an Appendix which is an extra chapter at the end The_mathematical_mind.asp. [4) loAN] AMES , Autism in mathematics, The Mathematical of a book which people can read if they want to. Intelligencer 25 (2003), 62-65. And that is what I have done." I guess most math­ [5] ALEX KAsMAN, Mathematical fiction, http: I /math. ematicians need some editorial help! His writing is cofc . edu/faculty/kasman/MATHFICT/. not as smooth in the proof as in the other mathe­ [6] SYLVIA NASAR, A Beautiful Mind: A Biography of john matical explanations, and I do not think any of us Forbes Nash ]r., Simon & Schuster, 1998. would complain about relegating the proof to the [7] DAVE WEICH, The curiously irresistible literary debut of appendix. Mark Haddon, http: I jwww. powells. com/ The book's chapters are numbered using prime authors/haddon.html . numbers, which is why the first chapter is Chap­ ter 2, and why Chapter 71 comes after only about fifty pages. Christopher of course includes a clear explanation of the Sieve of Eratosthenes. In addi­ tion there are nice discussions of the Monty Hall problem, the logistic equation, Conway's soldiers, chaos theory, and several other mathematics, logic, and physics topics. They are all explained in a clear and understandable way. I am passionately involved in mathematical out­ reach, and I would say that Christopher is a nat­ ural at it. Many mathematicians wax lyrical about the beauty of mathematics, but when asked to share that beauty with the general public, they are apt to use on phrases like "let X be a projective variety over a field of characteristic p". It really both­ ers me that some mathematicians do not care enough about our profession and the public to try to find some link between what they do and what the public can relate to. One of my main goals in mathematical outreach is to show that the beauty of mathematics is all around us. We should not be afraid of starting to talk about mathematics to anybody, anytime. Are you concerned about how to pick the right topics at the right level and pre­ sent them in a clear and understandable way? Then The Curious Incident is a good book for you. just do it the way Christopher does it! The Curious Incident of the Dog in the Night-time is not just another story about a "crazy" mathe­ matician, but a compassionate tale of mathemati­ cal outreach. And when was the last time there was an honest mathematical proof in a book on the New York Times fiction best seller list?

References [1] SIMON BARON-COHEN, SALLY WHEELWRIGHT, VALERIE STONE, and MELISSA RUTHERFORD, A mathematician, a physicist and a computer scientist with Asperger Syndrome: Per­ formance on folk psychology and folk physics tests, Neurocase 5 (1999), 475-483. [2] SIMON BARON-COHEN, The Essential Difference: The Truth about the Male and Female Brain, Perseus Books Group, 2003. Chapter 11 is available online at http: I I leitl.org/docs/a-professor-of-mathematics. pdf.

MARCH 2006 NOTICES OF THE AMS 345 Sullivan Receives 2004 National Medal of Science

On November 14, 2005, the of view. His geometric insights led to many im­ White House announced the portant results on the topology of manifolds. His names of the eight recipients of theory of real and rational homotopy types, based the 2004 National Medal of Sci­ on differential forms, has had profound applica­ ence. Among those honored is tions, for example, to the topology of complex mathematician DENNIS P. Sum­ al­ VAN. gebraic varieties. Sullivan According to the White has made important contributions to House news release, "The Na­ the study of foliations and dynamical systems. He tional Medal of Science honors has also proved foundational results on quasi­ individuals for pioneering sci­ conformal and Lipschitz manifolds, categories that entific research in a range of are intermediate between the topological and Dennis Sullivan fields, including physical, bio- smooth ones. During the 1980s and 1990s, he was logical, mathematical, social, responsible for the emergence of the field of con­ behavioral, and engineering sciences, that enhances formal dynamics as a lively and important branch understanding of the world and leads to innova­ of mathematics straddling the traditional borders tions and technologies that give the United States its global economic edge." The award was estab­ between pure and applied areas. In recent years, lished by Congress in 1959 and is administered by he launched the field of string topology. the National Science Foundation. A list of previous -Allyn jackson medalists is available at http: I l www. nsf. gov 1 nsblawardslnmsl medal.htm. Dennis Parnell Sullivan was born in Port Huron, Kenneth Arrow, Economist with Michigan, in 1941. He received his Ph.D. from Mathematical Outlook, Receives Princeton University in 1965, under the direction National Medal of William Browder. Sullivan has held positions at Princeton, the Massachusetts Institute of Tech­ One of the recipients of the 2004 National nology, and the Institut des Hautes Etudes Scien­ Medal of Science is the economist Kenneth ]. tifiques in Paris. Currently he holds the Einstein Arrow of Stanford University, whose work is Chair at the Graduate Center of the City University quite mathematical and treats such areas as of New York and is a professor of mathematics at the economics of information and organiza­ Stony Brook University. His honors include the tion, collective decision-making, and general AMS Veblen Prize in Geometry (1971), the Elie Car­ equilibrium theory. Arrow's bachelor's and tan Prix en Geometrie of the Academie des Sci­ master's degrees are in mathematics, and he ences de Paris (1981), and the King Faisal Prize switched to economics for his Ph.D. He has 120 (1994). He is a member of the U.S. National Acad­ publications listed in MathSciNet. Arrow re­ emy of Sciences. ceived the Nobel Prize in Economics in1972. Sullivan's early work was in homotopy theory and - A.]. surgery, to which he brought a new, geometric point

346 N OTICES OF THE AMS VOLUME 53, NUMBER 3 Donaldson and Narasimhan Receive 2006 King Faisal Prize

SIMON DONALDSON and M. S. NARASIMHAN have been jointly awarded the 2006 King Faisal International Prize for Science. The prize, presented by the King Faisal Foundation, consists of a gold medal and a cash prize ofUS$200,000, which the two recipients will share. Born in 195 7 in Cambridge, England, Simon Kir­ wan Donaldson received his Ph.D. in 1983 from Ox­ ford University, under the direction of Michael Atiyah. Donaldson was a professor at Oxford Uni­ versity and at Stanford University before becom­ ing a professor at Imperial College, London. He is now a Royal Society Research Professor at Imper­ ial and also serves as president of Imperial's In­ stitute of Mathematical Sciences. His many honors include the Fields Medal (1 986) and the Crafoord Prize (1994). He is a fellow of the Royal Society, Lon­ don. Donaldson's early research revolutionized Simon Donaldson M.S. Narasimhan four-dimensional differential topology, revealing The close connection between the research of surprising new phenomena through the application the two prize winners is illustrated by the fact that of ideas from gauge theory. He has also made foun­ one of Donaldson's earliest papers bears the title dational contributions to complex and symplectic "A New Proof of a Theorem of Narasimhan and geometry and to global analysis of partial differ­ Seshadri" Uournal of Differential Geometry, 1983), ential equations on manifolds. referring to the landmark paper "Stable and Unitary Mudumbai Seshachalu Narasimhan was born in Vector Bundles on Compact Riemann Surfaces", by 1932 in Thandarai, in the state of Tamiladu, in Narasimhan and C. S. Seshadri (Annals of Mathe­ India. He received his Ph.D. from the University of matics, 1965). Narasimhan's paper with S. Ramanan Bombay in 1960, under the direction of Komaravolu on universal connections ("Existence of universal Chandrasekharan. For many years Narasimhan was connections", American journal of Mathematics, a professor at the Tata Institute of Fundamental Re­ 1961 and 1963) has been very influential in the search in Mumbai. In 1992, he went to the Inter­ exchange of ideas between mathematics and theo­ national Centre for Theoretical Physics in Trieste, retical physics surrounding index theory and gauge where he headed the research group in mathe­ theory. This exchange of ideas is also the context for matics. He is now an Honorary Fellow of the Tata much of Donaldson's important work. Institute of Fundamental Research in India. In 19 7 5 The King Faisal Foundation was established he received the Bhatnagar Prize for Mathematics in 1976. Of its many philanthropic activities, the (1975), which is the most prestigious award given King Faisal International Prize is the most widely in India. He also received the Third World Acad­ known. Prizes for Service to Islam, Islamic Studies, emy Award for Mathematics in 1987 and is a Fel­ and Arabic Literature were established in 1977 low of the Royal Society, London. Narasimhan is a and first awarded in 1979. In 1981 the Prize Board pioneer of the study of moduli spaces of holo­ added Medicine; Science was included in 1982. morphic vector bundles on projective varieties. The Science subcategories cover a broad scope: His work on projectively flat connections was the physics, mathematics, chemistry, and biology. The starting point for the development of the so-called prizes are awarded during a ceremony in Riyadh, Kobayashi-Hitchin correspondence linking the dif­ Saudi Arabia, under the auspices of the King of ferential and algebraic geometry of vector bundles Saudi Arabia. over complex manifolds. - Allyn jackson

MARCH 2006 N OTICES OF THE AMS 347 Moment of Proof

Jonathan David Farley

I wondered if I was in the right place. The sun was Jodie Foster did not have math, but the novel by smiling, the skies a beautiful blue. L.A. The streets Carl Sagan had a chapter on transcendental num­ were long, wide, and almost deserted. The large bers), Arcadia (a play that begins with a conversa­ building in front of me looked fairly uninviting. tion about Fermat's Last Theorem), and, of course, Move along, it said, nothing to see here. Proof It was a short walk down an incline to a guard's Often in film mathematicians are absent-minded glass booth. ''I'm going to the set of the television eccentrics who can't balance a checkbook, extras show NUMB3RS," I said tentatively. I don't think the from Revenge of the Nerds, or erratic wunderkinds guard called anyone or even checked my ID. He just on the verge of some great discovery-and the gave me a badge, and directions. Soon enough I was verge of a nervous breakdown. with my contact, with whom I had only previously There is of course Russell Crowe in A Beautiful communicated via email. We were in an elevator Mind, the story of the mathematical genius who going up. One flight, two flights-the doors opened. talked to space aliens and wound up winning the Suddenly I was inside FBI headquarters .... Nobel Prize. Matt Damon in Good Will Hunting In recent years, math has broken out of the played a troubled kid from the streets who as­ classroom and onto the stage and silver screen. tounds MIT professors with his mathematical What's the image of mathematicians in pop culture? prowess. In the time travel story Sphere, Samuel L. Why are we seeing more and more of them in the Jackson is a mathematician who obtained both his movies? And can these movies help mathematics? Ph.D.'s before age twenty. The hit play Proof is about a woman who sees dead people-but just Fermat's Last Theatre may have solved the most important mathematics It certainly seems hard to deny that the number of problem in the world. Then there's Tom Stoppard's movies and plays featuring math and mathemati­ Arcadia, considered by some the best play of the cians is rising exponentially. The list includes 21 twentieth century, about a precocious teenager Grams (Sean Penn plays a mathematician, at one who discovers fractal geometry about 160 years point uttering the grammatical equivalent of before it was actually invented. Finally there's nails-on-the-chalkboard, "mathematics are"), Mean Pi, about a man on the brink of madness who Girls (teen queen Lindsay Lohan plays a "math­ thinks the secret of God can be found in math's lete" -solving a calculus problem is a miniature most famous number. climax in the story), Die Hard 3 (Bruce Willis and Sometimes in film mathematicians are unreal­ Sam Jackson have to solve a math problem in order istically suave. In previously unpublished inter­ to stop a mad bomber), Contact(the movie starring views I conducted for Time magazine, I asked several mathematicians and filmmakers for their jonathan David Farley is a Science Fellow of the Center thoughts. "Hollywood has a mythical idea of a for International Security and Cooperation at Stanford Uni­ mathematician," said Harvard professor Dick Gross, versity. His email address is l atti ce@stanford. edu. the technical advisor for It's My Turn (starring Jill

348 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Clayburgh and Michael Douglas). "Clayburgh's char­ I discussed an anecdote about Norbert Wiener with acter was a glamorous female mathematician," he the head writer; and I told Andy that the story continued. "When I had her get together with sev­ about Schrodinger's cat in one of the scripts (in eral actual women mathematicians, [Clayburgh] which a cat is poisoned) might offend cat lovers. said she found them too depressed, not what she (When I met him, the creator of the show himself had in mind for the character at all." In jurassic brought up the fact that this ·story might cause of­ Park, Jeff Goldblum tries too hard to be hip in his fense, although he didn't know it was I who had shades and black leather jacket. In Sneakers, the emailed Andy about this days before.) I told Andy mathematician wears all white. and one of the writers about "reflexive theory," a But there are exceptions: sometimes in film branch of mathematical psychology used by the So­ mathematicians are almost human. Jeff Bridges viet defense establishment; ten months later, Andy plays a mathematics professor who marries Barbra and another writer held a conference call with Streisand in The Mirror Has Two Faces, and Edward Vladimir Lefebvre, the Russian who created this the­ James Olmos is a teacher who inspires kids with ory. Tony Harkin, an applied mathematics fellow calculus in Stand and Deliver. In the view of for­ at Harvard1 and consultant with my company mer Cal tech number theorist Edray Goins, "One of Ho ll ywoodmath. com, told Andy about how math­ the first mathematicians ever to be featured in a ematics is being used to try to predict the paths of positive light was 'Dwayne Wayne' of [the sitcom] hurricanes. (The very next script we were sent was A Different World." In it, actor Kadeem Hardison entitled, "Whorlwind" (sic).) Tony also suggested plays a math major at a black college. that the mathematics professor in the show, Char­ In a society so innumerate and math phobic, why lie, might want to think twice about dating his are we seeing so much mathematics on film? "Math graduate student. is a wonderful storytelling tool, filled with Since the hit television crime drama NUMB3RS metaphors for the human condition," says Sofia premiered in early 2005, mathematicians have Quintero, creator of the hip-hop drama Picture Me been asking, "Can movies and television help math­ Rollin'. "It's a language that is both mystical and ematics?" I'll be a bit perverse, and ask the converse: universal." "Can mathematics help the movies?" It can, and in Adds British Academy Award-winning filmmaker two ways: by improving plots and by improving John Lynch, "Here's a world most people think of profits. as dull, as incomprehensible, and they can't un­ "Guys-this is really wonderful stuff," Andy told derstand why anybody would do it. But within it us later. "Unwittingly, you've sparked a creative are the most passionate drives and emotions that argument over here: Nick Falacci, one of the cre­ are fundamental to the human spirit." Indeed, ators, really liked the Secret Sharer stuff. Cheryl, Enigma, the World War II spy thriller starring Kate his wife and co-creator, really wants to embrace the Winslet, is as much a love story as it is a story about mathematics of a hurricane for better thematic mathematical codes. (It draws of course from the resonance. My guess is we'll use both, just save life of mathematician Alan Turing, whose true one for a coming episode." story is told in the film Breaking the Code, based Mathematicians can help movies by providing on the play by Hugh Whitemore.) Lynch co-directed writers with interesting ideas around which they Fermat's Last Theorem, a documentary about can weave their stories. (Right now Gary Lorden of the only theorem ever mentioned in a Star Trek Cal tech is the consultant for NUMB3RS.) The whole episode. mathematical community-if not the whole coun­ "Math in Hollywood films is great," adds Lynch's try-may benefit if NUMB3RS succeeds. co-director, Simon Singh, "but the real breakthrough For 'this reason Robert Barker of the U.S. Air would be prime-time TV." Force has crafted a program whereby mathemati­ cians and scientists can learn the art of screen­ Running the NUMB3RS writing. The hope is that more positive images of "Let me toss this out to you, see what you think." scientists on-screen today will result in more It was my contact, Andy. "We're developing a story American scientists in the laboratory tomorrow. around what our writer is calling 'mosaic theory'­ It may actually be a matter of national secu­ basically, is it possible to have only five pieces in rity-at least, it looks like the government thinks a 1,000-piece puzzle, yet know what that entire so. When I visited the NUMB3RS office and set, puzzle is? ... Is the puzzle metaphor a good one one of the people I bumped into was George S., a for mosaic theory? Is it realistic from an applied special agent for the FBI. math point of view?" We could even create our own opportunities. A I mentioned "secret sharing" schemes to Andy documentary based on Karl Sabbagh's readable (whereby a secret is divided up into pieces so that Dr. Riemann's Zeros might have popular appeal. it is impossible for it to be reconstructed from any proper subset of the pieces). When I visited the set 1 Now at the Rochester Institute of Technology.

MARCH 2006 NOTICES OF THE AMS 349 The AMS could work on the documentary been tapped into in the same fashion as mathe­ Incompleteness: The Life of Kurt G6del with Philipp matics organizations to encourage women and girls Steger of the Austrian embassy's Office of Science to see the film. Such organizations might have in­ and Technology. (The Austrian logician would cluded the National Organization of Women and the have been 100 in 2006 had he continued taking Association for Women in Mathematics. Moreover, his vitamin C supplements.) The Sloan Foundation well-placed newspaper editorials could have re­ and EUROPAWS (a European society for the public vived the Summers controversy to the benefit of awareness of science) could help fund these Proof. As the National Science Foundation (NSF) has projects. special initiatives for increasing the numbers of girls But there's another, perhaps more significant, who go into math-related careers, the NSF could way mathematics can help the movies. have been encouraged to support field trips for Proofof Principle schools to see the movie. Or, special screenings could have been set up in schools for a fixed sum. The movie Proof, starring Gwyneth Paltrow and Anthony Hopkins, cost an estimated US$20 million. The Primer Number Theorem In its first three weeks, it earned only US$2.5 mil­ lion- not exactly a large cardinal. This could have Shane Carruth studied applied mathematics as an adverse consequences on whether we see more undergradute. In 2004 his film Primer won the math in theatres. What might the mathematics Dramatic Grand Jury Prize at the Sundance Film Fes­ community have done? tival. At the request of Primer's public relations Firstly, Proofs base audience could have been firm, Tony Harkin (mentioned above) attended the mobilized. Over 30,000 people belong to the AMS Boston premiere in order to lead an audience dis­ and related organizations. (This figure does not cussion about the science in the film. The theatre even include math teachers at the high school level was sold out. or student mathematics organizations.) College This is an excellent way to create buzz for a film. mathematics classes typically have 20-30 students There could have been similar audience discus­ each. Add to this the approximately 15 million sions for Proof. These special screenings would high school students, and, even if only a fraction themselves have generated additional publicity for of these individuals were encouraged (or required Proof. by their teachers) to watch Proof, perhaps an additional US$1 0 million or more could have been Is Math the New Black? netted for the film. University mathematics What is the future of cinemath? departments and math organizations (at little or "The truth is that math is a difficult and lonely no expense to the film's distributor, Miramax) could business, conducted largely within the confines of have promoted the film and encouraged professors one's own mind," pines Harvard's Dick Gross. "This and student organizations to watch Proof. The AMS has little appeal to Hollywood." and other groups could also have asked their That is, it used to have little appeal. In the members to do so. summer of 2005 a producer who was adapting The In January 2005 we told Jill DeWinter of NUMB3RS Oxford Murders for the big screen contacted Hol­ that a "math literacy" campaign based around lywood Math and Science Film Consulting. Lattice NUMB3RS would be embraced by schools and theorist Guillermo Martinez penned The Oxford math teachers across the country, thus magnifying Murders, a detective novel set in the quaint English their audience enormously without expensive city where the spires dream and the dons are only advertising. We later explicitly suggested that slightly mad. they work with the National Council of Teachers of Mathematics (NCTM) to prepare lesson plans, sam­ Already under way is the shooting of The ple assignments, background notes and references Da Vinci Code, based on the book by Dan Brown. for topics that arise in the show, including explana­ The movie stars Tom Hanks and (1+) 5)/2, and is tions of formulas and other supplementary materi­ sure to be a blockbuster hit. (Incidentally, Dan als. On September 19, 2005, NUMB3RSlaunched Brown's father is a math teacher and part of the such an educational initiative in cooperation with reason why Fibonacci numbers appear so promi­ NCTM and Texas Instruments. nently in the book.) Harvard president Larry Summers' inept remarks And what's next for TV? "Maybe [TV sitcom about women's "innate ability" (or inability) to do psychiatrist] Frasier could switch jobs," jokes mathematics garnered a huge amount of publicity Simon Singh, "and turn to proving the Riemann over several weeks. This kind of controversy could hypothesis ... " have been turned to Proofs advantage, given the I guess you could call it the revenge of the geeks. subject matter of the film. Women's organizations, Or numerical chic. But for now, in Hollywood, it's both for adults and for college students, could have hip to be square-or some other geometric object.

350 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Impact Factor and How It Relates to Quality of Journals Vitali Milman

At the touch of a computer key, statistical infor­ is the number of published papers during these two mation is so readily available nowadays that many years (Y- 1 andY- 2) in this journal. of us may be tempted to overestimate its value. This may, perhaps, be a very appropriate ap­ I have been looking into the question of the va­ proach for, say, medical sciences or biology, where lidity of information gleaned from the Journals the influence of a publication is decided in the Impact Index (see definition below) and have noted first year or so after publication and, after three that, for the mathematical community at least, this or four years many results are already irrelevant. index gives, in some cases, a misleading impression However, what does this mean for mathematics? of the relative standing of journals. This data is Let P be a paper, published in the year Y - 2, often used by department heads in faculty as­ which influences the work of a group of researchers. sessments, and people have begun to check Impact Even if they work very quickly, it can take a few Factors (IF) of journals before submitting papers. months for them to produce results and write them So, editors of mathematical journals have no choice down. Then new papers, with references to P are but to try somehow to improve the IF ranking of submitted. It takes at least a year, and more often their journals, although they are aware that it has a year and a half or two, for these papers to be ac­ little to do with the real quality of the journal. Un­ cepted and published. However, it is now too late fortunately, most actions to increase the IF have an for the references in these papers to be included anti-scientific form, and we mathematicians should in the IF of year Y. The situation is even worse work to create a different index, more suitable for with papers published in the year Y - 1. research in mathematics. You may then ask, how do mathematical jour­ Below I would like to describe why the Impact nals show any positive index? The answer is that Factor, as it is calculated today, is not suitable for mathematics papers are very often distributed and mathematical journals and would like to open a dis­ widely known before publication. What can edi­ cussion of how to change the present method of tors do to influence this? Very simply, not to pub­ IF calculation to reflect the realities of mathemat­ lish the best papers too quickly! (Please don't take ics research. First, let me explain how the impact index is com­ this suggestion too seriously, but are you sure it doesn't enter their heads?) Also, the actual num­ puted: The IF of a journal for year Y (say, for ber of papers published each year is used for IF Y := 2003) is the ratio C I N, where C is the num­ ber of citations during year Y (i.e., 2003 in our ex­ computations, and not the number of pages. This ample) of papers published in a particular journal means that a paper of a hundred pages, with many in years Y - 1 and Y- 2 (i.e., 2002 and 2001). N parts, which may be of interest to people in dif­ ferent fields, has a much better chance of raising Vitali Milman is professor of mathematics at Tel Aviv the impact index of the journal than a short (and University. His email adress is mil man@post. tau. ac. i l. possibly brilliant) paper of, say, five pages long.

MARcH 2006 NOTICES OF THE AMS 351 journal No. arts. Citations Math. IF journals which publish around fifteen papers a ]. AMS 34 470 13.824 year (like Acta Math.) or publish around fifty pa­ pers (or over 100), should be judged differently. Annals of Math. 68 630 9.265 However, as I have already said, I am just trying to Acta Math. 19 164 8.667 correct the existing structure with the hope that our GAFA 34 266 7.834 mathematical community may influence the Thom­ Invent. 85 646 7.600 son Company, which regularly publishes the Impact Duke 94 570 6.009 Factor, to change the system for mathematics and ]. Funct. Anal. 141 666 4.723 create (and use) an Impact Factor suitable for math­ Adv. Math 66 274 4.660 ematics. Trans. AMS 216 724 3.352 The preceding analysis and comments were pre­ Isr.]. Math. 108 280 2.593 pared in November 2004. Recently one very im­ portant change has occurred. In September 2005 MathSciNet of the AMS has created its own Citation There are many more very "wrong" ideas which Database, where we see a citation index for may come to mind when we think about our im­ mathematics journals which takes into account pact index. However, I have learned recently of one the last five years' publications. (See "MathSciNet­ consequence of the current structure of the Index Matters", Notices, January 2006.) It immediately which is actually having a positive influence. Many introduces some adjustments to the order found journals have begun to make all accepted papers in Thomson's citation index, and the relative pic­ available online. This increases the exposure time ture is much closer to the table above. before year Y, the "critical" time! This is a very However, I would further suggest that the AMS crucial one-year period. include the accumulated impact factor and not In light of this, I would like to discuss a differ­ only a local one. What I mean is the following: on ent approach for computing the IF value for math­ a specific day, say 31 December (of each year) con­ ematical journals. The idea is simple. We know sider all citations to articles published in a partic­ that a few more than just two years are necessary ular journal in, say, 1999 (or a period of a number to estimate, appreciate, and understand real math­ of years, say, between 1999 and 2002). This accu­ ematical progress, real achievement. So why not cal­ mulated number of citations is then divided by culate data from a few more years back? the number of articles published during that time I checked a few examples, ten journals which (say, during 1999, or during 1999, 2000, 2001 and came to mind as being highly thought of, but of dif­ 2002), the resulting ratio being the cumulative im­ ferent levels. The examples include GAFA, of which pact factor. I am the managing editor. The table above was cal­ culated for articles published in the year 1999; the number of articles published in that year is shown in column "No. Arts". The "Citations" column shows the number of references, up to October 2004, to articles published in each journal in 1999. The Mathematical Impact Factor (Math. IF) is a ratio of number of citations to number of articles. I believe these examples reflect well our under­ standing of the status of journals. A surprising re­ sult can be seen for the journal of the AMS; there is a large gap between JAMS' Math. Impact Factor and that of the other journals. I believe this could be connected to the fact that some areas in math­ ematics have a much higher citation index, e.g., the­ oretical computer science or related discrete math­ ematics papers, firstly because of different styles of references acceptable in those fields, but also because many more mathematicians now work in these fields. JAMS publishes papers in these areas, whereas, Acta, for example, does not; nor do most of the other journals in our sample list (although I may be mistaken here). Many more ideas come to mind for how one may compute a meaningful index reflecting the relative value of different journals. Also, clearly,

352 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Programs that Make a Difference

University of Iowa Department of In 2005 the AMS Committee on the Profession Mathematics Graduate Program with the endorsement of the Council, (CoProf), The Department, the University, and the State. The initiated a project to recognize mathematics de­ department has forty-one tenure-track faculty mem­ strong partment programs that are exceptionally bers, five postdoctoral faculty, and averages five in mentoring women and underrepresented mi­ short-term visitor appointments each year. The de­ norities. Called "Programs that Make a Differ­ partment enrolls more than 7,000 students annu­ ence," the project will each year identify two ex­ ally. The department currently has 112 graduate emplary programs that: (1) aim to bring more students and 185 undergraduate majors. Typically, individuals from underrepresented minority the department graduates twelve Ph.D.'s per year. backgrounds into some portion of the pipeline An integral part of the department is the interdis­ beginning at the undergraduate level and lead­ ciplinary Ph.D. Program in Applied Mathematical ing to an advanced degree in mathematics, or and Computational Sciences (AMCS). The univer­ retain them in the pipeline; (2) have achieved sity enrolls 29,000 students annually and is a mem­ documentable success in doing so; and (3) are ber of the Big Ten Conference. The population of replicable models. Iowa is three million people, fewer than 5 percent For 2006 CoProf chose two departments to of whom are from underrepresented minority back­ recognize. The citation states: "This year the grounds. AMS recognizes the graduate program at the Uni­ Minority Graduate Student Enrollment in the versity of Iowa. and the Summer Institute in Department. In 1995 the department made a long­ Mathematics for Undergraduates (SIMU) Re­ term commitment to substantially increase the search Experiences for Undergraduates program number of its U.S. graduate students from under­ conducted at the Universidad de Puerto Rico, Hu­ represented minority groups (African American, macao, from 1998 to 2002. Both of these pro­ Latino/ a, Native American U.S. citizens, and per­ grams have made significant, successful efforts manent residents). Supported in part by five U.S. to encourage underrepresented minorities to Department of Education Graduate Assistance in continue in the study of mathematics." Areas of National Need (GAANN) grants in eight An important component of "Programs that years, the department's U.S. underrepresented mi­ Make a Difference" is a series of profiles of the nority graduate student population has grown from recognized departments that highlight their zero students in 1995 to twenty-four students cur­ successes and provide information on how the rently. U.S. minority students have accounted for programs operate. What follows are profiles, 20- 25 percent of the department's graduate stu­ prepared by CoProf, of the two programs rec­ dent population over the past six years. These per­ ognized in 2006. Information also appears at the centages are among the highest at U.S. majority in­ website http: I / www. ams. o rg/ employment/ stitutions. The department has done this while makeadi ff. html. maintaining high average Graduate Record Exam­ ination scores and grade point averages for enter­ - Ellen]. Maycock ing students. The Department's Programs. The department AMS Associate Executive Director has built its programs with care and foresight. for Meetings and Professional Services Although these programs were developed in the

MARCH 2006 NOTICES OF THE AMS 353 context of the department's minority graduate stu­ Outreach and Transformation. The department dent initiative, these programs have always been has built ties with several institutions in mathe­ open to majority students. As a consequence the matics at the same time that it has provided lead­ quality of education has been raised for all students. ership in minority graduate education in Iowa: Some of the department's programs are: • Working together with the department of math­ • A three-week intensive Summer Institute for in­ ematics at Florida A&M University, the depart­ coming students ment built a partnership-the Alliance for the • Intensive faculty mentoring for all students Production of African American Ph.D.'s in the • Intensive formal peer mentoring for first-year Mathematical Sciences-with the departments of courses mathematics at Florida A&M University, Alabama • Intensive formal peer mentoring for Ph.D. Com­ A&M University, Jackson State University, and prehensive Exams Benedict College, as well as departments of • A new course, "Introduction to the Graduate mathematics, statistics, and testing and mea­ Program" surement at the Iowa Regents institutions. The Changes in the composition of the graduate stu­ goal of the Alliance is to provide for a seamless dent population have created a significant change transition for graduates of Alliance undergrad­ in the department's culture. Friendships and close uate institutions who wish to study mathemat­ working relationships have formed between mi­ ics and statistics at majority Ph.D. granting in­ nority students and their majority counterparts, stitutions. These efforts were supported, in part, and ethnic and gender inclusion has become the by a three-year grant from the NSF (National norm. This new environment is a successful re­ Science Foundation) Division of Mathematical cruiting tool, not just for underrepresented mi­ Sciences and are now supported by an EMSW21-MCTP nority graduate students but for all students and grant (Enhancing the Mathe­ matical Sciences Workforce also faculty. For example, 41 percent of the grad­ in the 21st Cen­ tury-Mentoring uate students in the department are now women. Through Critical Transition Success. Eight of the department's U.S. minor­ Points). • The department entered ity students have been awarded the Ph.D. since into a partnership with the University 1998. Three of these students earned their degrees of Puerto Rico-Mayaguez. This in 2004-5, roughly 10 percent of the total number partnership includes several other campuses of the University of doctoral degrees awarded to U.S. minority stu­ of Puerto Rico, notably, the cam­ pus at Humacao. dents nationally in mathematics in this period. • The department has a close working These Ph.D. recipients have taken a wide range of relationship with the Mathematics and Theoretical Biology In­ positions, ranging from tenure-track positions at stitute for undergraduates run by Carlos Castilla­ four-year schools to postdocs at Purdue University Chavez of Arizona State University. Eleven par­ and Los Alamos National Laboratory. The depart­ ticipants in this Research Experiences for ment expects that on average it will award Ph.D.'s Undergraduates program have enrolled as grad­ to three U.S. minority students each year for the uate students in the department. foreseeable future. In addition, retention rates have • The department spearheaded the successful increased for all students. For its efforts, in May grant application to the NSF's AGEP (Alliance for 2005, the department received the Presidential Graduate Education and the Professoriate) pro­ Award for Excellence in Science Mathematics and gram in support of increasing the number of doc­ Engineering Mentoring (PAESMEM) in a White House torates awarded to U.S. underrepresented mi­ award ceremony attended by department chair nority students in science, technology, David Manderscheid and other members of the de­ engineering, and mathematics graduate pro­ partment. grams at the three Iowa State Regents Universi­ Administration. As part of the department's ties: Iowa State University, the University of goal to institutionalize its minority graduate stu­ Iowa, and the University of Northern Iowa. dent initiative, the department has made perma­ • The department led the formation of the Heart­ nent changes in its administration. A first step in land Mathematics Partnership. The Partnership this direction was taken in 199 5 when the depart­ is a consortium of the mathematics departments ment created a new committee, the departmental at twelve area colleges and universities and the Minority Student Recruitment and Development department of mathematics at the University Committee (MSRDC), which oversees all aspects of of Iowa. The goal of the partnership is to increase the department's minority program. As the program the number of U.S. citizens who earn a Ph.D. de­ has developed, the department chair and the di­ gree in the mathematical sciences. rector of the AMCS program have been given new Program Cost. In addition to the grant support responsibilities as has the director of graduate mentioned above, the department also gets support studies. from the Sloan Foundation Minority Ph.D. Program.

354 NOTICES OF THE AMS VOLUME 53, NUMBER 3 The College of Liberal Arts and Sciences at the uni­ SIMU: Summer Institute in Mathematics for versity provides some funds for recruiting, and Undergraduates the Graduate College has provided some fellowship Program Name and Location. Summer Institute in support. The principal cost of the program, how­ Mathematics for Undergraduates (SIMU) at the Uni­ ever, is faculty time. versity of Puerto Rico-Humacao. Replicability. The department has given a great Program Goal. To increase the number of Lati­ deal of thought about how to replicate its success. nos and Native Americans earning graduate de­ Members of the faculty have lectured widely on the grees and pursuing careers in the mathematical sci­ program and are very willing to work with other ences. departments to help them develop successful pro­ Primary Audience. Hispanic/ Latina and Native grams tailored to their needs. The department of American, U.S citizen, and permanent resident un­ mathematics at the University of Iowa sees five dergraduates (especially juniors and seniors) in­ phases to successful replication: terested in pursuing a graduate degree in the math­ 1. A core of three or four senior faculty dedicated ematical sciences. to the program is necessary to start. There Number of Participants Served. 1998: 27. should also be the support of the department 1999-2002: 24 per year. Total number served by chair. The senior faculty should be willing to do the program from 1998-2002: 115. the intensive mentoring necessary. Program Description. SIMU was a six-week re­ 2. A critical mass of minority graduate students search program in the mathematical sciences for should be established. This can be accomplished undergraduate sophomores, juniors, and seniors by intensive recruiting and, preferably, fellow­ from across the U.S. and Puerto Rico. The program Puerto Rico­ ship support to attract students. Numerous fel­ took place at the University of Foundation lowship grant opportunities, both internal and Humacao and was a National Science (NSF) Research Experience for Undergraduates external, are available, and most administra­ (REU) site from 2000-2002; it also received signif­ tions are very willing to provide matching funds, icant funding from the National Security Agency if needed. (NSA). SIMU was the largest (serving twenty-four 3. The department should have reasonable expec­ students per year) NSF REU program in mathe­ tations as there will no doubt be growing pains matics. as the program is established. SIMU was designed for Chicana/ Latino and Na­ 4. Cultural change within the department should tive American undergraduates who take place. The department should be willing to 1. had completed at least two years of university­ look at, over time, its practices in graduate ed­ level mathematics courses (e.g., Calculus I-III, ucation and compare them with best practices. Differential Equations, Linear Algebra, etc.); 5. The final stage is institutionalization. In this 2. wanted to conduct undergraduate research in the stage leaders of the effort reach out to other fac­ mathematical sciences; and ulty once the program is working. The whole de­ 3. were interested in pursuing a graduate degree partment is transformed and then reaches out in the mathematical sciences. to other departments. Each of the twenty-four students who partici­ pated in the 2002 SIMU program received a Contacts for further information: US$2,200 stipend; round-trip travel to Humacao, PR; and room and board for the duration of the six­ Phil Kutzko week program. Director, Iowa AGEP; Director, HBCU Alliance During the 2002 SIMU, each student Department of Mathematics 1. participated in a mathematical seminar under University of Iowa the direction of Ricardo Cortez or Victor Moll; Email: [email protected] . uiowa.edu 2. participated in a computational laboratory that was intertwined with the seminar; Eugene Madison 3. completed an undergraduate research project Chair MSRDC designed by the seminar leader and done in col­ Department of Mathematics laboration with other SIMU students; University of Iowa 4. gave a colloquium talk and wrote a technical re­ Email:[email protected] port on his or her research project; 5. attended a series of colloquium talks given by David Manderscheid mathematicians and scientists who are leading Department Chair researchers in their fields; Department of Mathematics 6. attended workshops whose focus was to de­ University of Iowa velop skills and techniques needed for research Email:[email protected] careers in the mathematical sciences;

MARCH 2006 NOTICES OF THE AMS 355 • Eight were accepted into mathematics, educa­ AMS Committee on the Profession tion, or computer science master's programs James G. Arthur (AMS President), University of Toronto • Three were accepted into other graduate pro- James W. Cannon, Brigham Young University grams Ruth M. Charney, Brandeis University • Three have completed a Ph.D. in mathematics Robert ]. Daverman (AMS Secretary), University of Tennessee • One has completed a Ph.D. in physics John H. Ewing (AMS Executive Director), AMS • Ten have finished master's degrees in math-re­ Carolyn S. Gordon, Dartmouth College lated programs Anne Greenbaum, University of Washington, Seattle • Three are still in mathematics Ph.D. programs Jim E. Haste, Pitzer College • Two are still in other Ph.D. programs Brian H. Marcus, University of British Columbia SIMU 1999 Helen Moore, American Institute of Mathematics • Twenty-four students participated Lior Pachter, University of California, Berkeley • 70 percent had no prior research experience Kimberly R. Pearson, Valparaiso University • Thirteen were accepted into Ph.D. programs in Javier Rojo, Rice University the mathematical sciences Claude L. Schochet, Wayne State University • Four were accepted into engineering, mathe­ CarolS. Wood (chair), Wesleyan University matics, or education master's programs • Seven have finished master's degrees in math­ ematics 7. learned techniques to maximize a student's like­ • Seven are still in mathematics Ph.D. programs lihood of admission to graduate programs as • Two are still in master's degree programs well as the likelihood of winning fellowships; and SIMU 2000 8. participated in cultural and recreational activi­ • Twenty-four students participated ties in Puerto Rico. • 75 percent had no prior research experience After SIMU, each student had: • Ten were accepted into mathematics Ph.D. pro­ 1. an opportunity to attend the conference of the grams Society for the Advancement of Chicanos and Na­ • Six were accepted into engineering, mathemat­ tive Americans in Science (SACNAS), the Joint ics, or education master's programs Mathematics Meetings, and other forums where • One was accepted into another graduate pro­ they were able to present their undergraduate gram research; • Four have finished master's degrees in mathe­ 2. the opportunity to continue the professor­ matics student and peer relationships developed with • Nine are still in mathematics Ph.D. programs their seminar leader and with other professional • Two are still in master's degree programs mathematicians and students involved in SIMU. SIMU 2001 Years of Operation and Current Status. The • Twenty-four students participated; one has not program operated 1998-2002. The option of yet graduated restarting the program is under consideration. • Over 70 percent hadno prior research experi­ Documentable Effectiveness. The figures below ence are as of 2005 and are not entirely complete due • Sixteen were accepted into Ph.D. programs in the to a few students not responding to questionnaires. mathematical sciences Please note that the total number of students served • Three were accepted into master's programs in by the program, l15, is not the same as the total mathematics, engineering, or computer science number enrolled for all five years of the program • Two have finished master's degrees in mathe­ because some of the students attended SIMU more matics than once. Also, the numbers for the same cohort • Fourteen are still in mathematics Ph.D. pro­ of students may not always agree because some stu­ grams dents stopped responding to the follow-up ques­ • Two are still in master's degree programs tionnaires, so that it might be known that a student SIMU 2002 entered a Ph.D. program, but the student's cur­ • Twenty-four students participated rent status is unknown. • Over 78 percent had no prior research experi­ SIMU 1998 ence • Twenty-seven students participated • Nineteen of twenty-four presented posters on • Over 70 percent had no prior research experi­ their research at the Mathematical Association ence of America (MAA) undergraduate poster ses­ • Twelve were accepted into mathematics Ph.D. sion at the Baltimore Joint Mathematics Meetings programs in 2003

356 NOTICES OF THE AMS VOLUME 53, NUMBER 3 • Twelve were accepted into Ph.D. programs in mathematics and related areas • Two were accepted into master's programs in mathematics, engineering, or education Other Evidence of Success. Post-program sur­ The Mathematical Moments program veys of participants showed that the students' ed­ is a series of illustrated "snapshots" ucational experience was positively affected by designed to promote appreciation and SIMU. For example, 78 percent of the students had understanding of the role mathematics not worked on undergraduate research prior to plays in science, nature, technology, and SIMU but 96 percent of the students wanted to human culture. work on undergraduate research after SIMU; 96 Download these and other percent of the students said that SIMU was either Mathematical Moments pdf files at "successful" or "very successful" in familiarizing www.ams.org/mathmoments. them with mathematics experimentation and research protocols and techniques; 92 percent of the students said that SIMU "increased signifi­ Compressing Data cantly" or "increased" their desire to pursue a grad­ uate education in mathematics or science. Through digltiution. films lh,,t require I 0.000 feet of mpe now fit on~ disk less than five Inches In diame ter. An Program Cost. Approximately US$9,000 per stu­ important part o( dlgkiution is dat;t tompreuion. whi' h involves converting a large file to a sm~ ll erversion , frorn dent per year for the six-week summer program and which dle original (or a dose approximation) can be recre.lted. Line;~ ralgebra, probabi l i ty.graph theory and abstract algebra are among the areas of m.u hematics at· the foundation o f the post-program experiences. various compression algorithms that make n1odern te~;hnolo­ Replicability. The program model is a well­ gies such as DVOs. HOTV and large dal.1ba$es. possible. defined one that could reasonably be replicated for different target audiences. The SIMU co-directors believe that the program model is especially suited to groups that are underrepresented in the math­ ematical sciences. Indeed, there is a program in the works at California State Polytechnic Univer­ sity-Pomona that will borrow from the SIMU model, as has the MAA's National Research Experiences for Undergraduates program. Issues of which Replicators Should Be Aware. www.am s, or g i m a ch moments The commitment to the program by the staff needs Speech to be very high in order for the program to be ef­ • Recognizing fective. In particular, potential program organizers • Compressing Data should be aware of the large amount of time and • Being a Better Sport effort required to organize and run a program such as SIMU. • Targeting Tumors Contacts for Further Information. See the SIMU • Defeating Disease website (http: I /www. uprh. edu/ - si mu/) or con­ • Getting Results on the Web tact the co-directors at the SIMU email address. The co-directors can also be contacted directly: • Designing Aircraft Herbert A. Medina • Eye-dentifying Yourself Department of Mathematics • Enhancing Your Image Loyola Marymount University Los Angeles, CA • Simulating Galaxies Email: hmedi na@l mu. edu • Revealing Nature's Secrets • Securing Internet Communication Ivelisse Rubio Department of Mathematics • Making Movies Come Alive University of Puerto Rico-Humacao • Listening to Music Email: [email protected] • Making Votes Count • Forecasting Weather (~~ AMS ~~~AN MATHEMATICAL SOCIETY

MARCH 2006 NOTICES OF THE AMS 357 Mathematics People

kernel function is smooth on D x D- 6.. In 1974 Feffer­ Hirachi Receives Bergman Prize man established an asymptotic expansion for the Bergman KENGo HlRAcHI of the University of Tokyo has been awarded kernel K on strongly pseudoconvex domains. Let r denote the 2006 Stefan Bergman Prize. Established in 1988, the a smooth defining function forD. There are smooth func­ prize recognizes mathematical accomplishments in the tions and 1./J on D such that K = r - n- l + f./Jlog(r). areas of research in which Stefan Bergman worked. The In 1979 Fefferman described an analogy between CR prize consists of one year's income from the prize fund. geometry and Riemannian geometry in which the Bergman Currently this income is about US$25,000 per year. kernel is analogous to the heat kernel. Since then many au­ The previous Bergman Prize winners are: David W. thors have studied the relationship between the singular­ Catlin (1989), Steven R. Bell and Ewa ligocka (1991), Charles ities of the Bergman kernel and CR geometry, relating the Fefferman (1992), Yum Tong Siu (1993), John Erik Forn~ss complex Monge-Ampere equation to the invariant theory (1994), Harold P. Boas and Emil J. Straube (1995), David E. developed by Chern-Moser-Tanaka, thereby developing Barrett and Michael Christ (1997), John P. D'Angelo (1999), the analogy. Bailey-Eastwood-Graham expressed the sin­ Masatake Kuranishi (2000), Laszlo Lempert and Sidney gularity of r - n- l in terms of CR invariants of the bound­ Webster (2001), M. Salah Baouendi and Linda Preiss ary of D, but the coefficient 1./J of the logarithmic term has Rothschild (2003), Joseph ]. Kohn (2004), and Elias M. remained more mysterious. Stein (2005). On the selection committee for the 2005 One of Hirachi's striking contributions is an expression prize were Michael Christ, John P. D'Angelo (chair), and for the singularity of f./Jlog(r) in Charles Fefferman. terms of so-called Weyl func­ Citation tionals of weight k. For weight k, with k .:s; n+2, he proved The Bergman prize for 2006 is awarded to Kengo Hirachi that all Weyl invariants of of the University of Tokyo for his deep work on the sin­ weight k are CR invariants and gularities of the Bergman and Szeg6 kernels and their re­ vice versa. lationship to CR geometry. Hirachi's work employs a wide Hirachi has also done im­ range of tools in geometry and analysis, including several Kengo Hirachi portant work relating the complex variables, the complex Monge-Ampere equation, Bergman and Szeg6 kernels. microlocal analysis, parabolic invariant theory, explicit The Szeg6 kernel is analogous to the Bergman kernel; its computations, and computer algebra packages. domain is square-integrable functions on the boundary with In a paper in the Annals of Mathematics (2000) Hirachi respect to some smooth surface measure dm, and it pro­ constructed CR invariants of strongly pseudoconvex bound­ jects to boundary values of holomorphic functions. There aries via a deep study of the logarithmic singularity of the is an asymptotic expansion for the Szeg6 kernel on strongly Bergman kernel. He has proved various results linking the pseudoconvex domains as well, where the exponent - n - 1 Bergman and Szeg6 kernels, and he has made significant aboveis replaced by - n . progress to a program in which the Bergman kernel func­ Consider fbv f./Jdm, where 1./J is now the coefficient of tion plays a role analogous to the heat kernel of Rie­ the log term in the formula for the Szeg6 kernel. Hirachi mannian geometry. proved that this integral is independent of the choice of The Bergman kernel function of a bounded domain D smooth measure dm, and that it is invariant under smooth in complex Euclidean space en is the integral kernel (strongly pseudoconvex) deformations of the domain D. for the orthogonal projection from L 2 (D) to the closed Any smooth convex domain whose boundary has nowhere subspace of holomorphic functions in L 2(D). Let D denote vanishing Gauss curvature can be deformed smoothly to the closure of D, and let 6. denote the boundary diagonal. a ball, whose Szeg6 kernel has no logarithmic terms. When D is strongly pseudoconvex (more generally when Hirachi's theorem therefore implies that fbD f./Jdm = 0, the a-Neumann problem is subelliptic) the Bergman and consequently 1./J must vanish somewhere. Hirachi

358 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics People shows in many other cases (e.g., in dimension two) that Society assisted Wells Fargo in interpreting the terms of fbv tJfdm = 0. the will to assure sufficient breadth in the mathematical It is often useful to provide links between the Bergman areas in which the prize may be given. Awards are made and Szeg6 kernels. Hirachi introduced a meromorphic every one or two years in the following areas: (1) the family of kernel functions including both the Bergman theory of the kernel function and its applications in real and Szeg6 kernels as special values, thereby relating their and complex analysis and (2) function-theoretic methods asymptotic expansions. Hirachi also formulated and proved in the theory of partial differential equations of elliptic type a transformation law for the Szeg6 projection on abstract with attention to Bergman's operator method. strongly pseudoconvex CR manifolds, generalizing work of Fefferman for boundaries of domains. -Allyn jackson In addition, Hirachi modified the microlocal methods of Kashiwara and used results of Webster to study the Bergman kernel on real ellipsoids. Hirachi proved, for ex­ Mathematics Wunderkind Wins ample, that for all small perturbations (as real ellipsoids) of the ball, the log term must be present (whereas both Siemens-Westinghouse kernels for the ball have no log term). With Komatsu, Hirachi developed a successful theory of local Sobolev­ Competition Bergman kernels (where one projects to the holomorphic "I don't think I have ever met anyone with such enthusi­ functions in various Sobolev spaces). Hirachi and Komatsu asm and raw talent for math," said Peter Ebenfelt of the used microlocal analysis to compute various universal University of California, San Diego. He was talking about constants in CR geometry. With Fefferman, Hirachi showed Michael Viscardi, an extraordinarily talented young man who that the coefficient of the log term in an asymptotic ex­ in December 2005 was named the top individual winner pansion of the Szeg6 kernel is a multiple of the CR cur­ in the Siemens-Westinghouse competition. Viscardi, sixteen vature. Also in that paper the CR curvature is expressed years old, wrote a paper on the Dirichlet problem that net­ in terms of the Fefferman-Graham metric. ted him the top prize of a US$100,000 In summary, Hirachi has contributed new and basic college scholarship. The paper has facts on the Bergman and Szeg6 kernels for strongly pseudo­ already been accepted for publica­ convex domains, and he has related the singularities of tion. In addition to excelling in math­ these kernels to polynomials in the Chern-Moser-Tanaka ematics, Viscardi is a talented musi­ invariants. His work establishes deep and important con­ cian and has won several music nections between complex analysis and CR geometry. prizes. Of the nineteen students winning Biographical Sketch scholarships in the prestigious Born on November 30, 1964, Kengo Hirachi received his Siemens-Westinghouse competition B.S. (1987), his M.S. (1989), and his Dr.Sci. (1994) from Osaka (six went to individuals and six to University. The advisor for his doctoral dissertation was teams), Viscardi was the only one Gen Komatsu. After holding a position as a lecturer at Michael Viscardi with a mathematics project. His suc- Osaka University, Hirachi took his present position as an cess was widely reported in the local associate professor at the University of Tokyo. He has and national media, including an interview on National Pub­ been a visitor at the Mathematical Sciences Research In­ lic Radio and an appearance on ABC World News Tonight stitute in Berkeley, at the Erwin Schrodinger Institute for as "Person of the Week". Mathematical Physics in Vienna, and at Princeton Univer­ Viscardi, whose father is a software engineer and whose sity. His honors include the Take be Senior Prize (1999) and mother holds a Ph.D. in neuroscience, is home-schooled. the Geometry Prize (2003), both given by the Mathemati­ He studied high school mathematics on his own, using cal Society of Japan. standard textbooks. When he reached eighth grade, he started taking mathematics courses at UCSD. His first About the Prize course there was calculus, taught by Ebenfelt, who re­ The Bergman Prize honors the memory of Stefan Bergman, called that Viscardi, then thirteen years old, was the best best known for his research in several complex variables, student in the class. He would often come to Ebenfelt's as well as the Bergman projection and the Bergman ker­ office hours, not so much to discuss the course material, nel function that bear his name. A native of Poland, he which he could study on his own, but to talk about taught at Stanford University for many years and died mathematics problems he found in the American Mathe­ in 1977 at the age of 82. He was an AMS member for matical Monthly. thirty-five years. When his wife died, the terms of her will The next year, Viscardi took an analysis course intended stipulated that funds should go toward a special prize in for junior and senior mathematics majors preparing for her husband's honor. doctoral work in mathematics. UCSD faculty member Linda The AMS was asked by Wells Fargo Bank of California, the managers of the Bergman Trust, to assemble a com­ Photo of Michael Viscardi is courtesy of the Siemens Foundation. mittee to select recipients of the prize. In addition, the Photograph by Peg Skorpinski.

MARcH 2006 NOTICES OF THE AMS 359 Mathematics People

Rothschild taught the course. "His performance was truly given a function f continuous in the plane and a bounded remarkable, and I would have been very happy to recom­ simply connected domain U. The integral off over U is zero, mend him then for a top Ph.D. program in mathematics," as are all integrals over rigid motions of U. Is f identically she recalled. The following fall he took a graduate class zero? The problem can be stated in other, equivalent, in complex analysis from faculty member Salah Baouendi, ways, in particular as a partial differential equations prob­ who also raved about Viscardi's performance, which was lem. Ebenfelt's and Viscardi's work is still in the very early at the level of the top graduate students in the course. "He stages, and it is not clear yet where it will lead. is simply in a class by himself, truly amazing," Baouendi When he is not doing mathematics, Viscardi loves to play commented. "I am convinced that he has an enormously music. If he does more of one and less of the other, he starts bright future ahead of him." Baouendi's course awakened in Viscardi a deep inter­ to feel "a little bit funny," he said. "I really need both in est in complex analysis, so much so that he decided to try order to feel somewhat balanced. But math and music are to do research in this area and asked Ebenfelt for a topic. in fact very similar-both are beautiful and elegant." He Ebenfelt proposed the topic that led to Viscardi's winning is an accomplished pianist and violinist, a composer, and paper on the Dirichlet problem for the Laplace operator concertmaster of the San Diego Youth Symphony. He is also in a smoothly bounded, simply connected planar domain the first violinist of the San Diego Youth Symphony String U. As Ebenfelt explained it, the problem is the following. Quartet. Among his honors in music are the first place prize Given a continuous data function h on the boundary au, in the 2005 H. B. Goodlin Foundation Scholarship Com­ find a harmonic function in the interior whose boundary petition in the Senior Piano Division, and the first place values are h. This problem is already well understood, prize in the 2005 Music Teachers' Association of Califor­ but Viscardi considered the case where the data function nia Concerto Competition, for violin. h is the restriction to aU of a rational function (of z, that Viscardi has applied to Harvard, the Massachusetts In­ is, a quotient of two holomorphic polynomials) and asked stitute of Technology, and Caltech. At the time of this for what U is it the case that all solutions are rational writing he had been accepted at Harvard, and it seems in­ harmonic functions? His paper completely characterizes evitable he will be accepted anywhere he applies. As far all such domains both geometrically and function­ as future career plans go, Viscardi said he would like to theoretically. The paper, which lists Ebenfelt and Viscardi become a mathematician, "while doing as much music as as joint authors, has been accepted by the journal Com­ putational Methods and Function Theory. Ebenfelt noted I can." Clearly this is a young man with a brilliant future that he is not usually a coauthor on his student's papers ahead of him. According to Baouendi, the work for which but that Viscardi insisted on it. Viscardi won the Siemens-Westinghouse competition could "Michael is a wonderful person," Ebenfelt commented. easily be a Ph.D. dissertation. "What more can I say?" "He is extremely pleasant and very enthusiastic about Baouendi declared. "Sixteen-year-old Michael is already math. It is a real treat to discuss math with him. It's not performing as a professional mathematician of high very often one has a student who gets so excited about a caliber. This is indeed truly extraordinary." theorem or lemma that his face practically glows." If his UCSD professors are effusive about his prodigious math­ -Allyn jackson ematical talent, they are equally charmed by his winning personality. According to Rothschild, Viscardi is "extremely likeable, sociable, and modest, with a great sense of Raman Awarded TWAS Prize humor." And what is he up to nowadays? Viscardi replied in an PARJMALA RAMAN of the Tata Institute of Fundamental Re­ email message that captures his astounding capacity for search has been awarded the 2006 prize of the Academy mathematics in the winsome voice of a sixteen-year-old: of Sciences for the Developing World (formerly the Third "When I started taking Calculus with Professor Ebenfelt World Academy of Science, TWAS). Raman was awarded back in 8th grade, I learned that he wrote a book with Pro­ the prize in mathematics "for her work on the quadratic fessor Baouendi and Professor Rothschild titled Real Sub­ analogue of Serre's conjecture, the triviality of principal manifolds in Complex Space and Their Mappings. I thought homogeneous spaces of classical groups over fields of that was a pretty cool title, and I also couldn't get past the cohomological dimension 2 and the u-invariant of p-adic first page! It's part of Differential Geometry, so my tem­ porary goal in math was to eventually take that subject. function fields." That's exactly what I'm taking now, which is very nice! I The Academy of Science for the Developing World an­ can now read a few more pages of the book." In fall 2005 nually awards prizes of US$1 0,000 each to scientists from he was taking graduate courses in differential geometry developing countries who have made outstanding contri­ and analytic number theory. butions in the fields of agricultural sciences, biology, Viscardi is also continuing to do joint research with chemistry, earth sciences, engineering sciences, mathe­ Ebenfelt, focusing on the Pompeiu problem, which was orig­ matics, medical sciences, and physics. inally proposed in the 1920s by D. Pompeiu as a problem in integral geometry. The problem is the following. One is -From a TWAS announcement

360 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics People

South Wales; Christiane Lemieux, University of Calgary; Polterovich and Tsai Awarded Peter Mathe, Weierstrass Institute for Applied Analysis Andre Aisenstadt Prize and Stochastics; Joseph F. Traub, Columbia University; and Hemyk Wozniakowski, Columbia University and Uni­ IOSIF POLTEROVICH of the University of Montreal and TAI­ versity of Warsaw. PENG TsAI of the University of British Columbia are the re­ cipients of the 2006 Andre Aisenstadt Prize of the Centre -joseph Traub, Columbia University de Recherches Mathematiques (CRM) of the University of Montreal. Polterovich was honored for his work in geo­ metric spectral theory, and Tsai was selected for his work Otto Receives Leibniz Prize in nonlinear partial differential equations. The Andre Aisenstadt Mathematics Prize consists of The Deutsche Forschungsgemeinschaft (DFG, German Re­ CA$3,000 and a medal. It is awarded to recognize talented search Foundation) has announced the winners of its 2006 young Canadian mathematicians in pure and applied math­ Gottfried Wilhelm Leibniz Prize. The DFG Grants Com­ ematics who have held a Ph.D. for no longer than seven mittee named eleven scientists and academics as recipi­ years. ents of the most valuable research prize in Germany. The award of up to 1.55 million € (approximately US$1.8 mil­ -From a CRM announcement lion) funds research work over a five-year period and can be used flexibly by the prizewinners. One mathematician was among the Leibniz prizewin­ Montalban Awarded ASL Sacks ners: FELIX Orro of the Universitat Bonn, whose area is analysis of partial differential equations. He received a grant Prize of 1. 55 million €. The research conducted by mathematician Felix Otto ANTONIO MONTALBAN of the University of Chicago has been focuses on the analysis of pattern-forming processes that awarded the 2005 Sacks Prize of the Association for Sym­ occur frequently in models used to describe a variety of bolic Logic (ASL). The prize is awarded annually for the physical phenomena and often have a multiscale nature. most outstanding doctoral dissertation in mathematical The mathematical tools he uses for his work include logic. According to the prize citation, Montalban's thesis advanced analytical methods and numerical simulation. "contains deep and major contributions to an impres­ He is particularly interested in micromagnetism, which is sively broad array of areas in logic, including computability especially important for the development of new data theory, reverse mathematics, and effective mathematics." storage technology, as well as coarsening and growth Montalban received his Ph.D. in 2005 from Cornell Uni­ processes, which are of particular importance in material versity. science. The Sacks Prize was established to honor Gerald Sacks After obtaining his doctorate from the Universitat Bonn, of Harvard University and the Massachusetts Institute of Otto first went to the USA, where he was a visiting scholar Technology for his contribution to mathematical logic. at the Courant Institute at New York University and then The prize consists of a cash award and five years' free mem­ went to the University of California at Santa Barbara. He bership in the ASL. returned to Bonn in 1999 to take up a chair at the Insti­ tute for Applied Mathematics. Amongst the national and -From an ASL announcement international prizes he has received are a Sloan Research Fellowship and the Max Planck Research Award.

Pillichshammer Wins -From a DFG news release Information-Based Complexity Young Researcher Award Rhodes Scholarships Awarded FRIEDRICH PILLICHSHAMMER of Johannes Kepler University in Four students in mathematical sciences are among the Linz, Austria, has been awarded the Information-Based thirty-two American men and women who have been Complexity Young Researcher Award for 2005. The award selected as Rhodes Scholars by the Rhodes Scholarship recognizes significant contributions to information-based Trust. The Rhodes Scholars were chosen from 903 appli­ complexity by researchers who have not reached their cants who were endorsed by 333 colleges and universities thirty-fifth birthdays by September 30 of the year of the in a nationwide competition. The names and brief award. The prize consists of US$1 ,000 and a plaque. The biographies of the mathematics scholars follow. award will be presented at the Schloss Dagstuhl Seminar ALISON CROCKER of Poughkeepsie, New York, majors in on Continuous Algorithms and Complexity in September physics and mathematics at Dartmouth College. She has 2006. The prize committee consisted of Josef Dick, Uni­ done summer research at the Cold Regions Research versity of New South Wales; Frances Kuo, University of New and Engineering Laboratory, where she built and tested a

MARCH 2006 NOTICES OF THE AMS 361 Mathematics People proof-of-concept ice detection system for aircraft. She is awards by the Carnegie Foundation for the Advancement an All-American cross-country skier and a rower on the U.S. of Teaching and the Council for Advancement and Support junior national rowing team. She plans to do a doctorate of Education (CASE), which cosponsor the awards. The in astrophysics at Oxford. Professor of the Year Awards are intended to reward out­ ADAM D. CHANDLER of Burlington, North Carolina, is a standing professors for their dedication to teaching, their senior at Duke University majoring in mathematics. He has commitment to students, and their innovative instruc­ conducted numerous research projects in applied com­ tional methods. putational mathematics, including work on computational The State Professors of the Year Award Program selects quantum chemistry, molecular evolution, and traffic mod­ outstanding educators in all fifty states, the District of eling. As a research intern at the National Security Agency, Columbia, Guam, Puerto Rico, and the U.S. Virgin Islands. he focused on problems in cryptography. He is a Goldwa­ Winners receive personalized award certificates as well as ter and Byrd scholar and managing editor of the journal national and local media recognition. State and national of Young Investigators. He is a cellist and president of the winners are chosen on the basis of their dedication to Duke Symphony Orchestra, and he has volunteered at an undergraduate teaching, determined by excellence in the orphanage in Tanzania. He plans to study for an M.Sc. in following four areas: impact on and involvement with applied and computational mathematics at Oxford. undergraduate students; scholarly approach to teaching RAHuL SATIJA of Potomac, Maryland, is a senior at Duke and learning; contributions to undergraduate education in University who majors in biology and music and minors the institution, community, and profession; and support in mathematics. He has conducted research in bioinfor­ from colleagues and current and former undergraduate matics and has won a Faculty Scholar Award from Duke students. University and a Meritorious Solution Award at the Inter­ national Mathematical Contest in Modeling. He is the -From a Carnegie Foundation announcement recipient of a Goldwater Scholarship and of Duke's only music performance scholarship. He is concertmaster for the Duke Symphony Orchestra and first violinist of a stu­ Szpiro a Finalist for Descartes dent string quartet; he also teaches violin to inner-city Prize youths and plays recreational tennis. He plans to work for a D.Phil. in bioinformatics at Oxford. GEORGE SZPIRO, a reporter and columnist for the Swiss news­ ELIANA HECHTER of Phoenix, Arizona, is an eighteen­ paper Neue Zurcher Zeitung, was named a finalist for the year-old senior mathematics major at the University of Descartes Prize for Science Communication, given by the Washington. She has done research on the neurobiologi­ European Union. Being named a finalist carries a cash cal basis of behavior in marine fauna and has been sup­ prize of 5,000 € (about US$6,000). Szpiro was cited for his ported through a Vertical Integration of Research and series of stories "Maths for Sunday Morning-50 Stories Education (VIGRE) grant in mathematical sciences from on Mathematics and Science". He writes a monthly math­ the National Science Foundation. She works as a research ematics column that appears on Sundays in the Neue assistant at the Center for Cell Dynamics and as a teach­ Zurcher Zeitung. Szpiro is also the author of the book ing assistant for advanced calculus. She is a Goldwater Kepler's Conjecture: How Some of the Greatest Minds in Scholar, a creative writer, and a long-distance runner. She History Helped Solve One of the Oldest Math Problems plans to study for a D.Phil. in mathematics at Oxford. in the World (John Wiley and Sons, January 2003), which Rhodes Scholarships provide two or three years of study was reviewed in the January 2005 issue of the Notices. at the University of Oxford in England. The value of the His latest book, The Secret Lives of Numbers: 50 Easy Rhodes Scholarship varies depending on the academic Pieces on How Mathematicians Work and Think, will appear field, the degree (B.A., master's, doctoral), and the Oxford in spring 2006. college chosen. The Rhodes Trust pays all college and university fees and provides a stipend to cover students' -Allyn jackson necessary expenses while they are in residence in Oxford, as well as during vacations, and transportation to and Authors Receive Chauvenet from England. The total value averages approximately US$40,000 per year. Prize for Notices Article -From a Rhodes Scholarship Trust announcement GONTER ZIEGLER and FLORIAN PFENDER have been honored with the 2006 Chauvenet Prize of the Mathematical Association of America (MAA) for an article that appeared in the Notices. Their article, "Kissing numbers, sphere packings, Professor of the Year Awards and some unexpected proofs", appeared in the September Announced 2004 issue. The Notices editors extend their congratula­ tions to the authors. Additional details about the prize will M. VALI SIADAT of Richard ]. Daley College in Chicago, Illi­ appear in the May 2006 Notices. nois, and JIM COYKENDALL of North Dakota State University have been chosen to receive State Professor of the Year -Allyn jackson

362 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics Opportunities

sense of fairness, good judgment, and a high degree of AMS Congressional Fellowship personal integrity. The AMS, in conjunction with the American Association Applicants should have a Ph.D. or equivalent training in for the Advancement of Science (AAAS), will sponsor a Con­ a field of the mathematical sciences, a broad knowledge gressional Fellow from September 2006 through August of one of the relevant disciplinary areas of the DMS, some 2007. The Fellow will spend the year working on the staff administrative experience, a knowledge of the general sci­ of a member of Congress or a congressional committee as entific community, skill in written communication and a special legislative assistant in legislative and policy areas preparation of technical reports, an ability to communicate orally, and several years of successful independent re­ that require scientific and technical input. The fellowship search normally expected of the academic rank of associ­ is designed to provide a unique public policy learning ex­ ate professor or higher. Skills in multidisciplinary research perience, to demonstrate the value of science and gov­ are highly desirable. ernment interaction, and to bring a technical background Qualified individuals who are women, ethnic/racial mi­ and external perspective to the decision-making process norities, and/or persons with disabilities are strongly in Congress. Deadline for applications is March 31, 2006. urged to apply. No person shall be discriminated against Applicants should have received a Ph.D. or an equiva­ on the basis of race, color, religion, sex, national origin, lent doctoral-level degree in the mathematical sciences age, or disability in hiring by the NSF. by the application deadline. For further information see Program director positions recruited under this the webpage http://www.ams.org/government/ announcement may be filled under one of the following congressfell owann. html or contact the AMS Washing­ appointment options: ton office at 202-588-1100; email: amsdc@ams. org. Visiting Scientist Appointment Appointment to this position will be made under the Excepted Authority of -AMS announcement the NSF Act. Visiting scientists are on unpaid leave status from their home institutions and appointed to NSF's pay­ roll as federal employees. NSF withholds Social Security Program Director Positions at taxes and pays the home institution's contributions to maintain retirement and fringe benefits (i.e., health bene­ NSF fits and life insurance) either directly to the home insti­ tution or to the carrier. Appointments are usually made The Division of Mathematical Sciences (DMS) announces for up to one year and may be extended for an additional a nationwide search for a number of program director year by mutual agreement. positions at the National Science Foundation (NSF). Intergovernmental Personnel Act (IPA) Assignment In­ NSF Program Directors bear the primary responsibility dividuals eligible for an IPA assignment with a federal for carrying out the NSF's overall mission: to support in­ agency include employees of state and local government novative and merit-reviewed activities in basic research and agencies or institutions of higher education, Indian tribal education that contribute to the nation's technical strength, governments, and other eligible organizations in instances security, and welfare. To discharge this responsibility re­ in which such assignments would be of mutual benefit quires not only knowledge in the appropriate disciplines to the organizations involved. Initial assignments under but also a commitment to high standards, a considerable IP A provisions may be made for a period of up to two years, breadth of interest and receptivity to new ideas, a strong with a possible extension for up to an additional two-year

MARCH 2006 NOTICES OF THE AMS 363 Mathematics Opportunities period. The individual remains an employee of the home institution, and NSF provides funding toward the assignee's International Mathematics salary and benefits. Initial IPA assignments are made for Competition for University a one-year period and may be extended by mutual agree­ ment. Students Temporary Excepted Service Appointment Appointment The Thirteenth International Mathematics Competition to this position will be made under the Excepted Authority for University Students (IMC) will be held July 20-26, 2006, of the NSF Act. Candidates who do not have civil service at Odessa I.I. Mechnikov National University in Odessa, status or reinstatement eligibility will not obtain civil Ukraine. The competition is coorganized by University service status if selected. Candidates currently in the College London and the Odessa I.I. Mechnikov National competitive service will be required to waive competitive University, with sponsorship from D. E. Shaw & Co. civil service rights if selected. Usual civil service benefits Participating universities are invited to send several (retirement, health benefits, life insurance) are applicable students and one teacher each. Individual students are for appointments of more than one year. Temporary welcome. The competition is intended for students appointments may not exceed three years. completing their first, second, third, or fourth years of For additional information on NSF's rotational pro­ university education and will consist of two sessions of grams, see "Programs for Scientists, Engineers and five hours each. Problems will be taken from the fields Educators" on the NSF website at http: I lwww. nsf. gov I of algebra, analysis (real and complex), and combinatorics. aboutlcareer_opps. The working language will be English. Applicants should send a letter of interest and vita to Deadline for registration is May 31, 2006. For more Deborah F. Lockhart, Executive Officer, Division of Mathe­ complete information, see the website http: I lwww. matical Sciences, National Science Foundation, 4201 Wilson i me-math. org or contact John E. Jayne, Department of Boulevard, Suite 1025, Arlington, VA 22230; phone: 703- Mathematics, University College London, Gower Street, 292-8870; fax: 703-292-9032; email: dl ockhart@nsf. gov. London WClE 6BT, United Kingdom; telephone +44-20-76 79 NSF is an Equal Opportunity Employer committed to em­ 7322;fax+44-20-7419 2812; email: j. jayne@imc-math. org. ploying a highly qualified staff that reflects the diversity If you would like a copy of the competition poster, please of our nation. This announcement can also be found at send your request with postal address to j. jayne@ http:llwww.nsf.govlpublicationslpub_summ. i me-math. o rg. jsp?ods_key=dms060L -From an IMC announcement - NSF announcement Call for Nominations for Summer Program for Women Information-Based Complexity Undergraduates Prize The 2006 Summer Program for Women in Mathematics The Prize for Achievement in Information-Based Com­ (SPWM) will take place at George Washington University in plexity is awarded every year for work done in a single year, Washington, DC, from July 1 to August 5, 2006. This is a in a number of years, or over a lifetime. The work can be five-week intensive program for mathematically talented published in any journal, in a number of journals, or in undergraduate women who are completing their junior monograph form. years and may be contemplating graduate study in math­ The prize consists of US$3,000 and a plaque. The ematical sciences. Goals of this program are to commu­ deadline for nominations is March 31, 2006. However, a nicate an enthusiasm for mathematics, to develop research person does not need to be nominated to win the award. skills, to cultivate mathematical self-confidence and in­ For more information, contact Joseph Traub at t raub@cs. dependence, and to promote success in graduate school. columbia. edu. Sixteen women will be selected. Each will receive a travel allowance, campus room and board, and a stipend of -Joseph Traub, Columbia University US$1,500. The deadline for applications to be received is March 1, 2006. Early applications are encouraged. For further information contact the director, Murli M. Gupta, Project NExT: New Experiences email: mmg@gwu. edu; telephone: 202-994-4857; or visit the program's website at http: I lwww. gwu. edul-spwml. in Teaching Application material may be printed from the website. Project NExT (New Experiences in Teaching) is a profes­ sional development program for new and recent Ph.D.s in -Murli M. Gupta, George Washington University the mathematical sciences (including pure and applied

364 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics Opportunities

mathematics, statistics, operations research, and mathe­ matics education). It addresses all aspects of an academic career: improving the teaching and learning of mathe­ matics, engaging in research and scholarship, and partic­ ipating in professional activities. It also provides the par­ ticipants with a network of peers and mentors as they assume these responsibilities. Each year, about sixty fac­ ulty members from colleges and universities throughout the country are selected to participate in a workshop pre­ ceding the Mathematical Association of America (MAA) summer meeting, in activities during the summer MAA meetings and the Joint Mathematics Meetings in January, and in an electronic discussion network. Faculty for whom the 2006-2007 academic year will be the first or second year of full-time teaching (post-Ph.D.) at the college or university level are invited to apply to become Project NExT Fellows. The application deadline is Aprill7, 2006. For more information, see the Project NExT website, http:// archives.math.utk.edu/projnext/ . Project NExT is a program of the MAA. It receives major funding from the ExxonMobil Foundation, with additional funding from the Dolciani-Halloran Foundation, the Amer­ ican Mathematical Society, the Educational Advancement Foundation, the American Statistical Association, the National Council of Teachers of Mathematics, Texas Instruments, the Association of Mathematics Teacher Educators, the Association for Symbolic Logic, and the Greater MAA Fund. -Christine Stevens, Project NExT --

J

MARCH 2006 NOTICES OF THE AMS 365 For Your Information

the 'real' mathematicians, the mathematics of Fermat and Mathematics Awareness Month, Euler and Gauss and Abel and Riemann, is almost wholly April2006 'useless'." Yet it is mathematics developed by those very mathematicians, along with Hardy himself, that keeps The AMS, the American Statistical Association (ASA), the today's Internet transactions secure. Mathematical Association of America (MAA), and the According to Bruce Schneier, one of the world's fore­ Society for Industrial and Applied Mathematics (SIAM) an­ most security experts and the author of the influential book nounce that the theme for Mathematics Awareness Month Applied Cryptography, "Cryptographic security comes 2006 is Mathematics and Internet Security. from mathematics, not from people and not from ma­ When you use your home computer to log on to your chines. Mathematical security is available to everyone, bank account and pay a bill, to buy a book from Amazon, both the weak and the powerful alike, and gives ordinary or to buy or sell people a very powerful tool to protect their privacy. That's something on the cryptographic ideal of security." ebay, you assume But number-theory-based encryption is not the only your personal de­ case where mathematics plays an important role in Inter­ tails-your Social net security. Several of the essays presented on the Math­ Security number, ematics Awareness Month website (http: I jwww. your bank ac­ mathaware. o rg) describe other examples. And if history count access is any guide, we can expect more in the years to come. password, or Each year in April the Joint Policy Board for Mathematics your credit card sponsors Mathematics Awareness Month to recognize the number-cannot importance of mathematics through written materials be read by an and an accompanying poster that highlight mathematical unauthorized developments and applications in one particular area. third party. What makes this possi­ Editor's Note: This issue of the Notices carries a feature ble is mathemat­ article, "Find Me a Hash", by Susan Landau, on the theme ics. of Mathematics Awareness Month. Pure mathe­ matics, in fact. -AMS announcement For, by a surpris­ ing twist of fate, today's Internet commerce makes heavy use of encryption techniques that depend upon re­ sults in number theory, a branch of mathematics that until relatively recently was thought of as strictly "pure mathematics", with no real-world applications. In his book A Mathematician's Apology the famous British number theorist G. H. Hardy declared, "The 'real' mathematics of

366 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Inside the AMS

AMS Sponsors NExT Fellows Deaths of AMS Members Each year the AMS sponsors six NExT Fellows who are ]ANIT L. ANDERSEN, associate professor, Hope College, Holland, affiliated with Ph.D.-granting institutions and who show MI, diedonNovember 24, 2005.BomonDecember 11,1957, promise in mathematics research. The names, affiliations, she was a member of the Society for 22 years. and areas of research of the 2005-2006 AMS Fellows are: ALBERT E. BABBITT ]R., retired, from Bethesda, MD, died in August 2005. Born on August 11, 1927, he was a mem­ MATTHEW G. BoYLAN, University of South Carolina, number ber of the Society for 54 years. theory; DAVID E. BROWN, Utah State University, graph the­ RAoUL Borr, professor emeritus, Harvard University, died ory; ANNE ComNs, Centre College, Danville, Kentucky, com­ on December 20, 2005. Born on September 24, 1923, he putational geometry/computational topology; RoBERT B. was a member of the Society for 55 years. Ems, Illinois Institute of Technology, combinatorics and HANs-HENNING BucHSTEINER, from Halle, Germany, died on graph theory; BoRBALA MAzzAG, Humboldt State University, July 7, 2004. Born on September 15, 1937, he was a mem­ mathematical biology; and VIOLETA VASILEVSKA, University of ber of the Society for 6 years. South Dakota, Vermillion, topology. ]AMES P. BuRLING, professor emeritus, State University of Project NExT (New Experiences in Teaching) is a pro­ New York, College at Oswego, died on July 7, 2005. Born fessional development program for new or recent Ph.D.'s on May 29, 1930, he was a member of the Society for 47 years. in the mathematical sciences (including pure and applied EDWIN L. CROW, retired, from Boulder, CO, died on Oc­ mathematics, statistics, operations research, and mathe­ tober 29, 2005. Born on September 15, 1916, he was a mem­ matics education). It addresses all aspects of an academic ber of the Society for 66 years. career: improving the teaching and learning of mathe­ WILUAM DILLWORTH, retired engineer from Beloit, WI, died matics, engaging in research and scholarship, and partic­ on March 27,2003. Born on October4, 1919, he was a mem­ ipating in professional activities. It also provides the ber of the Society for 9 years. participants with a network of peers and mentors as they DIANE DOWLING, retired, University of Manitoba, Canada, assume these responsibilities. Each year sixty to seventy died onApril29, 2005. Born on February 21, 1933, she was new Ph.D.'s receive Project NExT Fellowships, which allow a member of the Society for 48 years. them to attend special events at the summer MathFest of MURRAY S. KLAMKIN, professor emeritus, University of Al­ berta, Canada, died on August 6, 2004. Born on March 5, the Mathematical Association of America and at the Joint 1921, he was a member of the Society for 56 years. Mathematics Meetings. The AMS also holds activities for PETER A. NIELSEN, executive with IBM, Austin, TX, died on the AMS NExT Fellows at the Joint Mathematics Meetings. August 2, 2005. Born on November 16, 1960, he was a mem­ For further information about Project NExT, visit the ber of the Society for 22 years. website http: I /archives. math. utk. edu/proj next/. PEG-Foo SIEW, associate professor, Curtin University of Technology, Perth, Australia, died on September 1, 2005. -Elaine Kehoe Born on May 21, 1941, he was a member of the Society for 9 years. ABRAHAM SPITZBART, professor emeritus, University of Wisconsin-Milwaukee, died on July 6, 2005. Born in Octo­ Correction ber 1915, he was a member of the Society for 66 years. The "MathSciNet Matters" column in the January 2006 CHARLES B. THOMAS, professor, University of Cambridge, issue listed an incorrect URL for locating journals with England, died on December 16, 2005. Born on August 17, MathSciNet reference lists. The correct URL is 1938, he was a member of the Society for 39 years. http://www.ams.org/mrcitations/journal_list. html. -Norman Richert, Math Reviews

MARCH 2006 NOTICES OF THE AMS 367 Reference and Book List

The Reference section of the Notices Upcoming Deadlines February 28, 2005: Applications is intended to provide the reader with February 10, 2006: Applications for for Clay Mathematics Institute (CMI) frequently sought information in Math for America Foundation New­ Summer School. See http: I lwww. an easily accessible manner. New ton Fellowships. See the website claymath.orglsummerschool or information is printed as it becomes contact summerschoo http:llwww.mathforamerica.orgl. l @cl aymath. available and is referenced after the org; telephone: 617-995-2600. February 15, 2006: Nominations first printing. As soon as information March 1, 2006: Applications for for Clay Mathematics Institute is updated or otherwise changed, it Summer Program for Women in Math­ will be noted in this section. (CMI) Liftoff Program. See http: I I ematics (SPWM). See "Mathematics claymath.orglfaslliftoff_ Opportunities" in this issue. Contacting the Notices fell owsl; telephone 617-995-2600; March 1, 2006: Applications for The preferred method for contacting email:[email protected]. summer program of the Christine the Notices is electronic mail. The editor is the person to whom to send Where to Find It articles and letters for consideration. A brief index to information that appears in this and previous issues of the Notices. Articles include feature articles, AMS Bylaws-November2005, p. 1239 memorial articles, communications, AMS Email Addresses-February 2006, p. 251 opinion pieces, and book reviews. The AMS Ethical Guidelines-June/July 2004, p. 675 editor is also the person to whom to AMS Officers 2004 and 2005 (Council, Executive Committee, send news of unusual interest about Publications Committees, Board of Trustees)- May 2005, p. 564 other people's mathematics research. AMS Officers and Committee Members-October 2005, p. 1073 The managing editor is the person Conference Board of the Mathematical Sciences-September 2005, to whom to send items for "Mathe­ p. 892 matics People", "Mathematics Op­ Information for Notices Authors-]une/]uly 2005, p. 660 portunities", "For Your Information", Mathematics Research Institutes Contact Information-August 2005, "Reference and Book List", and "Math­ p. 770 ematics Calendar". Requests for National Science Board-January 2006, p. 62 permissions, as well as all other New Journals for 2004-]une/]uly 2005, p. 662 inquiries, go to the managing editor. NRC Board on Mathematical Sciences and Their Applications-March 2006, The electronic-mail addresses are p. 369 noti ces@math. ou. edu in the case of NRC Mathematical Sciences Education Board-April 2005, p. 465 the editor and noti ces@ams. org in NSF Mathematical and Physical Sciences Advisory Committee-February the case of the managing editor. The 2006,p. 255 fax numbers are 405-325-7484 for Program Officers for Federal Funding Agencies-October 2005, the editor and 401-331-3842 for the p. 1069 (DoD, DoE); November 2005, p. 1223 (NSF) managing editor. Postal addresses Stipends for Study and Travel-September 2005, p. 900 may be found in the masthead.

368 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Reference and Book List

Mirzayan Science and May 31, 2006: Registration for the Joyce R. McLaughlin, Rensselaer Technology Policy Graduate Fellow­ Thirteenth International Mathematics Polytechnic Institute ship Program of the National Acade­ Competition for University Students Charles Manski, Northwestern Uni­ mies. See the website http: I I (IMC). See "Mathematics Opportuni­ versity www7.nationalacademies.orgl ties" in this issue. Jill P. Mesirov, Broad Institute pol i cyfe ll ows or contact The Na­ June 1, 2006: Applications for fall Andrew Odlyzko, Digital Tech­ tional Academies Christine Mirzayan program of the Christine Mirzayan Sci­ nology Center, University of Minnesota Science and Technology Policy Grad­ ence and Technology Policy Graduate John Rice, University of Califor­ uate Fellowship Program, 500 Fifth Fellowship Program of the National nia, Berkeley Street, NW, Room 508, Washington, Academies. See the website http: I I Stephen M. Robinson, University of DC 20001; telephone: 202-334-2455; www7.nationalacademies.orgl Wisconsin, Madison fax: 202-334-1667. pol i cyfe ll ows or contact The Na­ Edward J. Wegman, George Mason March 1, 2006: Proposals for NSF tional Academies Christine Mirzayan University Program on Mathematical Sciences: Science and Technology Policy Gradu­ Detlof von Winterfeldt, University Innovations at the Interface with the ate Fellowship Program, 500 Fifth of Southern California Physical and Computer Sciences and Street, NW, Room 508, Washington, Lai-Sang Young, Courant Institute Engineering: Computer Science. See DC 20001; telephone: 202-334-2455; of Mathematical Sciences the website http: I lwww. nsf. gov I fax: 202-334-1667. The postal address for BMSA is: pubsl2005lnsf05622lnsf05622. October 1, 2006: Applications for Board on Mathematical Sciences and htm. AWM Travel Grants. See http: I lwww. Their Applications, National Academy March 1, 2006: Applications for awm-math.orgltravelgrants.html; of Sciences, Room K974, 500 Fifth EDGE Program. See the website telephone 703-934-0163; email: awm@ Street, NW, Washington, DC 20001; http:llwww.edgeforwomen.orglor math. umd. edu; or contact Associa­ telephone: 202-334-2421; fax: 202- contact the EDGE Program, Depart­ tion for Women in Mathematics, 334-242212101; email: ms@nas. edu; ment of Mathematics, Bryn Mawr 11240 Waples Mill Road, Suite 200, World Wide Web: http: I lwww7. College, 101 North Merion Avenue, Fairfax, VA 22030. nationalacademies.orglbmsiBMSA_ Bryn Mawr, PA 19010; email: edge@ Members. html. edgeforwomen. org; telephone 610- Board on Mathematical Sciences 876-3527. and Their Applications, Book list March 31, 2006: Applications for National Research Council The Book List highlights books that AMS Congressional Fellowships. See The Board on Mathematical Sciences have mathematical themes and are "Mathematics Opportunities" in this and Their Applications (BMSA) was aimed at a broad audience potentially issue. established in November 1984 to including mathematicians, students, March 31, 2006: Nominations for lead activities in the mathematical and the general public. When a book the Prize for Achievement in Infor­ sciences at the National Research has been reviewed in the Notices, a mation-Based Complexity. See "Math­ Council (NRC). The mission of BMSA is reference is given to the review. Gen­ ematics Opportunities" in this issue. to support and promote the quality erally the list will contain only books March 31, 2006: Nominations for and health of the mathematical sci­ published within the last two years, Third World Academy of Sciences ences and their benefits to the nation. though exceptions may be made in Prizes. See http: I lwww. twas. o rgl. The current BMSA members are: cases where current events (e.g., the April 7, 2006: Proposals for 2007 Tanya Styblo Beder, Citigroup death of a prominent mathematician, NSF-CBMS Regional Conferences. See Alternative Investments coverage of a certain piece of mathe­ the CBMS website, http: I lwww. Patrick L. Brockett, University of matics in the news) warrant drawing cbmsweb.orgiNSFI2007_call .htm Texas at Austin readers' attention to older books. Sug­ or contact: Conference Board of the Avarinda Chakravarti, Institute of gestions for books to include on the list Mathematical Sciences, 1529 Eigh­ Genetic Medicine, Johns Hopkins may be sent to noti ces-bookl i st@ teenth Street, NW, Washington, DC University ams .org. 20036; telephone: 202-293-1170; fax: Philip Colella, Lawrence Berkeley '''Added to "Book List" since the 202-293-3412; email: l kol be@maa. National Laboratory list's last appearance. org or rosi er@georgetown. edu. Lawrence Craig Evans, University May 1, 2006: Applications for AWM of California, Berkeley A 3 & His Algebra: How a Boy from Travel Grants. See http: I lwww. awm­ John E. Hopcroft, Cornell University Chicago's West Side Became a Force in math. orgltravel grants. html; tele­ C. David Levermore, University of American Mathematics, by Nancy E. phone 703-934-0163; email: wm@math. Maryland Albert. iUniverse, Inc., January 2005. umd. edu; or contact Association for Charles M. Lucas, American Inter­ ISBN 0-595-32817-2. (Reviewed De­ Women in Mathematics, 11240 Waples national Companies cember 2005.) Mill Road, Suite 200, Fairfax, VA David McLaughlin (chair), New Action This Day, edited by Michael 22030. York University Smith and Ralph Erskine. Random

MARCH 2006 NOTICES OF THE AMS 369 Reference and Book List

House of Canada, February 2003. Dark Hero of the Information Age: janos Bolyai, Euclid, and the ISBN 0-593-04910-1. In Search of Norbert Wiener, by Flo Nature of Space, by Jeremy]. Gray. ''Beyond Coincidence: Amazing Sto­ Conway and Jim Siegelman. Basic MIT Press, May 2003. ISBN 0-262- ries of Coincidence and the Mystery Books, December 2004. ISBN 0-738- 57174-9. (Reviewed October 2005.) and Mathematics behind Them, by 20368-8. john Pell (1611-1685) and His Martin Plimmer and Brian King. *Divine Proportions: Rational Correspondence with Sir Charles Thomas Dunne Books, December Trigonometry to Universal Geometry, Cavendish: The Mental World of an 2005. ISBN 0-312-34036-2. by N. ]. Wildberger. Wild Egg Books, Early Modern Mathematician, by Noel Beyond Reason: Eight Great Prob­ September 2005. ISBN 0-9757492-0-X. Malcolm and Jacqueline Stedall. lems That Reveal the Limits ofScience, The Equation That Couldn't Be Oxford University Press, second by A. K. Dewdney. Wiley, April 2004. Solved (How Mathematical Genius Dis­ edition, January 2005. ISBN 0-198- ISBN 0-471-01398-6. covered the Language of Symmetry), 56484-8. The Book of Presidents. London by Mario Livio. Simon and Schuster, The Knot Book: An Elementary Mathematical Society, 2005. ISBN 0- September 2005. ISBN 0-743-25820-7. Introduction to the Mathematical 950-27341-4. The Essential Turing, edited by Theory ofKnots, Colin C. Adams. AMS, A Brief History of Infinity, by Paolo B. Jack Copeland. Oxford University September 2004. ISBN 0-8218-3678-1. Zellini. Penguin Books (paperback), Press, September 2004. ISBN 0-198- (Reviewed September 2005.) March 2005. ISBN 0-141-00762-1. 25080-0. Knots and Links, by Peter R. The Calculus Gallery: Masterpieces Experimentation in Mathematics: Cromwell. Cambridge University Press, from Newton to Lebesgue, by William Computational Paths to Discovery, by October 2004. ISBN 0-521-83947-5. Dunham. Princeton University Press, Jonathan Borwein, David Bailey, and Luck, Logic, and White Lies: The December 2004. ISBN 0-691-09565-5. Roland Girgensohn. A K Peters, March Mathematics of Games, by Jorg Chance: A Guide to Gambling, Love, 2004. ISBN 1-56881-136-5. (Reviewed Bewersdorff. Translated by David the Stock Market and just About Every­ September 2005.) Kramer. A K Peters, November 2004. thing Else, by Amir D. Aczel. Thun­ The Fermat Diary, by C.]. Mozzochi. ISBN 1-56881-210-8. der's Mouth Press, October 2004. ISBN AMS, August 2000. ISBN 0-8218- M. C. Escher's Legacy: A Centen­ 1-56858-316-8. (Reviewed August 2670-0. nial Celebration, edited by Doris Schattschneider and Michele Emmer. 2005.) The Fermat Proof, by C. J. Mozzochi. Change Is Possible: Stories of Women Trafford Publishing, Inc., February Springer, September 2005 (paperback edition). ISBN 3-540-20100-9. and Minorities in Mathematics, by Pa­ 2004. ISBN 1-412-02203-7. Math and the Mona Lisa: The Art tricia Clark Kenschaft. AMS, Septem­ Geometry and Meaning, by Dominic and Science of Leonardo da Vinci, by ber 2005. ISBN 0-8218-3748-6. Widdows. Center for the Study of Lan­ Bulent Atalay. Smithsonian Books, Coincidences, Chaos, and All That guage and Information, November April 2004. ISBN 1-588-34171-2. Math jazz: Making Light of Weighty 2004. ISBN 1-575-86448-7. The Math Instinct: Why You're a Ideas, by Edward B. Burger and Michael God Created the Integers, by Stephen Mathematical Genius (Along with Lob­ Starbird. W. W. Norton, August 2005. Hawking. Running Press, October 2005. sters, Birds, Cats, and Dogs), by Keith ISBN 0-393-05945-6. ISBN 0-762-41922-9. Devlin. Thunder's Mouth Press, March The Colours of Infinity: The Beauty G6del's Theorem: An Incomplete 2005. ISBN 1-56025-672-9. and Power of Fractals, by Michael Guide to Its Use and Abuse, by Torkel Mathematical Adventures for Stu­ Barnsley, Nigel Lesmoir-Gordon, Franzen. AKPeters, May 2005. ISBN 1- dents and Amateurs, David F. Hayes Benoit B. Mandelbrot, Ian Stewart, 56881-238-8. and Tatiana Shubin, editors. Mathe­ Gary Flake, Robert Prechter, and Graphic Discovery: A Trout in the matical Association of America, 2004. Arthur C. Clarke. Clear Press, March Milk and Other Visual Adventures, by ISBN 0-88385-548-8. 2004. ISBN 1-904-55505-5. Howard Wainer. Princeton University Mathematical fllustrations: A Manual Complexities: Women in Mathe­ Press, October 2004. ISBN 0-691- of Geometry and PostScript, by Bill Cas­ matics, edited by Bettye Anne Case 10301-1. selman. Cambridge University Press, and Anne M. Leggett. Princeton Uni­ Incompleteness: The Proof and December 2004. ISBN 0-521-54788-1. versity Press, January 2005. ISBN Paradox of Kurt G6del, by Rebecca *Mathematical Musings: A Collection 0-691-11462-5. Goldstein. W. W. Norton, February ofQuotes, edited by Dan Sonnenschein. Converging Realities: Toward a 2005. ISBN 0-393-05169-2. Clarium Press, November 2005. ISBN Common Philosophy of Physics and The Infinite Book: A Short Guide to 0-9697688-8-5. Mathematics, by Roland Omnes. the Boundless, Timeless and Endless, by Mathematicians under the Nazis, by Princeton University Press, November John D. Barrow. Pantheon, August Sanford L. Segal. Princeton University 2004. ISBN 0-691-11530-3. 2005. ISBN 0-375-42227-7. Press, July 2003. ISBN 0-691-00451-X. The Curious Incident of the Dog in Introducing Game Theory and Its (Reviewed April2005.) the Night-time, by Mark Haddon. Vin­ Applications, by Elliott Mendelson. Mathematics by Experiment: Plau­ tage, May 2 004. ISBN 1-400-0 3 2 71-7. CRC Press, July 2004. ISBN 1-584- sible Reasoning in the 21st Century, by (Reviewed in this issue.) 88300-6. Jonathan Borwein and David Bailey.

370 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Reference and Book List

A K Peters, December 2003. ISBN The Road to Reality: A Complete 1-56881-211-6. (Reviewed September Guide to the Laws of the Universe, by 2005.) Roger Penrose. Knopf, February 2005. Mathematics in Nature: Modeling ISBN 0-679-45443-8. Patterns in the Natural World, by Saunders Mac Lane: A Mathemati­ John A. Adam. Princeton University cal Autobiography, by Saunders Press, November 2003. ISBN 0-691- Mac Lane. A K Peters, May 2005. ISBN 11429-3. (Reviewed June/July 2005.) 1-56881-150-0. (Reviewed December Meta Math! The Quest for Omega, by 2005.) Gregory Chaitin. Pantheon, October Science in the Looking Glass, by 2005. ISBN 0-375-42313-3. E. Brian Davies. Oxford University More Mathematical Astronomy Press, August 2003. ISBN 0-198- Morsels, by Jean Meeus. Willmann­ 52543-5. (Reviewed December 2005.) Bell, 2002. ISBN 0-943396-743. Sneaking a Look at God's Cards: Musings of the Masters: An An­ Unraveling the Mysteries of Quantum thology of Miscellaneous Reflections, Mechanics, by Giancarlo Ghirardi, edited by Raymond G. Ayoub. translated by Gerald Malsbary. Prince­ Mathematical Association of Amer­ ton University Press, revised edition, ica, June 2004. ISBN 0-88385-549-6. January 2005. ISBN 0-691-12139-7. New Mexico Mathematics Contest Spaceland, by Rudy Rucker. Tor Problem Book, by Liong-shin Hahn. Books, June 2002. ISBN 0-765-30366- University of New Mexico Press, No­ 3. (Reviewed August 2005.) vember 2005. ISBN 0-8263-3534-9. Stalking the Riemann Hypothesis: The Newtonian Moment: Isaac New­ The Quest to Find the Hidden Law of ton and the Making ofModem Culture, Prime Numbers, by Dan Rockmore. by Mordechai Feingold. New York Pantheon, April 2005. ISBN 0-375- Library and Oxford University Press, 42136-X. December 2004. ISBN 0-195-17735-5. A Tour through Mathematical Numbers, the Language ofScience, Logic, by Robert S. Wolf. Mathemati­ by Tobias Dantzig. Pi Press, fifth edi-· cal Association of America, January -<. ,,• • tion, March 2005. ISBN 0-131-85627-8. 2005. ISBN 0-88385-036-2. The Oxford Murders, by Guillermo The Transformation of Mathemat­ ics in the Early Mediterranean World: Martinez. Abacus, January 2005. ISBN From Problems to Equations, by Reviel 0-349-11721-7. (Reviewed November Netz. Cambridge University Press, 2005.) June 2004. ISBN 0-521-82996-8. The Pea and the Sun: A Mathe­ Using the Mathematics Literature, matical Paradox, by Leonard M. by Kristine K. Fowler. Marcel Dekker, Wapner. A K Peters, April2005. ISBN June 2004. ISBN 0-824-75035-7. 1-56881-213-2. ''The Visual Mind II, edited by ''Fiero della Francesca: A Mathe­ Michele Emmer. MIT Press, May 2005. matician's Art, by]. V. Field. Yale ISBN 0-262-05076-5. University Press, August 2005. ISBN. The Works of Archimedes: Trans­ 0-300-10342-5. lation and Commentary. Volume I: PopCo, by Scarlett Thomas. Har­ The Two Books On the Sphere and vest Books, October 2005. ISBN 0-156- The Cylinder. Edited and translated by 03137-X. (Reviewed February 2006.) Reviel Netz. Cambridge University Probability Theory: The Logic of Press, April2004. ISBN 0-521-66160- Science, by E. T. Jaynes. Edited by G. 9. (Reviewed May 2005.) Larry Bretthorst. Cambridge Univer­ sity Press, April 2003. ISBN 0-521- 59271-2. (Reviewed January 2006.) R. L. Moore: Mathematician and Teacher, by John Parker. Mathemati­ cal Association of America, 2004. ISBN 0-88385-550-X. Reality Conditions: Short Mathe­ matical Fiction, by Alex Kasman. Math­ ematical Association of America, May 2005. ISBN 0-88385-552-6.

MARCH 2006 NOTICES OF THE AMS 371 ------AMERICAN MATHEMATICAL SOCIETY AMS EXEMPLARY PROGRAM PRIZE

At its meeting in January 2004, the AMS Council approved the establishment of a new award called the AMS Award for an Exemplary Program or Achievement in a Mathematics Department. It is to be presented annually to a department that has distinguished itself by undertaking an unusual or particularly effective program of value to the mathematics community, internally or in relation to the rest of society. Examples might include a department that runs a notable minority outreach program, a department that has instituted an unusually effective industrial mathematics intern­ ship program, a department that has promoted mathematics so successfully that a large fraction of its university's undergraduate population majors in mathematics, or a department that has made some form of innovation in its research support to faculty and/or graduate students, or which has created a special and innovative envi­ ronment for some aspect of mathematics research.

The prize amount is $1,200. All departments in North America that offer at least a bachelor's degree in the mathematical sciences are eligible.

The Prize Selection Committee requests nominations for this award, which will be announced in late November, 2006. Letters of nomination may be submitted by one or more individuals. Nomination of the writer's own institution is permitted. The letter should describe the specific program( s) for which the department is being nominated as well as the achievements that make the program( s) an out­ standing success, and may include any ancillary documents which support the success of the pro­ gram(s). The letter should not exceed two pages, with supporting documentation not to exceed an additional three pages.

All nominations should be submitted to the AMS Secretary, Robert J. Daverman, American Mathematical Society, 312D Ayres Hall, University ofTennessee, Knoxville TN 37996-1330. Include a short description of the work that is the basis of the nomination, with complete biblio­ graphic citations when appropriate. The nominations will be forwarded by the Secretary to the Prize Selection Committee, which will make the final decision on the award.

Deadline for nominations is Aprill, 2006. AMERICAN MATHEMATICAL SOCIETY

M The selection committee for these prizes requests nominations for con­ sideration for the 2006 awards. Further information about the prizes c can be found in the November 2005 Notices, pp. 1251-1255 (also avail­ 0 able at http://www.ams.org/prizes-awards). Three Leroy P. Steele Prizes are awarded each year in the following cat­ ·- egories: (I) the Steele Prize for Lifetime Achievement: for the cumulative ~ influence of the total mathematical work of the recipient, high level of c research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) the ·- Steele Prize for Mathematical Exposition: for a book or substantial sur­ E vey or expository-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has 0 proved to be of fundamental or lasting importance in its field, or a model of important research. In 2007 the prize for Seminal Contribution to Research will be awarded for a paper in geometry/topology.

a. Nominations with supporting information should be submitted to the Secretary, Robert J. Daverman, American Mathematical Society, 312D 0 Ayres Hall, University ofTennessee, Knoxville, TN 37996-1330.1nclude a ... short description on the work that is the basis of the nomination, includ­ ing complete bibliographic citations. A curriculum vitae should be - included. The nominations will be forwarded by the Secretary to the - prize selection committee, which will, as in the past, make final decisions u" on the awarding of prizes. Deadline for nominations is March 3 I, 2006. Mathematics Calendar

The most comprehensive and up-to-date Mathematics Calendar information is available one-MATH at http://www.ams.org/mathcal/.

March 2006 of NSF funding. Organizers:Thomasivey(iveyt©cof c. edu),AlexKasman(kasmana© '' 2-3 DARPA Workshop: State-Dependent Delays in Regulatory cofc. edu). Networks, DIMACS Center, CoRE Bldg, Rutgers University, Piscat­ away, New Jersey. Deadlines: Those wishing to speak should contact the organizers as soon as possible, and in any case by February 15, 2006. Goal: To begin a dialogue between mathematicians with expertise in the dynamics of delay differential equations and control theory, 6-10 IMA Workshop: Natural Images, University of Minnesota, 207 and biologists with expertise in the mechanisms in signal trans­ Church St. SE, 400 Lind Hall, Minneapolis, Minnesota. (Jan. 2006, duction/gene regulatory networks. For the biologists the potential p. 82) benefit will be new models within which to understand, test, and control gene regulation. For the mathematicians the potential bene­ 7-11 Holomorphic Dynamics Workshop, Fields Institute, Toronto, fit will be a concrete set of problems around which the development Canada. (Nov. 2005, p. 1263) of this new theory can be focused. 1 0-1 2 Recent Developments in Higher Dimensional Algebraic Organizers: Tim Buchman, Washington University, buchman© Geometry, The Japanese American Mathematics Institute and the wustl. edu; Jon Lorsch, John Hopkins University, j lorsch©jhmi. Johns Hopkins University, Baltimore, Maryland. (Feb. 2006, p. 284) edu; Konstantin Mischaikow, Georgia Institute of Technology, mischaik©math. gatech. edu. 1 2-1 8 Workshop on "Special Geometries in Mathematical Information: M. Sohor, DIMACS Center, sohor©dimacs. rutgers . Physics", Kuhlungsborn, Germany. (Dec. 2005, p. 1379) edu, 1 2-~ 5 RDSES/ESI Educational Workshop on Discrete Probability, 732-445-5928;http://dimacs.rutgers.edu/Workshops/Delays. Erwm Schroedinger Institute (ESI), Vienna, Austria. (Jan. 2006, p. 66)

2-5 SouthEastern Analysis Meeting, SEAM XXII, University of 1 3-1 7 Anatomy of Integers, Centre de recherche mathematiques Florida, Gainesville, Florida. (Jan. 2006, p. 66) Universite de Montreal, Montreal, Quebec, Canada. (Apr. 2005, p. 478) '' 4-5 Southeast Geometry Conference, College of Charleston, Charleston, South Carolina. 1 3-1 7 Workshop on 3-manifolds after Perelman, International Description: The SEGC is a yearly regional conference covering all Centre for Mathematical Sciences, Edinburgh, United Kingdom. (Feb. 2006, p. 284) areas of geometry, and featuring 40-45 minute talks, bothinvited and contributed. Young researchers (postdocs and graduate students) '' 1 3-June 16 IPAM Long Program on Cells and Materials: At are especially encouraged to participate in the conference, which the .Interface Between Mathematics, Biology and Engineering, will include a Poster Session on the evening of March 4. Some Institute for Pure and Applied Mathematics, UCLA, Los Angeles, support for travel expenses may be available, pending the approval California.

This section contains announcements of meetings and conferences respect to participation in the meeting, thls fact should be noted. 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 meetings of national or international mathematical organizations. A or mathcal@ams. org. complete list of meetings of the Society can be found on the last page· of In order to allow participants to arrange their travel plans, organizers of each Issue. meetmgs are urged to sublnit 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 achleve thls, 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 it has been held and a reference will be given in parentheses to the December issues will include, along with new announcements, references month, year, and page of the issue in whlch the complete information to any previously announced meetings and conferences occurring withln appeared. Asterisks ('') mark those announcements containing new or the twelve-month period following the month of those issues. New 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 withln the speakers (or sometimes a general statement on the program), deadlines twelve-month period. for abstracts or contributed papers, and source of further information. The Mathematics Calendar, as well as Meetings and Conferences of Meetings held outside the North American area may carry more detailed the AMS, is now available electronically through the AMS website on the information. In any case, if there is any application deadline with World Wide Web. To access the AMS website, use the URL: http://www. ams .org/.

374 NOTICES OF TIIE AMS VOLUME 53, NUMBER 3 Mathematics Calendar

Organizing Committee: Tom Chou (UCLA, Biomathematics), Tra­ 27-31 Spectral Theory and Mathematical Physics, A Conference chette Jackson (University of Michigan, Mathematics), Mark Lewis in Honor of Barry Simon's 60th Birthday, California Institute of (U. of Utah, Mathematics), John Lowengrub (University of Minnesota Technology, Pasadena, California. (Jan. 2006, p. 67) I UC Irvine, Mathematics), Sharon Lubkin (North Carolina State '' 29-31 Interpolation Theory and Applications: Conference in University, Mathematics), Stanley Osher (IPAM, Mathematics), Bill Honor of Michael Cwikel, Holiday Inn, Coral Gables, Florida. Tawil (Baxter Biosciences, BioSurgery), Ben Wu (UCLA, Material Sciences and Engr: Biomed Engr). Main Speakers: Brudnyi (Technion), Cerda (Barcelona), Corach (Buenos Aires), Grafakos (Missouri), Junge (Illinois), Kalton (Mis­ Information: Further information and an online application for souri), Kenig (Chicago), Krugljak (Lulea), Martin (Barcelona), Mastylo funding as a long-term participant is available at http : I /www . (Poznan), M. Mitrea (Missouri), Pick (Prague), Pustylnik (Technion), ipam .ucla.edu/programs/cm2006/. Rochberg (Washington Univ.), Sagher (FAU), Shvarstman (Technion), Sadosky (Howard), Tadmor (Maryland), Weidl (Stuttgart). '' 14-1 71PAMWorkshop:Tutorials forCellsand Materials Program, Institute for Pure and Applied Mathematics, UCLA, Los Angeles, Organizers: J. Cerda (Barcelona), L. De Carli (FlU), N. Kalton California. (Missouri), M. Mastylo (Poznan), M. Milman (FAU), J. Peetre (Lund). Information: Further information and an online application for Contact: Mario Milman (extrapol@bellsouth .net). funding is available at http ://www .ipam.ucla.edu/programs/ Information: cmtut/. 30-April 5 CRM-CiaySchool on Additive Combinatorics, Centre de 1 7-19 Transport Properties of Random Schrodinger Operators, Recherches Mathematiques, Montreal, Quebec, Canada. (Jan. 2006, University of Kentucky, Lexington, Kentucky. (Feb. 2006, p. 284) p. 67) 21 - 24 Workshop on Valuation Theory and its Applications - In April 2006 Memoriam Otto Endler, State University of Campinas (UNICAMP), 1-2 AMS Southeastern Section Meeting, Florida International Campinas, SP/ Brazil. (Nov. 2005, p. 1263) University, Miami, Florida. (May 2005, p. 570) 24-25 Complex Geometry (in honor of Domingo Toledo's 60th 1-2 Graduate Student Topology Conference, Indiana University, birthday), University of Utah, Salt Lake City, Utah. (Feb. 2006, Bloomington, Indiana. (Feb. 2006, p. 285) p. 284) 3-7 IMA Workshop: Shape Spaces, University of Minnesota, 207 '' 24-26 Lie Algebra Workshop, University of Ottawa, Ottawa, Church St. SE, 400 Lind Hall, Minneapolis, Minnesota. (Jan. 2006, Ontario, Canada. p. 82) Speakers: G. Benkart (Wisconsin), I. Dimitrov (Queen's), Y. Gao (York), J. Greenstein (Riverside), J. Landsberg (Texas A&M), K. Misra 3-7 8th International Conference on Approximation and Op­ (North Carolina), J. Patera (Montreal), A. Pianzola (Alberta), H. Usefi timization in the Caribbean, Universidad Autonoma de Santo (Western Ontario), K. Zhao (Wilfrid Laurier). Domingo (UASD), Dominican Republic. (Dec. 2005, p. 1379) Information: http : I /www. fields. utoronto. ca/programs/ * 3-7 Profinite Arithmetic Geometry and Related Moduli Spaces, scientific/05-06/lie_algebra/. Red Lodge, Montana. Description: Our series will combine moduli space ideas with '' 2 5-30 Algebras and Coalgebras-Tools for Geometry, Physics profinite Galois and group theory. and Computer Science, Joint Meeting of the American University Organizers: Pierre Debes, Mike Fried, Hiroaki Nakamura and Ken in Cairo, Cairo University and the Mathematical Sciences Research Ribet. This is the first of three conferences in a series. Centre, Cairo, Egypt. Topics: Structured approaches to Geometric Galois Theory and Topics: Rings and Corings, Modules and Comodules, Algebras and related structure of profinite fundamental or Galois groups, The Coalgebras in Computer Science, Geometry, and Physics. arithmetic and/or geometry of moduli spaces (e.g., Hurwitz spaces Main Speakers: Tomasz Brzezinski, University of Wales, Swansea, or Shimura varieties), 1-adic representations or versions of Serre's UK; Heinz-Peter Gumm, University of Marburg, Germany; Pavel Open Image Theorem, Applications of profinite groups of finite Kolesnikov, Sobolev Institute, Novosibirsk, Russia; Juergen Fuchs, cohomological dimension like Demuskin and p-Poincare dual Karlstad University, Sweden; Dirk Kreimer, IHES, Paris, France; groups, Outcomes related to Shimura-Taniyama-Weil, Heights of Shahn Majid, Queen Mary University, London, UK; Robert Wis­ points on moduli spaces, Annabelian problems on the uniqueness bauer, University of Duesseldorf, Germany; Farahat Ali, Al-Azhar of the absolute Galois closures defined by moduli points. University, Egypt. Information: http : I /www. math. uci. edu;-mfried/ conferences/ Deadlines: Abstract: January 15th, 2006; Registration: January prof-geom-rims .html. 31st, 2006; Submission of paper for the conference proceedings: May 31st, 2006. 3-7 Workshop on Jordan Structures in Analysis and Geometry, Contact: Mohamed F. Yousif, The American University in Cairo, National Sun Yat-sen University, Kaohsiung, Taiwan. (Dec. 2005, 113 Kasr el Aini street, Cairo 11 511, Egypt; email: algebra@ p. 1379) aucegypt. edu; fax#: 20-2-795-7565; phone #: 20-2-797-5622 and 3-8 International Workshop on Multi-Rate Processes & Hys­ 20-2-797-5616. teresis, University College Cork, Cork, Ireland. (Jun/Jul. 2005, Information: Visit the website: http : I /www. aucegypt . edu/ p. 674) conferences/algebra/. * 6-9 Extreme forms of real algebraic varieties, AIM Research '' 27-31 IPAM Workshop: Membrane Protein Science and Engi­ Conference Center, Palo Alto, California. neering, Institute for Pure and Applied Mathematics, UCLA, Los Description: The workshop will bring together researchers from Angeles, California. diverse areas in an attempt to answer the ill-posed question "What Organizing Committee: Tom Chou, Chair (UCLA), Francisco is the best real form for an algebraic variety?" at least in some Bezanilla (UCLA), Ka Yee Lee (University of Chicago), Jacob Schmidt simple cases. (UCLA). Organizers: Ilia Itenberg, Grigory Mikhalkin, and Oleg Viro. Information: Further information and an online application for Focus: This workshop, sponsored by AIM and the NSF, will focus on funding is available at http://www.ipam.ucla.edu/programs/ real loci of algebraic varieties, and more specifically, their extreme cmws1/. geometrical and topological forms.

MARCH 2006 NOTICES OF THE AMS 375 Mathematics Calendar

Topics: Here are 3 groups of examples that have appeared partic­ 20-May 2 INFORMS Practice Conference: Applying Science to the ularly many times in recent research: Harnack and anti-Harnack Art of Business, Hotel Intercontinental Miami, Florida. (Aug. 2005, curves; maximal real projective hypersurfaces and complete in­ p. 787) tersections. Varieties with extremal behavior of their amoebas 29-May 1 2006 Barrett Lectures in Topology, University of or co-amoebas; varieties close to the tropical limit. Extreme real Tennessee, Knoxville, Tennessee. (Sept. 2005, p. 953) forms for some specific classes of varieties: Grassmannians, Farro, Calabi-Yau, etc. 30-May 5 Conference on Topology, Geometry and Physics, Deadline: February 6, 2006. Columbia University, New York, New York. (Jan. 2006, p. 67) Information: http: I /aimath. org/ ARCC/workshops/extremereal. May 2006 html. 1-June 30 Random Graphs and Large-Scale Real-World Net­ 6-1 2 Additive Combinatorics, Centre derecherchemathematiques, works, Institute for Mathematical Sciences, National University of Universite de Montreal, Montreal, Quebec. (Apr. 2005, p. 478) Singapore, Singapore. (Dec. 2005, p. 1380) 8-9 AMS Central Section Meeting, University of Notre Dame, Notre '' 3-5 DIMACS Workshop on Sequence, Structure and Systems Dame, Indiana. (May 2005, p. 570) Approaches to Predict Protein Function, DIMACS Center, CoRE '' 18-221PAMWorkshop: Microfluidic Flows in Nature and Microflu­ Bldg, Rutgers University, Piscataway, New Jersey. idic Technologies, Institute for Pure and Applied Mathematics, Goal: To bring together biologists, computer scientists and math­ UCLA, Los Angeles, California. ematicians who work on various aspects of protein function Organizing Committee: Andrea Bertozzi (UCLA, Mathematics), prediction. This workshop will provide both a venue for reviewing Michael Brenner (Harvard University, Applied Mathematics and the current state-of-the-art of diverse methods as well as a plat­ Applied Physics), Vittorio Cristini (University of California at form for further cross-fertilization and integration of sequence, Irvine, Biomedical Engineering and Mathematics), John Frangos (La structure and systems approaches. Jolla Bioengineering Institute), Howard Willet (UCLA, Orthopedic Organizers: Anna Panchenko, NIH, panch©mail . nih. gov; Teresa Surgery). Przytycka, NIH, przytyck©mail. nih. gov; Mona Singh, Princeton Information: Further information and an online application for University, mona©CS . Princeton. EDU. funding is available at http: I /www . ipam . ucla. edu/programs/ Information: M. Sohor, DIMACS Center, sohor©dimacs . rutgers. cmws2/. edu, 732-445-5928; http : I /dimacs. rutgers . edu/Workshops/ ProteinFunction. 20- 22 2006 SIAM International Conference on Data Mining, Hyatt Regency Bethesda, Bethesda, Maryland. (Dec. 2005, p. 1379) 5-1 0 Combinatorial and Geometric Group Theory, Vanderbilt University, Nashville, Tennessee. (Jun/ Jul. 2005, p. 674) '' 20-21 Workshop in Mathematical Physics, Stevens Institute of Technology, Hoboken, New Jersey. 6-11 (REVISED) International Conference on Fourier and Complex Analysis: Classical problems current view, Protaras, Cyprus. Description: The workshop in Mathematical Physics aims at bring­ (Oct. 2005, p. 1087) ing together the researchers in mathematical physics in order to understand the state of the art in mathematical physics and * 8-1 2 IPAM Workshop: Angiogenesis, NeoVascularization and discuss promising research directions in: Conservation Laws, Ki­ Morphogenesis, Institute for Pure and Applied Mathematics, UCLA, netic Equations, Nonlinear PDE's, Variational Principles; Riemann Los Angeles, California. Boundary-Value Problems and Singular Integral Equations; Ap­ Organizing Committee: Luisalruela-Arispe (UCLA), TrachetteJack­ plications in Hydrodynamics, Theory of Elasticity, Fiber Optics, son (University of Michigan), Howard Levine (Iowa State University), Magnetohydrodynamics (MHD), Contact Problems in Mechanics. John Lowengrub (University of California at Irvine), Bill Tawil Organizers: P.Dubovski and M. Zabarankin. (UCLA). Information: email: pdubovsk©stevens. edu; http: I /personal . Information: Further information and an online application for stevens.edu/-pdubovsk/mathphysics . html. funding is available at http: I /www. ipam. ucla. edu/programs/ cmws3/. 20-22 Logic, Models and Computer Science: LMCS06, Dipar­ timento di Matematica e Informatica, Universita di Camerino, 8-19 CANT 2006: Combinatorics, Automata and Number Theory, Camerino, Italy. (Oct. 2005, p. 1087) University of Liege, Belgium. (Jun/ Jul. 2005, p. 674) 2 2-2 3 AMS Eastern Section Meeting, University of New Hampshire, 9-11 DIMACS Workshop on Clustering Problems in Biologi­ Durham, New Hampshire. (May 2005, p. 570) cal Networks, DIMACS Center, CoRE Bldg, Rutgers University, Piscataway, New Jersey. (Dec. 2005, p. 1380) 22-27 Design Theory, Graph Theory, and Computational Meth­ ods, IPM, Tehran, Iran. (Jan. 2006, p. 67) 1 0-1 2 Workshop on Numerical, Mathematical and Modeling Analysis Related to Fluid Dynamics in Hydrogen Fuel Cells, 23-May 6 Rigidity and Flexibility (ESI-Program 2006), The Inter­ University of Ottawa, Ottawa, Ontario, Canada. (Feb. 2006, p. 285) national Erwin Schrbdinger Institute for Mathematical Physics (ESI), Vienna, Austria. (Feb. 2006, p. 285) 1 0-1 3 CSNA06. Annual meeting of the Classification Society of North America, DIMACS Center, Rutgers University, Piscataway, * 24-28 XIV Brazilian Logic Conference, Itatiaia, Rio de Janeiro, New Jersey. (Dec. 2005, p. 1380) Brazil. Topics: Algebraic logic, logic and computation, history of logic, 1 3-1 8 Analytic methods for Diophantine equations, Banff In­ logic and education, model theory, non-classical logics, philosophy ternational Research Station, Banff, Alberta, Canada. (Apr. 2005, of formal sciences, and set theory. This event will also celebrate p. 478) the centennial of the birth of Kurt Godel. 14-20 International Symposium on Functional Equations, Uni­ Deadline: For submission of contributed talks is January 15, 2006. versity of Louisville, Louisville, Kentucky. (Jan. 2006, p. 67) Information: email: XIVEBL©inf . puc-rio. br; http : I /XIVEBL. 1 5-1 7 SIAM Conference on Imaging Science, Radisson Hotel inf . puc-rio. brI. Metrodome, Minneapolis, Minnesota. (Oct. 2005, p. 1087) 29-30 AMS Western Section Meeting, San Francisco State Univer­ 1 5-1 7 The First International Conference on Mathematical sity, San Francisco, California. (May 2005, p. 570) Sciences, Al-Azhar University, Gaza, The Palestinian Authority.

376 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics Calendar

(Feb. 2006, p. 285) 28-30 Annual Meeting of the Canadian Society for History and 1 5-1 7 Workshop on Probabilistic Symmetries and their Appli­ Philosophy of Mathematics (CSHPM), York University Toronto, Ontario, Canada. (Jan. 2006, p. 68) cations, University of Ottawa, Ottawa, Ontario, Canada. (Feb. 2006, p. 285) 29-June 3 International Conference on Toric Topology, Osaka City University, Osaka, Japan. (Dec. 2005, p. 1381) 1 6-1 8 LMS Workshop on Cluster Algebras and Teichmi.iller The­ ory, University of Leicester, Leicester, United Kingdom. (Feb . 2006, 30-June 2 DAYS on DIFFRACTION-2006, St. Petersburg University p. 285) & St. Petersburg Branch Steklov Math. Inst., St. Petersburg, Russia. (Jan. 2006, p. 68) 16-20 5th Conference on Function Spaces at SlUE, Southern Illinois University, Edwardsville, Illinois. (Oct. 2005, p. 1087) 30-June 2 Geometry and Representation Theory: A conference 17-19 Conference of Applied Statistics in Ireland, University in honor of George Lusztig, M.I.T., Cambridge, Massachusetts. College Cork, Cork, Ireland. (Sept. 2005, p. 953) (Feb. 2006, p. 285) '' 1 7-20 Nonlinearity: from Turbulent to Magic, NORDITAandNiels 30-June 3 2006 USENIX Annual Technical Conference, Boston, Bohr Institute, Copenhagen, Denmark. Massachusetts. (Dec. 2005, p. 1381) Topics: Wave chaos, quantum chaos, turbulence, exceptional magic, 30-June 6 NAFSA 8-8th International Spring School on Nonlinear statistical mechanics, ergodic theory. ANalysis, Function Spaces and Applications, Czech University of Oganizing committee: B. Lautrup, A. Brandenburg, M. H. Jensen. Agriculture, Prague, Czech Republic. (Oct. 2005, p. 1088) International committee: G. Tanner, R. Artuso, P. Cvitanovic', S. '' 31-June 4 MASSEE International Congress on Mathematics: Creagh, T. Guhr, G. Huber, H. H. Rugh. MICOM-2006, Cyprus, Greece. Information: http: I I cats. nbi. dk. Information: All corresondence concerning the MASSEE Interna­ tional Congress on Mathematics should be sent to the following 17-21 ASL Annual Meeting, Montreal, Canada. (Juni Jul. 2005, address: Organizing Committee MICOM-2006 Cyprus Mathemati­ p. 674) cal Society 36 Stasinou Street, Office 102 Strovolos 2003, Nicosia, ,., 21-24 Workshop on the universal Urysohn metric space, Ben­ Cyprus; fax: +357-22352059; tel: +357-223 78101; email: xms©cms. Gurion University of the Negev, Beet-Sheva, Israel. org.cy; [email protected]; makrides . greg©usa. Principal Invited Speakers: Vladimir Uspenskiy (Ohio University, net;http : //www.cms.org.cy. USA) and Anatoly Vershik (St. Petersburg branch of Steklov Institute, Russia). June 2006 Scientific Committee: Alexander Kechris (Caltech), Vitali Milman (Tel-Aviv University), Stevo Todorcevic (University of Toronto, and 1-July 31 Algorithmic Biology: Algorithmic Techniques in Com­ CNRS). putational Biology, Institute for Mathematical Sciences, National Organizers: Matatyahu Rubin (Chair), Vladimir Fonf and Arkady University of Singapore, Singapore. (Dec. 2005, p. 1381) Leiderman (all Ben-Gurion University, Israel), Vladimir Pestov (Uni­ 4-1 0 Workshop on Commutative Rings, Cortona, Italy. (Feb. 2006, versity of Ottawa, Canada). p. 286) Information: http: I lwww .math . bgu . ac. ilrarkadyiWorkshop 5-9 Poisson 2006: Poisson Geometry in Mathematics and _2006lmain_page.html. Physics, National Olympic Memorial Youth Center, Tokyo, Japan. 22-26 13th International Conference on Gambling & Risk Taking, (Jan. 2006, p. 68) Harrah's Lake Tahoe, Stateline, Nevada. (Jan. 2006, p . 67) 5-9 Selfsimilar groups and conformal dynamics, AIM Research 22-26 IMA Workshop: Visual Learning and Recognition, Univer­ Conference Center, Palo Alto, California. (Dec. 2005, p. 1381) sity of Minnesota, Minneapolis, Minnesota. (Jan. 2006, p. 82) 5-l 2 Motives and Periods, University of British Columbia, Van­ '' 22-261PAM Workshop: Systems Biology and Molecular Modeling, couver, BC, Canada. (Dec. 2005, p. 1381) Institute for Pure and Applied Mathematics, UCLA, Los Angeles, 7-10 Symposium in Complex Analysis, Kranjska Gora, Slovenia. California. (Nov. 2005,p. 1264) Organizing Committee: James Glazier (Indiana University, Rio­ 9-14 Eight International Conference on Geometry, Integrability complexity Institute) Dan Kamei (UCLA, Bioengineering) Douglas and Quantization, Sts. Constantine and Elena resort (near Varna), Lauffenburger (Massachusetts Institute of Technology, Biological Bulgaria. (Oct. 2005, p. 1088) Engineering) Ben Wu (UCLA, Bioengineering and Weintraub Center for Reconstructive Biotechnology). 11 -1 4 ICMSE 2006- International Conference in Mathematics, Information: Further information and an online application for Sciences and Science Education, University of Aveiro, Aveiro, funding is available at http: I lwww. ipam. ucla. edulprogramsl Portugal. (Jan. 2006, p. 68) cmws41. 1 2-1 5 2006 International Conference on Applied Mathematics and Interdisciplinary Research-Nankai, Nankai University, Tianjin, 22-26 Low Eigenvalues of Laplace and Schrodinger Operators, P.R. China. (May 2005, p. 570) AIM Research Conference Center, Palo Alto, California. (Oct. 2005, p. 1087) 1 2-1 5 Journees Peter Shalen, Centre de Recherches Mathema­ tiques, Montreal, Quebec, Canada. (Feb. 2006, p. 286) 22-27 "Banach spaces and their applications in Analysis", in honor of Nigel Kalton's 60th birthday, Miami University, Oxford, 12-1 6 EMS mathematical weekend in Pays de Loire, Universite Ohio. (Jan. 2006, p. 67) de Nantes, Nantes, France. (Feb. 2006, p. 286) 23-2 7 Hyperbolic Geometry Workshop, Fields Institute, Toronto, 1 2-1 6 Function Theories in Higher Dimensions, Tampere Univ. Canada. (Nov. 2005, p. 1264) of Technology, Tampere, Finland. (Juni Jul. 2005, p. 674) 24-26 International Workshop on Post-Quantum Cryptography, 1 2-1 6 The Twenty-Second Annual Mathematical Problems in Katholieke Universiteit Leuven, Belgium. (Sept. 2005, p. 953) Industry Workshop, Needham, Massachusetts. (Dec. 2005, p. 1381) 25-27 Complex and Harmonic Analysis, an international con­ ,., 1 2-1 7 Steklov Mathematical Institute International Workshop ference, Thessaloniki, Greece. (Oct. 2005, p. 1088) on Mathematical Hydrodynamics, Steklov Mathematical Institute,

MARCH 2006 NOTICES OF THE AMS 377 Mathematics Calendar

Moscow, Russia. 20-22 International Conference on Mathematical, Statistical, Program: The workshop will cover recent research advances in the and Computer Methods in HIV/AIDS, Regal Kowloon Hotel, East theory of both viscous and inviscid fluids,including the study of the Kowloon, Hong Kong. (Feb. 2006, p. 286) Euler equations, the Navier-Stokes equations, and the equations 21-231CNPAA-2006: Mathematical Problems in Engineering and governing the evolution of waves on the surface of fluids. The Aerospace Sciences, Budapest, Hungary. (May 2005, p. 570) meeting will include leading senior researchers in this area and younger scientists, and will attempt not only to cover past research 2 2-24 2nd International Workshop on Graph-Theoretic Concepts accomplishments but point out new directions of investigation. in Computer Science, Bergen, Norway. (Dec. 2005, p. 1382) Organizing Committee: Walter Craig, Andrei Fursikov, Patrick 23-26 2006 International Conference on Topology and its Gerard, Sergei Kuksin, Armen Sergeev, Eugene Wayne. Applications, Aegion, Greece. (Jan. 2006, p. 69) Contact Information: Contact Ms. M.A. Pribyl, (preferably by e-mail). email: mh2006@mi. ras. ru. Mail Address: Steklov Mathematical 2 5-2 8 SIAM Conference on Discrete Mathematics, University of Institute, Gubkina 8, Moscow, 119991, Russia; Tel.: +7 (095) 135- Victoria, Victoria, British Columbia, Canada. (Dec. 2005, p. 1382) 22-91; Fax: +7 (095) 135-05-55. 25-28 INFORMS International Hong Kong 2006, Sheraton Hotel & Towers Hong Kong, Hong Kong, China. (Jun/Jul. 2005, p. 674) 1 3-1 7 The Fifteenth International Workshop on Matrices and Statistics, Uppsala, Sweden. (Jan. 2006, p. 68) 25-28 The Sixth AIMS Conference on Dynamical Systems, Differential Equations and Applications, University of Poitiers, '' 1 6-1 8 NKS 2006 Wolfram Science Conference, The Fairmont, Poitiers, France. (Jun/ Jul. 2005, p. 674) Washington, DC. Description: Stephen Wolfram's book A New Kind of Science 26-30 Algebraic Combinatorics: An International Conference introduced a new paradigm for doing science. From this has now in honor of Eiichi Bannai's 60th birthday, Sendai International grown an energetic community developing the paradigm not only Center, Sendai, Japan. (Dec. 2005, p. 1382) in science but also in technology, business, and the arts. The 26-30 Calibrations, AIM Research Conference Center, Palo Alto, NKS 2006 Wolfram Science Conference is a unique opportunity to California. (Sept. 2005, p. 954) connect with the leaders in the NKS community, explore exciting 27-July 3 International Commission on Mathematical Instruc­ developments now underway, and learn how to apply the latest tion: Challenging Mathematics In and Beyond the Classroom, NKS ideas and methods to your work. In addition to a wide range of Trondheirn, Norway. (Apr. 2005, p. 478) general and specialized presentations, the conference will feature real-time computer experiments by Stephen Wolfram, problem­ 29-July 4 21st International Conference on Operator Theory, solving labs, an art gallery, opportunities for students, and much West University, Timisoara, Romania. (Feb. 2006, p. 286) more. * 30-July 5 Computability in Europe 2006 (CiE 2006): Logical Host: Wolfram Research, Inc. Approaches to Computational Barriers, Swansea, Wales. Information: Phone: 1-800-WOLFRAM (965-3726) or +1-217-398- Organizer: Within the network "Computability in Europe" (CiE). 0700; email: nksconference@wolframscience. com; http: I /www . Focus: Is on aspects of logical approaches to computational wo1framscience.com/conference/2006. barriers. There will be tutorials on Logical Approaches to the P vs. NP problem (S. Buss) and "Quantum Algorithms: Standard Tools 16-20 CAIMS-MITACS Joint Annual Conference, York University, and New Models" (]. Kempe). Toronto, Ontario, Canada. (Jan. 2006, p. 68) Invited Speakers: J. Bergstra, L. Cardelli,]. W. Dawson,]. Krajicek, 19-23 Formal Power Series and Algebraic Combinatorics 2006, E. Mayordomo Camara, I. Nemeti, H. Schwichtenberg, and A. Catamaran Resort Hotel, Pacific Beach, San Diego, California. Weiermann. (Jan. 2006, p. 69) Special Sessions: Are planned on Challenges in Complexity, Com­ 19-23 Free Analysis, AIM Research Conference Center, Palo Alto, putable Analysis, Foundations of Programming, Godel Centenary: California. (Jun/Jul. 2005, p. 674) His Legacy for Computing, Mathematical Models of Computers and Hypercomputers, and Proofs and Computation. '' 19-23 Harmonic Analysis and Related Problems 2006 (HARP Information: For more information, visit http : I /www. cs. swansea. 2006), Zaros, Crete, Greece. ac . uk/ cie06/. Organizer: This meeting is organized in the context of the European Research Network HARP. July 2006 Plenary Speakers: Jon Bennett, Michael Christ, Guy David, M. Burak Erdogan, Alex Iosevich, Sergei Konyagin, Michael Lacey, Pertti 2-7 ICOTS 7, Working Cooperatively in Statistics Education, Mattila, Antonios Melas, Giancarlo Mauceri, Alan Mcintosh, Frank Salvador (Bahia), Brazil. (Mar. 2004, p. 361) · Merle, Detlef Mueller, Alexander Olevskii, Fulvia Ricci, Cristoph 3-7 Inverse Problems in Applied Sciences-toward break Thiele, Ana Vargas, Luis Vega, Jim Wright. through, University Conference Hall, Hokkaido University, Sap­ lnformation:Pleasevisithttp: I /fourier. math. uoc. gr;-harp2006. poro, Japan. (Nov. 2005, p. 1264) 1 9-23 Modern stochastics: theory and applications,KyivNational *5-8 Numerical Analysis and Approximation Theory, NAAT 2006, Taras Shevchenko University, Kyiv, Ukraine. (Sept. 2005, p. 953) Department of Applied Mathematics of Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania. 1 9-23 The International Summer School in Several Complex Topics: Functions approximation, Integral operators, Differential Variables, Szczyrk, Poland. (Dec. 2005, p. 1382) operators, Numerical analysis and stability methods, Positive oper­ 1 9-24 Hodge Theory, Venice International University, Venice­ ators, Rate of convergence, Splines, Wavelets, Stochastic processes, Island of San Servolo, Italy. (Feb. 2006, p. 286) Approximation of linear functionals, Honorary Chair: D. D. Stancu. Invited Speakers: Preliminary list: Francesco Altomare (Bary, Italy), 19-July 7 Computational Number Theory and Applications George Anastassiou (Memphis, USA), Borislav Bojanov (Sofia, Bul­ to Cryptography, University of Wyoming, Laramie, Wyoming. garia), Heiner Gonska (Duisburg, Germany), Gradirnir Milovanovic (Feb. 2006, p. 286) (Nis, Serbia), Jozsef Szabados (Budapest, Hungary), Peter Vertesi 1 9-3 0 SMS 2006/NATO Advanced Study lnstitute-Combi natorial (Budapest, Hungary). Optimization: Methods and Applications, Universite de Montreal, Scientific Committee: 0. Agratini, F. Altomare, P. Blaga, Gh. Coman, Montreal (QC) Canada. (Jan. 2006, p. 69) H. Gonska, G. Milovanovic.

378 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Mathematics Calendar

7-8 Second International Conference on Nonsmooth/Nonconvex 24-December 22 Noncommutative Geometry, Isaac Newton In­ Mechanics with Applications in Engineering, Faculty of Engineer­ stitute for Mathematical Sciences, Cambridge, England. (Aug. 2004, ing, Aristotle University, Thessaloniki, Greece. (Apr. 2005, p. 478) p. 835) 9-1 5 DIMACS Reconnect Conference 20Q6: Reconnecting Teach­ 2 5-2 7 International Conference on Mathematics, Institute of ing Faculty to the Mathematical Sciences Research Enterprise, Mathematics, National University of Mongolia, Ulaanbaatar, Mon­ Morgan State University, Baltimore, Maryland. (Jan. 2006, p. 69) golia. (Feb. 2006, p. 287) 1 0-1 2 SIAM Conference on Analysis of Partial Differential Equa­ 27-August 2 ASL European Summer Meeting (Logic Colloquium tions, Boston Park Plaza Hotel and Towers, Boston, Massachusetts. '06), Nijmegen, Netherlands. (Jun/Jul. 2005, p. 675) (Dec. 2005, p. 1382) '' 30-August 4 Institute of Mathematical Statistics (IMS) Annual 1 0-1 41nternational Conference on Analytic Topology, Lake Plaza Meeting and X Brazilian School of Probability (XEBP), Instituto Hotel, Rotorua, New Zealand. (Apr. 2005, p. 478) Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil. 1 0-14 Ninth International Conference on p-adic functional Topics: Statistics program (July 30-August 2), Fifth International analysis, University of Concepcion, Concepcion, Chile. (Aug. 2005, Symposium on Probability and its Applications (August 2-4). p. 787) IMS Special Invited Speakers: IMS Wald Lectures: Peter Hall; IMS Le 17-21 Classification theory for abstract elementary classes, Cam Lecture: Stephen Stigler; IMS Medallion Lectures: Greg Lawler, AlM Research Conference Center, Palo Alto, California. (Feb. 2006, Michael Woodroofe, Thomas Mountford. p. 286) EBP Invited Speakers: Vincent Beff ara (ENS Lyon); ]. van den Berg (CWI); Stella Brassesco (IVIC); Donald Dawson (Carleton); Paul 1 ?-August 4 International Conference and Instructional Work­ Dupuis (Brown Univ.); Vlada Limic (UBC); Jim Pitman (Univ. Calif., shop on Discrete Groups, The Morningside Center of Mathematics, Berkeley) Beijing, People's Republic of China. (Dec. 2005, p. 1382) Information: http : I lwww. imstat. orglmeetingsiiMS2006I. 1 ?-August 11 Clay Mathematics Institute 2006 Summer School: Arithmetic Geometry, Mathematisches Institut, Georg-August­ 30-August 5 International Conference on Radicals (ICOR-2006), Universitii.t, Gottingen, Germany. (Feb. 2006, p. 287) Kiev National Taras Shevchenko University, Kiev, Ukraine. (Dec. 2005, p. 1383) 1 ?-August 11 Spectral Theory and Partial Differential Equations, Isaac Newton Institute for Mathematical Sciences, Cambridge, 30 ~ August 24 Bayesian Nonparametric Regression: Theory, England. (Aug. 2004, p. 835) Methods and Applications, Isaac Newton Institute for Mathe­ matical Sciences, Cambridge, United Kingdom. (Oct. 2004, p. 1097) ,., 1 8-21 International Conference on Interdisciplinary Social Sci­ ences, University of the Aegean, Island of Rhodes, Greece. 31-August 2 2006 Symposium on Performance Evaluation of Description: The conference will examine the nature of disciplinary Computer and Telecommunication Systems, Calgary, Canada. and interdisciplinary practices across the social sciences, as well as (Dec. 2005, p. 1383) the relation of the social to the natural sciences, applied sciences 31-August 4 Numerical invariants of singularities and higher­ and the professions. dimensional algebraic varieties, AIM Research Conference Center, Information: For full details of the conference, including an online Palo Alto, California. (Jan. 2006, p. 70) callforpapersform, visithttp: I lwww. SocialSciencesConference. com. August 2006 '' 1-5 Ninth Meeting of New Researchers in Statistics and Proba­ '' 1 8-21 Thirteenth Workshop on Logic, Language, Information bility, University of Washington, Seattle, Washington. and Computation (WoLLIC'2006), Stanford, California. Description: The IMS Committee on New Researchers is organizing Description: This is the thirteenth ina series ofworkships intended a meeting of recent Ph.D. recipients in Statistics and Probability. to foster interdisciplinary research in pure and applied logic. The purpose of the conference is to promote interaction among new Deadline: For submission of papers is March 1, 2006. researchers primarily by introducing them to each other's research Information: http: I lwww. cin. ufpe. br;-wolliclwollic2006l. in an informal setting. As part of the conference, participants will present talks and posters on their research and discuss interests and 23-27 The Ninth International Conference on Integral Methods professional experiences over meals and social activities organized in Science and Engineering (IMSE-2006), Sheraton Fallsview Hotel through the meeting as well as by the participants themselves. The and Conference Centre, Niagara Falls, Ontario, Canada. (Aug. 2005, relationships established in this informal collegiate setting among p. 787) junior researchers are ones that may last a career (lifetime?!) The 24-28 2nd SIPTA School on Imprecise Probabilities, Rey Juan meeting is to be held prior to the 2006 Joint Statistical Meetings in Carlos University, Madrid, Spain. (Feb. 2006, p. 287) Seattle, WA. Information: http: I lwww. stat. ohio-state. edu;-pf ciNRCI. 24-28 Brazilian Operator Algebras Conference, Florianopolis, Brazil. (Feb. 2006, p. 287) 2-4 31st Sapporo Symposium on Partial Differential Equations, Department of Mathematics, Hokkaido University, Sapporo, Japan. '' 24-28 New Trends in Viscosity Solutions and Nonlinear POE, (Jan. 2006, p. 70) Instituto Superior Tecnico, Lisbon, Portugal. Information: http: I lwww. math. ist. utl. pt;-dgomesl 7-11 Partial Differential Equations on Noncom pact and Singular newtrendsl. Manifolds, University of Potsdam, Potsdam, Germany. (Feb. 2006, p. 287) 24-28 The Eleventh International Conference on Difference * 9-12 The Third International Conference on Neural, Parallel & Equations and Applications, Kyoto University, Kyoto, Japan. Scientific Computations, Morehouse College, Atlanta, Georgia. (Jan. 2006, p. 69) Topics: Computational methods on all aspects of Neural, Parallel, 24-August 4 Pan-American Advanced Studies Institute (PASI): and Scientific Computing such as Computational Methods of Nonlin­ Analysis and Probability in Quantum Physics, San Joaquin ear Systems, Algorithm Designs, Hardware/ Software Engineering, campus, Pontificia Catolica de Chile, Santiago, Chile. (Feb. 2006, Computer Modeling, Networking Dynamics, Neurodynamics, Pat­ p. 287) tern Recognition, Performance Measurements, Computer Vision,

MARCH 2006 NOTICES OF THE AMS 379 Mathematics Calendar

Imaging, Cognition, Speech Modeling, Computational Mathematics, * 4-8 Stochastic Analysis in Mathematical Physics, CI Univ. Lisbon, Biomedical Engineering, Artificial Intelligence, Systolic Algorithms, Lisbon, Portugal. . Evaluation and Prediction of Computer Complexes, Cluster Com­ Description: To illustrate the versatility of modern stochastic puting, VLSI Design, Computer Architectures, Simulation, ODL analysis in various fields and applied mathematics, with a special (Open Distance Learning) Systems, Systems Security, Combina­ emphasis on Mathematical Physics. Among the themes covered torics, Graph Theory, Fuzzy Systems, and Simulation. in this meeting are: Hydrodynamics and turbulence, Conformal Deadlines: Submission of article (on or before): March 31, 2006. invariance, Random environment, Applications of large deviations, Acceptance of article: April 30, 2006. Camera-Ready paper: May 15, Quantum field theory. 2006. Pre-Registration: (on or before April 30, 2006) US $225.00, Scientific Committee: G. Ben Arous (Lausanne), A. B. Cruzeiro (Students: US $125.00). Registration: (after May 1, 2006) US $250.00, (Lisbon), Y. Le Jan (Paris), ]. C. Zambrini (Lisbon). (Students: US $150.00). Banquet: August 10, 2006. Motel Cut off Information: http : I lgfm. cii. fc. ul. ptlevents-enlsamp2006l. Date: July 1, 2006. Information: Contact M. Sambandham, ICNPSC3, Department of 4-29 The Painleve Equations and Monodromy Problems, Issac Mathematics, Morehouse College, Atlanta, Georgia 30314; email: Newton Institute for Mathematical Sciences, Cambridge, England. icnpsc3~yahoo.com;http : llwww . dynamicpublishers.comiNPSC3. (Jan. 2006, p. 70) 9-12 SIAM Conference on Nonlinear Waves and Coherent Struc­ * 1 2-1 5 Twenty-firstAnnuallEEESymposium on Logic in Computer tures, University of Washington, Seattle, Washington. (Jan. 2006, Science (LICS 2006), Seattle, Washington. p. 70) Description: The UCS Symposium is an annual international forum 1 1-1 5 Groups of Diffeomorphisms 2006, University of Tokyo, on theoretical and practical topics in computer science that relate Tokyo, Japan. (Jan. 2006, p. 70) to logic in a broad sense. LICS 2006 will be held as part of the Fourth Federated Logic Conference (FLoC'06). * 12-1 ?International Conference on Differential Equations, Dedi­ Invited Speakers: A. Blass, A. Gordon, and 0. Kupferman. There cated to the 1 OOth Anniversary of Ya. B. Lopatynsky, Ivan Franko will also be a joint LICSI RTAI SAT plenary lecture given by R. Bryant. National University of Lviv, Lviv, Ukraine. lnformation:http: I lwww .lfcs. informatics. ed. ac. ukllics.For Topics: General theory of differential equations, Algebraic meth­ further information about FLoC'06, see http: I lwww. informatik. ods in the theory of differential equations, Analytical theory of hu-berlin.dellicsl. differential equations, Applications of the theory of. differential equations. * 1 3-1 9 Workshop on Triangulated Categories, University of Leeds, Invited Speaksers: V. Barbu (Romania), H. Beresticki (France), M. United Kingdom. Chi pot (Switzerland), j.l. Diaz (Spain), H. Engl (Austria), A. Friedman Aim: To bring together researchers from many parts of mathe­ (USA), A. M. Filirnonov (Russia), Y. Giga (Japan), A. L. Gladkov matics who all use triangulated categories and related structures. (Belarus), M. L. Gorbachuk (Ukraine), S.D. Ivasyshen (Ukraine), N. V. Hopefully, the event will promote interactions across traditional Jitaraeu (Moldova), R. Kersner (Hungary), E. Ya. Khruslov (Ukraine), borders. Moreover, it should give postgraduate students and post V. A. Kondratiev (Russia), A. I. Kozhanov (Russia), A. Lorenzi (Italy), docs the opportunity to learn about exciting modern developments. F. Murat (France) A. D. Myshkis (Russia), A. Prilepko (Russia), V. Organizers: Thorsten Holm, Peter Jorgensen, Raphael Rouquier. Pukhnachov (Russia), A. Shishkov (Ukraine), L. Veron (France), H. Preliminary list of speakers (to be confirmed): Paul Balmer Zoladek (Poland). (Zurich), John Greenlees (Sheffield), Srikanth Iyengar (Lincoln), lnformation:email:ICL100~franko . lviv . ua;http: I lwww. franko. Bernhard Keller (Paris), Henning Krause (Paderborn), Ralf Meyer lviv.ualfacultylmechmatiDepartmentsiConflindex.htm. (Muenster), Arnnon Neeman (Canberra), Dmitri Orlov (Moscow), jeremy Rickard (Bristol), Stefan Schwede (Bonn). December 2006 Information: http: I lwww .maths . leeds. ac. uklpurelalgebral 1 5-1 7 International Conference on Computer & Information TriCat06. html. Science (ICCIS'2005), Fort Panhala, Kolhapur, India. (Dec. 2004, p. 1379) 14-1 8 International Conference on Spectral Theory and Global Analysis, Carl von Ossietzky University, Oldenburg, Germany. 16-20 DION 2005: An International Conference on Diophantine (Feb. 2006, p. 287) · Equations: in honour of Professor T. N. Shorey on his 60th Birthday, Tata Institute of Fundmental Research, Mumbai, India. 1 4-1 8 Seventh International Conference on Monte Carlo and (Juni Jul. 2005, p. 675) Quasi-Monte Carlo Methods in Scientific Computing (MCQMC 2006), Ulm, Germany. (Nov. 2005, p. 1264) 26-january 6, 2006 CIMPA School on Commutative Algebra, Institute of Mathematics, Hanoi, Vietnam. (Juni Jul. 2005, p. 675) 1 6-1 9 First announcement: Satellite Conference on Algebraic I Geometry, Segovia, Spain. (Jan. 2006, p. 70) January 2007 1 6-1 9 Trends and Challenges in the Calculus of Variations and 8-June 29 Analysis on Graphs and its Applications, Isaac Newtoh its Applications, Toledo, Spain. (Jan. 2006, p. 70) Institute for Mathematical Sciences, Cambridge, UK. (Nov. 2005, p. 1264) . ! 22-30 International Congress of Mathematicians, Madrid, Spain. (Nov. 2005,p. 1264) 1 5-July 6 Highly Oscillatory Problems: Computation, Theofv and Application, Isaac Newton Institute for Mathematical ciences, September 2006 Cambridge, UK. (Nov. 2005, p. 1264 (Dec. 2005, p. 1383))' i 1-4 Topics in Mathematical Analysis and Graph Theory, Univer­ sity of Belgrade, Faculty of Electrical Engineering, Department of Applied Mathematics, Serbia and Montenegro. (Dec. 2005, p. 1383) 4-8 Barcelona Analysis Conference, University of Barcelona, Barcelona, Spain. (Jan. 2006, p. 70) 4-8 Satellite Conference on Differential Equations and Singulari­ ties, in honor of J. M. Aroca's 60th birthday, Tordesillas(Valladolid, Spain). (Juni Jul. 2005, p. 675) '

380 NOTICES OF 1HE AMS VOLUME 53, NUMBER 3 New Publications Offered by the AMS

vector fields, dessins d'enfants, and circle packings; Analysis 1- T. Rogers, Jr., Siegel disks whose boundaries have only two complementary domains; K. A. Roth, Non-uniform porosity for a subset of some Julia sets; B. Skorulski, The existence of ------Complex Dynamics conformal measures for some transcendental mer.omorphic CONTEMPORARY functions. ------MATHEMATICS------Twenty-Five Years after Contemporary Mathematics, Volume 396 Complex Dynamics the Appearance of the March 2006, approximately 208 pages, Softcover, ISBN 0-8218- Twenty-Five Years Mandelbrot Set 362 5-0, 2000 Mathematics Subject Classification: 37F45, 37F50; after the Appearance 37F10, 30005, 37035, 54Fl5, All AMS members US$47, List of the Mandelbrot Set Robert L. Devaney and US $ 59, Order code CONM/ 396 Robert L. Devaney Undo Keen linda Keen, Editors Editors @ Chaotic behavior of (even the A Categorical ------simplest) iterations of polynomial M EMOIRS "' r '~•' ~JI!I\l"'<~«:-~ &; t, maps of the complex plane was of the \m~ru~n ~1<>th~m•tt< •' 'w< tdy Approach to known for almost one hundred years due to the pioneering work of Farou, Julia, and their A Categorical Approach Imprimitivity contemporaries. However, it was only twenty-five years ago to Imprimitivity Theorems for C*- Dyna mical Systems that the first computer generated images illustrating Theorems for Siegfried Echlerhoff properties of iterations of quadratic maps appeared. These S. Kali szewsk i C'''-Dynamical John Quigg images of the so-called Mandelbrot and Julia sets immediately lain Raeburn resulted in a strong resurgence of interest in complex Systems dynamics. The present volume, based on the talks at the Siegfried Echterhoff, conference commemorating the twenty-fifth anniversary of the S. Kaliszewski, John Quigg, appearance of Mandelbrot sets, provides a panorama of current research in this truly fascinating area of mathematics. and lain Raeburn Contents: D. K. Childers, J. C. Mayer, H. M. Tuncali, and Contents: Introduction; Right-Hilbert bimodules; The E. D. Tymchatyn, Indecomposable continua and the Julia sets categories; The functors; The natural equivalences; of rational maps; E. Bedford and 1- Smillie, The Henan family: Applications; Appendix A. Crossed products by actions and The complex horseshoe locus and real parameter space; coactions; Appendix B, The imprimitivity theorems of Green R. L. Devaney, Baby Mandelbrot sets (!.domed with halos in and Mansfield; Appendix C. Function spaces; Appendix. families of rational maps; R. L. Devaney, M. Holzer, and Bibliography. I D. Uminsky, Blowup points and baby Mandelbrot sets for Memoirs of the American Mathematical Society, Volume 180, singularly perturbed rational maps; R. Dujardin, Some Number 850 remarks on the connectivity of Julia sets for 2-dimensional diffeomorphisms; S. L. Hruska, Rigorous numerical studies of · March 2006, 169 pages, Softcover, ISBN 0-8218-3857-1, LC the dynamics of polynomial skew products of ([ 2 ; L. Keen, 2005057160, 2000 Mathematics Subject Classification: 46L55, Open problems; L. Keen and N. Lakic, Accumulation points of Individual member US$39, List US$65, Institutional member iterated function systems; L. Keen and S. Yuan, Parabolic US$52, Order code MEM0/ 180/ 850 perturbations of the family A tan z; K. M. Pilgrim, Polynomial

MARCH 2006 NOTICES OF THE AMS 381 New Publications Offered by the AMS

Recent Advances in Applications Operator-Related

Q.~~\ $...~ ~~~"' fu,u.c.ti<\U. 1:\.\.~(\~)\ in Operator-Related Discrete Methods in Function Theory Alec L. Matheson, Lamar DIMACS ~ fu l)hotmoe: M a ~g Alec L. MathesOn University, Beaumont, TX, and~ IConiputff~ Epidemiology Michael I. Stessln Rlchold M. Timoney Michael I. Stessin, State Editors James Abello and University of New York (SUNY), Diacrete Graham Cormode, Editors Albany, NY, and Methods in ------Epidemiology Studies of the spread and containment A p J '-'~ r ~~ fi ~ "''"'"I Richard M. Timoney, Trinity College, Dublin, Ireland, of disease rely at heart on a variety of mathematical and computational Editors techniques. This collection aims to The articles in this book are based on talks at a conference introduce the fundamentals of devoted to interrelations between function theory and the epidemiology and to showcase contemporary theory of operators. The main theme of the book is the role of work using discrete Alexandrov-Clark measures. Two of the articles provide the mathematical techniques. Introductory chapters explain the introduction to the theory of Alexandrov-Clark measures and fundamental concepts of epidemiology, the basic tools to its applications in the spectral theory of linear operators. provided by mathematics and computer science, and some of the outstanding open The remaining articles deal with recent results in specific problems in the area. Contributed articles then directions related to the theme of the book. highlight particular problems in monitoring disease outbreaks, vaccination strategies, and modelling Contents: A. Poltoratski and D. Sarason, Aleksandrov-Clark disease survival factors, and successfully apply techniques measures; A. Matheson and M. Stessin, Applications of such as formal concept analysis, support vector machines, spectral measures; J. Agler and J. E. McCarthy, Parametrizing random graph models, and systems of differential equations. distinguished varieties; j. T. Anderson and J, Wermer, Contents: J. Abello, G. Cormode, D. Fradkin, D. Madigan, Approximation by CR functions on the unit sphere in 1(2 ; s. M. Buckley and D. VukotiC , Superposition operators and 0. Melnik, and I. Muchnik, Selected data mining concepts; D. Schneider, Descriptive the order and type of entire functions; K. Dyakonov and epidemiology: A brief introduction; D. Shannon, Biostatistical D. Khavinson, Smooth functions in star-invariant subspaces; w. challenges in molecular data analysis; L. Hirschman and L. s. R. Garcia, Conjugation and Clark operators; D. Girela, A E. Damianos, Mining online media for global disease class of conformal mappings with applications to function outbreak monitoring; D. Ozonoff, A. Pogel, and T. Harman, Generalized contingency tables and spaces; H. Koo and W. Smith, Composition operators betwe~n concept lattices; j. Abello A. Bergman spaces of functions of several vanables; M. J. Martm and Pogel, Graph partitions and concept lattices; K. Desai, and D. VukotiC , Isometries of some classical function spaces M.-C. Boily, B. Masse, and R. M. Anderson, Using transmission dynamics models to among the composition operators; A. Montes-Rodriguez a~d validate vaccine efficacy measures prior to conducting HIV s. A. Shkarin, New results on a classical operator; J. Pau, Stze vaccine A. conditions to be in a finitely generated ideal of H oo ; efficacy trials; Vazquez, Causal tree of disease transmission and the spreading w. T. Ross, The classical Dirichlet space; E. Saksman and of infectious diseases; Eubank, V. S. Anil Kumar, A. C. Sundberg, Comparing topologies on the space of s. M. V. Marathe, Srinivasan, and N. Wang, Structure of composition operators; W. Smith, Brennan's conjecture for social contact networks and their impact weighted composition operators. on epidemics; J. Abello and M. Capalbo, Random graphs (and the spread of infections in a social network); Contemporary Mathematics, Volume 393 S. G. Hartke, Attempting to narrow the integrality gap for the February 2006, 214 pages, Softcover, ISBN 0-8218-3925-X, LC firefighter problem on trees; j. Li, I. Muchnik, and 2005053622, 2000 Mathematics Subject Classification: 46£15, D. Schneider, Influences on breast cancer survival via SVM , 46£20, 47B33, 47B38, 47B32, 30E20, 30£25, All AMS members classification in the SEER database; D. Fradkin, I. Muchnik, US$55, List US$69, Order code CONM/ 393 P. Hermans, and K. Morgan, Validation of epidemiological models: Chicken epidemiology in the UK; Index. DIMACS: Series in Discrete Mathematics and Theoretical' Computer Science, Volume 70 ! April 2006, approximately 246 pages, Hardcover, ISBN 0-8218- 3754-0, 2000 Mathematics Subject Classification: 92D30; 68R01, All AMS members US$71, List US$89, Order code DIMACS/ 70

382 NOTICES OF THE AMS VOLUME 53, NUMBER 3 New Publications Offered by the AMS

May 2006, approximately 241 pages, Hardcover, ISBN 0-8218- Disease Evolution 3753-2, 2000 Mathematics Subject Classification: 34C60, 35Q80, Models, Concepts, and 62P10,92B05,92C50,92C60,92D10,92D15,92D25,92D30, Data Analyses All AMS members US$71, List US$89, Order code DIMACS/ 71 Zhilan Feng, Ulf Dieckmann, and Simon Levin, Editors The Radon Proceedings or Symposia in Infectious diseases are continuing to APPLIED M ATHEMATI~ S Transform, Inverse threaten humankind. While some Volume63 diseases have been controlled, new The Raddn'Transform, Problems, and diseases are constantly appearing. Inverse Problems, Tomography Others are now reappearing in forms and Tomography that are resistant to drug treatments. A capacity for continual Gestur 6Jafsson Gestur Olafsson and re-adaptation furnishes pathogens with the power to escape Eric Todd Quinto Eric Todd Quinto, Editors our control efforts through evolution. This makes it Editors imperative to understand the complex selection pressures that Since their emergence in 191 7, are shaping and reshaping diseases. Modern models of tomography and inverse problems evolutionary epidemiology provide powerful tools for creating, remain active and important fields expressing, and testing such understanding. that combine pure and applied mathematics and provide strong interplay between diverse Bringing together international leaders in the field, this mathematical problems and applications. The applied side is volume offers a panoramic tour of topical developments in best known for medical and scientific use, in particular, understanding the mechanisms of disease evolution. The medical imaging, radiotherapy, and industrial non-destructive volume's first part elucidates the general concepts underlying testing. Doctors use tomography to see the internal structure models of disease evolution. Methodological challenges of the body or to find functional information, such as addressed include those posed by spatial structure, stochastic metabolic processes, noninvasively. Scientists discover defects dynamics, disease phases and classes, single- and multi-drug in objects, the topography of the ocean floor, and geological resistance, the heterogeneity of host populations and tissues, information using X-rays, geophysical measurements, sonar, and the intricate coupling of disease evolution with between­ or other data. host and within-host dynamics. The book's second part shows how these methods are utilized for investigating the dynamics This volume, based on the lectures in the Short Course The and evolution of specific diseases, including HIV I AIDS, Radon Transform and Applications to Inverse Problems at the tuberculosis, SARS, malaria, and human rhinovirus infections. American Mathematical Society meeting in Atlanta, GA, January 3-4, 2005, brings together articles on mathematical This volume is particularly suited for introducing young aspects of tomography and related inverse problems. The scientists and established researchers with backgrounds in articles cover introductory material, theoretical problems, and mathematics, computer science, or biology to the current practical issues in 3-D tomography, impedance imaging, local techniques and challenges of mathematical evolutionary tomography, wavelet methods, regularization and approximate epidemiology. inverse, sampling, and emission tomography. All contributions Contents: Model infrastructure: M. Boots, M. Kamo, and are written for a general audience, and the authors have A. Sasaki, The implications of spatial structure within included references for further reading. populations to the evolution of parasites; T. Day and Contents: E. T. Quinto, An introduction to X-ray tomography S. Gandon, Insights from Price's equation into evolutionary and radon transforms; A. K. Louis, Development of algorithms epidemiology; R. D. Holt and M. Barfield, Within-host pathogen in computerized tomography; A. Faridani, Fan-beam dynamics: Some ecological and evolutionary consequences of tomography and sampling theory; P. Kuchment, Generalized transients, dispersal mode, and within-host spatial transforms of radon type and their applications; heterogeneity; J. K. Kelly, Evolutionary and dynamic models of P. Massopust, Inverse problems in pipeline inspection; infection with internal host structure; W. M. Getz and L. Borcea, Robust interferometric imaging in random media; J. 0. Uoyd-Smith, Basic methods for modeling the invasion Index. and spread of contagious diseases; Applications to specific diseases: W. M. Getz, J. 0. Uoyd-Smith, P. C. Cross, S. Bar­ Proceedings of Symposia in Applied Mathematics, Volume 63 David, P. L. Johnson, T. C. Porco, and M.S. Sanchez, Modeling March 2006, 158 pages, Hardcover, ISBN 0-8218-3930-6, LC the invasion and spread of contagious diseases in 2005057107, 2000 Mathematics Subject Classification: 92C55, heterogeneous populations; M. A. Charleston and 44Al2, 65R32, 65R10, 94A20, 42C40, All AMS members A. P. Galvani, A cophylogenetic perspective on host-pathogen US$31, List US$39, Order code PSAPM/63 evolution; Z. Feng and L. Rong, The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens; W. J. Koppelman and F. R. Adler, Do rhinoviruses follow the neutral theory? The role of cross-immunity in maintaining the diversity of the common cold; A. L. Uoyd and D. Wodarz, Drug resistance in acute viral infections: Rhinovirus as a case study; D. L. Smith, M. F. Boni, and R. Laxminarayan, Dynamics and control of antibiotic resistance in structured metapopulations. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, Volume 71

MARCH 2006 NOTICES OF THE AMS 383 New Publications Offered by the AMS

Introduction to the Differential Equations Introduction to Mathematics of the Mathematics Finance of Finance Holder Continuity of R. J. Williams, University of ME¥,;9IRS California, San Diego, La jolla, American ~lathem a tlcal Soe1ety Weak Solutions to California, USA Holder Continuity Subelliptic Equations of Weak Solutions The modern subject of mathematical to Subelliptic Equations with Rough lEi---- finance has undergone considerable with Rough Coefficients development, both in theory and ErlcT. Sawyer Coefficients Richard L. Wheeden practice, since the seminal work of Eric T. Sawyer and Black and Scholes appeared a third of a century ago. This book is intended as an introduction to some elements of the Richard L. Wheeden AmerleanMatllemallcalS<>clety theory that will enable students and researchers to go on to Contents: Introduction; Comparisons read more advanced texts and research papers. of conditions; Proof of the general The book begins with the development of the basic ideas of subellipticity theorem; Reduction of the proofs of the rough hedging and pricing of European and American derivatives in diagonal extensions of Hi.irmander's theorem; Homogeneous the discrete (i.e., discrete time and discrete state) setting of spaces and subrepresentation inequalities; Appendix; binomial tree models. Then a general discrete finite market Bibliography. model is introduced, and the fundamental theorems of asset Memoirs of the American Mathematical Society, Volume 180, pricing are proved in this setting. Tools from probability such Number 847 as conditional expectation, filtration, (super)martingale, equivalent martingale measure, and martingale representation March 2006, 157 pages, Softcover, ISBN 0-8218-3826-1, LC are all used first in this simple discrete framework. This 2005057158, 2000 Mathematics Subject Classification: 35B65, provides a bridge to the continuous (time and state) setting, 35Dl0, 35H20, Individual member US$38, List US$63, which requires the additional concepts of Brownian motion Institutional member US$ 50, Order code MEM0/ 180/ 847 and stochastic calculus. The simplest model in the continuous setting is the famous Black-Scholes model, for which pricing and hedging of European and American derivatives are developed. The book concludes with a description of the fundamental theorems for a continuous market model that generalizes the simple Black-Scholes model in several General and directions. Interdisciplinary This item will also be of interest to those working in probability. Contents: Financial markets and derivatives; Binomial model; Finite market model; Black-Scholes model; Multi-dimensional Black-Scholes model; Conditional expectation and LP -spaces; Mathematical Discrete time stochastic processes; Continuous time stochastic Sciences Professional processes; Brownian motion and stochastic integration; Bibliography; Index. Directory, 2006 Graduate Studies in Mathematics, Volume 72 This annual directory provides a handy reference to various April 2006, 152 pages, Hardcover, ISBN 0-8218-3903-9, LC organizations in the mathematical 2005055575, 2000 Mathematics Subject Classification: 60G42, sciences community. Listed in the 60G44, 60]65, 62P05, 91B28; 58]65, 60]60, All AMS members directory are the following: officers of US$31, List US$39, Order code GSM/ 72 over thirty professional mathematical organizations; addresses of selected government agencies; academic departments in the mathematical sciences; and alphabetic listings of colleges and universities. March 2006, approximately 160 pages, Softcover, ISBN 0-8218- 3899-7, List US$ 55, Institutional member US $44, Order code PRODIR/2006

384 NOTICES OF THE AMS VOLUME 53, NUMBER 3 New Publications Offered by the AMS

groups, asymptotic cones, Thompson's group, Nielsen fixed CONTEMPORARY '?:, 150 Years of point theory, homology, groups acting on trees, groups ------' MATHEMATICS·'· '~i Mathematics at generated by finite automata, iterated monodromy groups, random walks on finitely generated groups, heat kernels, and 150 Years Washington currents on free groups. of Mathematics at Washington University University in Contents: G. N. Arzhantseva, A dichotomy for finitely generated in St. Louis subgroups of word hyperbolic groups; L. Bartholdi and Z. Sunili:, Gory R. Jensen St. Louis Steven G. Krontz Some solvable automaton groups; A. V. Borovik and Editors Gary R. Jensen and A. G. Myasnikov, Quotient tests and random walks in Steven G. Krantz, Editors computational group theory; K. S. Brown, The homology of Richard Thompson's group F; K.-U. Bux and R. Perez, On the Articles in this book cover a wide growth of iterated monodromy groups; A. N. Dranishnikov, On range of important topics in Bestvina-Mess formula; K. Dykema, Symmetric random walks on mathematics and are based on talks given at the conference certain amalgamated free product groups; D. Farley, Homology commemorating the 150th anniversary of Washington of tree braid groups; R. Geoghegan and F. Guzman, Associativity University in St. Louis. The volume is prefaced by a brief and Thompson's group; V. S. Guba, Traveller salesman property history of the Washington University Department of and Richard Thompson's group F; S. V. Ivanov, On groups with Mathematics, a roster of those who received the PhD degree periodic products of commutators; I. Kapovich, Currents on free from the department, and a list of the Washington University groups; A. Karlsson and M. Neuhauser, Heat kernels, theta Department of Mathematics faculty since the founding of the identities, and zeta functions on cyclic groups; M. R. Kelly, The university. Nielsen fixed point structure for homotopy idempotents on Contents: S. G. Krantz, An anecdotal history of the surfaces; T. Napier and M. Ramachandran, Thompson's group F Washington University Mathematics Department; G. R. Jensen is not Kahler; A. Yu. Ol'shanskii and M. V. Sapir, Groups with and S. G. Krantz, Dissertation list; G. R. Jensen and non-simply connected asymptotic cones; D. V. Osin, Relative S. G. Krantz, History of the Mathematics faculty; R. L. Bryant, Dehn functions of amalgamated products and HNN-extensions; Geometry of manifolds with special holonomy: "100 years of A. M. Brurmer and S. N. Sidki, Endomorphisms of the finitary holonomy"; R. R. Coifman, Geometric harmonic analysis in group of isometries of the binary tree. high dimensions: Challenges and opportunities; J.P. D'Angelo, Contemporary Mathematics, Volume 394 An analogue of Hilbert's seventeenth problem in one complex March 2006, 234 pages, Softcover, ISBN 0-8218-3756-7, LC dimension; J. J. Kohn, Hypoelliptic second order equations 2005057078, 2000 Mathematics Subject Classification: 20-06; that lose derivatives; H. B. Lawson, Jr., Stokes' theorem and 55-06, 57-06, All AMS members US$55, List US $69, Order minimal surfaces; J, D. McNeal, L 2 estimates on twisted code CONM/ 394 Cauchy-Riemann complexes; Y. Meyer, From wavelets to atoms; R. Schoen, The mathematics of general relativity: Problems and progress; R. R. Coifman, A. Bonami, R. Bryant, and J.P. D'Angelo, Panel discussion. Twisted Tensor MEIYJ.9IRS Contemporary Mathematics, Volume 395 Americo.n l\lathemat lcalSodety Products Related to March 2006, 145 pages, Softcover, ISBN 0-8218-3603-X, LC Twist ed Ten sor Products the Cohomology of 2005057233, 2000 Mathematics Subject Classification: 00-02; Rela ted to the Cohomology 32T27, 32T05, 35N15, 53C29, 83C05, 49Q05, 81Q70, 83C10, of the Classify ing Spaces the Classifying 42B20, 42B99, All AMS members US$39, List US$49, Order of Loop Groups Katsuhiko Kuribayashi Spaces of Loop code CONM/ 395 Mamon1 Mimura Tetsu Nishimoto Groups Katsuhiko Kuribayashi, Mamoru Mimura, and Geometry and Topology Tetsu Nishimoto Contents: Introduction; The mod 2 cohomology of BLS 0 (n) ; The mod 2 cohomology of BLG for G = Spin(n) (7 .::: n .::: 9); CONTEMPORARY ;. Topological and The mod 2 cohomology of BLG for G = Gz,F4; A MATHEMATICS - •• Asymptotic Aspects multiplication on a twisted tensor product; The twisted tensor ------product associated with H * (Spin(N) ; 7L/ 2); A manner for Topological and of Group Theory calculating the homology of a DGA; The Hochschild spectral Asymptotic Aspects Rostislav Grigorchuk, sequence; Proof of Theorem 1.6; Computation of a cotorsion of Group Theory product of H * (Spin(10); 7L/ 2) and the Hochschild homology Rostlslav Grlgorchuk Michael Mihalik, Mark Sapir, Michael Mihalik of H * (BSpin(lO); 7L/ 2); Proof of Theorem 1.7; Proofs of Mark Sapir Zoran Su nil< and Zoran SuniK., Editors Proposition 1.9 and Theorem 1.10; Appendix; Bibliography. Editorsn '10c' The articles in this volume are based Memoirs of the American Mathematical Society , Volume 180, ------on the talks given at two special Number 849 Amellcll'l fAIUhemalleo.! Soci~ 'r: ~·1 sessions at the AMS Sectional meetings ""'~ --"'"*~- "-']: YL""-.!0»~~~"""01 March 2006, 85 pages, Softcover, ISBN 0-8218-3856-3, LC held in 2004. The articles cover various 2005057159, 2000 Mathematics Subject Classification: 55T20, topological and asymptotic aspects of 57T30, 55S05, Individual member US$33, List US $ 55, group theory such as hyperbolic and relatively hyperbolic Institutional member US $44, Order code MEM0/ 180/ 849

MARCH 2006 NOTICES OF THE AMS 385 New Publications Offered by the AMS

Equivalences of MEMOIRS Mathematical Physics of the Amulco.nMathemat>culSoeooty Classifying Spaces

Equivalences Completed at the of Classifying Spaces The Principle of the Completed Prime Two at the Prime Two Bob Oliver Fermionic Projector Bob Oliver Felix Finster, Universitdt Contents: Introduction; Higher limits over orbit categories; Reduction to Regensburg, Germany AmerlconMnthemotle<>lSocoety simple groups; A relative version of The "principle of the fermionic A -functors; Subgroups which contribute projector" provides a new to higher limits; Alternating groups; mathematical framework for the Groups of Lie type in characteristic two; Classical groups of formulation of physical theories and Lie type in odd characteristic; Exceptional groups of Lie type is a promising approach for physics in odd characteristic; Sproadic groups; Computations of beyond the standard model. This book lim1 (Zc); Bibliography. begins with a brief review of relativity, Memoirs of the American Mathematical Society, Volume 180, relativistic quantum mechanics, and classical gauge theories, Number 848 emphasizing the basic physical concepts and mathematical foundations. The external field problem and Klein's paradox March 2006, 102 pages, Softcover, ISBN 0-8218-3828-8, LC are discussed and then resolved by introducing the fermionic 2005058189, 2000 Mathematics Subject Classification: 55R37; projector, a global object 55P60, 20D05, Individual member US$35, List US$ 59, in space-time that generalizes the notion of the Dirac sea. At the mathematical Institutional member US$47, Order code MEM0/180/848 core of the book is a precise definition of the fermionic projector and the use of methods of hyperbolic differential equations for detailed analysis. The fermionic projector makes it possible to On Maps from Loop formulate a new type of variational principle in space-time. MEMOIRS The mathematical tools are developed for the analysis of the of ttte American Mnthemoheal Soeody Suspensions to Loop corresponding Euler-Lagrange equations. A particular variational principle is proposed that gives rise to an effective On Maps from Spaces and the Loop Suspensions interaction which shows many similarities to the interactions to Loop Spaces and Shuffle Relations on of the standard model. the Shuffle Relations on the Cohen Groups the Cohen Groups The main chapters of the book are easily accessible for Jle Wu beginning graduate students in mathematics or physics. Jie Wu Several appendices provide supplementary material, which will be useful to the experienced researcher. Contents: Introduction; Maps from loop suspensions to loop spaces; Titles in this series are copublished with International Press, Shuffle relations; Bibliography. Cambridge, MA. Memoirs of the American Mathematical Society, Volume 180, Contents: The principle of the Fermionic projector-A new Number 851 mathematical model of space-time; Preliminaries; The Fermionic projector in the continuum; The principle of the March 2006, 64 pages, Softcover, ISBN 0-8218-3875-X, LC Fermionic projector; The continuum limit; The Euler-Lagrange 2005058190, 2000 Mathematics Subject Classification: 55Pxx; equations in the vacuum; The dynamical gauge group; 18G20, 20C05, 55Qxx, 55Uxx, Individual member US$30, List Spontaneous block formation; The effective gauge group; US$ 50, Institutional member US$40, Order code Connection to the Fock space formalism; Some formulas of MEM0/ 180/851 the light-cone expansion; Normalization of chiral fermions; The regularized causal perturbation theory; Linear independence of the basic fractions; The commutator [P, Q]; Perturbation calculation for the spectral decomposition of P(x,y)P(y,x); Bibliography; Index; Notation index. AMS/IP Studies in Advanced Mathematics, Volume 35 February 2006, 302 pages, Hardcover, ISBN 0-8218-3974-8, LC 2005053035, 2000 Mathematics Subject Classification: 83-02, 81-02; 83C47, 81T20,81T25, 81T27,81T13, 83£99, 83C60,Ail AMS members US$52, List US$65, Order code AMSIP/35

386 NOTICES OF THE AMS VOLUME 53, NUMBER 3 New Publications Offered by the AMS

Number Theory fMiMBWft M New AMS-Distributed Valuations, Publications Valuations, Orderings, and Orderings, Milnor K-Theory and Milnor K-Theory Ido Efrat Algebra and Algebraic

ldo Efrat This monograph is a comprehensive exposition of the modern theory of Geometry valued and ordered fields. It presents the classical aspects of such fields: their arithmetic, topology, and Galois Cohomologie locale des faisceaux theory. Deeper cohomological aspects are studied in its last part in an coherents et theoremes de elementary manner. This is done by means of the newly developed theory of generalized Milnor K-rings. The book Lefschetz locaux et globaux emphasizes the close connections and interplay between (SGA 2) valuations and orderings, and to a large extent, studies them in a unified manner. Seminaire de geometrie algebrique du The presentation is almost entirely self-contained. In particular, Bois Marie 1962 the text develops the needed machinery of ordered abelian Alexander Grothendieck groups. This is then used throughout the text to replace the more classical techniques of commutative algebra. Likewise, This volume is a new updated edition of the book, the book provides an introduction to the Milnor K-theory. Cohomologie locale des faisceaux coherents et theonfmes de The reader is introduced to the valuation-theoretic techniques Lefschetz locaux et globaux (SGA 2), Advanced Studies in Pure as used in modern Galois theory, especially in applications to Mathematics 2, North-Holland Publishing Company, birational anabelian geometry, where one needs to detect Amsterdam, 1968, by A. Grothendieck, et al. valuations from their "cohomological footprints". These In this monograph are given necessary and sufficient powerful techniques are presented here for the first time in a conditions for the finiteness of the local cohomology sheaves unified and elementary way. of coherent sheaves. These results provide algebraization Contents: Part I. Abelian Groups: Preliminaries on abelian theorems leading in particular, with the help of purity results groups; Ordered abelian groups; Part II. Valuations and also proved in the text, to Lefschetz's theorem for both the orderings; Valuations; Examples of valuations; Coarsenings of fundamental group and the Picard group. valuations; Orderings; The tree of localities; Topologies; A publication of the Societe Mathematique de France, Marseilles (SMF), Complete fields; Approximation theorems; Canonical distributed by the AMS in the U.S., Canada, and Mexico. Orders from valuations; Valuations of mixed characteristics; Part III. Galois other countries should be sent to the SMF. Members of the SMF receive theory: Infinite Galois theory; Valuations in field extensions; a 30% discount from list. Decomposition groups; Ramification theory; The fundamental Contents: Introduction; Les invariants cohomologiques equality; Hensel's lemma; Real closures; Coarsening in globaux et locaux relatifs a un sousespace ferme; Application algebraic extensions; Intersections of decomposition groups; aux faisceaux quasi-coherents sur les preschemas; Invariants Sections; Part IV. K-rings: K-structures; Milnor K-rings of fields; cohomologiques et profondeur; Modules et foncteurs Milnor K-rings and orderings; K-rings and valuations; K-rings dualisants; Dualite locale et structure des Hi(M); Les of wild valued fields; Decomposition of K-rings; Realization of foncteurs Ext'z(X; F, G) et Extz(F, G); Criteres de nullite, K-structures; Bibliography; Glossary of notation; Index. condituions de coherence des faisceaux Ext~ (F, G); Le Mathematical Surveys and Monographs, Volume 124 theoreme de finitude; Geometrie algebrique et geometrie formelle; Application au groupe fondamental; Application au March 2006, 288 pages, Hardcover, ISBN 0-8218-4041-X, LC groupe de Picard; Applications aux schemas algebriques 2005057091, 2000 Mathematics Subject Classification: 12]10, projectifs; Problemes et conjectures; Profondeur et theoremes 12]15; 12E30, 12]20, 19F99, All AMS members US$60, List de Lefschetz en cohomologie etale; Index des notations; Index US $75, Order code SURV/ 124 terminologique. Documents Mathematiques, Number 4 October 2005, 208 pages, Hardcover, ISBN 2-85629-169-4, 2000 Mathematics Subject Classification: 14-02, 14A15, 14D15, 14E20, 14F35, 14H30, Individual member US$52, List US$ 58, Order code SMFDM/4

MARCH 2006 NOTICES OF THE AMS 387 Classified Advertisements Positions available, items for sale, services available, and more

of-the-art campus in New Cairo beginning immediately. The American University in NEW JERSEY Fall Semester, 2007 (see the New Campus Cairo is an Equal Opportunity Employer. website at http: I /www. aucegypt. edu/ 000232 INSTITUTE FOR ADVANCED STUDY ncd/New\%2 0Campus. html ). AUC's degree School of Mathematics programs are accredited by the Commis­ Princeton, New jersey 08540 sion on Higher Education of the Middle TAIWAN States Association of Colleges and Schools. The Institute for Advanced Study invites For more information see our website at applications for the position of professor NATIONAL TAIWAN UNIVERSITY www. aucegypt. edu. One-, two-, or three­ in the School of Mathematics. Candidates Department of Mathematics year appointments subject must have distinguished records of to mutual achievement in mathematical research. agreement will begin September 2006. Re­ The Department of Mathematics at the Na­ The professors in the Institute's School of newal of an appointment depends upon in­ tional Taiwan University invites applica­ Mathematics are collectively responsible stitutional needs and/or the appointee's tions for several openings at the levels of for maintaining a strong program of re­ performance. The normal teaching load is tenure-track assistant, associate and full search through their own work and also by three courses per semester and English is professors in the research areas of com­ selecting a number of visiting members the language of instruction. Salary and putational and applied mathematics, prob­ every year. Applications, including cur­ rank are according to scale based on qual­ ability and financial mathematics, statis­ riculum vitae and list of publications, ifications and professional experience. For tical sciences and other related areas should b e addressed to M. Hayes, School expatriates, benefits include housing, an­ beginning August 1, 2006. The depart­ Administrative Officer (FP), School of Math­ nual round-trip air travel for appointee ment also invites applications for several ematics, Institute for Advanced Study, and qualifying dependents, plus schooling postdoctoral positions in all research areas Princeton, NJ 08540. An Equal Opportunity for the equivalent of up to two children at beginning August 1, 2006. Applicants for Employer: women and minority group Cairo American College. In view of AUC's all positions should send a letter of ap­ members are encouraged to apply. protocol agreement with the Egyptian Gov­ plication, an up-to-date curriculum vitae, recent publications, and three letters 000231 ernment, which requires specific propor­ of tions of Egyptian, U.S., and third-country recommendation to: citizen faculty, at this time preference will The Dean's Office College of Sciences EGYPT be given to qualified applicants who are U.S. citizens. APPLICATION INSTRUC­ National Taiwan University TIONS: E-mail a letter of intent specifying 1 Roosevelt Road, Section 4 AMERICAN UNIVERSITY IN CAIRO Taipei, Taiwan 106 Position# MATH-1/2 with a Department of Mathematics current C.V. Tel: (02) 33664188 to [email protected] and Revised announcement Fax: (02) 23622005 arrange to have three letters of recom­ email: [email protected]. mendation mailed to: Dr. Earl (Tim) Sulli­ The Department of Mathematics antici­ Applicants should specify clearly on the pates two vacancies. The successful can­ van, Provost, American University in Cairo, 420 Fifth Avenue, Fl. 3, New York, N.Y. letter of application the position(s) they are didate will teach all levels of undergradu­ applying for. Complete applications re­ 10018-2729. For full consideration, can­ ate mathematics courses, as well as ceived before March 15, 2005, will receive didates must also complete the Personnel participate in research. Candidates with ex­ full consideration. For more information pertise in Actuarial Sciences will be given Information Form provided at http: I I about the department please consult the special consideration. Founded in 1919, forms.aucegypt.edu/ provost/ pif3. Web at http: I /www. math. ntu. edu. tw. AUC's campus is currently located in Cairo, html . Applications accepted until position 000233 Egypt, but will be moving to a new, state- is filled; review of candidates will begin

Suggested uses for classified advertising are positions available, books issue-April 28, 2006, August 2006 issue-May 26, 2006; September 2006 or lecture notes for sale, books being sought, exchange or rental of houses, issue-June 27, 2006, October 2006 issue-July 31, 2006. and typing services. U.S. laws prohibit discrimination in employment on the basis of color, age, The 2006 rate is $100 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 multi­ accompanied by a statement that the institution does not discriminate on ple 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 spe­ announcements can be placed anonymously. Correspondence will be cific wording may be found on page 13 73 (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 401-455-4084 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 There are no member discounts for classified ads. Dictation over the cl assads@ams. org. 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: April2006 billed upon publication. issue-January 31, 2006; May 2006 issue-February 28, 2006; June/July 2006

388 NOTICES OF THE AMS VOLUME 53, NUMBER 3 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 continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http: I jwww. ams. org/meeti ngs/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting.

Special Sessions Miami, Florida Approximation Theory and Orthogonal Polynomials, Doron Florida International University S. Lubinsky, Georgia Institute of Technology, and Edward B. Saff, Vanderbilt University. April1-2, 2006 Commutative Algebra and Algebraic Geometry, Laura Saturday - Sunday Ghezzi, Florida International University, Huy Tai Ha, Tu­ lane University, and Aron Sirois, University Federal de Meeting #1 015 Pernambuco. Southeastern Section Composition Operators and Complex Dynamical Systems, Associate secretary: Matthew Miller Brian P. Kelly, University of Louisiana, Monroe, and Christo­ Announcement issue of Notices: January 2006 pher N. B. Hammond, Connecticut College. Program first available on AMS website: February 16, 2006 Financial Mathematics, Alec N. Kercheval and Craig A. Program issue of electronic Notices: April 2006 Nolder, Florida State University. Issue of Abstracts: Volume 27, Issue 2 Geometry of Banach Spaces and Connections with Other Areas, Edward W. Odell, University of Texas at Austin, Deadlines Thomas B. Schlumprecht, Texas A&M University, and For organizers: Expired Stephen Dilworth, University of South Carolina. For consideration of contributed papers in Special Sessions: Geometry of Riemannian Manifolds with Additional Struc­ Expired tures, Tedi C. Draghici, Gueo V. Grantcharov, and Philippe For abstracts: Expired Rukimbira, Florida International University. Harmonic Analysis and Partial Differential Equations, Mario The scientific information listed below may be dated. For Milman, Florida Atlantic University, and Marius Mitrea, Uni­ the latest information, see www. ams. org/amsmtgs/ versity of Missouri. sectional.html. History of Mathematics, Karen H. Parshall, University of Invited Addresses Virginia. Imaging, Homogenization, and Shape Optimization, Michael Andrea R. Nalunod, University of Massachusetts, Amherst, S. Vogelius, Rutgers University, and Shari Moskow, Uni­ Bilinear operators in analysis and PDEs. versity of Florida. Edward Odell, University of Texas at Austin, Embeddings Interpolation Theory and Applications, Michael Cwikel, in Banach space theory. Technion, Laura De Carli, Florida International Univer­ Karen V. H. Parshall, University of Virginia, The British de­ sity, and Mario Milman, Florida Atlantic University. velopment of the theory of invariants, 1841-1895. Invariants of Low-Dimensional Manifolds, Thomas G. MichaelS. Vogelius, Rutgers University, Electromagnetic Lennes, Florida International University, and Nikolai N. imaging- An applied analyst's perspective. Saveliev, University of Miami, Coral Gables.

MARCH 2006 NOTICES OF THE AMS 389 Meetings & Conferences

Mathematical Models in Image and High-Dimensional Data Invited Addresses Analysis, Hanna E. Makaruk and Robert M. Owczarek, Los Douglas N. Arnold, Institute for Math and Applications, Alamos National Laboratory, and Nikita Sakhaneko, Uni­ University of Minnesota, Title to be announced. versity of New Mexico and Los Alamos National Labora­ tory. Bela Boliobas, University of Memphis and Cambridge Uni­ versity, Inhomogeneous random graphs (Erdos Memorial Monomials and Resolutions, Joseph P. Brennan, North Lecture). Dakota State University, and Heath M. Martin, University of Central Florida. Steven C. Hofmann, University of Missouri, Title to be an­ nounced. Nonlinear Waves, Andrea R. Nahmod, University of Mass­ achusetts, Amherst, and Sijue Wu, University of Michigan Michael Larsen, University of Indiana, Title to be an­ at Ann Arbor. nounced. Partial Differential Equations and Several Complex Vari­ Christopher M. Skinner, University of Michigan, Title to ables, Shiferaw Berhanu, Temple University, and Hamid be announced. Meziani, Florida International University. Special Sessions Qualitative Analysis ofPartial Differential Equations, Con­ gming Li, University of Colorado, and Wenxiong Chen, Algebraic Structures of Exactly Solvable Models (Code: SS Yeshiva University. 9A), Michael Gekhtman, University of Notre Dame, Mikhail Shapiro, Michigan State University, and Alexander Stolin, Recent Developments on Fluid and Geophysical Fluid Dy­ University of Gothenburg. namics, C. Cao and T. Tachim Medjo, Florida Interna­ tional University, and X. Wang, Florida State University. Analysis and Geometry of Non-linear Evolution and Equa­ tions (Code: SS 19A), Alexandrou A. Himonas and Gerard Singular Integrals, Geometric Analysis, and Free Boundary K. Misiolek, University of Notre Dame. Problems, Marianne Korten and Charles N. Moore, Kansas State University. Combinatorial Algebraic Geometry (Code: SS 2A), Juan C. Migliore, University of Notre Dame, and Uwe R. Nagel, Uni­ Spectral Geometry ofManifolds with Boundary and Singular versity of Kentucky. Spaces, Juan B. Gil, Pennsylvania State University, Altoona, and Patrick T. McDonald, New College, University of South Commutative Algebra and Algebraic Geometry (Code: SS Florida. lA), Alberto Corso, University of Kentucky, Claudia Polini, University of Notre Dame, and Bernd Ulrich, Purdue Uni­ Structure of Function Spaces and Applications, Jan Lang, versity. The Ohio State University, and Osvaldo Mendez, Univer­ sity of Texas at El Paso. Developments and Applications in Differential Geometry (Code: SS 4A), Jianguo Cao, Xiaobo Liu, and Brian Smyth, University of Notre Dame. Notre Dame, Indiana Dynamical Systems (Code: SS lOA), Francois Ledrappier, University of Notre Dame, and Arnie Wilkinson, North­ University of Notre Dame western University. April8-9, 2006 Harmonic Analysis, PDE and Geometric Function Theory (Code: SS 14A), John L. Lewis, University of Kentucky, and Saturday - Sunday Steve C. Hofmann, University of Missouri. Meeting #1 016 Holomorphic Methods and Heat Kernels in Harmonic Analy­ Central Section sis and Quantization Theory (Code: SS 16A), Brian Hall and Associate secretary: Susan]. Friedlander William Kirwin, University of Notre Dame. Announcement issue of Notices: January 2006 Mathematical Biology (Code: SS llA), Mark Alber and Bei Program first available on AMS website: February 2 3, 2006 Hu, University of Notre Dame. Program issue of electronic Notices: April 2006 Model Theory and Computability (Code: SS SA), Steven Issue of Abstracts: Volume 27, Issue 2 Allen Buechler and Julia Knight, University of Notre Dame, Steffen Lempp, University of Wisconsin, and Sergei Deadlines Starchenko, University of Notre Dame. For organizers: Expired New Developments in Optimization (Code: SS lSA), Leonid For consideration of contributed papers in Special Sessions: Faybusovich, University of Notre Dame. Expired For abstracts: February 14, 2006 Nonlinear Waves (Code: SS 13A), MarkS. Alber and Pavel Lushnikov, University of Notre Dame, and Ildar Gabotiv The scientific information listed below may be dated. For and Vladimir E. Zakharov, University of Arizona. the latest information, see www. ams. org/amsmtgs/ Number Theory (Code: SS 21A), Scott T. Parsell and sectional . html. Jonathan P. Sorenson, Butler University.

390 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Meetings & Conferences

Numerical Solution of Polynomial Systems (Code: SS 7A), Konstantina Trivisa, University of Maryland, College Park, Christopher S. Peterson, Colorado State University, and Title to be announced. Andrew J. Sommese, University of Notre Dame. Special Sessions PDEs and Geometric Analysis (Code: SS 22A), Matt Gursky and Qing Han, University of Notre Dame. Algebraic Groups (Code: SS SA), George]. McNinch, Tufts Several Complex Variables (Code: SS 6A), Nancy K. Stan­ University, and Eric Sommers, University of Massachusetts­ ton and Jeffrey A. Diller, University of Notre Dame. Amherst. Special Functions and Orthogonal Polynomials (Code: SS SA), Arithmetic Geometry and Modular Forms (Code: SS 6A), Paul Diego Dominici, State University of New York at New E. Gunnells and Farshid Hajir, University of Massachusetts, Paltz. Amherst. Topics in Representation Theory (Code: SS 17A), Sam Evens, Arrangements and Configuration Spaces (Code: SS lOA), University of Notre Dame, and Jiu-Kang Yu, Purdue Uni­ Graham C. Denham, University of Western Ontario, and versity. Alexander I. Suciu, Northeastern University. TopologyandPhysics(Code: SS 18A), StephanA. Stolz and Banach Lattices, Regular Operators, and Applications (Code: Bruce Williams, University of Notre Dame. SS 3A), A. K. Kitover, Community College of Philadelphia, Undergraduate Mathematical Research (Code: SS 20A), M. Orhon, University of New Hampshire, and A. W. Wick­ Francis X. Connolly, University of Notre Dame, and stead, Queen's University of Belfast. Zsuzsanna Szaniszlo, Valparaiso University. Banach Spaces of Analytic Functions (Code: SS 2A), Rita A. Water Waves (Code: SS 12A), David Nicholls, University of Hibschweiler, University of New Hampshire, and Thomas Illinois at Chicago. H. MacGregor, SUNY Albany and Bowdoin College. Discrete and Convex Geometry (Code: SS lA), Daniel A. Klain, University of Massachusetts (Lowell), Barry R. Mon­ Durham, New son, University of New Brunswick, and Egon Schulte, Northeastern University. Hampshire Galois Theory in Arithmetic and Geometry (Code: SS 8A), University of New Hampshire Florian Pop and David Harbater, University of Pennsyl­ vania, and Rachel J. Pries, Colorado State University. April22-23, 2006 Geometric Methods in Group Theory and Topology (Code: Saturday - Sunday SS 9A), Kim Ruane, Tufts University, Jennifer Taback, Bowdoin College, and Peter N. Wong, Bates College. Meeting #1 017 Global Perspectives on the Geometry of Riemann Surfaces Eastern Section (Code: SS 14A), Eran Makover and Jeffrey K. McGowan, Associate secretary: Lesley M. Sibner Central Connecticut State University. Announcement issue of Notices: February 2006 Program first available on AMS website: March 9, 2006 HopfAlgebras and Galois Module Theory(Code: SS 4A), Tim­ Program issue of electronic Notices: April 2006 othy Kohl, Boston University, and Robert G. Underwood, Issue of Abstracts: Volume 27, Issue 2 Auburn University Montgomery. Mathematical Challenges in Physical and Engineering Sci­ Deadlines ences (Code: SS 13A), Marianna A. Shubov, University of For organizers: Expired New Hampshire. For consideration of contributed papers in Special Sessions: Quantum Invariants of Knots and 3-Manifolds (Code: SS Expired llA), Charles D. Frohman, University of Iowa, and Raz­ For abstracts: February 21, 2006 van Gelca, Texas Tech University. The scientific information listed below may be dated. For Symplectic and Contact Topology (Code: SS 7A), Weimin the latest information, see www. ams. o rg/amsmtgs/ Chen, Michael G. Sullivan, and Hao Wu, University of sectional . html. Massachusetts, Amherst. Topological Algebras and Applications (Code: SS 12A), Invited Addresses Alexander A. Katz, St. John's University, and Genady Y. Ailana M. Fraser, University of British Columbia, Title to Grabarnik, IBM T. ]. Watson Research Center. be announced. Dmitri Nikshych, University of New Hampshire, Algebraic theory of tensor categories. Florian Pop, University of Pennsylvania, From topological covers to algebraic numbers.

MARcH 2006 NOTICES OF THE AMS 391 Meetings & Conferences

Geometric Dynamics and Ergodic Theory (Code: SS llA), San Francisco, Yitwah Cheung and Arek Goetz, San Francisco State Uni­ versity, and Slobodan Simic, San Jose State University. California Geometry of Grabner Bases (Code: SS 2A), Bernd Sturm­ fels, University of California Berkeley, and Alexander San Francisco State University Yong, University of Minnesota and Fields Institute. Hilbert Functions April29-30, 2006 and Resolutions (Code: SS 12A), Ben­ jamin Richert, California Polytechnic State University, and Saturday - Sunday Sean Sather-Wagstaff, California State University, Dominguez Hills. Meeting #1 018 History and Philosophy of Mathematics Western Section (Code: SS 1A), Shawnee L. McMurran, California State University, San Associate secretary: Michel L. Lapidus Bernardino, and James J. Tattersall, Providence College. Announcement issue of Notices: February 2006 Program first available on AMS website: March 16, 2006 Homological and K-theoretical Trends in Algebraic Com­ Program issue of electronic Notices: April 2006 binatorics (Code: SS 6A), Joseph Gubeladze and Serkan Issue of Abstracts: Volume 27, Issue 2 Rosten, San Francisco State University. Liapunov Exponents and Nonuniform Hyperbolicity (Code: Deadlines SS 7A), Anton Gorodetski and Vadim Kaloshin, Califor­ For organizers: Expired nia Institute of Technology. For consideration of contributed papers in Special Sessions: Lie Algebras and Applications (Code: SS 9A), Dimitar Expired Grantcharov, San Jose State University, Vera Serganova, For abstracts: March 7, 2006 University of California Berkeley, and Arturo Pianzola, University of Alberta. The scientific information listed below may be dated. For Partial Differential Equations and Their Applications (Code: the latest information, see www. ams. org/amsmtgs/ SS 14A), Steve Shkoller, University of California Davis. sectional.html. Probability and Statistical Physics (Code: SS SA), Marek Biskup, University of California Los Angeles, Noam Berger, Invited Addresses California Institute of Technology and University of Cali­ Uncoln Chayes, University of California Los Angeles, Title fornia Los Angeles, and Balint Virag, University of Toronto. to be announced. Q-series and Partitions (Code: SS SA), Neville Robbins, San C. Robin Graham, University of Washington, Title to be an­ Francisco State University. nounced. Vadirn Kaloshin, California Institute of Technology, Non­ local instabilities of the planar three body problem. Salt Lake City, Utah Benoit B. Mandelbrot, Yale University, The Nature of University of Utah Roughness in Mathematics, Science, and Art(Einstein Pub­ lic Lecture in Mathematics). October 7-8,2006 Yuval Peres, University of California Berkeley, Hex, ran­ Saturday - Sunday dom-turn games, and the infinity Laplacian Meeting #1 019 Special Sessions Western Section Associate secretary: Michel L. Lapidus Computational Arithmetic Geometry (Code: SS 13A), Ken­ Announcement issue of Notices: August 2006 neth A. Ribet, University of California Berkeley, and Kristin Program first available on AMS website: August 24, 2006 Estrella Lauter, Microsoft Corporation. Program issue of electronic Notices: October 2006 Elliptic Methods in Geometry (Code: SS 3A), C. Robin Gra­ Issue of Abstracts: Volume 27, Issue 3 ham, University of Washington, and Rafe Mazzeo, Stan­ ford University. Deadlines Enumerative Aspects of Polytopes (Code: SS 1 OA), Federico For organizers: March 7, 2006 For consideration of contributed papers in Special Ardila and Matthias Beck, San Francisco State University. Sessions: June 20, 2006 Fractal Geometry: Connections to Dynamics, Geometric For abstracts: August 15, 2006 Measure Theory, Mathematical Physics and Number The­ ory (Code: SS 4A), Michel L. Lapidus and Erin P. Pearse, The scientific information listed below may be dated. For University of California Riverside, and Machiel van the latest information, see www. ams. o rg/ amsmtgs/ Frankenhuijsen, Utah Valley State College. sectional.html.

392 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Meetings & Conferences

Invited Addresses Special Sessions William Arveson, University of California Berkeley, Title Analysis and Potential Theory on Metric Spaces (Code: SS to be announced. 4A), Thomas Bieske, University of South Florida, and Zair Alexei Borodin, California Institute of Technology, Title Ibragimov and Nageswari Shanmugalingam, University to be announced. of Cincinnati. Izabella Joanna Laba, University of British Columbia, Title Applied Algebraic Geometry and Cryptography (Code: SS to be announced. 3A), Jintai Ding, Jason Eric Gower, and Timothy J. Hodges, Darren Long, University of California Santa Barbara, Title University of Cincinnati, Lei Hu, Chinese Academy of Sci­ to be announced. ences, and Dieter S. Schmidt, University of Cincinnati. Birational Geometry(Code: SS 2A), Mirel Constantin Caibar Special Sessions and Gary P. Kennedy, Ohio State University. Commutative Algebra (Code: SS 3A), Paul Roberts, Anurag Ergodic Theory(Code: SS 1A), Nikos Frantzikinakis, Penn­ K. Singh, and Oana Veliche, University of Utah. sylvania State University, Bryna R. Kra, Northwestern Uni­ Harmonic Analysis: Trends and Perspectives (Code: SS 1A), versity, and Mate Wierdl, University of Memphis. Alex Iosevich, University of Missouri, and Michael T. Nonlinear Functional Analysis and Applications (Code: SS Lacey, Georgia Institute of Technology. 5A), S. P. Singh and Bruce Watson, Memorial University Nonlinear Differential Equations: Methods and Applica­ of Newfoundland. tions (Code: SS 2A), David G. Costa, University of Nevada, and Zhi-Qiang Wang. Storrs, Connecticut Cincinnati, Ohio University of Connecticut University of Cincinnati October 28-29, 2006 Saturday - Sunday October 21-22, 2006 Saturday - Sunday Meeting #1 021 Eastern Section Meeting #1 020 Associate secretary: Lesley M. Sibner Central Section Announcement issue of Notices: August 2006 Associate secretary: Susan]. Friedlander Program first available on AMS website: September 14, Announcement issue of Notices: August 2006 2006 Program first available on AMS website: September 7, 2006 Program issue of electronic Notices: October 2006 Program issue of electronic Notices: October 2006 Issue of Abstracts: Volume 27, Issue 4 Issue of Abstracts: Volume 27, Issue 3 Deadlines Deadlines For organizers: March 28, 2006 For organizers: March 21, 2006 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: July 11, 2006 July 5, 2006 For abstracts: September 6, 2006 For abstracts: August 29, 2006

The scientific information listed below may be dated. For The scientific information listed below may be dated. For the latest information, see www. ams. o rg/amsmtgs/ the latest information, see www. ams. org/amsmtgs/ sectional . html. sectional . html .

Invited Addresses Invited Addresses Suncica Canic, University of Houston, Title to be an­ Changfeng Gui, University of Connecticut, Storrs, Title to nounced. be announced. Bryna R. Kra, Northwestern University, Title to be an­ Niranjan Ramachandran, University of Maryland, College nounced. Park, Title to be announced. Ezra N. Miller, University of Minnesota, Title to be an­ Karman Soundararajan, University of Michigan, Title to be nounced. announced. jon G. Wolfson, Michigan State University, Title to be an­ Katrin Wehrheim, Institute for Advanced Study, Title to nounced. be announced.

MARCH 2006 NOTICES OF THE AMS 393 Meetings & Conferences

Special Sessions Boundary Operators in Real and Complex Domains (Code: SS 3A), Loredana Lanzani, University of Arkansas, Fayet­ Analysis and Probability on Fractals (Code: SS 3A), Robert teville, and David E. Barrett, University of Michigan, Ann S. Strichartz, Cornell University, and Alexander Teplyaev, Arbor. University of Connecticut, Storrs. Combinatorial Methods in Equivariant Topology (Code: SS 1A), Tara Holm, University of Connecticut, Storrs, and Tom C. Braden, University of Massachusetts, Amherst. New Orleans, Number Theory (Code: SS 2A), Keith Conrad, University of Connecticut, Storrs, David Pollack, Wesleyan University, Louisiana and Thomas A. Weston, University of Massachusetts, New Orleans Marriott and Sheraton New Amherst. Orleans Hotel January 4-7,2007 Fayetteville, Thursday - Sunday Arkansas Meeting #1 023 ]oint Mathematics Meetings, including the 113th Annual University of Arkansas Meeting of the AMS, 90th Annual Meeting of the Mathe­ November 3-4, 2006 matical Association ofAmerica (MAA), annual meetings of Friday - Saturday the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the Meeting #1 022 winter meeting of the Association for Symbolic Logic (ASL), Southeastern Section with sessions contributed by the Society for Industrial and Associate secretary: Matthew Miller Applied Mathematics (SIAM). Announcement issue of Notices: September 2006 Associate secretary: Susan J. Friedlander Program first available on AMS website: September 21, Announcement issue of Notices: October 2006 2006 Program first available on AMS website: November 1, 2006 Program issue of electronic Notices: November 2006 Program issue of electronic Notices: January 2007 Issue of Abstracts: Volume 27, Issue 4 Issue of Abstracts: Volume 28, Issue 1 Deadlines Deadlines For organizers: April 3, 2006 For organizers: April1, 2006 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: July 18, 2006 August 1, 2006 For abstracts: September 12, 2006 For abstracts: September 26, 2006

The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Davidson, North sectional.html. Invited Addresses Carolina Richard P. Anstee, University of British Columbia, Title to Davidson College be announced. Arun Ram, University of Wisconsin, Title to be announced. March 3-4, 2007 Saturday - Sunday Donald G. Saari, University of California Irvine, Title to be announced. Southeastern Section Associate secretary: Matthew Miller Andras Vasy, Massachusetts Institute of Technology, Title Announcement issue of Notices: To be announced to be announced. Program first available on AMS website: To be announced Special Sessions Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Analytic Number Theory and Modular Forms (Code: SS 2A), Matthew Boylan, University of Illinois, Urbana­ Deadlines Champaign, and Gang Yu, University of South Carolina. For organizers: August 3, 2006 Dirac Operators in Analysis and Geometry (Code: SS 1A), For consideration of contributed papers in Special Sessions: John Ryan, University of Arkansas, Marius Mitrea, Uni­ To be announced versity of Missouri, and Mircea Martin, Baker University. For abstracts: To be announced

394 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Meetings & Conferences Oxford, Ohio Tucson, Arizona Miami University University of Arizona March 16-1 7, 2007 April21-22, 2007 Friday - Saturday Saturday - Sunday Central Section Western Section Associate secretary: Susan]. Friedlander Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program first available on AMS website: To be announced 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: To be announced For organizers: September 21, 2006 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced

The scientific information listed below may be dated. For the latest information, see www. ams. org/amsmtgs/ Warsaw, Poland sectional.html. University of Warsaw Special Sessions July 31-August 3, 2007 Finite Geometry and Combinatorics (Code: SS 3A), Mark A. Tuesday - Friday Miller, Marietta College. Associate secretary: Susan ]. Friedlander Geometric Topology (Code: SS 2A), jean-Francois LaFont, Announcement issue of Notices: To be announced SUNY Binghamton and Ohio State University, and Ivonne Program first available on AMS website: To be announced J. Ortiz, Miami University. Program issue of electronic Notices: To be announced Large Cardinals in Set Theory (Code: SS 1A), Paul B. Lar­ Issue of Abstracts: To be announced son, Miami University, Justin Tatch Moore, Boise State Uni­ versity, and Ernest Schimmerling, Carnegie Mellon Uni­ Deadlines versity. For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced Hoboken, New Jersey For abstracts: To be announced Stevens Institute of Technology Albuquerque, New April14-15, 2007 Saturday - Sunday Mexico Eastern Section Associate secretary: Lesley M. Sibner University of New Mexico Announcement issue of Notices: To be announced Program first available on AMS website: To be announced October 1 3-14, 2007 Program issue of electronic Notices: To be announced Saturday - Sunday Issue of Abstracts: To be announced Western Section Associate secretary: Michel L. Lapidus Deadlines Announcement issue of Notices: To be announced For organizers: September 14, 2006 Program first available on AMS website: To be announced For consideration of contributed papers in Special Sessions: Program issue of electronic Notices: To be announced To be announced Issue of Abstracts: To be announced For abstracts: To be announced Deadlines For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced

MARCH 2006 NOTICES OF THE AMS 395 Meetings & Conferences For abstracts: To be announced Baton Rouge, Murfreesboro, Louisiana Tennessee Louisiana State University, Baton Rouge March 28-30,2008 Middle Tennessee State University Friday - Sunday November 3-4, 2007 Southeastern Section Associate secretary: Matthew Miller Saturday - Sunday Announcement issue of Notices: To be announced Southeastern Section Program first available on AMS website: To be announced Associate secretary: Matthew Miller Program issue of electronic Notices: To be announced Announcement issue of Notices: To be announced Issue of Abstracts: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: To be announced For organizers: August 28, 2007 For consideration of contributed papers in Special Sessions: Deadlines To be announced For organizers: April 4, 2007 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced For abstracts: To be announced Bloomington, Indiana Indiana University San Diego, California April4-6,2008 Friday- Sunday San Diego Convention Center Central Section Associate secretary: Susan]. Friedlander January 6-9, 2008 Announcement issue of Notices: To be announced Sunday - Wednesday Program first available on AMS website: To be announced ]oint Mathematics Meetings, including the 114th Annual Program issue of electronic Notices: To be announced Meeting of the AMS, 9lst Annual Meeting of the Mathe­ Issue of Abstracts: To be announced matical Association of America (MAA), annual meetings of Deadlines the Association for Women in Mathematics (A w.M) and the For organizers: September 4, 2007 National Association of Mathematicians (NAM), and the For consideration of contributed papers in Special Sessions: winter meeting of the Association for Symbolic Logic (ASL), To be announced with sessions contributed by the Society for Industrial and For abstracts: To be announced Applied Mathematics (SIAM). Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2007 Claremont, California Program first available on AMS website: November 1, 2007 Program issue of electronic Notices: January 2008 Claremont McKenna College Issue of Abstracts: Volume 29, Issue 1 May 3-4, 2008 Deadlines Saturday - Sunday Western Section For organizers: April1, 2007 Associate secretary: Michel Lapidus For consideration of contributed papers in Special Sessions: Announcement issue of Notices: To be announced To be announced Program first available on AMS website: To be announced For abstracts: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: To be announced

396 NOTICES OF THE AMS VOLUME 53, NUMBER 3 Meetings & Conferences

For consideration of contributed papers in Special Sessions: Deadlines To be announced For organizers: March 24, 2008 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Rio de Janeiro, Brazil For abstracts: To be announced Instituto Nacional de Matematica Pura e Aplicada (IMPA) Shanghai, People's june 4-7,2008 Republic of China Wednesday - Saturday Fudan University First ]oint International Meeting between the AMS and the Sociedade Brasileira de Matematica. December 17-21,2008 Associate secretary: Lesley M. Sibner Wednesday - Sunday Announcement issue of Notices: To be announced First ]oint Interntional Meeting Between the AMS and the Program first available on AMS website: Not applicable Shanghai Mathematical Society Program issue of electronic Notices: Not applicable Associate secretary: Susan ]. Friedlander Issue of Abstracts: Not applicable Announcement issue of Notices: To be announced Program first available on AMS website: Not applicable Deadlines Program issue of electronic Notices: Not applicable For organizers: To be announced Issue of Abstracts: Not applicable For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced Vancouver, Canada For abstracts: To be announced University of British Columbia October 4-5, 2008 Washington, District Saturday - Sunday Western Section of Columbia Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Marriott Wardman Park Hotel and Omni Program first available on AMS website: To be announced Shoreham Hotel Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced January 7-10,2009 Wednesday - Saturday Deadlines ]oint Mathematics Meetings, including the llSth Annual For organizers: March 9, 2008 Meeting of the AMS, 92nd Annual Meeting of the Mathe­ For consideration of contributed papers in Special Sessions: matical Association of America (MAA), annual meetings of To be announced the Association for Women in Mathematics (A WM) and the For abstracts: To be announced National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Huntsville, Alabama Applied Mathematics (SIAM). Associate secretary: Lesley M. Sibner University of Alabama, Huntsville Announcement issue of Notices: October 2008 Program first available on AMS website: November 1, 2008 October 24-26, 2008 Program issue of electronic Notices: January 2009 Friday - Sunday Issue of Abstracts: Volume 30, Issue 1 Southeastern Section Associate secretary: Matthew Miller Deadlines Announcement issue of Notices: To be announced For organizers: April1, 2008 Program first available on AMS website: To be announced For consideration of contributed papers in Special Sessions: Program issue of electronic Notices: To be announced To be announced Issue of Abstracts: To be announced For abstracts: To be announced

MARCH 2006 NOTICES OF THE AMS 397 Meetings & Conferences San Francisco, Boston, California Massachusetts Moscone Center West and the San Fran­ john B. Hynes Veterans Memorial Conven­ cisco Marriott tion Center, Boston Marriott Hotel, and January 6-9, 201 0 Boston Sheraton Hotel Wednesday - Saturday January 4-7,2012 ]oint Mathematics Meetings, including the 116th Annual Wednesday - Saturday Meeting of the AMS, 93rd Annual Meeting of the Mathe­ ]oint Mathematics Meetings, including the 118th Annual matical Association ofAmerica (MAA), annual meetings of Meeting of the AMS, 95th Annual Meeting of the Mathe­ the Association for Women in Mathematics (A VVM) and the matical Association ofAmerica, annual meetings of the As­ National Association of Mathematicians (NAM), and the sociation for Women in Mathematics (A VVM) and the National winter meeting of the Association for Symbolic Logic (ASL), Association of Mathematicians (NAM), and the winter meet­ with sessions contributed by the Society of Industrial and Applied Mathematics (SIAM). ing of the Association for Symbolic Logic (ASL), with sessions Associate secretary: Matthew Miller contributed by the Society for Industrial and Applied Math­ Announcement issue of Notices: October 2009 ematics (SIAM). Program first available on AMS website: November 1, 2009 Associate secretary: Michel L. Lapidus Program issue of electronic Notices: January 2010 Announcement issue of Notices: October 2011 Issue of Abstracts: Volume 31, I:;;sue 1 Program first available on AMS website: November 1, 2011 Program issue of electronic Notices: January 2012 Deadlines Issue of Abstracts: Volume 33, Issue 1 For organizers: April1, 2009 Deadlines For consideration of contributed papers in Special Sessions: To be announced For organizers: April1, 2011 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced New Orleans, For abstracts: To be announce-d Louisiana San Diego, California New Orleans Marriott and Sheraton New San Diego Convention Center and San Orleans Hotel Diego Marriott Hotel and Marina January 5-8,2011 January 9-12,2013 Wednesday - Saturday Wednesday - Saturday ]oint Mathematics Meetings, including the 117th Annual ]oint Mathematics Meetings, including the 119th Annual Meeting of the AMS, 94th Annual Meeting of the Mathe­ Meeting of the AMS, 96th Annual meeting of the Mathe­ matical Association ofAmerica, annual meetings of the As­ matical Association ofAmerica, annual meetings of the As­ sociation for Women in Mathematics (A VVM) and the National sociation for Women in Mathematics (A VVM) and the National Association ofMathematicians (NAM), and the winter meet­ Association ofMathematicians (NAM), and the winter meet­ ing of the Association for Symbolic Logic (ASL), with sessions ing of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Math­ contributed by the Society for Industrial and Applied Math­ ematics (SIAM). ematics (SIAM). Associate secretary: Susan]. Friedlander Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2010 Announcement issue of Notices: To be announced Program first available onAMS website: November 1, 2010 Program first available on AMS website: To be announced Program issue of electronic Notices: January 2011 Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 32, Issue 1 Issue of Abstracts: To be announced Deadlines Deadlines For organizers: April1, 2010 For organizers: April1, 2012 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced

398 NoTICEs oF THE AMS VOLUME 53, NUMBER 3 Meetings and Conferences of the AMS

Associate Secretaries of the AMS Western Section: MichelL. Lapidus, Department of Math­ Eastern Section: Lesley M. Sibner, Department of Mathe­ ematics, University of California, Sproul Hall, Riverside, CA matics, Polytechnic University, Brooklyn, NY 11201-2990; 92521-0135; e-mail: l api dus@math. ucr. edu; telephone: 951- e-mail: lsi bner@duke. poly. edu; telephone: 718-260-3505. 827-5910. Southeastern Section: Matthew Miller, Department of Math­ Central Section: Susan J. Friedlander, Department of Math­ ematics, University of South Carolina, Columbia, SC 29208- ematics, University of Illinois at Chicago, 8 51 S. Morgan (M/C 0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690. 249), Chicago, lL 60607-7045; e-mail: susan@math. nwu. edu; tele­ phone: 312-996-3041.

The Meetings and Conferences section of the Notices April4-6 Bloomington, Indiana p. 396 gives information on all AMS meetings and conferences May 3-4 Claremont, California p. 396 approved by press time for this issue. Please refer to the page June 4-7 Rio de Janeiro, Brazil p. 397 numbers cited in the table of contents on this page for more October 4-5 Vancouver, Canada p. 397 detailed information on each event. Invited Speakers ;md October 24-26 Huntsville, Alabama p. 397 Special Sessions are listed as soon as they are approved by December 17-21 Shanghai, People's the cognizant program committee; the codes listed are needed Republic of China p.397 for electronic abstract submission. For some meetings the list may be incomplete. Information in this issue may be dated. 2009 Up-to-date meeting and conference information can be January 7-10 Washington, DC p.397 found at www. ams. org/meeti ngs/. Annual Meeting 2010 Meetings: January 6-9 San Franciso, California p. 398 2006 Annual Meeting April1-2 Miami, Florida p. 389 2011 April8-9 Notre Dame, Indiana p. 390 January 5-8 New Orleans, Louisiana p. 398 April 22-23 Durham, New Hampshire p. 391 Annual Meeting April29-30 San Francisco, California p. 392 2012 October 7-8 Salt Lake City, Utah p. 392 January 4-6 Boston, Massachusetts p. 398 Annual Meeting October 21-22 Cincinnati, Ohio p. 393 2013 October 28-29 Storrs, Connecticut p.393 January 4-6 San Diego, California p. 398 November 3-4 Fayetteville, Arkansas p. 394 Annual Meeting 2007 Important Information Regarding AMS Meetings January 4-7 New Orleans, Louisiana p. 394 Potential organizers, speakers, and hosts should refer to Annual Meeting page 296 in the February 2006 issue of the Notices for gen­ March 3-4 Davidson, North Carolina p. 394 eral information regarding participation in AMS meetings and conferences. March 16-17 Oxford, Ohio p. 395 April14-15 Hoboken, New Jersey p. 395 Abstracts April21-22 Tuscan, Arizona p. 395 Speakers should submit abstracts on the easy-to-use interactive July 31-August 3 Warsaw, Poland p. 395 Web form. No knowledge of 1;tf_EX is necessary to submit an October 13-14 Albuquerque, New Mexico p. 395 electronic form, although those who use LAf_E)( may submit November 3-4 Murfreesboro, Tennessee p.396 abstracts with such coding, and all math displays and simi­ larily coded material (such as accent marks in text) must be typeset in 1;tf_EX. Visit http: I /www. ams. org/cgi -bin/ 2008 abstracts/abstract. pl. Questions about abstracts andre­ January 6-9 San Diego, California p. 396 quests for paper forms may be sent to abs-i nfo@ams. org. Annual Meeting Close attention should be paid to specified deadlines in this March 28-30 Baton Rouge, Louisiana p. 396 issue. Unfortunately, late abstracts cannot be accommodated. Conferences: (see http: I /www. ams. org/meeti ngs/ for the most up-to-date information on these conferences.) June 4-June 29, 2006: Joint Summer Research Conferences, Snowbird, Utah (see November 2005 Notices, page 1296). Co-sponsored conference: 22nd Annual Workshop on Mathematical Problems in Industry, June 12-16, 2006, Olin College, Need­ ham, MA. For details see http: I /projects. olin. edu/mpi 2006/. Recent advances in nonlinear partial differential equations and applications: A conference in honor of Peter D. Lax and Louis Niren­ berg, June 7-10, 2006, Toledo, Spain. For more details see http: I /www. mat. ucm. es/-1 n06/. Poisson 2006: Poisson Geometry in Mathematics and Physics, June 5--9, 2006, Tokyo, Japan. For more details see http://www.ams.org/mathcal/info/2006_jun5-9_tokyo.html.

MARCH 2006 NOTICES OF THE AMS 399 CAMBRIDGE OUTSTANDING SCHOLARSHIP

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