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Research in Collegiate Mathematics Education. V Annie Selden, Tennessee Technological University, Cookeville, Ed Dubinsky, Kent State University, OH, Guershon Hare I, University of California San Diego, La jolla, and Fernando Hitt, C/NVESTAV, Mexico, Editors This volume presents state-of-the-art research on understanding, teaching, and learning mathematics at the post-secondary level. The articles are peer-reviewed for two major features: (I) advancing our understanding of collegiate mathematics education, and (2) readability by a wide audience of practicing mathematicians interested in issues affecting their students. This is not a collection of scholarly arcana, but a compilation of useful and informative research regarding how students think about and learn mathematics. This series is published in cooperation with the Mathematical Association of America. CBMS Issues in Mathematics Education, Volume 12; 2003; 206 pages; Softcover; ISBN 0-8218-3302-2; List $49;AII individuals $39; Order code CBMATH/12N044

MATHEMATICS EDUCATION Also of interest .. RESEARCH: AGul

Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom Graduate Student Edition Faculty Edition Solomon Friedberg, Boston College, Chestnut Hill, MA This series is published in cooperation with the Mathematical Association of America. CBMS Issues in Mathematics Education, Graduate Student Edition: Volume I 0; 200 I; 67 pages; Softcover; ISBN 0-8218-2823-1 ; list $16;AII individuals $13; Order code CBMATH/1 ON044 CBMS Issues in Mathematics Education, Faculty Edition: Volume I 0; 200 I; 158 pages; Softcover; ISBN 0-8218-2875-4; list $29; All individuals $23; Order code CBMATH/1 O.FN044

The Mathematical Education ofTeache rs This series is published in cooperation with the Mathematical Association of America. CBMS Issues in Mathematics Education, Volume II; 2001; 145 pages; Softcover; ISBN 0-8218-2899-1 ; List $24;AII individuals $19; Order code CBMATH/1 I N044

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Editors Website: http://imrn.hindawi.com Morris Weisfeld IMRN provides very fast publication of research articles of high current interest in all Managing Editor areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics. Issues are published as frequently as Dan Abramovich necessary. IMRN is expected to publish 80± issues in volume 2004. The articles ofiMRN are Enrico Arbarello reviewed/indexed in COMPUMATH Citation Index, Current Contents, lSI Alerting Services, Joseph Bernstein Mathematical Reviews, Science Citation Index, SciSearch, and Zentralblatt fUr Mathematik. Enrico Bombieri There are no page charges. Submissions are made by email to [email protected]. Richard E. Borcherds New print subscribers shall receive a free copy of all back volumes, i.e., volumes 1991- 2003. Alexei Borodin New and current online subscribers shall receive "perpetual" online access to volumes Jean Bourgain 1991- 2004. Contact [email protected] for more details. Marc Burger ARTICLES Tobias Golding FORTHCOMING Corrado DeConcini • A Cuspidality Criterion for the Functorial Product on GL(2) x GL(3) PercyDeift with a Cohomological Application, Dinakar Ramakrishnan and Song Wang Robbert Dijkgraaf • A New Proof of Steinberg's Fixed-Point Theorem, G. I. Lehrer S. K. Donaldson • A Stacky Semistable Reduction Theorem, Martin Olsson Yakov Eliashberg • Associated Primes of Local Cohomology Modules and of Frobenius Powers, Edward Frenkel Anurag K . Singh and Irena Swanson Emmanuel Hebey • Bounded Geodesics in Moduli Space, Dmitry Kleinbock and Barak Weiss Dennis Hejhal • Gap Probability in the Spectrum of Random Matrices and Asymptotics of Polynomials Helmut Hofer Orthogonal on an Arc of the Unit Circle, I. V. Krasovsky Gerhard Huisken • l -Functions of Exponential Sums over One-Dimensional Affi.noids: Newton over Yasutaka Ihara Hodge, Hui June Zhu Michio Jimbo • Minimality and Nondegeneracy of Degree-One Ginzburg-Landau Vortex as a Hardy's Kurt Johansson Type Inequality, Manuel del Pi no, Patricio Felmer, and Michal Kowalczyk Masaki Kashiwara Kiran S. Kedlaya Carlos Kenig Sergiu Klainerman Toshiyuki Kobayashi Maxim Kontsevich Igor Krichever Shigeo Kusuoka Gilles Lebeau Joachim Lohkamp Nikolai Makarov Yu. l. Manin Barry Mazur Haynes Miller Shinichi Mochizuki Fabien Morel Michael Rapoport N. Yu. Reshetikhin Igor Rodnianski Peter Sarnak Freydoon Shahidi Stanislav Smirnov Michael Struwe G. Tian John Toth Takeshi Tsuji David Vogan Dan Voiculescu Hindawi Publishing Corporation , 410 Park Avenue, 15th Floor, #287 pmb, New York, Andrei Zelevinsky HINDAWI NY 10022, USA; Fax 1-866-446-3294 (USA, toll-free) Maciej Zworski NEW f6 FORTHCOMING from Birkhiiuser

Mathematical Theory Regularity Theory for Linear Algebra Second Edition of Diffraction Mean Curvature Flow J. H. KWAKand S. HONG, both, Pohang University of A. SOMMERFELD (Author) K. ECKER, Freie Universitiit, Berlin, Germany Science and Technology, Pohang, Republic of Korea R. J. NAGEM, Boston University, Boston, MA; M. ZAMPOLLI, NATO SACLANT Undersea Research Centre, La Spezia, Italy; and This work focuses on the motion of surfaces whose ':41ogical development of the subject .. . all the G. SANDRl,Boston University, Boston, MA (Translators) normal velocity at every point is given by the mean curvature at that point; this geometric heat flow process important theorems and results are discussed in A. Sommerfeld's Mathematische Theorie der Diffraction terms of simple worked examples. The student's is called mean curvature flow. Mean curvature flow and marks a milestone in optical theory, full of insights that related flows are important tools in mathematics and understanding is tested by problems at the end of are still relevant today. In a stunning tour de force, each subsection, and every chapter ends with mathematical physics. This work presents techniques in Sommerfeld derives the first mathematically rigorous the study of singularities of mean curvature flow and is exercises. " -CURRENT SCIENCE solution of an optical diffraction problem. This com­ (REVIEW OF THE FIRST EDITION) largely based on the work of K. Brakke, although more plete translation, reflecting substantial scholarship, is recent developments are presented as well: for exam­ Acornerstone of undergraduate mathematics, the first publication in English of Sommerfeld's original ple, relations to regularity theory for minimal surfaces, science, and eHgineering, the fundamentals of work. The extensive notes by the translators are rich in as in Allard's and de Giorgi's work. linear algebra are given a clear and rigorous historical background and provide many technical presentation in this text. With emphasis on both details for the reader. 2004/144 PP., 16 ILLUS./HARDCOVER/$ 129.00 ISBN 0-8176-3243-3 theory and computation, the authors cover linear 2004/190 PP., 381LLUS./HARDCOVER/$54.95 PROGRESS IN NONLINEAR DIFFERENTIAL EQUATIONS AND THEIR equations, matrices, determinants, and vector ISBN 0-8176-3604-8 APPLICATIONS, VOL 57 spaces, and continue on to complex vector spaces, PROGRESS IN MATHEMATICAL PHYSICS, VOL 35 inner products, quadratic forms, and Jordan canonical form. Approximation Theory Semiconcave Functions, From Taylor Polynomials to Wavelets KEY TOPICS AND FEATURES: Thorough treatment of the basics, with a particular focus on linear Hamilton-Jacobi 0. CHRISTENSEN and K. L. CHRISTENSEN, both, recurrence, difference, and differential equations Equations, and Technical University of Denmark, Lyngby, Denmark • Avariety of applications of linear algebra to This concisely written book gives an elementary intro­ diverse fields, such as economics, physics, and Optimal Control duction to a classical area of mathematics-approxima­ computer science • Rich selection of examples P. CANNARSA and C. SINESTRARI, both, Universitii di Roma tion theory-in a way that naturally leads to the modern and explanations, with a wide range of exercises at 'Tor Vergata', Roma, Italy field of wavelets. The exposition demonstrates the the end of every section. This text is a exposition of the theory of semiconcave dynamic nature of mathematics and how classical This second edition includes substantial revisions, functions, and of the role they play in optimal control disciplines influence many areas of modern mathemat­ new material on minimal polynomials and diagonal­ and Hamilton-Jacobi equations. The first part of the ics and applications. The emphasis is on ideas rather ization, and several illuminating figures and book covers general theory, encompassing all key than technical details and proofs. diagrams. Valuable for theoretical and applied results and illustrating them with significant examples. KEY TOPICS AND FEATURES: Description of wavelets in courses, this work is also ideal for self-study. The latter part is devoted to applications concerning the words rather than mathematical symbols • Numerous Bolza problem in calculus of variations and optimal exit­ 2004/APPROX 436 PP., 40 ILLUS./SOFTCOVER classical, illustrative examples and constructions • $49.95 (TENT.)/ISBN O-B176-4294-3 time problems for nonlinear control systems. Practical applications of wavelets to digital signal 2004/APPROX. 278 PP./HARDCOVER/$129.00 processing and data compression • Excercises. ISBN 0-817 6-4084-3 PROGRESS IN NONLINEAR DIFFERENTIAL EQUATIONS AND THEIR 2004/APPROX 168 PP., 40 ILLUS./SOFTCOVER/$29.95 (TENT.) APPLICATIONS, VOL 58 ISBN 0-8176-3600-5 Complex Numbers APPLIED AND NUMERICAL HARMONIC ANALYSIS from A to ••• Z Spatial Patterns T. ANDREESCU, University of Wisconsin, Whitewater, WI,· and An Introduction AMC, do University of Nebraska, Lincoln, NE; and Higher Order Models in D. AND RICA, Babes-Bolyai University, C/uj~Napoca, Romania Physics and Mechanics to Continuous· Time Complex Numbers from A to ... Z is an introduction to this L. A. PELETIER, University of Lei den, The Netherlands; and Stochastic Processes fascinating subject, with a particular emphasis on key W. C. TROY, University of Pittsburgh, PA Theory, Models, and Applications to concepts and elementary results concerning these "This book is about a new family of higher-order scalar Finance, Biology, and Medicine numbers. The reader learns how complex numbers may model equations recently proposed in order to gain V. CAPASSO and D. BAKSTEIN, both, MIRIAM Research be used to solve algebraic equations, as well as the insight into the dynamics of complex spatiotemporal Center, Italy geometric interpretation of complex numbers and the patterns in a wide range of physical and mechanical This book is a rigorous and self-contained introduction operations involving them. problems ... One finds a large amount of exercises and open problems that can serve as a starting point for to the theory of continuous-time stochastic processes. The theoretical parts of the book are augmented with Abalance of theory and applications, the work presents further research ... The authors have produced a well­ rich exercises and problems at various levels of difficul­ concrete examples of modeling ·real-world problems ty. Aspecial feature of the book is a selection of out­ written book, which gives a good picture of what is known about the canonical equations. " from engineering, biomathematics, industrial mathe­ standing Olympiad and other mathematical contest matics, and finance using stochastic methods. problems solved by employing the methods presented -NIEU ARCHIEF VOOR WiSKUNDE 2004/APPROX 300 PP., 10 ILLUS./HARDCOVER/$64.95 (TENT.) in the text. 2001/360 PP., 2891LLUS./HARDCOVER/S82.95 ISBN 0-8176-3234-4 ISBN 0-8176-4110-6 2004/APPROX. 340 PP., 150 ILLUS./SOFTCOVER/$49.95 (TENT.) MODELING AND SIMULATION IN SCIENCE, ENGINEERING AND ISBN 0-8176-4326-S PROGRESS IN NONLINEAR DIFFERENTIAL EQUATIONS AND THEIR TECHNOLOGY APPLICATIONS, VOL 45

CALL: 1-800-777-4643 • FAX: {201) 348-4505 E-MAIL: [email protected] • VISIT: www.birkhauser.com Please mention promotion #Y9658 when ordering. Prices are valid in the Americas only and are subject to change without notice. For price and ordering information outside the Americas, Birkhiiuser please contact Birkhauser Verlag AG by E-mail: [email protected] Boston · Basel · Berlin 4/04 Promotion #Y9658 Feature Articles 394 Quantum Game Theory Steven E. Landsburg Communication over quantum channels would allow superposition of messages, so games in which players use such channels would allow superposition of strategies. The author looks at a game theory whose strategy spaces allow such quantum moves.

403 Research Mathematicians and Mathematics Education: A Critique Anthony Ralston The author reviews the contributions of research mathematicians in K-12 mathematics over the past decade and gives his assessment of positives and negatives. Communications Commentary 392 Networks 389 Opinion Bill Casselman 390 Letters to the Editor 400 A Photographic Look at the Joint 41 7 A Mathematician's Survival Guide: Meetings, Phoenix 2004 Graduate School and Early Career 412 WHAT IS ... a Billiard? Development- A Book Review YakovSinai Reviewed by Elizabeth Theta Brown 4 14 The Great Prime Number Record Races Gunter M. Ziegler 421 2004 Steele Prizes 426 2004 Veblen Prize 428 2004 Wiener Prize 431 2004 Moore Prize 433 2004 Conant Prize 435 2004AwardforDistinguished Public Service 438 2003 Morgan Prize Notices Departments of the American Mathematical Society Advertisers' Forum ...... 402 Mathematics People ...... 440 EDITOR: Andy Magid Dawson Awarded CRM-Fields Prize, AWM Essay Contest Winners ASSOCIATE EDITORS: Announced. Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert J. Daverman, Nathaniel Dean, Rick Durrett, Mathematics Opportunities ...... Susan Friedlander, Robion Kirby, Steven G. Krantz, 441 Elliott H. Lieb, Mark Saul, Karen E. Smith, Audrey New NSF Funding Opportunities, Call for Nominations for Terras, Lisa Traynor CRM-Fields Prize, AP Calculus Readers Sought, News from the SENIOR WRITER and DEPUTY EDITOR: Clay Mathematics Institute, News from the International Centre Allyn Jackson for Mathematical Sciences, Training in Biological and MANAGING EDITOR: Sandra Frost Mathematical Sciences, Mathematical Social and Behavioral CONTRIBUTING WRITER: Elaine Kehoe Sciences. PRODUCTION ASSISTANT: Muriel Toupin PRODUCTION: Marcia Almeida, Kyle Antonevich, Inside the AMS ...... 443 Stephen Moye, Lori Nero, Karen Ouellette, Donna A Summer at Voice of America, Corrections to the 2003 Annual Salter, Deborah Smith, Peter Sykes Survey of the Mathematical Sciences (First Report), Correction: ADVERTISING SALES: Anne Newcomb von Neumann Awards, Deaths of AMS Members.

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Graphics Update My employer, the University of Oklahoma, is home to are­ requires good illustrations and computers in our age markable collection of original editions of science books. have ... made it possible for amateurs to produce their own Founded by an eminent oilman and book lover, Everette L. illustrations of professional quality. Possible, but not easy, DeGolyer, who in 1947 proposed "to give [the university] and certainly not as easy as it is to produce their own math­ a blank check to buy all of the important books in printed ematical writing with .. .TpC ..All you can hope for is that editions, from Aristotle forward, significant in the history the amount of work involved in producing an illustration of science," the university's History of Science Collections is in proportion to the intrinsic difficulty of what you contains some remarkable treasures. For example, it has want to do. And the intrinsic difficulty of producing a a first edition of Galileo, with notes by the author for a good mathematical illustration inevitably means that you second edition. Mathematical should expect to do some rarities in the collection include interesting mathematics as the first printed Euclid and the well as solve interesting first Euclid in English. The lat­ computational problems ter has the following interesting along the way. Mathemat­ feature: there are little fold-out ical illustrations are a spe­ paper models in the margins cial breed-good mathe­ of the solid geometry sections.1 matical illustration almost According to the collection's always requires mathe­ curator from the 1950s through matics in the process of the 1990s, Duane Roller, who making it." (For more in­ first showed me this Euclid vol­ troductory material from ume thirty years ago, these fig­ Casselman on mathemati­ ures were there because the cal illustration, see "Pic­ English translation was done tures and Proofs", Notices, before perspective drawing had November 2000, pp. made its way to the British Isles. 1257-66.) This is such a nice story that I We hope that having a have never checked to see if graphics editor is one way Roller's theory was correct. (Nor to ease the production of have I ever seen a modern pop-up Euclid.) mathematical illustrations for Notices articles, and Those fold-out figures in the antique Euclid volume are we invite Notices authors and potential authors to explore notable as state-of-the-art (for their time and place) math­ graphical possibilities with the graphics editor. ematical graphics. Uke all good mathematical graphics, they As readers will also note, this issue contains color show the reader things which are much easier to see than graphics. Beginning with this issue, color graphics will to visualize, in this case literally adding a dimension. be a periodic (roughly quarterly) feature of the Notices. As attentive readers of the masthead will have noted, That the use of color adds a dimension to mathematical with this issue Associate Editor Bill Casselman is now the graphics seems to be widely recognized, although the Notices graphics editor. Bill will continue with his previ­ conventions and practical economics of journal publish­ ous responsibility as covers editor and in addition will work ing often preclude the use of color in much of the archival with authors of features and communications to generate literature. We hope that the opportunity to publish color images for those articles. Bill will also contribute occasional graphics in the Notices will not only add to the value of graphics articles, like the one in this issue (pages 392-393). the Notices to our readers but also inspire Notices authors As Bill writes in the preface to his book Mathematical Il­ to use the color dimension when appropriate. lustrations:2 "To explain mathematics well often -Andy Magid

1 A modern reproduction can be found on page 16 of Edward R. Tufte, Envisioning Information, Graphics Press, Cheshire, CT, 1990. The photo with this article is from a slightly later English Euclid, now in the Thomas L. Fisher Rare Book Library, University of Toronto. 2 Cambridge Untversity Press, to appear, and online at http: I jwww. math.ubc.ca/-cass/graphics/manual/.

APRIL 2004 NOTICES OF THE AMS 389 Letters to the Editor The Notices invites readers to submit letters and opinion pieces Remembering Max Shiffman on topics related to mathematics. Don Spencer was a close friend and Electronic submissions are pre­ sometime collaborator of Max Shiff­ ferred (noti ces-1 etters@ man. I would like to add some of ams. org); see the masthead for his words to Peter Lax's obituary postal mail addresses. Opinion ["Max Shiffman (1914-2000)," De­ pieces are usually one printed cember 2003, page 1401]. Shiffman's page in length (about 800 words). friends and colleagues, according to Letters are normally less than Spencer, contributed financially to one page long, and shorter letters keep him at Chestnut Lodge when all are preferred. other sources of funding had dried up. He was transferred back to a Califor­ nia public institution only after the burden became too great. When Shiff­ man then convinced a jury that he Where do Notices Covers Come was mentally competent, they found From? him an apartment and a job (on a Usually Notices covers come grant). Shiffman recovered and was from the articles inside the No­ appointed to a professorship at Cali­ tices. fornia State at Hayward, but accord­ In practice, most covers are ing to Spencer some quirks remained. created specifically to relate to One was that Shiffman thought he articles that have been accepted should get royalties for his theorems. for publication. Sometimes the Another was that in the heat of argu­ cover image will come straight ment Shiffman might pull a crumpled from the article, but more often document from his pocket saying, it takes off from an idea in the ar­ "Here's a paper that proves I'm sane. ticle, even sometimes a mere hint. Show me yours." But it is possible to go the other In some ways, perhaps, Shiffman way-occasionally articles have was no more mad than most of us. At been written to go with an idea for one of Spencer's gatherings in Prince­ a cover. ton, the conversation turned to his If you have an image that you continuing collaboration with Shiff­ think would make a good cover, man while Shiffman was institution­ before submitting it you should alized. Spencer said that he could write an article to go with the work with him in his lucid times but cover. The article should be of had to recognize when he was "deep­ sufficient quality to be accepted ening", because "then you couldn't on its own. But it need not be trust his theorems." When asked long; in principle, it would be rea­ how he could tell, Spencer's list of sonable for the article to simply symptoms ended with "Talking with explain the picture. him was like dropping a stone in a The simplest criterion to keep bottomless pool- you would ask him in mind is that the picture should a question at nine and get an answer try to explain something mathe­ at noon." Silence, as every wife looked matical. If the picture is striking at her husband. enough that it could stand on its If anything mitigated the tragedy own, then the accompanying ar­ of a brilliant career devastated by ticle could explain the picture, mental illness, it is that the deep, rather than the picture illustrat­ affectionate bond between Shiffman ing the article. and his colleagues was finally broken only by their deaths. -Bill Casselman Covers/Graphics Editor -Murray Gerstenhaber ([email protected]) University of Pennsylvania [email protected] (Received December 9, 2003)

390 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Mathematical Word Processing • 15\TEX Typesetting • Computer Algebra

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Every year a consortium of mathematics organi­ distribution for small p. The mean degree of nodes zations chooses a theme for the month of April, is J1 = p(N - 1). The best-known phase transition is designated "Math Awareness Month" (see also that as J1 passes through 1, the largest connected ht tp: I /www. ams. org/ams/mam. html ). This year's component starts to grow rapidly. For J1 < 1 and theme is networks. The importance of networks in large Nit contains a very small fraction of all nodes, mathematics at the moment can be gauged by the but as J1 increases it takes up a sizeable part; for fact that the Notices has recently published two J1 > 1 the fraction f taken up is the unique root of reviews of books on the topic and will publish an f = 1 - e- llf. The sizes of the other components article by Peter Sarnak on a related topic in the near decrease dramatically as well. In the following two future. figures a sequence of randomly constructed graphs "Network" is the word everybody- physicists, is shown for p = 0.8 and 1.5. The colored nodes are biologists, sociologists, engineers-except mathe­ those in the five largest components. Also shown is maticians and a few computer scientists, uses for the empirical distribution of nodal degrees. what this minority calls "graphs", i.e., a set of nodes linked together by edges. Variants exist: the edges might be oriented or labeled, or the vertices clas­ • 22 sified. If there is a connotation to "network" as 13 opposed to "graph", it is that networks arise from real life and are concerned with relations between real objects. Important examples include meta­ bolic interactions of chemicals in living things, hard-wired connections among servers on the Internet, links between World Wide Web pages, citations of references in scientific papers, conta­ gion, and electric power grids. Networks-very, very large networks- are a routine and important part of modern life. The World Wide Web has much over 1 billion live nodes. How can we make sense of something that complex? Mean 0.8 The traditional way to analyze a graph is just to Size 400 draw it and look at it, but for large networks this is ridiculous. One has to condense the information available into a small package- in other words, to compile statistics. • 258 What does a typical graph look like? In the beau­ : : tiful theory originating mostly with Alfred Renyi and 8 ~ Pal Erdos, a random graph is obtained by starting with N nodes fixed in advance and then adding edges between random pairs with probability p. There are an astonishing number of interesting results known about such graphs, the most remarkable being that certain phase transitions occur as p increases. The degree of a node in a network is the number of edges leading to it. In the random graph, distribution of degrees is binomial, well approximated by a Poisson

-Bill Casselman is a professor of mathematics at the Univer· Mean 1.5 sity of British Columbia and the graphics editor of the Size 400 Notices. His email address is cass@math. ubc. ca.

392 NOTICES OF THE AMS VOLUME 51, NUMBER 4 increasing x, de Solla Price's distribution is reasonably approximated by a power law where the 1.0 N = 400 number of nodes of degree k is proportional to = 2500 some power 1/ k 01 with IX > 2, here 2 + 1/m. The = 00 following figure shows a degree distribution aris­ relative size ing from a sample network grown by cumulative advantage, along with the theoretical beta distrib­ 0.0 ution, all plotted on a log-log graph so as to show 0 2 3 up the power law indicated by a straight line. Mean degree 2 4 8 16 64 256 1024 The figure above shows how the largest com­ Degree (log) ponent grows as 11 increases through a sequence of values for graphs of sizes 400 and 2 5 00 constructed randomly by a computer. The graph for N = oo shows the theoretical fraction taken up by the • Actual frequency largest component for p > 1, a graph which Theoretical frequency the true fraction approximates more closely for Power law large N. The traditional theory of random graphs is math­ ematically impressive and plays an important role in estimates of the efficiency of many algorithms dealing with graphs. But its role in explaining what networks encountered in real life look like is min­ imal. Real networks do not spring out of nowhere; they grow in one way or another, and the structure they acquire depends on that growth. In particu­ lar, this causes them generally to look very differ­ ent from one of the Renyi-Erd6s random graphs. As far as I know, the first account of how real networks seem to grow dates to 1965, in a paper • by the remarkable polymath Derek de Solla Price. He was interested in the statistics of the science citation network, where directed edges lead from De Solla Price's work seems to have been long a paper to each paper it refers to. It had recently neglected, and his results were rediscovered become possible to scan efficiently through large apparently independently, by Albert Barabasi i~ amounts of data based on the Science Citation much cited recent work. Barabasi introduced the Index, and de Solla Price showed that a principle now popular term "preferential attachment" for of formation he called cumulative advantage led what de Solla Price calls cumulative advantage and the term "scale free" networks for those that sat­ to a stable degree probability distribution Pk = (1 + 1/ m)B(k + 1, 2 + 1/ m). Here Pk is the probabil­ isfy a power law of degree distribution, although ity of a node having degree k, m is the mean it seems more reasonable to apply the term only effective number of journal references in papers, to those which, like the networks of de Solla Price retain a stable degree distribution as they grow. ' and B is Legendre's beta function B(x, y) = [(x)[(y) j [(x + y). He also showed that there was Further Read ing some agreement between his theory and the nature of citations in the real world. Cumulative There is a great deal of pseudoscience as well as science in this field, which is enjoying a rapid advantage can be succinctly su~marized as growth at the moment. The most difficult problem "them what has, gets" - a paper that is frequently is to fit theory with practice. A very good reference referred to will likely get more citations than one for mathematicians is the recent article 'The struc­ referred to less often. (A warning: de Solla Price's ture and function of complex networks" by Mark paper will be sobering to a naive mathematician Newman in volume 45 of the SIAM Review (2003), who would like to think that a paper's intrinsic recommended also by Rick Durrett in his March quality plays a more important role than mere Notices review of Barabasi' s book Linked. It contains popularity.) What is perhaps surprising is that the a large and useful list of references on the topic. stable degree distribution is well approximated You can find listed there the papers of de Solla Price even for small networks growing in this way, at as well as an interesting introduction to his least away from the tail end of highly cited research on citation networks. · papers. Because B(x, y) ~ [(y) j xY for fixed y and

APRIL 2004 NOTICES OF THE AMS 393 Quantum Game Theory Steven E. Lands burg

An n-person game consists of Nash equilibrium either is or is not a good predic­ 1. n sets S; (i = 1, ... , n). tor of actual behavior depending on exactly what 2. n real-valued functions sort of behavior is being modeled. Turning now to normative criteria (criteria P; : sl X ... X Sn ~ R (i=1, ... ,n). intended to judge the desirability of outcomes), The set S; is called Player i 's strategy space. The the least controversial is the criterion of Pareto­ function P; is called Player i 's payoff function. Optimality. Given two n-tuples of strategies This formulation is general enough to model s = (SI, ... , Sn) and s' = (s1, .. . , Sn). we say that s pretty much any real-world strategic interaction: we (weakly) Pareto dominates s' if P;(s) ~ P;(s') for all take S; to be the set of actions available to i; it is easy to verify that Pareto dominance is a the character called Player i, we imagine partial order, and we say that s is Pareto optimal that each player must choose some ac­ if it is maximal for this order. tion, we imagine that those actions have In general, we have no guarantees of existence some joint consequence and that P; or uniqueness for Nash equilibria or for Pareto op­ measures Player i's assessment of the tima. When they do exist, Nash equilibria and Pareto value of that consequence. Given such a optima might or might not coincide. The single model, one can ask questions that tend to most famous example in the history of game fall into two broad categories: First, what do we theory is the Prisoner's Dilemma, one form of that think the players will do (assuming, ordinarily, which can be represented by the following matrix: they are sublimely selfish and sublimely rational)? Second, what do we think the players ought to do Player Two (according to some standard of fairness or justice or morality)? Questions of the first sort call for so­ c D lution concepts; questions of the second sort call for normative criteria. The most thoroughly studied solution concept (1) (3, 3) (0, 5) is Nash equilibrium, an outcome that results when c players maximize their own payoffs, taking other Player One players' behavior as given. More precisely, an n-tuple of strategies (SI, ... , Sn) is a Nash equilib­ D (5, 0) (1,1) rium if for every i and for every s E S;, P;(SI, ... ,S;- I,Si,Si+l···Sn) Here the rows are indexed by Player 1's strategy set ~ P;(SI, . . . ,Si- I,S,Si+lo ... ,Sn). {C, D}, the columns are indexed by Player 2's strat­ egy set {C,D}, and the (i,J) entry is (PI(i,j),Pz(i,j)). The associated story, which the Steven E. Landsburg is adjunct associate professor of reader can see enacted every Tuesday night on -economics at the University of Rochester. His email N. Y.P.D. Blue, goes as follows: Two prisoners have address is 1 andsbur@troi. cc . rochester. edu. jointly committed a crime. They are separated and

394 NOTICES OF THE AMS VOLUME 51, NUMBER 4 invited to testify against each other. Each receives and Even with probability 1 - p ". It is then natural the same menu of options: You're facing a ten­ to define new payoff functions year sentence for this crime. You can either coop­ erate with your buddy (strategy C) by refusing to Pj(p, q) = pqP;(Odd, Odd) testify against him or defect from your criminal + p(1 - q)Pi(Odd, Even) partnership (strategy D) by testifying. If you both + (1 - p)qP;(Even, Odd) cooperate (i.e., if you both stay silent), we'll have to convict you of a lesser crime, which will take + (1- p)(l- q)Pi(Even, Even). three years off each of your sentences. But if you More generally, given any game G with finite defect while your buddy cooperates, we'll take five strategy sets 5;, we define a new game G+ as fol­ years off your sentence (while he serves a full lows: Let Sj be the set of all probability distribu­ term). And if you both defect, we'll take one year tions on Si, and define off each of your sentences. As the reader can (and should) verify, the unique pt:St x ·· ·X S~~R Nash equilibrium (D, D) is also the unique out­ come that is not Pareto optimal. Rational selfish by prisoners always choose the one strategy pair that both can agree is undesirable-in the sense that (3) Pj(st •... ,s~) they would both prefer (C, C).l = J P;(sl, ... , Sn)dsi(sl) ... ds~(sn). The simplest game without a Nash equilibrium St X .. ·XSn is "odds and evens", represented by the game rna- (The restriction to games with finite strategy sets trix Player Two is so that we do not have to worry about conver­ gence issues in (3).) One proves via standard fixed 0 E point theorems that the game G+ has at least one Nash equilibrium, the key point being that each Sj, unlike the original Si, can be identified with a con­ (2) vex subset of a Euclidean space on which the Pj 0 (1, -1) (-1, 1) are continuous. Player One Thus in the case of Odds and Evens, G+ is a better model of reality than G is. I want to argue E (- 1, 1) (1, - 1) that the same thing is true more generally: If G is any game with finite strategy spaces intended to model some real-world interaction, then G+ is always a better model of that same interaction. Suppose there is a Nash equilibrium in which Here's why: In the real world, players must com­ Player 1 plays Even. Then Player 2 plays Odd, so municate their strategies either to each other or Player 1 plays Odd- contradiction; and similarly to a referee or to an interrogating detective, who with Odd and Even reversed. Thus there is no Nash then computes the payoffs. And as a practical equilibrium, and without an alternative solution matter, it is quite impossible for a referee or any­ concept we are unable to predict anything other one else to prohibit the use of mixed strategies. than paralysis on the part of both players. But of Player 1 announces "I defect!" How can the referee course anyone who has ever played this game know whether Player 1 arrived at this strategy knows what actually happens: players randomize through a legal deterministic process or an illegal their strategies, and each wins half the time. random one? One might be tempted to conclude that Nash Because there is no way to prohibit mixed equilibrium is the wrong solution concept for this strategies in practice, we might as well allow game. A better conclusion is that the mathemati­ them in the model. More generally, whenever cal structure (2) is a poor model for the real-world the real world imposes limits on referees' ability game of odds and evens. A better model would to observe and/or calculate, we should improve allow for mixed (i.e. randomized) strategies. So we the model by adjusting the strategy spaces and replace the strategy space 5; = {Odd, Even} with payoff functions accordingly. the unit interval Sj = [0, 1], using p E [0, 1] to Quantum game theory begins with the obser­ model the strategy "play Odd with probability p vation that the technology of the (near?) future is likely to dictate that much communication will 1 In fact, things are even worse than that. Each player has occur through quantum channels. For example, D as a dominant strategy, which means that D is the op­ timal play regardless of whether the other player chooses players might communicate their strategies to a ref­ C or D. This is a far more powerful reason to anticipate eree via email composed on quantum computers. an outcome of (D, D) than the mere fact that (D, D) is a Such communication creates the prospect of new Nash equilibrium. strategies whose use the referee cannot detect and

APRIL 2004 NOTICES OF THE AMS 395 therefore cannot prohibit: Instead of cooperating on his first turn and the matrix u- 1 on his second. or defecting (or randomizing between the two), a Here's why that's clever: If Player Two fails to flip, player might send a message that is some quantum then the net result of the three operations is superposition of the messages "I cooperate" and "I defect". To read the message, the referee must (4) u-1 oloU=I; destroy the superposition, along with any evidence that the superposition ever existed, which makes whereas if Player Two flips, the net result is superpositions effectively impossible to prohibit. - -./2+1 -./2 What cannot be prohibited must be ( (5) u- 1 oF o u = ~ allowed; therefore, if we want to model accurately the behavior of games in which players have access to "quantum moves", we should expand our This is great for Player One, because both (4) strategy spaces accordingly. and (5) map the state represented by H into itself One might guess that a quantum move is just (remember that any scalar multiple of H is equiv­ one more way to implement a mixed strategy, so alent to H). Thus whether or not Player Two flips that there is nothing new here for game theory. The the coin, Player One is guaranteed a win. physicist David Meyer [M] was the first to publish Meyer's example shows that quantum moves a counterexample to that guess. In Meyer's exam­ can be more powerful than mere mixed strategies, ple a single coin is passed back and forth between at least in a context where quantum moves are two blindfolded players. The coin starts out heads available to only one player. But of course it is up (call this state H). Player One has the option ei­ more natural to ask what happens if both players ther to flip the coin or to return it unflipped. Then are given access to a full set of quantum moves. Player Two (still blindfolded so he doesn't know The first example of a full-fledged quantum Player One's first move) has the same option: Flip game is due to Jens Eisert, Martin Wilkens, and or don't flip. And finally, Player One gets another Maciej Lewenstein [EWL]. Let G be a two-by-two turn. If the coin ends up in its initial state H, Player One wins. If it ends up in the opposite state T (tails game, that is, a game with two players, each of up), Player Two wins. whom has a two-point strategy space, say {C, D}. Here Player One has four strategies ("flip, flip", (The reader will have no difficulty extending this "flip, don't flip", etc.). Player Two has two strate­ construction to n-by-m games.) Each player is given gies ("flip" and "don't flip"). In any mixed strategy a coin that he returns to the referee either in its Nash equilibrium, Player Two flips with probabil­ original state (to indicate a play of C) or flipped (to ity . 5, Player One flips an even number of times with indicate a play of D). A player with access to quan­ probability .5, and each wins half the games. tum moves can act on his coin with any unitary Now suppose we treat the coin as a quantum matrix and therefore return it in a state object. Its state is an equivalence class of nonzero lXH + {3T vectors in the complex vector space spanned by H ("heads") and T ("tails"); two vectors are equivalent where lX and {3 are arbitrary complex numbers, if one is a scalar multiple of the other. A physical not both zero. (Here and in everything that follows operation on the coin corresponds to a unitary I will freely abuse notation by writing lXH + {3T operation on the state space; in particular, we can both for an element of the vector space C2 and for set things up so the operation "not flip" is repre­ the state represented by that vector.) When the sented by the identity transformation and the referee observes the coin, it will appear to be un­ operation "flip" is represented (with respect to flipped or flipped with probabilities proportional the basis {H, T}) by the unitary matrix to I £X 12 and I{31 2 . As long as the coins can be treated as independent quantum entities, then indeed all we have is a fancy way to implement a mixed strat­ egy-in other words, nothing new for game theory. Meyer's example was more interesting because both players acted on a single coin. Eisert, Wilkens, Now suppose that Player One (but not Player Two) and Lewenstein (referred to henceforth as EWL) has access to a full array of quantum moves; that make their example more interesting by assuming is, instead of simply flipping or not flipping, he can the players' coins are entangled so that there apply any unitary matrix he chooses. In particular, if he is clever, Player One will choose the matrix is a single state space for the pair of coins. Explicitly, let C2 be the two-dimensonal complex - 1- i - .j2 ) vector space spanned by symbols H and T; then the ( 2 2 u-- -./2 - 1+i state space for an entangled pair is 2 -2- (6)

396 NOTICES OF THE AMS VOLUME 51, NUMBER 4 where ~ is the equivalence relation that identifies a vector with all its nonzero scalar multiples. As before, I will write, for example, H 0 H both for so we can just as well define Player One's strategy a vector in the space C2 0 C2 and for the state it space to be SUz I { ± 1}. Moreover, the group SUz can represents. A physical operation on the first coin be identified with the group s3 of unit quaternions is represented by a two-by-two unitary matrix U via the map acting on the state space S as U 0 1 . A physical SUz- s3 operation on the second coin is represented by a two-by-two unitary matrix V acting on the state (~ ~) ~A+Bj. space as 1 0 vT. Now EWL conjure the following scenario: A pair The same analysis applies to Player Two. Thus we of coins starts in the state2 define the strategy spaces (H 0 H) + (T 0 T). (8) sf= s~ = s3 1{±1} = RP3. As before, each player is handed one of the Using language loosely, I will often identify a coins and invited to indicate a play of C by strategy with either of the two quaternions that applying the identity matrix ("leaving the coin represent it. untouched") or to indicate a play of D by applying Next we motivate definitions for the payoff the matrix functions. Start with the game G: F=(o-z. 01) Player Two

D ("flipping the coin"). As long as the players restrict c themselves to the two-point strategy space {C, D}, the pair of coins lands in one of the four states (9) c (XI. Y1) (Xz, Yz) (7a) CC = (H 0 H)+ (T 0 T), Player One (7b) CD= (H 0 T)- i(T 0 H), D (X3, Y3) (X4, Y4) (7c) DC=(H 0 T)+i(T 0 H),

(7d) DD = (H 0 H) - (T 0 T). Suppose Player One plays the quaternion p and Player Two plays the quaternion q. Write the The referee now performs an observation to product as determine which of the states (7a- d) the coins occupy and makes appropriate payoffs. (10) pq = rr1(pq) + rrz(pq)i + rr3(pq)j + rr4(pq)k If players cannot be trusted to restrict them­ where the rrex are real numbers unique up to a the two strategies C and D, then the selves to sign, because p and q are defined up to a sign. Using modeler should replace the game G mathematical the notation of (7a-d), a chase through the iso­ with a new game GQ that expands the strategy morphisms reveals that the coin is transformed spaces accordingly. Player One's strategy space from the initial state CC to a final state should consist of the operations that can be effected on the state spaceS (defined in (6)) via the rr1 (pq)CC + rrz(pq)CD + rr3(pq)DC + rr4(pq)DD. action of unitary matrices on the first variable. Let When the referee observes the coins' joint state, he Uz be the group of two-by-two unitary matrices. The of the four outcomes with proba­ matrices that fix S are the scalar matrices, which observes each form a subgroup S1 c Uz. Therefore, we define bilities Player One's strategy space to be the group Uz /S1. Prob (CC) = rr1 (pq)2 Prob (CD) = rrz (pq)2 Let SUz c Uz be the subgroup of matrices with Prob (DC) = rr3(pq)2 Prob (DD) = rr4(pq)2 : determinant one; then inclusion induces an iso­ morphism Thus we should define the payoff functions by 2 depends heavily on the assumption Everything to follow 4 that the coins start in the "maximally entangled" state (11) Pf(p , q) = 7Ti(pq)2 Xi' (H ® H) + (T ® T). If they were to start in a state of the form I C<=l s ® t, then the construction that follows would only re­ construct the classical mixed strategy game G+. [EWL] 4 studies families ofquantum games parameterized by the P~(p,q) = I 7Ti(pq)2 Yi. choice of initial state. C<=l

APRIL 2004 NOTICES OF THE AMS 397 Equations (8) and (11) define the quantum game Pt(s!, ... , s~) GQ associated to the game G of equation (9). The quantum game GQ is not at all the same as the = J Pt(Sl, ... , Sn)ds!(sl) ... ds~(sn). S1 X · ·· X Sn mixed strategy game G+. Using mixed strategies, the players can jointly effect some but not all (We identify the original strategy space St with a probability distributions over the four possible subset of sr by identifying the points with the prob­ outcomes; there is, for example, no pair of ability measure concentrated ons .) Now if we equip mixed strategies that can effect the probability RP3 with its Borel cr-algebra, then GQ acquires, quite distribution naturally, the structure of a measurable game. Thus Prob (CC) = 112 Prob (CD)= 0 equation (12) becomes a meaningful definition. The payoff functions in G' should be called Prob (DC)= 0 Prob (DD) = 112. Pj or PtQ+, but I will just call them Pt. By contrast, in the game GQ any probability dis­ In the game G' there is always at least one Nash tribution at all is realizable; in fact, more is true: equilibrium, namely (J.l, J.l) where J.1 is the uniform taking Player One's strategy as given, Player Two probability distribution on RP3 . There are usually can choose a strategy that effects any desired prob­ more interesting equilibria as well. For example, let ability distribution. (Proof: Let Player One choose us return to the Prisoner's Dilemma (1). It is easy strategy p, and let r be an arbitrary unit quaternion; to verify that the following pair of mixed strategies then Player Two can play p - 1r .) constitutes a Nash equilibrium: Thus Nash equilibria must be a great rarity in quantum games; in fact, Nash equilibria exist only J.1: Player 1 plays the quaternions 1 and k, each when there exists Oi E { 1, 2, 3, 4} that maximizes with probability 1 I 2. both Xcx and Ycx; in other words, Nash equilibria (13) exist only when there is no conflict between the v: Player 2 plays the quaternions i and j, each players. with probability 1 I 2. But nobody would ever actually play the game GQ anyway. Just as there is nothing to stop our Indeed, for any unit quaternion q =a+ bi + cj + dk, players from adopting quantum strategies, there we have is also nothing to stop them from adopting mixed 5 5 quantum strategies. So we really want to study the Pz(J.l,q) = 2a2 + 2b 2 + 2c2 + 2d2 , game so that q is an optimal response to J.1 if and only if (12) a = d = 0. Thus q = i, q = j are optimal responses; So far, we have defined the game G+ only when whence so is the strategy v. Sinlilarly with the play­ the game G has finite strategy spaces, which is cer­ ers reversed. tainly not the case for GQ. So to turn (12) into a In the Nash equilibrium (13), each player's pay­ definition, we must first give a more general defi­ off is 5/2, so (13) Pareto dominates the unique nition of G+. classical equilibrium (D, D) (where the payoffs are both 1 ). Nevertheless, (13) is still Pareto subopti­ Definition. A measurable game consists of mal, being dmninated by (C, C). More generally, we would like to classify the 1. n measure spaces (St, Vt) (i = 1, ... , n). (That Nash equilibria G' G is, St is a set and Vt is a cr-algebra on St.) in where is an arbitrary two­ by-two game. The results 2. n bounded measurable functions that follow are from the forthcmning article [L). Pt : S1 X · · · X Sn - R (i = 1, ... , n). Given a strategy J.l, we define the optimal response sets Definition. Let G be a measurable game. Then a (J.l) = 3 mixed strategy for Player i consists of a probability 01 {p E RP IP1 (p, J.l) is maximized}, measure on the space (St, Vt). Oz(J.l) = {q E RP3 1Pz(J.1, q) is maximized}. Definition. Let G be a measurable game. Then we Thus (J.l, v) is a Nash equilibrium if and only if v define a game G+ as follows: Let Sj be the set of all is supported on Oz(J.l) J.1 (v). probability measures on St and define and is supported on 01 This leads us to ask: Which subsets of RP3 can occur as Ot(J.1)? The answer is: Pt : S! X · · · X S~ - R Theorem. For any J.l, each Ot(J.l) is a projective by hyperplane in RP3 .

398 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Proof. We have That cuts the set of (equivalence classes of) potential Nash equilibria way down to size. But (14) pl (p, Jl) = f pl (p, q)dtJ(q), we can cut it down much further. Call a pair of weighted frames (Jl, v) realizable if it is a Nash which, for fixed Jl, is a quadratic form in the coef­ equilibrium for some quantum game. A reason­ 3 ficients rroc(P) and hence maximized (over S ) on able first guess is that every pair (Jl, v) is realizable. the intersection of s3 with the linear subspace of But the truth is quite the opposite: the main R 4 corresponding to the maximum eigenvalue of theorem of [L) establishes highly restrictive (and that form. 01 (Jl) is the image in RP3 of that linear easily checkable) necessary conditions for realiz­ subspace. Similarly, of course, for Oz. ability. Modulo some minor technical provisos, The theorem (or more precisely, the proof of the the theorem implies that for each J1 there are that (Jl, v) is theorem) has some immediate corollaries of con­ only a small finite number of v such finite number siderable interest. realizable. In many cases, that small is zero; in the remaining cases, the relevant strate­ Corollary. Let G be the arbitrary two-by-two game gies v are easy to describe. Given a particular game, of expression (9). Then in any mixed strategy quan­ this makes the search for Nash equilibria a tractable tum Nash equilibrium, Player One earns a payoff problem. of at least (X1 +Xz +X3 +X4)/4, and Player Two One can view quantum game theory as an exercise earns a payoff of at least (Y1 + Yz + Y3 + Y4)/4. in pure mathematics: Given a game G, we create a new game G! and we study its properties. But game Proof. The quadratic form (14) has trace theory has historically been interesting primarily X1 + Xz + X3 + X4 and hence a maximum eigen­ for its applications. As with quantum computing, value of at least one fourth that amount. Similarly, the applications of quantum game theory lie in the of course, for Player Two. future.3 The immediate task is to prove theorems that we expect will be useful a generation from now. mean­ Corollary. Suppose the game (9) is zero-sum, References ing that Xoc + Yoc = 0 for all 01. Then in any mixed [EWL] ]. EISERT, M. WILKENS, and M. LEWENSTEIN, Quantum strategy quantum equilibrium, Player One earns games and quantum strategies, Phys. Rev. Lett. 83 exactly(X1 + Xz + X3 + X4)/4 and Player Two earns (1999), 3077. exactly (Y1 + Yz + Y3 + Y4)/4. [L] S. LANDSBURG, Nash equilibria in quantum games, preprint available at http: I /www . l andsbu rg . To describe Nash equilibria in general, we need com/qgt.pdf. 3 to describe probability measures on RP in general. [M] D. MEYER, Quantum strategies, Phys. Rev. Lett. 8 (1999), Of course, there are a huge number of such mea­ 1052. sures, but fortunately they fall quite naturally into large equivalence classes. In particular, we say that two mixed quantum strategies J1 and tJ' are equiv­ alent if for all mixed quantum strategies v and for all 01 E {1, 2, 3, 4}, we have r 7Toc(pq)2dtJ(p)dv(q) J RP3 x RP3

= r 7Toc(pq)2dtJ'(p)dv(q) JRP3 x RP3 where the rroc are the coordinate functions on the quaternions as in (10). That is, two strategies are equivalent if one can be substituted for the other without changing either player's payoffs in any game. It turns out that equivalence classes of strate­ gies are quite large. I proved in [L) that every mixed quantum strategy is equivalent to a strategy sup­ ported on (at most) four mutually orthogonal points (that is, four points that are the images of mutu­ S3 ). Thus an equivalence ally orthogonal points in 3 It has been suggested that quantum games might have class of measures can be identified with a "weighted immediate applications in the theory of evolution, frame" in R4 , that is, a set of mutually orthogonal assuming that genetic mutations are driven by quantum vectors, each assigned a positive weight in such a events. To my knowledge there is no evidence to support way that the weights add up to one. this admittedly intriguing speculation.

APRIL 2004 NOTICES OF THE AMS 399 2004 Joint Mathematics Meetings in Phoenix

400 NOTICES OF THE AMS VOLUME 51, N UMBER 4 The 2004 joint Mathematics Meetings were held in Phoenix, AZ, january 7- 10. Some "snapshots" of Meetings events, people, and places are shown on these pages, including: registration desk, exhibits hall, speakers, prizewinners, Employment Center, email lab, Who Wants To Be A Mathematician game, Mathematics and the Arts exhibit, receptions, and meeting areas. For more details, see photo index on page 487.

APRIL 2004 N OTICES OF THE AMS 401 ~~AMS AMERICAN MATHEMATICAL SOCIETY CAMBRIDGE UNIVERSITY PRESS

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Not since the New Math period of the 1960s had university mathematicians played such important roles in K-12 education as in California during the 1990s. -David Klein, "A brief history of American K-12 mathematics education in the 20th century" [1]

ince at least the publication of A Nation at those in the Third International Mathematics Study Risk [2] in 1983, there has been ferment [4], as well as scores on various tests, together about precollege mathematics education in with a plethora of anecdotal evidence, suggest that Sthe U.S. Since then, but particularly since far from achieving (the first) President Bush's aim 199 3, research mathematicians have been that U.S. mathematics education should be second more active on the precollege mathematics scene to none by 2000, mathematics education in the than at any time since the days of the New Math U.S. is still nowhere near "second to none". Thus, in the 1960s. Indeed, the pages of the Notices have the efforts of research mathematicians, working to­ regularly had articles, opinion pieces, and letters gether with the other constituencies in math edu­ on the subject of school mathematics. This seems, cation, will be needed if the current situation is to therefore, a good time to review the impact of re­ show improvement. As noted in [5], "one of the search mathematicians on school mathematics most important ways mathematicians can be so­ over the past ten years. In this article I will consider cially responsible [is] by working to improve pre­ where the intervention of research mathematicians college math education." in school mathematics has had favorable results But instead of cooperation, we have had for the and where the results have been less than favor­ past decade, although recently at a lower decibel able. level, the Math Wars [6], [7], which pit (mainly) re­ Just about everyone agrees that research math­ search mathematicians against (mainly) college ematicians have the knowledge and expertise to and university mathematics educators and school make important contributions to the improvement mathematics teachers. No matter which side, if ei­ of school mathematics in the U.S. Indeed, it has been ther, of these wars you are on, it is clear that they stated by a prominent mathematics educator that have, at least, prevented more improvement in U.S. "American mathematics education has benefited school mathematics education than might other­ from a virtually continual stream of support from wise have been achieved. Throughout this article I prominent research mathematicians" [3]. Equally, will use the terms "traditional" and "progressive" just about everyone believes that school mathe­ (or "reform") to designate the two sides in the Math matics is in great, some would say dire, need of im­ Wars, because, whether you like these terms or provement. International comparisons, such as not, they have become traditional (!) in the litera­ ture. Anthony Ralston is professor emeritus of computer science My aim here is not to refight or continue to fight and mathematics at the State University of New York at Buffalo and an academic visitor in the Department of the Math Wars, at least insofar as their mathemat­ Computing at the Imperial College of Science, Technology ical substance is concerned. However, I will not hes­ and Medicine in London. His email address is itate to criticize the tactics of the math warriors [email protected]. when I think these have been counterproductive.

APRIL2004 NOTICES OF THE AMS 403 On the Plus Side difficult because of the errors, sometimes consid­ Why is there agreement among most university erable numbers of them, that have been found by mathematicians and mathematics educators that research mathematicians after publication. The er­ the potential contribution of university mathe­ rors found are often of a trivial and easily fixable maticians to school mathematics education is great? nature, but their existence must serve to weaken Is it because their knowledge of mathematics is su­ the effect of a curriculum proposal. perior to that of mathematics educators? Partly, Teaching Methods even though most mathematics educators, although On the mathedlistserv of which I am a member, they know far less mathematics than research I am often impressed by the insight of other mem­ mathematicians, are knowledgeable enough about bers, often research mathematicians, about how school mathematics to design and implement cur­ particular topics of school mathematics might be ricula for school mathematics. More important, I approached in novel and useful ways. Sometimes think, is the generally deep knowledge that uni­ the ideas may be impractical because of aspects of versity mathematicians have of mathematics, be­ school mathematics not fully understood by uni­ cause this gives them useful insight into what top­ versity mathematicians, but even when this is the ics are particularly important in school mathematics case, the ideas may suggest changes in approach and sometimes into good approaches for teaching to mathematics educators and teachers. Indeed, re­ these topics. With that said, here is a list of areas search mathematicians should be used much more where research mathematicians can make and have than they have been as a source of ideas for teach­ made useful contributions to the debate about ing the content of school mathematics. school mathematics. Another aspect of this concerns methods that Topics have become popular in school mathematics be­ The division of fractions was not included in the cause they are easy to teach but are counterpro­ original version of one of the more popular reform ductive to the understanding of the underlying curricula (Connected Mathematics Program). What­ mathematics. One such is the "infamous" [12, p. 7] ever the reason for this omission, knowledge of FOIL algorithm to multiply two linear polynomials how to divide fractions, while of limited value by bypassing the distributive law entirely. Research when the fractions consist entirely of numbers, is mathematicians have an important role to play in clearly important in much of more advanced math­ apprising mathematics educators and teachers of ematics. When this omission was noticed by a when the methods they teach, although perhaps well-known university mathematician, the devel­ narrowly effective and efficient, give the wrong opers of the curriculum in question promised to mathematical lesson and may, therefore, ill serve remedy this in the next version of the curriculum the student. and did so [8]. In-Service Education of Teachers A less clear-cut example was the postponement Although there may be doubts about whether of teaching the quadratic formula until the twelfth university mathematicians understand enough grade in another popular curriculum (Interactive about school pedagogical issues for them to con­ Mathematics Program) [9], [10]. Surely this is idio­ tribute much about these, there can be no doubt syncratic, and even if this delay is defensible, it is that they can contribute to improving the mathe­ useful to have university mathematicians question matical understanding of elementary and secondary choices of this kind. school teachers. Even if it is true that in-service In general, it is valuable for curriculum devel­ courses for teachers in summers and on weekends opers to have advice and criticism from university will probably never reach a significant fraction of mathematicians about what to include or omit, if teachers and often reach just those teachers who only because it is all too easy to make some bad are already among the most competent, these decisions when immersed in the details of a cur­ courses are an important way of getting mathe­ riculum to be used for all of elementary or all of matical knowledge to teachers that their preservice secondary school mathematics. education did not. Even if the subject matter of such Errors courses is not directly applicable to the grade Lengthy curriculum documents like books or taught by a teacher, it may nevertheless provide long papers are almost certain to contain mathe­ breadth and insight that will improve teaching im­ matical errors, no matter how competent the au­ mediately or in subsequent years. Considerable thors. This is particularly true when the documents numbers of research mathematicians, but surely not have many authors, as is almost always the case enough, have been involved in in-service courses for curriculum documents. Review of curriculum for teachers in recent years (e.g. [13, p. 535]). While documents by research mathematicians before pub­ such activities can only scratch the surface of the lication would always be salutary. Instead, the au­ problem of mathematics teaching in the U.S., they thors of documents such as the NCTM Standards are nevertheless valuable and deserving of [11] have made their lives considerably more applause.

404 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Pre service Education of Teachers Among the handful of nonmathematicians were Another matter on which there appears to be al­ some eminent scientists, including several Nobel most universal agreement is that the mathematics prizewinners. (It was noted by one observer [7, education of prospective teachers of school math­ p. 201] that while the letter listed affiliations, chairs ematics is, with plenty of exceptions of course, held, and prizes won, there was no mention of any woefully inadequate. Indeed, in some elementary teaching awards.) education programs, prospective teachers take no The gist of the letter was a criticism of the De­ mathematics at all. Moreover, even when mathe­ partment of Education for designating five reform matics courses for prospective teachers are taught mathematics curricula as "exemplary" and five oth­ in mathematics departments, they are normally ers as "promising". (While of course criticism of the taken much less seriously by the instructors than designations of any of these ten programs was even the standard lower-division mathematics quite appropriate, the Department of Education courses. There will probably not be much dis­ had been mandated to make such designations by agreement with the claim [13, p. 535] that "uni­ Congress, which also chose the "exemplary" and versity mathematics departments must do a bet­ "promising" categories.) The six signers were, I as­ ter job of teaching their students" or with the sume, familiar with all ten programs, but it is nearly stronger statement that both mathematicians and certain that at most a handful of the endorsers were mathematics educators have [14, p. 127] "largely familiar with all ten and highly probable that most failed to help teachers learn the mathematics they were familiar with none. need in pre-service." A recent attempt at redress No doubt many readers of this article have at one of this situation is [15] a collaboration of mathe­ time or another signed advertisements in news­ matics educators and mathematicians. In any case, papers on strictly political matters whose text was more--preferably many more--research mathe­ distributed to you by someone well known to you. maticians need to get involved with the mathe­ In such cases, the text of what you are signing is matics education of prospective elementary and the whole story, and you are not endorsing opin­ secondary teachers. ions about documents not read nor about issues On the Minus Side you know nothing about directly. The Riley letter was quite different. Most Relations between the research mathematics of the community and the university mathematics edu­ endorsers were expressing an opinion about doc­ cation community have never been worse than uments they had not seen. No doubt they decided they were in the late 1990s, although they appear to do so because they trusted the opinions of those to be less vicious now than they were then. The fault who distributed the letter, and perhaps they were is surely not all on one side. However, my belief is influenced by the inclusion in the letter of an in­ that much of the fault--and most of the bitter­ flammatory quote written five years previously by ness that has resulted--is indeed the fault of the someone who served on the expert panel (of fifteen research mathematics community, almost entirely members) that made the exemplary and promising the pure mathematics research community. In the recommendations. Math Wars the research mathematics community No reader of the Notices would, I think, express has departed from its own high intellectual stan­ an opinion about a research paper he or she had dards for research and has displayed an arrogance not read, even if the paper was by a trusted col­ that has made things much worse than they need league and even if it was also vouched for by other have been. Of course, neither of these strictures ap­ trusted colleagues. This would not be intellectually plies to every research mathematician who has respectable. (Yes, there was a caveat in the open been involved with the Math Wars, but it applies letter to the effect, "While we do not necessarily to too many and particularly to many of those who agree with each of the criticisms of the programs have been most vocal. Note that these remarks are described above ... ," but that does not negate my not meant to imply that the positions taken by re­ point.) search mathematicians have necessarily been wrong An interesting contrast to the Riley letter was a or unwise, but just that they have often been ex­ letter from sixty-five mathematicians published in pressed in most unfortunate ways. the American Mathematical Monthly and The Math­ The Riley Letter ematics Teacher in 1962 [17] in which they ex­ If there is one single incident that epitomizes the pressed concern about the New Math ("Mathe­ Math Wars, it is the open letter to then Secretary maticians, reacting to the dominance of education of Education Richard Riley published in the Wash­ by professional educators who may have stressed ington Post on 18 November 1999 [16]. The letter pedagogy at the expense of content, may now stress was signed by 6 mathematicians and "endorsed" content at the expense of pedagogy and be equally by 219 others, almost all of whom were research ineffective"). The signers of this letter knew whereof mathematicians, among them many eminent ones. they spoke, but also their language was restrained,

APRIL 2004 NoTICES OF THE AMS 40:> as evidenced by the two appearances of "may" in more than we do about how to represent ideas so the quotation above. that they make sense to their students. Note that I express no opinion whatever about The folly of using standardized tests to assess the quality of the ten exemplary/promising pro­ the value of a curriculum is best illustrated (to me) grams. How good or bad they may be is not my by a story that Steve Willoughby tells in his book point here. It is certainly quite appropriate for The Other End of the Log [24]. When he was jointly mathematicians to involve themselves with the appointed to the Department of Mathematics and politics of mathematics education, but when they the School of Education at the University of Wis­ do so using the techniques of the average politi­ consin in Madison in 1960, he simultaneously be­ cian, we are all worse off. came head of the mathematics department at Wis­ Test Scores consin High School, the university's laboratory In late 2002 I wrote an oped piece [18] that sug­ school. After his first year as the high school de­ gested that one result of the No Child Left Behind partment head, a faculty member in the university Act [19] would be an epidemic of rising test scores department of mathematics noted to his colleagues in the U.S. and that this would mask a continuing that the Wisconsin High School scores in a "state­ decline in school math education in the U.S. In re­ wide school mathematics test had deteriorated se­ sponse, one of the most zealous of the math war­ riously from the previous year." Willoughby, who riors wrote a letter to the newspaper attacking my believed that the test was "hogwash," said he would claim but in effect supporting it, because his ar­ arrange it so that the scores in the high school "will gument consisted of giving tables of increasing be the highest in the state this year." And that was test scores in Sacramento. In a similar vein, when what happened, because Willoughby knew how to I visited another prominent math warrior a couple give the students the kind of test-taking skills that of years ago and expressed doubts about the 1999 assured much improved performance on the California Framework [20], his response was to statewide test. give me a file of rising test scores in California Research mathematicians do their reputation schools. as trenchant thinkers no good whatsoever when I am constantly amazed that research mathe­ they use sloppy standards to judge whether a given maticians place any faith whatever in the results curriculum is improving math education or not. The of standardized tests, much less make them the ar­ fact is that judging how well or how poorly a new biter of success or failure of a curriculum ("if the curriculum is faring is damnably difficult and ex­ test scores in these programs don't go up, California cept in very rare cases impossible, except over a isn't interested" [21]). True, mathematicians are considerable number of years. In particular, those not statisticians, but surely they generally know that who tout test scores as a measure of how well or experiments (i.e. standardized tests) with a plethora how badly a new curriculum is performing do a dis­ of uncontrolled variables cannot possibly yield service to the entire mathematics education en­ meaningful results. (I suppose the uncontrolled terprise. variables are pretty obvious: classroom preparation Calculators time for the test, teaching to the test generally, ac­ Probably no issue in math education has gen­ countability pressures on teachers to get them to erated as much heat and as little light over the past show results, year-to-year experience with the test two decades as that of the use of calculators in and the kind of questions asked, to name but a few. mathematics education. Research mathematicians, And there is the accumulating evidence that schools particularly those who oppose the so-called re­ in some states are manipulating their dropout rates form curricula, are generally opposed to the use of to assure that students who would get low scores calculators in elementary school and wary about on tests do not take the tests. See also [22, p. 645].) their use in secondary school (and, for that mat­ The number of states where so-called "high­ ter, in university mathematics). Still, despite the stakes" tests are being used is increasing rapidly. claim that "a clear majority [of academic mathe­ In Massachusetts the MCAS (Massachusetts Com­ maticians] oppose the new trends in math educa­ prehensive Assessment System) is a continuing tion" [25], of which the use of calculators is per­ source of controversy. It has, claims the 1998 Mass­ haps the most prominent, there is precious little achusetts Teacher of the Year, put teachers under evidence about how the university mathematics "enormous pressure" to prepare students for the community as a whole feels about this issue. MCAS with the result that "teachers are doing Mathematics educators, even those who favor the things that are developmentally inappropriate with use of calculators in schools, would generally admit students" [23]. Now I do not suppose readers of this that there is no conclusive evidence about the effect article know much more than I do about what is of using them in school mathematics. However, there "developmentally appropriate" for schoolchildren, are numerous studies that purport to show that the but perhaps you will agree that good elementary use of calculators in schools at worst does no harm and secondary school teachers are likely to know to the learning of traditional mathematics and may

406 NOTICES OF THE AMS VOLUME 51, NUMBER 4 at best enhance that learning. Only a very few stud­ adults. No longer. This skill itself is of essentially ies seem to contradict these results. Few of the stud­ no value any more. Devoting a considerable amount ies on either side are compelling, but there is no of the instruction in elementary school to pencil­ question about where the weight of the evidence and-paper arithmetic can only be justified in the lies [26]. twenty-first century by arguing its value in instill­ Then there is the large amount of anecdotal ev­ ing essential understanding of numbers themselves idence ("horror stories" [12, p. 9]) and oracular (e.g. place value) or for its value in preparing stu­ pronouncements that support the position that dents for the further study of mathematics. If you calculator use in school is likely to rot the brain. wish to argue that something like traditional in­ This evidence is not compelling at all, not just be­ struction in pencil-and-paper arithmetic is a nec­ cause anecdotal evidence seldom can be, but also essary part of elementary school mathematics and because it is almost always used to support a pre­ also that the use of calculators should be banned determined position. from elementary school classrooms, you need to There is certainly a widespread belief, which I argue that the use of calculators at all will inhibit share, that students arriving at American colleges sufficient learning of pencil-and-paper arithmetic. and universities have been steadily more poorly pre­ Or you need to argue that teachers will so misuse pared for college mathematics over a period ex­ calculators if they are present that adequate learn­ tending back now at least a quarter of a century. ing of pencil-and-paper arithmetic will not ensue. This belief is consonant with the scores of Amer­ However, although there is no plausible evi­ ican students on international comparisons and on dence that in the hands of good teachers calcula­ college entrance examinations such as the SATs. tors produce bad effects in elementary school class­ Some of this poor preparation of incoming college rooms, this seems to have had no effect whatever and university students may perhaps be attribut­ on the research mathematics community.1 Oh yes, able to their use of calculators in elementary or sec­ in the hands of poor teachers or in classrooms in ondary school. But at most a miniscule amount schools where very little learning of any subject could be from this cause, since the problem pre­ takes place, children may become totally dependent dates the time when there was any substantial use on calculators, resulting in what has been called of calculators in American schools, and even today "Computer-Assisted Mathematical Incompetence" such usage is far, far from universal. Nevertheless, [31]. But it seems to me to be fantasy to believe that calculators are the standard scapegoat for the poor banning calculators in such classrooms or such preparation of students in basic mathematics (e.g., schools will have any noticeable effect on the arith­ "beguiled by ever fancier calculators and comput­ metic and mathematical abilities of students. ers, teachers appear less and less able to produce The ex cathedra statements of research mathe­ students who are masters of these basic topics" [27, maticians about school arithmetic are particularly p. 868]). Too often in the debate about calculators, unhelpful, because even when the arguments they as in much of the debate about mathematics edu­ use are plausible in themselves, they often try to bias cation, research mathematicians are wont to use the debate in ways that can only antagonize the post hoc ergo propter hoc reasoning (e.g., "many are mathematics education community. Long division ignorant due to a miseducation which involves provides a case in point. At least since the publica­ heavy use of calculators" [28, p. 459]). tion of the Cockcroft Report in 1983 [32, It may be that the teaching of pencil-and-paper p. 114], which stated a belief "that it is not profitable arithmetic, which has been the gateway to the study for pupils to spend time practising the traditional of school mathematics for more than a century, is method of setting out long division on paper, but as important as it has ever been. But this position that they should normally use a calculator," there can be supported only if there is recognition that has been an ongoing debate about whether the tra­ the terms of the debate, although not necessarily ditional long division algorithm (LDA hereafter) the conclusions, about what is important to teach should be taught at all and if it is taught, what level in mathematics have been changed forever by cal­ of proficiency pupils should attain. culators. Thus This is an important debate, because it gets to the heart of the question of what is still important Even if everything had been fine with U.S. math education, we would have to 1 What appears to be almost visceral opposition to tech­ pay attention now to how the availabil­ nology sometimes leads otherwise reasonable people to ity of sophisticated calculational tools make ridiculous claims such as "A computer cannot teach changes what is important to teach [29, any more effectively than an oscilloscope can bring about p. 244]. world peace" [30, p. 991]. I am not an enthusiast for com­ puters in classrooms or in teaching, but this comment is Before calculators became ubiquitous and cheap, rubbish and evinces no understanding whatever for what it was easy to argue that some skill in pencil-and­ has been accomplished in computer-related teaching in the paper arithmetic was necessary for just about all past quarter century.

APRIL 2004 NOTICES OF THE AMS 407 in school mathematics in the calculator age. A va­ which to beat the proponents of reform curricula. riety of positions, from attaining traditional pro­ (See the next section.) ficiency in LDA to not teaching it at all, can be rea­ At one extreme it means anything whatever to sonably argued, but what is not reasonable is to do with any nontraditional school mathematics argue, as research mathematicians are wont to do curriculum [37]. Almost as extreme is the charge [33], that there is only one right answer, namely that nontraditional curricula tolerate wrong an­ teaching the traditional LDA. In [33] a variety of ben­ swers or at least make no attempt to correct wrong efits of teaching LDA are argued. All are defensi­ answers. This charge is of course false [14] and can­ ble, but the impression that only LDA can achieve not be laid correctly at the door of any developer these benefits is, frankly, nonsense. One example of a nontraditional curriculum. Less extreme but of this is the claimed value of LDA for converting also false is the claim that there are some (who?) proper fractions to decimals. Actually this con­ who believe that "there is always more than one cor­ version is more easily accomplished by an algorithm rect answer to a math problem" [38, p. 869]. Least equivalent to LDA that writes the quotient as extreme but probably most numerous are those A/B = .abcdef... and computes the successive dig­ who reject the notion that some math problems can its of the quotient by multiplying by 10 and sub­ have more than one correct answer. This last po­ tracting. Not only is this method easier to repre­ sition is not "fuzzy" in any sense; it is true. Here sent in algorithmic notation than LDA, but in are two examples: (1) Given two parallel lines and addition it suggests immediately the well-known al­ a segment longer than the distance between the two gorithm for converting repeating decimals to frac­ lines, marked out on one of them, draw an isosce­ tions. The importance of LDA is linked in [33], as les triangle with the segment as one side and the it often is, to the similar problem of the division third vertex on the other parallel line. (2) Discover of polynomials. But division of polynomials is in and explain all you can about numbers that can be fact a much simpler computation than LDA and is written as the sum of consecutive natural numbers; easily expressed in algorithmic notation by a sim­ e.g. 9=4+5,9=2+3+4;15=1+2+3+4+5. pler algorithm than LDA. 2 Now of course each of these could be transformed The penchant of research mathematicians for into a problem with a single right answer. But to claiming that there is one right way to teach a par­ do so would be to miss the point that for the first ticular aspect of school mathematics is virtually problem it is instructive for students to see that never correct. Another example is the reported other students have solved the problem in differ­ claim that there is "one right way to teach odd and ent ways. The second, more advanced, problem il­ even numbers" [7, p. 199]. lustrates the general lesson that mathematics is More generally, harking back to the papers of about investigation, conjecture, refutation or proof, Benezet in the 1930s [34], it has been suggested by and explanation. a prominent research mathematician that perhaps Of course, precision is crucial in mathematics, the teaching of the algorithms of arithmetic "should but an emphasis on precision does not require be postponed ... until grade 6" [35]. Indeed, a use­ that all problems have only one correct answer. This ful exercise for all mathematicians and mathe­ notion, which seems obvious to me, appears to be matics educators would be the gedanken experi­ anathema to many research mathematicians. It car­ ment of wondering what would happen if ries over all too easily to the notion that curricu­ mathematics education did not exist now and you lum should be rigidly specified ("one right cur­ had to invent it. How much pencil-and-paper arith­ riculum") and that the method of teaching that metic would there be? How much mental arith­ curriculum should also be rigidly specified. metic? How much use of calculators [36]? Arrogance Fuzzy Math and the One Right Answer Syndrome The "fuzzy math" epithet implies that the tra­ Those who advocate the reform of school math­ ditional math side of the Math Wars has not made ematics curricula are often accused by their de­ much attempt at the "civil, constructive discourse" tractors-almost always research mathematicians that Suzanne Wilson [7, pp. 216-29] thinks is the or journalists influenced by research mathemati­ way to end the Math Wars (see also [39, p. 488]). cians or parents influenced by journalists-of ad­ Indeed, if there is one reason more than any other vocating "fuzzy math". What is fuzzy math? It is why the Math Wars may be expected to rumble on itself a fuzzy concept, meaning whatever the crit­ into the future, it is because all too many "mathe­ ics of new school curricula want it to mean at a matical scientists have tended to look upon edu­ given time. Of course, even if the term bears no re­ cation professionals with doubts bordering on ill­ lation at all to any truth, it is a wonderful club with disguised contempt" [40, p. 21]. Indeed,"sarcasm and ridicule" [41] and "caricatures" [14, pp. 127-9], 2 The implicit assumption here, of course, is that the co­ while noticeable on both sides of the Math Wars, efficient arithmetic in the division of polynomials is, in vir­ appear to be much more prevalent on the traditional tually all cases of interest, easily done mentally. side espoused by many research mathematicians.

408 NOTICES OF THE AMS VOLUME 51, NUMBER 4 The arrogance of mathematicians towards math­ have no credentials whatever in school mathe­ ematics education manifests itself in various ways. matics education. Here are two: Not all research mathematicians are so arro­ 1. Research in Mathematics Education. At the ex­ gant with respect to their abilities as math educa­ tremist end are those research mathematicians tors. For example, Wilfrid Schmid has written, "We who believe that research in math education is [i.e. mathematicians] are not qualified to do their (virtually?) an oxymoron (e.g., "mathematics ed­ [i.e. mathematics educators'] work" [25]. And he ucation research as described in this book is, in quite correctly goes on, "Yet we are qualified as crit­ fact, inferior to descriptive discussion because ics (emphasis added) of reforms in math education." it is descriptive discussion without humility" Often the attitude of research mathematicians [42, p. 282]).3 This conclusion is reached by about the mathematical abilities of mathematics ed­ defining research as a word that can be used only ucators is manifested in their tendency to nitpick when applied to theorem/proof mathematics the documents, such as [11], of reform mathe­ research or the scientific method paradigm of matics curricula. Of course finding errors and com­ the physical sciences. From this perspective just municating them to the authors of these docu­ about everything done in education or the so­ ments is a valuable service, even when the errors cial sciences is not research. Even those who take are little more than typographical. But to use such a less extreme view often subscribe to the dic­ nit-picking as a club to try to discredit these doc­ tum that "teaching must be an art and not a sci­ uments and their authors is unworthy. I dare say ence" [42, p. 284]. This is very convenient, be­ that most of the readers of this article have writ­ cause if teaching is an art, then anyone's opinion ten books and/or long papers in which inevitably is equally valid ("I know what I like"). This au­ minor errors remained. But they surely did not ex­ tomatically gives the opinions of research math­ pect to be pilloried for these. The publication in ematicians about math education the same 2000 of Principles and Standards of School Mathe­ standing as those of mathematics educators. matics [4 7], the revision of the NCTM Standards, But although some research mathematicians has damped down the criticism of [11] to some ex­ have thought long and hard and effectively about tent but has by no means done away with it. mathematics education, too many are in the If research mathematicians would engage in grips of "the questionable belief that, just be­ "civil, constructive" criticism rather than, more cause mathematicians are good at mathematics, often than not, arrogant putdowns, the result of the they should also be able to contribute to the ef­ Math Wars would not be an endless battle to the fective presentation of elementary mathematics detriment of school mathematics education in the to an often unmotivated and umesponsive pub­ U.S. (Of course, many (most?) mathematicians are lic" [44, p. 180]. not arrogant, but those who doubt the prevalence 2. The Mathematics Credentials of Mathematics Ed­ of intellectual arrogance among mathematicians ucators. Many mathematics educators who have might care to look at a recent study of seventy been involved with the development of reform mathematicians in Britain [46].) mathematics programs have Ph.D.'s in mathe­ matics and thus might a priori be considered competent in school mathematics. But, no: My conclusion is that although a number of re­ "There is a distinction between math educators search mathematicians have contributed positively who are primarily interested in questions in­ to school mathematics education in recent years volving education, and mathematicians who (see, for example, [40, p. 18]), the research mathe­ know about mathematics" [45]. This is, of course, matics community has largely squandered an op­ ironic, because more often than not the research portunity to have a significant positive impact on mathematics critics of mathematics educators American mathematics education. Too many have used a "scattershot approach" that often takes the 3 If humility is needed in the Math Wars, a bit from the re­ form of "unsubstantiated claims and random anec­ search mathematics community might be in order. As dotes" [3]. Too often the result has been that when Igor Sharygin [43] has noted: "... professionals in pure they have become active in mathematics education, mathematics usually consider school mathematics as an research mathematicians have not lived up to the integral part of mathematical science .... And it is a grave high standards that they normally bring to their fallacy. Having indisputable authority and enormous own professional work. knowledge in their own field, these prominent scientists How could research mathematicians make a have frequently invaded a domain of public education in which they are not always competent. (Another false be­ more positive contribution to school mathematics lief is that it is quite enough to be a good mathematician education in the future? The most important way to solve the problems of mathematical education.)" Al­ would be for research mathematicians to see their though Sharygin was referring to Russia, this quote is role as colleagues of mathematics educators and equally applicable to the U.S. constructive critics of work in mathematics

APRIL 2004 NOTICES OF THE AMS 409 education. In addition, (almost all) research math­ [8) G. LAPPAN, Connected Mathematics: 6th Grade, Prentice ematicians, who have had little or no experience in Hall School Group, 2004. elementary or secondary classrooms, should learn [9) D. KLEIN, Letter in Notices Amer. Math. Soc. 44 (March to be rather more humble about school mathe­ 1997) 310. matics, since their experience in college and uni­ [10) D. FENDEL, Letter in Notices Amer. Math. Soc. 44 versity classrooms is seldom relevant to elemen­ (June/ July 1997), 656-657. tary and secondary education. [11) Curriculum and Evaluation Standards for School Mathematics, National Council Another way for research mathematicians to of Teachers of Mathe­ matics, Reston, VA, 1989. make a positive contribution to school mathemat­ [12) H. Wu, The mathematics education reform: What is ics education would be to help in recruiting more it and why should you care? http: I /math. berkeley. intellectually able people to teach mathematics edu/-wu/reform3. pdf. rather than throwing brickbats about curriculum [13) _ · _ , Professional development of mathematics matters. This is a crucial problem [48] that badly teachers, Notices Amer. Math. Soc. 46 (May 1999), needs to be addressed, and it is one where both 535-542. mathematicians and mathematics educators should [14)]. ROITMAN, Beyond the Math Wars, Contemporary Is­ find it easy to sing from the same hymn book. It sues in Mathematics Education (E. A. GAvosTo, S. G. is past time that the intellectual leaders in Amer­ KRANTZ, W. McCALLUM, eds.), Cambridge University ican mathematics started to make the case that Press, 1999, pp. 123-134. those we attract to elementary and secondary [15) The Mathematical Education of Teachers, Confer­ school teaching need to be the "brightest and best" ence Board of the Mathematical Sciences, Washington, [24, p. 57], as intellectually able as those attracted DC, 2001. to professions such as law and medicine and, yes, [16) An Open Letter to the United States Secretary of to the academic world. Education, Richard Riley, Advertisement in the Washington Post, November 18, 1999; http: I /www. Pending such an epiphany by research mathe­ mathematicallycorrect.com/riley. htm. maticians, the Math Wars, which once were avoid­ [17) L. V. AHLFORS et al., On the mathematics curriculum able, will continue to be part of the school mathe­ of the high school, Amer. Math. Monthly 69 (1962), matics landscape. The research mathematics 189-193; Math. Teacher 55 (1962), 191-195. community, through its hubris, has by and large [18) A. RALsTON, Next disaster in American education: Ris­ contributed-and continues to contribute-to a ing test scores, The Sacramento Bee (1 December worsening situation in school mathematics in the 2002). U.S., a situation that shows no signs of getting [19) No Child Left Behind Act of 2001, H.R. 1 of 2001. much better in the foreseeable future. The lesson [20) Mathematics Framework for California Public Schools, of the New Math has not been learned. California Department of Education, Sacramento, CA, 1999. Acknowledgment [21) R.]. MILGRAM, as quoted in '"Exemplary' texts with­ I would like to thank H. Burkhardt, L. Burton, drawn from California adoption process", by D. ]. E. Dubinsky, B. Jacob, G. Rising, and M. Saul for help­ HOFF, Education Week 20 (October 18, 2000). ful comments on an early draft of this paper. [22) A. H. ScHOENFEW, Purposes and Methods of Research in Mathematics Education, Notices A mer. Math. Soc. 4 7 References (June/ July 2000), 641-649. [23) M. GINLEY, as quoted in "Mass teachers blast state tests [1) D. KLEIN, A brief history of American K-12 mathe­ in new TV ads", by]. GEHRING, Education Week 20 (No­ matics education in the 20th century, in Mathemati­ vember 22, 2000). cal Cognition: Current Perspectives on Cognition, Learn­ [24) S. S. WILLOUGHBY, The Other End of the Log, Vantage ing and Instruction, edited by]. M. Royer, Information Press, 2002. Age Publishing, 2003. [25) W. SCHMID, New battles in the Math [2) A Nation at Risk: The Imperative for Educational Re­ Wars, The Har­ form, U.S. Department of Education, Washington, DC, vard Crimson, May 4, 2000. 1983; http: I /www. goa 11 i ne. org/Goa l %20L i ne/ [26) P. H. DUNHAM, Hand-held calculators in mathematics NatAtRisk.html#anchor791390. education: A research perspective, in Hand-Held Tech­ [3)]. KILPATRICK, Confronting reform, Amer. Math. Monthly nology in Mathematics and Science Education: A Col­ 104 (1997), 955-962. lection of Papers, http: I /mathcs. muh len berg. [4] W. H. ScHMIDT (ed.), Facing the Consequences: Using edu/Standards_2000_paper.htm. TIMSS for a Closer Look at U.S. Mathematics and Sci­ [2 7) G. ANDREWS, Views on high school mathematics edu­ ence Education, Kluwer, 1999. cation, Notices Amer. Math. Soc. 43 (August 1996), [5) N. KosuTZ, Letter in Notices Amer. Math. Soc. 44 (May 866-873. 1997), 543. [28) G. E. ANDREws, Commentary on assessment standards [6) A. JACKSON, The Math Wars: Parts I and II, Notices Amer. for school mathematics, Notices Amer. Math. Soc. 44 Math. Soc. 44 (June/ July 1997), 619-702; (August (April1997), 458-462. 1997) 817-827. [29) R. HoWE, The AMS and mathematics education: The [7) S. WILSON, California Dreaming: Reforming Mathe­ revision of the "NCTM Standards", Notices Amer. Math. matics Education, Yale University Press, 2003. Soc. 45 (February 1998), 243-247.

410 NOTICES OF THE AMS VOLUME 51, NUMBER 4 [30] S. G. KRANTz, Two reactions to The Mathematical Ed­ ucation of Teachers, Notices A mer. Math. Soc. 48 (Oc­ About the Cover tober 2001), 985-991. [31] R. H. EsCOBALES ]R., Letter in Notices A mer. Math. Soc. The Internet on Aprill, 2003 44 (May 1997), 542-543. This month's cover was contributed by Bill [32] Mathematics Counts, Her Majesty's Stationery Of­ Cheswick of the Lumeta Corporation and di­ fice, London, 1983. agrams the Internet as it was on Aprill of last [33] D. KLEIN and R.]. MILGRAM, The role of long division year. It nicely fits in with the topic of this in the K-12 curriculum, http: I /math. stanford. edu/ ftp/milgram/long-division-try-again.doc, year's theme for "Math Awareness Month", February 2000. which is networks. Color coding in the figure [34] L. P. BENEZET, The story of an experiment, ]. Nat. indicates distance from the scanning host. Assoc. Ed. 24 (November, December 1935), 241-244, Layout in the diagram was determined by an 301-303; 25 (January 1936), 7-8. algorithm that considers the graph as a phys­ [35] A. GLEASON, Delay the teaching of arithmetic, un­ ical system of springs and determines its min­ published manuscript. imal energy configuration. Thus, in a sense the [36] A. RALsTON, A zero-based mathematics curriculum: graph interprets itself. More information on What it is and how it might be used, Proc. Working the Lumeta Internet mapping project can be Group, Eighth Internat. Con( Math. Ed., Seville, 1996, found at http: I I research. l umeta. com/ pp. 1-8. [3 7] Glossary of Terms, http: I I ches/map/ and also in the paper "Mapping mathematicallycorrect.com/glossary.htm. and visualizing the Internet" by Cheswick, [38] H. Wu, Views on high school mathematics education, Hal Burch, and Steve Branigan in the pro­ Notices Amer. Math. Soc. 43 (August 1996), 866-872. ceedings of the 2000 USENIX Annual Techni­ [39] R. W. RILEY, The state of mathematics education: cal Conference. Designing graph layout is an Building a strong foundation for the 21st century, No­ interesting problem. A good place to start in tices Amer. Math. Soc. 45 (Aprill998), 487-491. the huge literature on this topic might be the [40] H. BAss, Mathematicians as educators, Notices Amer. proceedings publication "Graph Drawing", Math. Soc. 44 (January 1997), 18-21. Lecture Notes in Computer Science, Springer [41] R.]. MAHER, Letter in Notices Amer. Math. Soc. 44 (Au­ 1984. gust 1997), 789-790. [42] G. E. ANDREWS , Review of Mathematics Education Re­ -Bill Casselman search: A Guide for the Research Mathematician, Amer. ([email protected]) Math. Monthly 108 (March 2001), 281-285. [43]1. SHARYGIN, Mathematical education and society (an outlook from Russia and into Russia), unpublished manuscript. [44] H. CLEMENS, The Park City Institute: A mathemati­ cian's apology, Notices Amer. Math. Soc. 39 (March 1992), 180-181. [45] R.]. MILGRAM, Written testimony to the hearing of the House of Representatives Committee on Education and the Workforce; Subcommittee on Early Childhood, Youth and Families; and Subcommittee on Postsec­ ondary Education, Training and Lifelong Learning on "The Federal Role inK-12 Mathematics Reform", Feb­ ruary 2, 2000. [46] L. BURTON, Mathematicians As Enquirers-Learning about Learning Mathematics, Kluwer (to appear 2004). [4 7] Principles and Standards for School Mathematics, Na­ tional Council of Teachers of Mathematics, Reston, VA, 2000; http://standards . nctm.org. [48] V. TROEN AND K. C. BoLEs, Who's Teaching Your Chil­ dren? Why the Teacher Crisis Is Worse Than You Think and What Can Be Done about It, Yale University Press, 2003.

APRIL 2004 NOTICES OF THE AMS 411 W H A T s a Billiard? YakovSinai

Billiards are dynamical systems. In the simplest case, between geodesic flows and flows generated by a "billiard table" is a compact domain Q c Rd with quadratic differentials where the curvature is con­ a piecewise smooth boundary. For a large part of the centrated at isolated points. theory the class of smoothness plays no role. The The most thoroughly studied billiards are the two­ reader is invited to think about components of the dimensional billiards. Some are integrable, meaning boundary aQ as subsets of C"" -submanifolds of that the phase space M, minus some submanifolds codimension 1. The phase space M of the billiard is of smaller dimension, can be decomposed into two­ the unit tangent bundle of Q with the natural iden­ dimensional invariant tori, and the dynamics on each tification at the boundary torus is described by quasi-periodic functions. Ex­ amples of such billiards include: (1) billiards inside (1) v' = V- 2(v, n(q))n(q), q E aQ parallelograms, (2) billiards inside equilateral trian­ where n(q) is the inward-pointing unit normal vec­ gles, and (3) billiards inside ellipses. According to a tor at q. The reflection law (1) is not used at intersec­ popular conjecture, the set of integrable billiards tions of the several components of the boundary, can be fully described and is not much wider than this where it has no meaning. list. The dynamics {sq of a billiard is the uniform Billiards in general strictly convex smooth do­ motion with unit speed inside Q of a point repre­ mains have some properties of integrable billiards. senting the billiard ball and with the reflections off A curve ;y c Q is called a caustic if any tangent ray the boundary given by (1), -oo < t < oo . The group to ;y after reflection remains tangent to ;y. Caustics {St} preserves the Liouville measure dqdwq(v) play an important role, because some semiclassi­ where Wq is the uniform measure on the unit cal approximations of eigenfunctions of Laplacians sphere of tangent vectors to Q with given q. are described in terms of caustics. V. F. Lazutkin Mechanical systems with elastic collisions often has shown that the set of tangent vectors to all caus­ give rise to billiards. For example, a system of tics is a set of positive measure in the phase space, one-dimensional point particles with arbitrary accumulating near the boundary. In spirit this re­ masses moving freely between elastic collisions sult belongs to KAM-theory.]. Mather proved that can be described as a billiard system inside a sim­ if the curvature of aQ is zero at one or several plex whose dimension equals the number of par­ points and negative otherwise, the billiard has no ticles. The system of N hard balls of radius p in a caustics (remember the orientation of n(q)). volume V is reduced to the billiard in the domain If Q is a polygon whose angles are rational mul­ V x V x · · · x V \ C where C is the union of the tiples of rr, then the velocity along each trajectory N times of the billiard can take finitely many values. Fixing cylinders these values we get a vector field on a two­ dimensional surface whose trajectories can be Cij = {q

412 NOTICES OF THE AMS VOLUME 51, NUMBER 4 lETs are closely connected with billiards in poly­ gons. The theory of such billiards is now an actively studied topic in the theory of dynamical systems. Almost nothing is known if the angles of a poly­ gon are incommensurate with rr. Even the theory of billiards in triangles with two irrational angles awaits development. Much can be said if the curvature of oQ is strictly positive on some components of the boundary and is zero on the others. Such billiards are called hy­ perbolic. This condition links the theory of hyper­ A typical form of a stable manifold. bolic billiards with geodesic flows on manifolds of negative curvature, Anosov flows, and the general some time in the past. A typical form of srn is theory of hyperbolic dynamical systems. A similar given in the accompanying figure. definition can be easily given in the multidimen­ sional case. Probably the first person who noticed This existence of srn and urn is a manifestation the analogy between systems with elastic colli­ of the intrinsic instability of the dynamics. There­ sions and related billiards and geodesic flows on fore, hyperbolic billiards are among the most pop­ manifolds of negative curvature was the Soviet ular models of deterministic chaos. physicist N. S. Krylov. In the two-dimensional case The main problem related to hyperbolic billiards the simplest examples of hyperbolic billiards are is the problem of their ergodicity. There is a squares from which one or several strictly convex general argument due to E. Hopf that gives the scatterers are cut out. ergodicity if srn and urn have a property called L. Bunimovich showed that the billiard inside a "local transitivity", meaning that for any two close "stadium" is in a natural sense also a hyperbolic points x, y E M one can construct a continuous billiard. A stadium is a domain bounded by two path from x to y which consists of finitely many semicircles and two parallel straight segments. components such that each component belongs Later, Bunimovich and Donnay extended this result either to srn or to urn. For smooth systems where to a wider class of domains in which semicircles the sum of dimensions of srn and urn is 2d - 2, can be replaced by general strictly convex curves local transitivity follows directly from their general and the straight segments are allowed to be non­ properties. It is not so for billiards, because smooth parallel. The motion of a billiard ball on a table that components of srn or urn can be arbitrarily small. is the complement of a random or periodic con­ This difficulty can be overcome with the help of the figuration of strictly convex scatterers is called a Fundamental Theorem for hyperbolic billiards, Lorentz gas and is one of the most popular mod­ which has several versions. The first one says that els in nonequilibrium statistical mechanics. in an arbitrary small neighborhood U of a typical Bunimovich stadia and similar billiards are pop­ point and an srn c U, the probability (in a natural ular models in the theory of quantum chaos, which sense) of points in srn for which the size of urn is studies the connections between eigenfunctions of not smaller than the size of the initial srn is greater Laplacians and ergodic properties of underlying clas­ sical dynamical systems. One of the reasons is their than some constant. This property is enough to simplicity and amenability to numerical methods. carry out Hopf's argument. In the hyperbolic theory of dynamical systems In the second version, based on the so-called a stable (unstable) manifold of a point x E M is Chernov-Sinai Ansatz, it is shown that local tran­ a local submanifold y 0 (t < 0), for with the property of local transitivity; therefore, this all y E y 2). In the case of hyperbolic (Y. G. Sinai, ed.), Encyclopedia of Mathematical Sciences, billiards, almost every point also has a stable (srn) vol. 100, Springer-Verlag, Berlin, 2000. and unstable (urn) manifold. This statement is a [2] Hard Ball Systems and the Lorentz Gas (D. Szasz, ed.), particular case of the general Hadamard-Perron Encyclopedia of Mathematical Sciences, vol. 101, theorem and is a relatively simple part of the Springer-Verlag, Berlin, 2000. theory. The new feature compared to the smooth [3] Selected Works of Eberhard Hopf with Commentaries situation is the appearance of cusp-type singular­ (Cathleen S. Morawetz, James B. Serrin, and Yakov G. ities on these manifolds, which are created by Sinai, eds.), Collected Works, vol. 17, Amer. Math. Soc., trajectories that were tangent to the boundary at Providence, Rl, 2002.

APRIL 2004 NoTICES oF THE AMS 413 The Great

2 3 5 7 11 13 17 19 23 29 Prime Number 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 Record Races 283 293 307 311 31 3 3 17 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 43 3 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 Giinter M. Ziegler 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 12 31 12 37 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811

The year 2003 ended with several prime Internet, get the GIMPS computer programs as well number records. For example, an effort as "their" numbers for testing, who have their per­ headed by ]ens Franke (Bonn Univer­ sonal computers do slave labor, and who report sity) led to the solution of the RSA-576 their results back to the project via the Internet. decoding problem: the factorization of Michael Shafer got the number n = 20996011 to a 174-digit decimal number. test whether 2n - 1 is prime. His PC "did it" with the GIMPS software, and it turned out that the We also have a new "largest known anwer is "yes!!" for this n. Mathworld reports that prime number": a Mersenne prime upon this success he performed a victory dance, number with 6320430 digits, called his wife and friends and began to celebrate. M = 220996011 - 1. The media attribute Let us recall: In honor of the French monk Marin the discovery to Michael Shafer, a chem­ Mersenne (1588-1648) the numbers of the form ical engineering student at Michigan Mn = 2n - 1 are called Mersenne prime numbers­ State University-but that is only part if they are prime. For this it is necessary (a nice ex­ of the story. ercise from elementary number theory) th at n it­ self is a prime. But this is not sufficient: n = 11 is Mersenne Numbers the first counter-example. In 1644 Mersenne The GIMPS project ("Great Internet Mersenne Prime claimed that Mn is a prime for n = 2, 3, 5, 7, 13, 17, Search", http: I /www. mersennse. org) was started 19, 31, 67, 127 and 257, but for no other prime in 1996. Its purpose is to search for larger and larger number smaller than 257 (and thus he got it wrong Mersenne prime numbers. The distributed com­ for exactly five cases). Mersenne prime numbers are puting project recruited volunteers who, via the rather rare: It is not known whether there are infi­ nitely many. Only the first 38 of them are known, Giinter M. Ziegler is professor of mathematics at the Tech­ plus only two more, including the newly discovered nical University, Berlin. His email address is M2o995011 which is now also the largest known [email protected]. This article was trans­ prime number. lated by the author from his article "Primzahi-Rekordjagd, Mitteilungen der Deutschen Mathematiker-Vereinigung It is quite remarkable that numbers with more 2003-4, S. 5-7. He acknowledges support from the DFG Re­ than six million digits can effectively b e tested for search Center FZT-86 "Mathematics in Key Technologies" primality. This is the genuine scientific (and in Berlin and from a DFG Leibniz grant. programming) achievement on which the new

414 NOTICES OF THE AMS VOLUME 51, N UMBER 4 record is based-that the number n = 20996011 that seems to slightly must be prime is only a little warm-up exercise for stretch the facts. (The con­ the new record. nection they can legiti­ mately make is that Cran­ PrimalityTests dell has worked on It has been shown only recently that there are exact implementing his method prime number tests that work in polynomial time­ for the prime number tests see the report in the May 2003 Notices, pp. 545-5 52. in Mathematica.) Indeed, This was a theoretical breakthrough, but it is not GIMPS works with a highly yet suitable for use "in practice." The GIMPS pro­ optimized assembly code. ject applies for each prime n a sequence of more They use floating point classical tests, which are nicely described at arithmetic because this is http: I lwww .mersenne. orglmath. htm: 1 more effective on Intel Pen­ In Phase I one looks for small prime divisors q tium processors, but this of 2n - 1. These have to satisfy (again a nice exer­ also means that the errors cise) q = 1 mod 2n and q = ±1 mod 8. Using a of floating point arithmetic modified "Sieve of Eratosthenes" adapted to such have to be detected and Marin Mersenne, 1 588-1648. factors, prime divisors of Mn up to approximately eliminated separately. 40000 (if any) are found. For this one can exploit the fact that divisibility tests for numbers of the Primality and Factoring form zn - 1 can be performed very effectively in Phases I and II of the GIMPS-sequence really do pro­ binary arithmetic. duce divisors in the case of a decomposable Mn, if In Phase II one then uses a special case of the they find any, but the third and decisive phase so-called "(p -I)-method" ofPollard(l974), which doesn't. In that case the answer will only be "de­ can be used to find factors of the form q = 2kn + 1, composable!" or not, without an explicit prime di­ for which q - 1 = 2kn consists of many small prime visor as a certificate. Thus a complete primality test factors, or (in an improved version) are highly de­ is performed, but no factorization is produced. composable except that one prime factor may be And there are good reasons for this: Not even a bit larger: To find q such that all prime factors in the special case of Mersenne numbers does one are smaller than B, one forms the product know an effective method for factoring. A method E := 0pRon Rivest, Adi and then one works with a special version of a Shamir and Leonard Adleman published in 1978. Fast Fourier Transform (FFT), in this case with an FFT with respect to an irrational basis that was in­ troduced by Richard Crandell and Barry Fagin (1994). On the web pages http: I lmathworl d. wolfram. com of the Mathematica project, which ad­ vertise the new record, it is suggested that GIMPS worked with a Mathematica implementation, but

1 For algorithmic prime number theory the experts rec­ ommend Richard Crandell and Carl Pomerance: "Prime Numbers. A Computational Perspective", Springer-Ver­ 40th Known Mersenne Prime Found!! lag, New York 2001. For a computer algebra perspective On November 17, 2003 Michael Shafer's comp.ner found the 40th ltn()Wn Mersenne prime, 220996•011 .1! on primality tests (and lots of other interesting topics) see This number ''weighs in" at a who ppi ng Q320 430 decimal digill_! This U also the larges t known prime Joachim von zur Gathen and ]urgen Gerhard: "Modern number, surpassing GIMPS' Iastdiscoverybyover 2 million digits . You can d.)\VIl\('1 ~ tite cllo;.!l! for }'OIS chance at findi~ the next world record prime! A r,,rum f0.L!!.'~!£Q.!!',sr_~ Is Computer Algebra", Cambridge University Press, second avallabl etoansweranyquestions)JOUmay h av~ edition, 2003. ~r~ IG!lo ns to Michael and every GIMPS contributor (or their p

APRIL 2004 NOTICES OF THE AMS 415 More Records And there are even more current records with re­ spect to factorization: Indeed, people are not only trying to test Mersenne numbers for primality, but Whal~lhellunob e oFieldSieve?~lovlewa ~clo n of out hq\1~1\UI' a~bd ques~on•

W9' o o:~ cunennywo•~ngon2"6 11 · 1 ,acolnpode n u mbeo also to decompose them completely into prime Wi!h24~d~l ..aldlgL \SiO'ltl ju•tolleiJ1ownU(IOt,326GZJ ~~~~':~l:o~:~l~ lclh-~· ~:;l!~:;~raan~ew~~~=on~ factors. NFSNET (http: I /www. nfsnet. org) is an­ SNFStecood

Vo ucOIObrowsee>1ark!d. Relations Received eooooo tolat21002742 cently (success announced on December 2, 2003) December2:nd2003 On O t ~embto 2nd, 2 ~ 03, NFSNET COIIplf~ ll>e managed in Internet joint work to factorize the fic!oonza!jonof2•757-1 Th t C213eOII'Ipcol~eel)fatkl r wu tound l obetn11produc1of p~ llnet\\11tl179~tnd 1 l4d~ l mal 19 Sep 2UOS 12 Dec' ~ 003 I ~lgl l l. SOelhe~foomor ; U~I_ U an~~to~ l Mersenne number 2757 - 1 completely. The prime ~aot~~~dpaJed In !his ptolecl Mrtl ieep vp tle 11ev1ng on I' AcUveMadlhlu,.. t 1' factors 9815263 and 561595591 of this number I ~,,~~,=.,.,------~"~~,~~ were known before, but the 212 digits of the other two factors made for a hard piece of work: It was This RSA method is currently treated in every new now decomposed into the prime factors elementary number theory text book, and also used 57221370220020678242482279750958577491 extensively in practice-see the manual pages for 51312827809388406962346253182128916964 ssh on your PC, or the homepage http: I I 593 www. rsasecurity. com of Rivest, Shamir andAdle­ and man's company. The security of their method 24033821640983508088736273403005965446 against unauthorized decoding depends on the 68900235634433213056506664319381390111 fact that with current technology it is very difficult 97710904242694120545430727149147426656 to decompose products with 150 or 200 decimal 77774247325292327559. digits into their prime factors. The company "RSA This achievement is based on the "Special Num­ Securities" has even offered prizes for factoring ber Field Sieve (SNFS)" -a faster specialized version their challenge problems.2 The first of them is/was of the GNFS, which is applicable only for special a prize of $10,000 for factoring the number "RSA- numbers, such as those of the form bn ± 1. 576" 18819881292060796383869723946165043980 Record Chase 71635633794173827007633564229888597152 The chase for new records will continue. In 2000, 34665485319060606504743045317388011303 the "Electronic Frontier Foundation" (http: I I 39671619969232120573403187955065699622 www. eff. org/) paid their first prize, $50,000, for 1305168759307650257059 the first prime number with more than one million with 17 4 decimal digits, that is, 5 76 binary digits digits. For the identification of a prime number with (bits). This problem has been cracked by ]ens Franke more than ten million decimal digits they have of­ of Bonn University, as reported by Heise Online fered a prize of $100,000. This adds to the excite­ news service on December 8, 2003: The number has ment, and the GIMPS project is looking for fellow factors combatants who would enlist their computers for 39807508642406493739712550055038649119 the record chase. 90643623425267084063851895759463889572 At the same time, the larger RSA challenge prob­ 61768583317 lems wait to be attacked. RSA-640, a number with and 193 decimal digits, is the next one on the list: 47277214610743530253622307197304822463 $20,000 has been offered for it. 29146953020971164598521711305207112563 And the next Mersenne number on the hit list 63590397527 of NFSNET is 2811 - 1. This project is looking for­ (87 digits each) and these are prime--which again collaborators as well. with the current methods is very easy to verify. For So, many poor little PCs will be fed numbers and his factorization Franke has used the "General tortured with prime number tests and decompo­ Number Field Sieve (GNFS)." This method was in­ sition methods, in the hope that their owners might troduced by Lenstra, Lenstra, Manasse and Pollard be able to cash in on a part of the fame and glory (1990). It has a running time of exp(O( ..V n log n)) (and the prize money) for the next record, which for n-digit numbers; this is not quite polynomial, surely is soon to be achieved. but is nearly so. The GNFS had already been used to factor the smaller test problems, ranging from RSA-100 to RSA-512, the latter in August 1999.

2 http: //www.rsasecurity.com/rsalabs/challenges/ factoring/ numbers.html.

416 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Book Review

A Mathematician's Survival Guide: Graduate School and Early Career Development

Reviewed by Elizabeth Theta Brown

A Mathematician's Survival Guide: Graduate studies from around School and Early Career Development the country remi­ Steven G. Krantz nisce over the effort­ AMS, 2003 less recruitment of 222 pages, $28.00, ISBN 0-8218-3455-X top-caliber students in days gone by. The serious undergraduate considering a Ph.D. Mathematics faces in mathematics is almost entirely ignorant of what stiff competition it is he or she really proposes to undertake. Even from other scientific a very sophisticated student who has had advanced and technical fields; a coursework and perhaps research experience is mathematically tal­ usually not in a position to foresee that graduate ented student can school and an academic career in mathematics will pursue an intellectu­ require patience, toughness, and social graces, in ally rewarding career addition to the obviously necessary qualities of without becoming a diligence, insight, and luck. Although the current Ph.D. mathematician. fashion of undergraduates doing original research Of course, the increased use of mathematics in has helped, it is still the case that most prospective these other areas provides delightful benefits for graduate students in mathematics who have been mathematics. Funding for, and progress in, our educated in the United States have no idea of the field is robust precisely because the public derives vicissitudes of mathematical research or even of the practical benefits from the applications of mathe­ broad outlines of academic careers in mathemat­ matics. Nevertheless, the increased competition ics. The resulting situation is bad for everyone: for students has brought new pressure on mathe­ some students who would be good mathematicians matics graduate programs to find, retain, and grad­ do not apply, while their peers enroll at institutions uate quality students. It is therefore in the inter­ to which they are ill suited. Students lose or diminish est of the mathematics community that students their potential careers, departments waste their enter graduate school with a realistic picture of limited resources on students who do not fit their what they will need to do in order to succeed and programs, and the mathematics community as a what they can expect to be the fruits of their suc­ whole loses by the squandering of nascent talent. cess. This issue has become more important than ever There are many things that a student may not over the past ten years, as directors of graduate know, though to readers of the Notices these may appear self-evident. The basic structure of a Elizabeth Theta Brown is assistant professor of mathe­ graduate program- coursework, exams, thesis, and matics at ]ames Madison University. Her email address is defense-the advisor/student relationship, and [email protected]. the practice of graduate funding through teaching

APRJL 2004 NOTICES OF THE AMS 417 assistantships and research support are among for mathematics Ph.D.'s, and describes the first the aspects of graduate education that a prospec­ few years of an academic job. In addition, it dis­ tive student may be ignorant of, especially if he or penses advice regarding standard pitfalls and she is not on familiar terms with any current math­ strategies for avoiding them. The final chapter pro­ ematics graduate students. vides a concrete overview of qualifying exam-level The transition from undergraduate to graduate mathematics, while the appendices describe vari­ status is accompanied by a change in social status ous aspects of academy structure. They form a from consumer of education to a position of re­ unique and very useful component of the book. sponsibility as a professional in training with a The Guide is divided into five sections of sev­ teaching or research assistantship. Most students eral chapters each. There is a preface that outlines grow into their new roles without serious trouble, the contents of the book, establishes Mr. Krantz's but there are many opportunities for mishap along conversational approach, and summarizes the the way. The naive view that some instructional book's goal as helping aspiring mathematicians to gaffes are too obviously wrong- succeed by educating them headed to be committed by any about the graduate school well-meaning beginning in- process. A typical comment: structor, however clueless, is "It is my fervent belief.. .that regularly met with hearty guf­ Krantz takes the main reason that people faws from senior faculty in often fail at tasks or programs charge of teaching, followed by the time that they set for themselves is agonizing counterexamples that they never figure out what drawn from their own depart­ to mention many it was that they were supposed ments. Outside the realm of small but to be doing" (p. xiv). In the pref­ teaching, every graduate pro­ ace he also comments on his gram has had mathematically essential points own qualifications for writing talented students whose prob­ such a book. This sincere con­ lems adjusting to their new set­ that would not all cern with helping the reader­ ting undermined their mathe­ understood as a prospective matical productivity, some­ be included doctoral student-realize his times leading to their depar­ or her full mathematical po­ ture from the program and in oral tential helps to redeem the mathematics altogether. Guide from some grating as­ A Mathematician's Survival advice one might pects of the author's style. Guide, by Steven G. Krantz, is give a student We will discuss the style of an omnibus of information and the Guide and the substance advice relating to every aspect of its advice below, after a sum­ of doctoral study in mathe- mary of the factual informa­ matics. Drawing on his almost tion that is the heart of the thirty years as a research mathematician at three book. The first section, "Getting Ready for Gradu­ major universities and experience as current chair­ ate School", discusses how to get into graduate man of the mathematics department at Washing­ school. It is mostly advice, although there is a flow ton University in St. Louis, Krantz sets out to give chart of graduate study (p. 6) and important in­ prospective mathematics doctoral students a de­ formation about English as a second language and tailed and realistic picture of what graduate school graduate funding. entails and how they can succeed in academia. The second and third sections, Chapters 3 through This book is in the same vein as his earlier books, 6, give a thorough outline of what to expect as one A Primer ofMathematical Writing and How to Teach progresses through a doctoral program, from qual­ Mathematics, as well as Tom Rishel's book, Teach­ ifying exams and their preparatory courses, through ing First; it is a conversational, direct, professional finding an advisor and writing a thesis. The various development guide to the business of being and be­ important players in a graduate student's life are coming a mathematician. The book's target audi­ described. Krantz takes time to mention many small ence is graduate and pregraduate students of math­ but essential points that would not all be included ematics, though pretenure faculty can also find in oral advice one might give a student, like the useful information here. It is surprising that a short importance of attending colloquia, of maintaining book can effectively treat such a big topic, but the a relationship with more faculty than just the thesis Guide is largely successful in its goals. It outlines advisor, and of respecting the departmental staff. the roughly standard procedures for applying to One of the strengths of the book is the way it col­ and completing a Ph.D. program in mathematics, lects this advice in an organized written format that touches on the variety of employment possibilities students can refer to as needed. These chapters also

418 NOTICES OF THE AMS VOLUME 51, NUMBER 4 discuss some of the uncomfortable dilemmas that might be so repelled that they will not read on to will eventually arise for any student, such as com­ the informative later chapters. The comment in peting with other graduate students, handling the preface that he aims to present information in conflicts with faculty, and difficulties with the "an accessible but authoritative tone" (p. xiv) would thesis. The sections give sound, straightforward have annoyed this reader as a pregraduate, as advice in a mentorlike and encouraging manner. would the statement in Chapter 1, "You are now This is followed by a chapter about finding a job an adult and you should look after yourself" (p. 4). as a newly minted Ph.D. The Guide takes seriously For readers familiar with Krantz's Primer and How the possibility of industrial employment, though to Teach, his style in the Guide echoes the old boy it does not do much beyond acknowledging that tone that surfaced in the earlier books, as when he this is a legitimate source of jobs. Most of the job quotes Woody Allen's remark about a liaison with advice is oriented to the academically employed a minor, "The heart wants what it wants" (p. 89), Ph.D., and again the advice is solid. Finer points of in the book's discussion of sexual conduct in grad- life as a mathematician, for in- uate school. Although the book stance the tensions of research goes on to point out the short­ and teaching, the importance of comings of this attitude for having collaborators, and par­ people who want to be em­ ticulars of how the tenure sys­ Saving ployed in universities, it is a tem interacts with academic the best for last, jarring choice of source. These ranks, are addressed in Chap­ remarks and others like them ter 8, "Afterthoughts". The spe­ Krantz ends give the impression that Krantz cific information about acad­ has not thought as carefully emy structure and history is with the about his readers' sensibilities especially worthwhile, since not as he has about what they need many graduate students will "Elements of to know. This is unfortunate. know it all, and it is nice to The text will needlessly alien­ have everything laid out to­ Mathematics", ate some students. gether in one place. A careful a lovely chapter Advice about graduate review of this section would school is a subject on which help in a job search. addressed to reasonable minds can differ, Saving the best for last, regardless of delivery. The Krantz ends with the "Elements early-stage thoughtful faculty reader will of Mathematics", a lovely chap­ probably find that he or she ter addressed again to early­ graduate students would not offer the same opin­ stage graduate students. In the ions as does the Guide on var- first sections he gives an ious points. This does not de­ overview of the mathematics tract from the book's value, that is roughly standard for since where one disagrees, one qualifying exams. Then he ends with an even bet­ is spurred to reflect on what advice would be more ter section, 9.5, "How Do All of These Subjects Fit appropriate. In an ideal situation, this book would Together?" Nowhere does he claim or try for detail be given or recommended to students by a faculty or completeness; instead, he provides a big-pic­ member (undergraduate mentor, director of grad­ ture gloss that most graduate students do not bring uate studies, thesis advisor, etc.) who has read the to their program and will rarely hear articulated by book, with comments on those passages where he the people around them. Last come appendices or she disagrees with the text. that outline the organizational structure of a math­ The Guide's advice strays outside the realm ematics department and a university, and the aca­ of reasonable difference, however, in one case. demic ranks. Again this is information that most Section 2.6, "Special Concerns for Underrepresented graduate and prospective graduate students will nei­ Groups", is poorly thought out. (One might also feel ther have nor know that they should have but will that the subtleties confronting such groups are need. The third appendix is a checklist of questions given short shrift in section 6.4, "Intimacy with students should ask when considering a graduate Members of the Mathematics Department".) "Special program. Concerns" is a very short section that opens with Together with nuts and bolts information, Krantz a credible lament that some groups are underrep­ dispenses commentary and advice throughout the resented among mathematicians. The prospective text, especially in the preface and early chapters. He student is then urged to put a premium on finding is not at his best when giving advice to pregradu­ a program in which he or she is not the only mem­ ates. His style is at times condescending, to the ber of the relevant group, if possible a program in point that one worries that self-respecting students which there are faculty from that group. The text

APRIL 2004 NOTICES OF THE AMS 419 says that the goal is to find a program where one will "be comfortable and happy and feel that you PAPERBACK NOW are supported and valued in your new environ­ ment" (p. 25), but the main specific advice about how to achieve this is to find a program with peo­ AVAILABLE! ple who are from the same underrepresented group, on the theory that they will provide the most natural role models. Other specific sugges­ The Honors Class: tions, to look for "special housing" and "special counselors", are bizarre. A charitable reader might Hilbert's Problems and Their Solvers conclude that the first applies to people with physical disabilities, but the second is difficult to The interpret. Honors Class Whatever is meant by this advice, it is inaccurate. Ben Yandell The most successful mathematicians from under­ 2003; 496 pp.; represented groups that this reviewer is aware of ISBN 1-56881-216-7 are those who went to highly rated graduate schools, like Harvard, Yale, and Michigan, and worked with Paperback; $19.95 first-rate faculty. Integration of mathematics being what it is, almost all of these women and minori­ ties had peers, professors, and thesis advisors who were Caucasian males. Indeed, this is the pattern for any group's entry into the ranks of mathemat­ ical researchers. Today it is not surprising to meet a prominent American mathematician who is Jewish, but there was a time when Jews were not welcome at elite U.S. universities. Social values changed, and talented Jewish graduate students found forward-thinking non-Jewish mentors. Of course it is important to find a program in which the faculty and other graduate students take one seriously. It is insulting to all parties, though, to suggest that this is best accomplished by sticking with members of one's own group. On the evidence, the best plan for a student from an underrepresented group who wants a career as an academic mathematician is essentially the same as for anyone else: find an honest and powerful advisor at a highly ranked program. In summary, A Mathematician's Survival Guide is an informative and useful reference for aspiring mathematics Ph.D.'s and faculty who mentor them. It would make an attractive component of an intro­ ductory package for incoming graduate students or a counseling book for senior undergraduates con­ sidering their options. It is also a worthwhile read for any faculty member who is interested in gradu­ ate education as a starting point for his or her own reflections, and as a window onto another thinker's point of view.

References [1] STEVEN G. KRANTZ, How to Teach Mathematics, second A K Peters, Ltd. edition, Amer. Math. Soc., Providence, Rl, 1999. 63 South Ave., Natick, MA 01760-4626 [2] _ _ , A Primer of Mathematical Writing, Amer. Math. Tel: (508) 655-9933 Fax: (508) 655-5847 Soc., Providence, Rl, 1997. www.akpeters.com • [email protected] [3] THOMAS W. RISHEL, Teaching First: A Guide for New Mathematicians, Math. Assoc. Amer., Washington, DC, 2000.

420 NOTICES OF THE AMS VOLUME 51, NUMBER 4 2004 Steele Prizes

The 2004 Leroy P. Steele Prizes were awarded at the The 2004 Steele Prizes were awarded to JoHNW. 11 Oth Annual Meeting of the AMS in Phoenix in MILNOR for Mathematical Exposition, to LAWRENCE C. January 2004. EvANs and NICOLAI V. KRYLOV for a Seminal Contri­ The Steele Prizes were established in 1970 in honor bution to Research, and to CATHLEEN SYNGE MORAWETZ of George David Birkhoff, William Fogg Osgood, and for Lifetime Achievement. The text that follows William Caspar Graustein. Osgood was president of presents, for each awardee, the selection commit­ theAMS during 1905-06, andBirkhoffservedin that tee's citation, a brief biographical sketch, and the capacity during 1925-26. The prizes are endowed awardee's response upon receiving the prize. under the terms of a bequest from Leroy P. Steele. Up to three prizes are awarded each year in the follow­ Mathematical Exposition: john W. Milnor ing categories: (1) Lifetime Achievement: for the Citation cumulative influence of the total mathematical work The Leroy P. Steele Prize for Mathematical Exposi­ of the recipient, high level of research over a period tion is awarded to John W. Milnor in recognition of time, particular influence on the development of of a lifetime of expository contributions ranging a field, and influence on mathematics through Ph.D. across a wide spectrum of disciplines including students; (2) Mathematical Exposition: for a book topology, symmetric bilinear forms, characteristic or substantial survey or expository-research paper; classes, Morse theory, game theory, algebraic K­ (3) Seminal Contribution to Research (limited for theory, iterated rational maps ... and the list goes on. 2004 to analysis): for a paper, whether recent or not, The phrase "sublime elegance" is rarely associated that has proved to be of fundamental or lasting im­ with mathematical exposition, but it applies to all portance in its field or a model of important research. of Milnor's writings, whether they be research or Each Steele Prize carries a cash award of $5,000. expository. Reading his books, one is struck with The Steele Prizes are awarded by the AMS Council the ease with which the subject is unfolding, and acting on the recommendation of a selection it only becomes apparent after reflection that this committee. For the 2004 prizes the members of ease is the mark of a master. Improvement of the selection committee were: M. Salah Baouendi, Milnor's treatments often seems impossible. Andreas R. Blass, Sun-Yung Alice Chang, Michael G. A portion of Kauffman's review of Symmetric Crandall (chair), Craig L. Huneke, Daniel J Kleitman, Bilinear Forms by Milnor and Husemoller conveys Tsit-Yuen Lam, Robert D. MacPherson, and Lou P. the beauty evident in all of Milnor's expository Van den Dries. work: " ... Appendix 4, where this result is proved, The list of previous recipients of the Steele Prize is alone worth the price of the book. It contains may be found in the November 2003 issue of the Milnor's proof of a Gauss sum formula (due to Notices, pages 1294-8, or on the World Wide Web, R. J Milgram) that uses elegant combinatorics and http://www. ams.org/pri zes- awards. Fourier analysis to produce an argument whose

APRIL 2004 NOTICES OF THE AMS 421 John W. Milnor Lawrence C. Evans Nicolai V. Krylov

corollaries include the divisibility theorem, the Response law of quadratic reciprocity and its equivalent It is a great pleasure to receive this award, and I in the language of forms over Z: the Weil reci­ certainly want to thank the members of the Selec­ procity theorem. The proof is short, beautiful, and tion Committee for their consideration. It is of mysterious." course also a tribute to my many coauthors: let me Milnor's many expository contributions to the mention Dale Husemoller, Larry Siebenmann, mathematical literature have influenced more than Jonathan Sondow, Mike Spivak, Jim Stasheff, and one generation of mathematicians. Moreover, the Robert Wells. examples that they provide have set a standard of I have always suspected that the key to the most clarity, elegance, and beauty for which every math­ interesting exposition is the choice of a subject that ematician should strive. the author doesn't understand too well. I have the Biographical Sketch unfortunate difficulty that it is almost impossible John Milnor was born in Orange, New Jersey, in for me to understand a complicated argument unless 1931. He spent his undergraduate and graduate stu­ I try to write it down. Over the years I have run into a great many difficult dent years at Princeton, working on knot theory bits of mathematics, and thus I keep finding myself writing things down. (And also under the supervision of Ralph Fox, and also dab­ rewriting, since I never get things right the first bling in game theory with his fellow students John few times. Years ago, I was the despair of secretaries Nash and Lloyd Shapley. However, like his mathe­ who would produce beautifully typed manuscripts, matical grandfather, Solomon Lefschetz, he had only to have them repeatedly cut, pasted, and great difficulty sticking to one subject. Under the scribbled over. Computers have eliminated this inspiration of Norman Steemod and later John particular problem, but it still makes life difficult Moore, he branched out into algebraic and differ­ for coauthors.) ential topology. This led to problems in pure al­ I am very happy to report that as mathematics gebra, including algebraic K-theory and the study keeps growing, there are more and more subjects of quadratic forms. More recently, conversations that I have to fight to understand. with William Thurston and Adrien Douady led to studies in real and complex dynamical systems, Seminal Contribution to Research: which have occupied him for the last twenty years. Lawrence C. Evans and Nicolai V. Krylov But he is still restless: one current activity is an Citation attempted exposition of problems of complexity The Steele Prize for Seminal Research is awarded in the life sciences. to Lawrence C. Evans and Nicolai V. Krylov for the After many years at Princeton at the university "Evans-Krylov theorem" as first established in the and also at the Institute for Advanced Study, and papers: after shorter stays at the University of California, Lawrence C. Evans, "Classical solutions of fully Los Angeles, and the Massachusetts Institute of nonlinear convex, second order elliptic equations", Technology, Milnor moved to the State University Communications in Pure and Applied Mathematics of New York at Stony Brook, where he has been the 35(1982),no. 3, 333-363;and director of the Institute for Mathematical Sciences N. V. Krylov, "Boundedly inhomogeneous ellip­ since 1989. tic and parabolic equations", Izvestiya Akad. Nauk

422 NOTICES OF THE AMS VOLUME 51, NUMBER 4 SSSR, ser. mat. 46 (1982), no. 3, 487-523; and trans­ I was not especially afraid to look at so-called "fully lated in Mathematics of the USSR, Izvestiya 20 nonlinear" elliptic equations in the late 1970s and (1983), no. 3, 459-492. early 1980s. Fully nonlinear elliptic equations are of interest These are important PDEs, examples of which in many subjects, including the theory of controlled are the Monge-Ampere equation and Hamilton­ diffusion processes and differential geometry. It is Jacobi-Bellman equations in stochastic optimal therefore of great interest to understand when these control theory. And they are really, really nonlin­ equations have classical solutions. The first results ear. But their solutions satisfy maximum principles, of any generality exhibiting classical solutions of the and this was a clue. It turns out that (i) when the subclass of uniformly elliptic equations under suit­ nonlinearity is convex, we can get "one-sided" con­ able convexity conditions are due to the recipients trol on second derivatives; and that then (ii) the in the cited works. These authors, independently PDE itself provides a functional relationship among and with different arguments, established the Holder the various second derivatives, yi.elding thereby continuity of second derivatives in the interior, via "two-sided" control. (Earlier Calabi had derived a priori estimates, a result now known as the Evans­ third derivative bounds for the Monge-Ampere Krylov theorem. The Evans-Krylov theorem was equation, and Brezis and I had treated the very both a capstone on fundamental contributions of special case of the maximum of two linear elliptic the recipients and others and a harbinger of things operators.) to follow from the community. All success in mathematics turns largely upon While the Steele Prize for Seminal Research is persistence and luck; and while I can take some explicitly awarded for the named works, it is noted credit for the persistence, the luck was, well, luck­ that both recipients have made a variety of distin­ chiefly in that, quite unknown to me, one N. V. guished contributions to the theory of nonlinear Krylov in the Soviet Union had turned his attention partial differential equations. to these same problems at about the same time. Biographical Sketch: Lawrence C. Evans And Nick's contributions to the subject have been Lawrence C. Evans was born November 1, 1949, in extraordinary, including not only the interior Holder Atlanta, Georgia. He received his B.A. from Van­ second derivative estimates, for which indepen­ derbilt University in 1971 and his Ph.D. from the dent discovery we are being honored, but also his University of California, Los Angeles, in 1975; his previous, and great, work with Mikhail Safonov on advisor at UCLA was M. G. Crandall. Evans held Holder bounds and the Harnack inequality for non­ positions at the University of Kentucky from 1975 divergence structure second-order elliptic equa­ to 1980, at the University of Maryland from 1980 tions with discontinuous coefficients. We needed to 1989, and is currently professor of mathemat­ these to carry out step (i) mentioned above. Nick ics at the University of California at Berkeley, a also later derived boundary second derivative position he has held since 1989. He has been a estimates, something at which I completely failed. visiting professor at Northwestern University So it is really an honor to share this prize with (1977-78) and at the Institute for Advanced Study Nick and to have seen over the past twenty years (1988). Noteworthy publications include Weak the magnificent work of Caffarelli, Guan, Li, Convergence Methods for Nonlinear Partial Differ­ P.-L. Lions, Nirenberg, Spruck, Trudinger, Urbas, ential Equations (CBMS Regional Conference Series Wang, and many other researchers vastly extend­ in Mathematics, volume 74, AMS, 1990), Measure ing these ideas. Theory and Fine Properties ofFunctions, coauthored Biographical Sketch: Nicolai V. Krylov with R. F. Gariepy (Studies in Advanced Mathe­ Nicolai Vladimirovich Krylov was born in Soudogda, matics, CRC Press, 1992), and Partial Differential the region of Vladimir, Russia, on June 5, 1941. Equations (Graduate Studies in Mathematics, vol­ He received his Ph.D. in 1966 and his doctorate of ume 19, AMS, 1998). science in 1973 from Moscow State University; his Response: Lawrence C. Evans scientific advisor was E. B. Dynkin. Krylov taught It is a wonderful honor to share with Nick Krylov at Moscow State University from 1966 to 1990; he this year's Steele Prize for a Seminal Contribution has taught at the University of Minnesota since to Research. When I was Mike Crandall's graduate 1990, and currently holds the position of Samuel G. student at UCLA and at Wisconsin over thirty years Ordway Professor of Mathematics. He has super­ ago, I learned from him the then startling lesson vised the graduate degrees of fifteen students. that nonlinear analysis need not be solely based Krylov has given invited addresses at the Interna­ upon linearization, meaning small perturbation tional Congress of Mathematicians in Helsinki (19 78) theory from linear approximations. Brezis, Brow­ and Berkeley (1986) and has given fifty-eight invited der, Crandall, ].-L. Lions, and many others in the lectures, has written nearly two hundred research 1970s pioneered the analysis of various sorts of articles, and has published five monographs. A strongly nonli.near operators, a theory in which member of many journal editorial boards, Krylov linearity played little role at all. I think this was why was elected a Fellow of the American Academy of Arts

APRIL 2004 NOTICES OF THE AMS 423 and Sciences in 1993, received a Humboldt Research extra smoothness assumptions on the data. A major Award for Senior U.S. Scientists in 2001, and has been step forward in this respect was achieved by a recipient of numerous National Science Foundation M. Safonov in 1984 when he showed that the esti­ grants. mate holds only under "natural" conditions. Response: Nicolai V. Krylov It is a great honor to share with Craig Evans this Lifetime Achievement: Cathleen Synge year's Steele Prize for a Seminal Contribution to Morawetz Research. Citation In the times when I was an undergraduate Cathleen Morawetz has greatly influenced mathe­ student in Moscow State University, all kinds of matics in the broad sense throughout her long and control theory became popular. My scientific advi­ distinguished career. Her fundamental research sor, E. B. Dynkin, became interested in stochastic has resulted in seminal contributions to a number control theory, and being a brilliant lecturer, he of areas. These contributions include her early easily attracted many people, including me, into it. work on equations of mixed type, with its striking As often happens in probability theory, it was consequences for the theory of flow around airfoils, very easy to understand why certain probabilistic her work on local energy decay for waves in the quantities should satisfy Bellman equations, but complement of an obstacle, and her results con­ discouragingly for quite a while there were no ideas cerning the existence of transonic flow with shocks. on how to prove this. Bellman equations are fully Throughout Professor Morawetz's work one finds nonlinear possibly degenerate second-order partial the theme of deep, creative mathematics used in differential equations with convex nonlinearity, the treatment of problems selected because of of which the Monge-Ampere equation is the most their interest in applied areas. She has not only famous example. When in about 1963 I asked contributed greatly to mathematics but also to the 0. A Oleinik what was known about such equations, vitality of the interaction between mathematics the answer was very short: "Nothing." This boosted and its applications. even further my desire to prove the solvability of Cathleen Morawetz's influence on mathematics Bellman equations by probabilistic means. However, extends well beyond her research contributions. In it took seven years before I realized how to prove a residence at the Courant Institute of Mathematical basic estimate, and after that the theory was com­ Sciences for almost all of her career, she provided pleted in 1971-72. guidance and inspiration to the stream of visitors It took even longer to develop an analytic and postdoctoral appointees, as well as to her own approach. Working on some very natural questions students. Her works include a number of influen­ from stochastic control theory, M. Safonov and I tial contributions written in collaboration with were lucky enough to obtain in 1978 Holder norm younger mathematicians. estimates for solutions of linear equations with Beyond these mathematical contributions, com­ possibly rough coefficients. These estimates prove, manding in themselves, Cathleen Morawetz has in particular, the continuity of harmonic functions provided strong leadership for and representation corresponding to diffusion processes with mea­ of the mathematical community via her remarkable surable coefficients. An automatic consequence and generous service. The AMS has benefited from of this fact is the lower semicontinuity of super­ her membership on many committees, from her ten harmonics. On the other hand, it is trivial to see years of service as a Trustee of the Society, and her that the second order directional derivatives of service as President of the Society. She dispatched solutions of Bellman equations are superharmon­ her duties in these roles with excellence and did ics for certain diffusions. Thus they should be not merely serve; she provided leadership. The upper semicontinuous. But the equation itself says larger community benefited from her wisdom in that a certain function of these directional deriva­ positions such as that of a Trustee of Princeton tives is continuous. In addition, the function is University and a Trustee of the Sloan Foundation; monotone, and this yields the continuity of second­ mathematics also benefited from being represented order derivatives. by her in these roles. Among her pioneering "firsts", Remarkably, Craig Evans obtained similar results one notes that she was the first woman to direct at about the same time. Since then I have become a an institute of mathematics in the U.S. and she great admirer of Craig's talent, and I am very honored was the first woman to receive the National Medal to share the prize with him. of Science for work in mathematics. Our results opened up the area to analytic treat­ Thank you, Cathleen, for all you have done. ment, and since then very many mathematicians Biographical Sketch have made amazing contributions. I want to men­ Cathleen Synge Morawetz was born in Toronto, tion only one directly related to our prize. A weak Canada, on May 5, 1923. She received a B.A. in ap­ point in Craig's and my argument is that we need plied mathematics from the University of Toronto to differentiate the equation twice, which led to in 194 5, an M.Sc. from the Massachusetts Institute

424 NOTICES OF THE AMS VOLUME 51, NUMBER 4 of Technology in 1946, and a Ph.D. from New York deal in my career ideas, working as a chronographer University in 1951. From 1950 to 1951 she was a during World War II, seriously contemplating teach­ research associate at MIT working on hydrody­ ing in India (a chance meeting with Cecilia Krieger namic stability with C. C. Lin. From 1951 on she sent me off to graduate school worked with the group at NYU that became the instead), trying out and fail­ Courant Institute, mainly at first with L. Bers, ing at electrical engineering at K. 0. Friedrichs, and H. Grad. MIT. There was also a consid­ Bers and Friedrichs introduced her to the fasci­ erable amount of external so­ nating problems of transonic flow; Harold Grad cial pressure to abandon my introduced her to problems in magnetohydrody­ career, but such ideas did not namics, especially the mathematical problem enter the minds of Courant associated with very thin plasmas; and from Joe and his colleagues-nor for Keller she learned the open problems of wave that matter of my husband, propagation. Herbert. She became an assistant professor at the insti­ Among the many people at tute in 1958. Always involved in some adminis­ the Courant Institute who ed­ tration, she eventually served as director of the ucated, mentored, and helped Courant Institute from 1984 to 1988. She retired me in the vast literature of in 1993. mathematics (I have a bad Cathleen Morawetz gave the AMS Gibbs Lecture memory) were not only in 1981. During much of her career she received Friedrichs but Lipman Bers, support from the Office of Naval Research. Joe Keller, Harold Grad, Fritz She served the Society as a member of the Coun­ John, Paul Garabedian, Peter cil from 1973 to 1975, as a member of the Execu­ Lax, and Louis Nirenberg. Let Cathleen Synge Morawetz tive Committee in 1975 and from 1994 to 1998, as me add the names of my col- a trustee from 1975 to 1985, and was the second laborators who taught me so much: Walter Strauss, woman president of the Society from 199 5 to 199 7. Jim Ralston, and Ralph Phillips. She is still a member of two committees. She Lastly, and by no means least, I am forever received the National Medal of Science in 1998. indebted to my mother for instilling in me the Cathleen Morawetz was a trustee of Princeton idea of ambition (then very unladylike) and to my University, a trustee of the Sloan Foundation, a mem­ father for the idea of intellectual achievement (not ber of the board of NCR, and a founding director to mention the introduction to Courant). of JSTOR (1995-98). In addition, she served on the board of the Mathematical Sciences Research Insti­ tute and chaired the board for theoretical physics of the Dublin Institute for Advanced Studies. She has received numerous honorary degrees. She first studied the nonlinear wave propagation of shock wave theory as a student and later, at the suggestion of I. Segal, of semilinear equations. This resulted in fundamental work with Walter Strauss. Both her transonic theories and her work in wave propagation involved finding special identities and inequalities for the relevant equations. Response Receiving the Steele Prize for Lifetime Achievement is not only a huge honor but a stunning surprise for which I am very grateful. But I can never be quite as grateful as I am to those people who mentored and encouraged me in a lifetime of mathematics which, somewhat to my surprise, still goes on. The person to whom I am most grateful is Richard Courant, who steadfastly employed me in real research as I struggled to get a Ph.D. and to bear and raise four children between 1946 and 1958. He claimed it was Kurt Friedrichs who constantly rec­ ommended me to him, but Courant was surely the only person with the authority to follow this non­ standard path. Before that time I wavered a great

APRIL 2004 NOTICES OF THE AMS 425 2004 Veblen Prize

The 2004 Oswald Veblen Prize Citation in Geometry was awarded at The 2004 Veblen Prize in Geometry is awarded to the 11 Oth Annual Meeting of David Gabai of Princeton University in recognition the AMS in Phoenix in Janu­ of his work in geometric topology, in particular, the ary 2004. topology of 3-dimensional manifolds. The Veblen Prize is awarded Since its beginnings in the early twentieth cen­ every three years for a notable tury, 3-dimensional topology has occupied a cen­ research memoir in geometry tral role in geometric topology. It is tantalizingly or topology that has appeared close to what we can directly visualize, yet with its during the previous five years knotting phenomena it is an incredibly complex and in a recognized North Ameri­ difficult subject. For the last twenty years, Gabai can journal (until 2001 the has been one of the leading figures in this field. He David Gabai was usually awarded prize has led many of the main avenues of development, every five years). Established in developing tools in order to solve some of its most 1964, the prize honors the memory of Oswald Ve­ important problems himself, tools that have turned blen (1880- 1960), who served as president of the central to the further development of the AMS during 1923- 24. It carries a cash award of out to be $5,000. subject. One aspect of 3-dimensional topology greatly in­ The Veblen Prize is awarded by the AMS Coun­ fluenced by Gabai is the study of surfaces inside a cil acting on the recommendation of a selection 3-manifold and the intersection patterns of two or committee. For the 2004 prize the members of the of these. His introduction of the notion of thin selection committee were: Andrew]. Casson, Yakov more which he used to resolve the question Eliashberg, and John W. Morgan (chair). position, Previous recipients of the Veblen Prize are: Chris­ known as "Property R" about when surgery on a knot tos D. Papakyriakopoulos (1964), Raoul H. Bott in the 3-sphere can yield a 3-manifold homeomor­ (1964), Stephen Smale (1966), Morton Brown and phic to the product of the 2-sphere and a circle, has Barry Mazur (1966), Robion C. Kirby (1971), Den­ found application far beyond Gabai's original use, nis P. Sullivan (1971), William P. Thurston (1976), for example, in the proof that knots are determined James Simons (1976), Mikhael Gromov (1981), by their complements. Gabai's study of surfaces is Shing-TungYau (1981), Michael H. Freedman (1986), achieved in large part through the study of more Andrew]. Casson (1991), Clifford H. Taubes (1991), general objects, codimension-one laminations, in Richard Hamilton (1996), Gang Tian (1996), Jeff a 3-manifold. In a sequence of papers beginning Cheeger (2001), Yakov Eliashberg (2001), and in the 1980s and summarized in his talk at the Michael]. Hopkins (2001). 1990 International Congress of Mathematicians in The 2004 Veblen Prize was awarded to DAVID Kyoto entitled "Foliations and 3-manifolds", Gabai GABAI. The text that follows presents the selection developed the theory of these objects. In Gabai's committee's citation, a brief biographical sketch, hands, they have served to help unlock some of the and the awardee's response upon receiving the topological mysteries of 3-dimensional topology. prize. This theory of these laminations has now grown to

426 NOTICES OF 1HE AMS VOLUME 51, NUMBER 4 the extent that it is a subdomain of 3-dimensional which matched my talents. Being in a field then con­ topology in its own right. sidered by many experts to be either finished or More recently, Gabai has investigated 3-dimen­ peripheral, I could slowly and happily build my in­ sional hyperbolic manifolds. Hyperbolic 3-manifolds tuition and technical skills without the distraction are a rich and much studied class. Conjecturally at of noise or the danger of being trampled. It turned least, they are by far the richest and most interest­ out my constructions of foliations were useful and ing class. One of the central problems in 3-manifold of contemporary interest, so I got a great job at Cal­ topology is how to tell when a 3-manifold is hyper­ tech. In 1991 I attempted to teach a topics course bolic, i.e., has a hyperbolic structure. By strikingly on the hyperbolic geometry that I should have original arguments in a series of papers [listed below], learned as a graduate student. I started with Chap­ Gabai has answered this question in a special case, ter 1 of Thurston's 1977-78lecture notes but could by showing that every irreducible 3-manifold with not get past Chapter 5. There he discussed why the homotopy type of a hyperbolic manifold has a Mostow's rigidity theorem implies that a manifold hyperbolic structure. Further developments of his finitely covered by a hyperbolic 3-manifold is methods led to a proof by Gabai of the Smale Con­ homotopy equivalent to a hyperbolic 3-manifold. jecture for hyperbolic 3-manifolds-describing the I got stuck trying to prove the converse. Ultimately, homotopy type of the space of self-diffeomor­ the converse became the start of work cited here. phisms-and also to new estimates for the volumes It is a great pleasure to thank Ulrich Oertel, with of hyperbolic 3-manifolds. whom I introduced essential laminations in 1986, "Homotopy hyperbolic 3-manifolds are virtually and Rob Meyerhoff and Nathaniel Thurston, who hyperbolic", J Amer. Math. Soc. 7 (1994), no. 1, 193-8. worked with me on the homotopy hyperbolic pro­ "On the geometric and topological rigidity of ject. I also thank my long-time collaborator Will hyperbolic 3-manifolds", ]. Amer. Math. Soc. 10 Kazez and my collaborators Peter Milley and (1997), no. 1, 37-74. Valentin Poenaru. Victor Guillemin's beautiful dif­ (jointly with Robert Meyerhoff and Nathaniel ferential topology course during my last semester Thurston) "Homotopy hyperbolic 3-manifolds are at MIT kept me from opting out of mathematics for hyperbolic", Ann. of Math. (2) 157 (2003), no. 2, medical school. Bill Thurston's influence has been 335-431. immense. I am in debt to my teachers, and I count my students among them. I very much appreciate Biographical Sketch the many mathematicians who have encouraged me David Gabai received his B.S. from the Massachu­ over the years. Finally, I thank my many hosts in setts Institute of Technology (1976), his M.A. from China, England, France, Israel, and Japan for pro­ Princeton University (1977), and his Ph.D. from viding quiet environments during short visits so Princeton (1980) under the direction of William that I could hide out and nurture my thoughts. Thurston. After positions at Harvard and the Uni­ Much of what I know was done or inspired by versity of Pennsylvania, he spent most of the years former prizewinners. It is humbling to be the 1986-2001 at the California Institute of Technol­ recipient of the 2004 Veblen Prize. ogy and has been at Princeton University since 2001. He held visiting positions at the Institute for Advanced Study, Princeton (1982-83, fall1989); the Mathematical Sciences Research Institute, Berke­ ley (1984-85, 1996-97); and the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France (1985-86). He has received a National Science Foun­ dation Postdoctoral Fellowship, a Sloan Fellowship, and an AMS Centennial Fellowship. He gave a 45-minute invited talk at the 1990 International Congress of Mathematicians in Kyoto and an hour invited talk at the 1995 joint meeting of the AMS and the Mexican Mathematical Society in Guana­ juato, Mexico. He also gave the 1996 Porter Lectures (Rice University), the 2001 Marston Morse memo­ rial lectures (lAS), and the 2002 Unni Namboodiri memorial lectures (University of Chicago).

Response I have been incredibly lucky all my life. As a grad­ uate student at Princeton, Bill Thurston suggested an area of mathematics, foliations on 3-manifolds,

APRIL 2004 NOTICES OF THE AMS 427 2004 Wiener Prize

The 2004 AMS-SIAM committee's citation, a brief biographical sketch, Norbert Wiener Prize in and the awardee's response upon receiving the Applied Mathematics was prize. awarded at the llOth Annual Meeting of the AMS Citation in Phoenix in January 2004. The Norbert Wiener Prize in Applied Mathematics The Wiener Prize is is awarded to James A. Sethian of the University awarded every three years of California at Berkeley for his seminal work on to recognize outstanding the computer representation of the motion of contributions to applied curves, surfaces, interfaces, and wave fronts, and mathematics in the high­ for his brilliant applications of mathematical and est and broadest sense computational ideas to problems in science and (until 2001 the prize was awarded every five years). engineering. Established in 196 7 in His earliest work included an analysis of the honor of Norbert Wiener motion of flame fronts and of the singularities they James A. Sethian (1894-1964), the prize was develop; he found important new links between endowed by the Depart­ the motion of fronts and partial differential ment of Mathematics of the Massachusetts Insti­ equations, and in particular found that the correct tute of Technology. The prize is given jointly by the extension of front motion beyond a singularity AMS and the Society for Industrial and Applied follows from an entropy condition as in the theory Mathematics (SIAM). The recipient must be a mem­ of nonlinear hyperbolic equations. These connec­ ber of one of these societies and a resident of the tions made possible the development of advanced United States, Canada, or Mexico. The prize carries numerical methods to describe front propagation a cash award of $5,000. through the solution of regularized equations on The recipient of the Wiener Prize is chosen by fixed grids. a joint AMS-SIAM selection committee. For the 2004 In a subsequent work (with S. Osher) Sethian prize the members of the selection committee extended this work through an implicit formulation. were: Alexandre]. Chorin (chair), Martin Grbtschel, The resulting methodology has come to be known as and Philip ]. Holmes. the "level set method", because it represents a front The previous recipients of the Wiener Prize are: propagating in n dimensions as a level set of an ob­ Richard E. Bellman (1970), Peter D. Lax (1975), ject in (n + 1) dimensions. Next, Sethian tamed the Tosio Kato (1980), Gerald B. Whitham (1980), Clif­ cost of working in higher dimensions by reducing ford S. Gardner (1985), Michael Aizenman (1990), the problem back down to its original dimensional­ Jerrold E. Marsden (1990), Hermann Flaschka ity. This set of ideas makes possible the solution of (1995), Ciprian Foias (1995), Alexandre ]. Chorin practical problems of increasing importance and (2000), and Arthur T. Winfree (2000). sophistication and constitutes a major mathemati­ The 2004 Wiener Prize was awarded to ]AMES cal development as well as an exceptionally useful SETHIAN. The text that follows presents the selection computational tool with numerous applications.

428 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Among the practical problems solved by Sethian has received SIAM's I. E. Block Community Lecture are: the tracking of interfaces and drops in fluid Prize. mechanics with applications to inkjet design for He is an associate editor of SIAM Review, the high-speed printers; the analysis of crystal growth journal of Mathematical Imaging and Vision, and (with]. Strain); motion under mean curvature, con­ the journal on Interfaces and Free Boundaries. struction of minimal surfaces, and knot recognition in computational geometry; design of optimal struc­ Response tures under loads (with A. Wiegmann); and the In the course of a normal day, a letter from the AMS analysis of anisotropic front propagation and mixed appears and jolts one out of a busy routine. I am discrete-continuous control. Each of these appli­ grateful for this unexpected shock: It is a wonder­ cations required extensions and modifications of ful honor to be the recipient of the Norbert Wiener the basic tools as well as new understanding of the Prize. problems under investigation. My interest in front propagation began with Sethian's mathematical description of etching the suggestions of Alexandre Chorin at Berkeley. and deposition in the manufacture of computer Starting with his cell fraction-based Huyghens chips has illuminated processes such as ion-milling, propagation algorithm, he artfully led me toward visibility, resputter, and material-dependent etch unanswered questions about interface evolution, the rates; the resulting algorithms are now an indis­ Landau instability, and ill-posedness. The appeal pensable part of industrial semiconductor fabri­ of alternative approaches stemmed from the sheer cation simulations throughout the world. His frustration of attempting to elevate existing nu­ models of implicit surface motion together with merical front propagation schemes beyond simple fast Eikonal solvers are standard fare in medical two-dimensional problems. I have a fond memory and biomedical shape extraction and in fields such of buying building blocks in 19 7 8 from a local as shape-based image interpolation, shape-from­ toy store in an optimistic attempt to visualize the shading, stereoscopic vision, and texture segmen­ various cases involved in a three-dimensional tation; they are used in hospital electron beam version of the Volume-of-Fluid algorithm. The scanners to quantify cardiac motion and efficiency. visual aids were not enough, and my thesis instead focused on developing and analyzing a mathe­ Recently, Sethian (with S. Fomel) developed efficient matical model of flame and front propagation. numerical methods for simulating multiple-arrival A Danforth Fellowship at Berkeley, followed by wavefront propagation by solving Liouville-type a National Science Foundation [NSF] Postdoctoral equations; this work has direct applications in Fellowship at the Courant Institute, and then a Sloan seismic imaging and geophysical inverse problems Foundation Fellowship back at Berkeley, coupled and has already been put to use by the petroleum to support from the U.S. Department of Energy industry. [DOE], generously allowed me time for subsequent A particularly noteworthy aspect of Sethian's work work on entropy conditions for front propagation, is that he pursues his ideas from a first formulation as well as links between the regularizing effects of a mathematical model all the way to concrete of curvature on Hamilton-Jacobi equations for front applications in national laboratory and industrial propagation and viscosity in conservation laws, and settings; his algorithms are currently distributed opened up the strategy of applying shock schemes in widely available packages. Sethian's work is a to interface problems. shining example of what applied mathematics can Indeed, this work on casting front propagation accomplish to benefit science as a whole. in the language of differential geometry and partial differential equations benefitted from a collection of Biographical Sketch disparate ideas and tools that were bubbling together James A. Sethian was born on May 10, 1954, in in the late 19 70s and early 1980s. The work of Washington, DC. He received a B.A. in mathemat­ M. Crandall and P.-L. Lions, and then L. C. Evans, on ics from Princeton University in 1976 and a Ph.D. viscosity solutions for Hamilton-Jacobi equations, in applied mathematics from the University of Cal­ G. Earles' analysis of that Berkeley flame model, the ifornia, Berkeley, in 1982. After a National Science maturation of numerical schemes for hyperbolic Foundation Postdoctoral Fellowship at the Courant conservation laws, and fresh ideas about curve Institute of Mathematical Sciences, he joined the evolution by M. Gage and M. Grayson all formed faculty at UC Berkeley, where he is now professor part of the landscape in those early years. of mathematics as well as head of the mathemat­ These ideas have led to several algorithms based ics department at the Lawrence Berkeley National on a partial differential equations view of evolving Laboratory. He has been a plenary speaker at the fronts. The first such algorithm, which relied on International Congress of Industrial and Applied embedding the front as a particular level set of a Mathematicians, has been an invited speaker at higher-dimensional function and employed high­ the International Congress of Mathematicians, and order schemes for the underlying Hamilton-Jacobi

APRIL 2004 NOTICES OF THE AMS 429 equation, became known as the "Level Set Method". I have been lucky to have had pivotal teachers The work is joint with Stanley Osher at UCLA, who stood as "transformers" along the way, in­ whose enthusiasm is a force unto itself, and I have vesting their own energy to raise the voltage and warm memories of that collaboration. then graciously passing it on. At a public junior high As first laid out, that version of the Level Set school in Virginia, W. Taylor was the first to tell me Method was mathematically appealing, numeri­ to study mathematics. A high school teacher offered cally robust, and unnecessarily slow. I was fortu­ similar encouragement, adding that I was almost nate to have D. Chopp, now at Northwestern, as as good as the kid sitting next to me. I don't feel my first Ph.D. student, with whom the ideas of too bad, since that kid, Eric Schmidt, is now CEO reinitialization and adaptivity were developed, of Google. W. K Allard, then at Princeton, brought pointing the way towards making these methods me to PDEs. 0. Hald at Berkeley introduced me practical and efficient. The resulting Narrow Band to numerics. A. Chorin is a wise and skilled thesis Level Set Methods were honed with an equally advisor; I am fortunate to be in the large commu­ talented Ph.D. student, D. Adalsteinsson, now at nity of his former students. the University of North Carolina, Chapel Hill, who Finally, I am grateful to the Department of helped put these methods on a competitive foot­ Energy for its long-term support of these efforts, the ing with other methods of the day. National Science Foundation, and the ongoing op­ In this short space I cannot do justice to the large portunity to interact at Berkeley and the Lawrence amount of work done on level set methods and the Berkeley National Laboratory with people of singu­ surprising areas to which they have been applied. lar talent, warmth, and support, including G. I. Many efforts, including large-scale projects at the Barenblatt, A. Grunbaum, 0. Hald, and R. Malladi. DOE National Laboratories, in particular at the Lawrence Berkeley National Laboratory, semester­ long programs at IPAM [Institute for Mathematics and its Applications] at UCLA, and focused teams such as those in the semiconductor industry, have all contributed to pushing these techniques forward. I would like to make mention of a few of my most recent collaborators. R. Malladi developed ground­ breaking work while at the University of Florida, applying these algorithmic ideas to image seg­ mentation, and I was fortunate that he chose to take his NSF Computational Sciences Postdoctoral Fel­ lowship at Berkeley. He continues to be a leader in applying PDE-based techniques for medical and biomedical applications, and I am grateful for the ongoing collaboration. A. Vlad:imirsky, a former student now at Cornell, was instrumental in extending PDE-based front propagation techniques to produce extraordinarily fast methods for optimal control and anisotropic front propagation. S. Fomel, a former postdoc now at Texas, was pivotal in devising PDE front schemes for multiple "nonviscosity" arrivals with the same computa­ tional efficiency. And]. Wilkening, a former student now at Courant, tackled the difficult problem of front propagation and void motion in the context of electromigration. It is a joy to work with such able talents. Two other endeavors deserve mention, in part because of the extensive work done long after I left the scene. The work with A. Majda on zero Mach number combustion has been honed and melded into complex combustion calculations by ]. Bell and P. Colella. And the work on mathematical botany, anisogamy, and chemotaxis continues to be pioneered by P. Cox, director of the National Tropical Botanical Garden.

430 NOTICES OF THE AMS VOLUME 51, NUMBER 4 2004 Moore Prize

The 2004 E. H. Moore Research Article Prize was Citation awarded at the 11 Oth Annual Meeting of the AMS "Hilbert schemes, polygraphs, in Phoenix in January 2004. This was the first time and the Macdonald positivity the prize was awarded. conjecture", journal of the The Moore Prize will be awarded every three AMS 14 (2001), 941-1006. years for an outstanding research article that ap­ Mark Haiman's ground­ breaking paper proves both peared in one of the AMS primary research jour­ the n! conjecture and Mac­ nals: journal of the AMS, Proceedings of the AMS, donald's positivity conjecture, Transactions of the AMS, AMS Memoirs, Mathe­ both long-standing open prob­ matics of Computation, Electronic journal of Con­ lems in algebraic combina­ formal Geometry and Dynamics, and Electronic torics, through the develop­ journal ofRepresentation Theory. The article must ment of remarkable new have appeared during the six calendar years end­ notions in algebraic geometry ing a full year before the meeting at which the and a tour-de-force derivation Mark Haiman in commutative algebra. prize is awarded. The prize carries a cash award The last step concerns the defining equations of of $5,000. the polygraph, an arrangement of linear subspaces The prize honors the extensive contributions of connected with the geometry of the Hilbert scheme E. H. Moore (1862-1932) to the AMS. Moore founded of n points in the plane. This is then shown to the Chicago section of the AMS, served as the coincide with the Hilbert scheme of regular orbits Society's sixth president (1901-02), delivered the of the symmetric group acting on labeled config­ Colloquium Lectures in 1906, and founded urations of n points. The key result states that and nurtured the Transactions of the AMS. the isospectral Hilbert scheme is normal, Cohen­ The Moore Prize is awarded by the AMS Coun­ Macaulay, and Gorenstein. cil acting on the recommendation of a selection Haiman's paper has within the last two years already led to numerous other new developments committee. For the 2004 prize the members of the at the interface of combinatorics, algebraic geom­ selection committee were: Bela Bollobas, Lawrence etry, and representation theory. Craig Evans (chair), Grigorii A. Margulis, George C. These include: M. Haiman, "Vanishing theorems Papanicolaou, and Andrew ]. Wiles. and character formulas for the Hilbert scheme of The 2004 Moore Prize was awarded to MARK points in the plane", Invent. Math. 149, no. 2 (2002), HA!MAN. The text that follows presents the selection 3 71-407; I. Gordon, "On the quotient ring by diag­ committee's citation, a brief biographical sketch, onal invariants", Invent. Math. 153 (2003), 503-518; and the awardee's response upon receiving the and]. Haglund, "A proof of the q, t-Schroder con­ prize. jecture", Internat. Math. Res. Notices (2004).

APRIL 2004 NOTICES OF THE AMS 431 Biographical Sketch ,-----~--~- ~- ~ -~ -~-- - - ~ - ~- -~ ~ I Bndgrng Support tor Ph)"' JI/ComputJtwnal Srrentrst' Entenn~ Bwlogv ~ Mark Haiman received the Ph.D. in 1984 from the ' 2005 Career Awards at the Scientific Interface ! Massachusetts Institute of Technology, under the direction of Gian-Carlo Rota. He continued at MIT Deadline: May 3, 2004 as an applied mathematics instructor and then $500,000 award over five years for postdoctoral fellows assistant professor until1991, when he moved to

• These portable awards support up to two years of advanced post~ the University of California, San Diego, becoming doctoral training and the first three years of a faculty appointment full professor in 1997. • Candidates must hold a Ph.D. in mathematics, physics, biophysics, Haiman is presently professor of mathematics chemistry (physical, theoretical, or computational), computer at the University of California, Berkeley, where he science, statistics, or engineering and must not have accepted a faculty appointment at the rime of application moved in 2001 following a semester there as a

• Candidates should propose innovative approaches to answer visiting Miller professor in fall 2000. He serves on important biological questions the editorial board of Algebra Universalis and on • BWF encourages proposals that include experimental validation the scientific advisory board of the Centre de of theoretical models Recherches Mathematiques in Montreal, Canada. • Degree-granting institutions in the U.S. and Canada may Haiman's research interests encompass a mix of nominate up to two candidates combinatorics, algebraic geometry, and represen­ • Complete program information, eligibility guidelines, and appli­ cation forms are available on BWF's website at www.bwfund.org tation theory, with an additional occasional inter­ est in lattice theory. Response It is an honor and a special pleasure to be chosen as the first recipient of the E. H. Moore Research Article Prize. The publication activity of the AMS is, I think, the most important of all its contribu­ tions to mathematical life. Since the journal of the AMSwas started in 1988, I have always thought first of submitting there on those occasions when I be­ lieved a paper I had just completed to be one of my best. It's a bit of good fortune that by doing so I became eligible for a prize which did not yet exist. The cited work bears the name of a sole author, but it was not done alone. The conjectures that started it all came out of a long and rewarding collaboration with Adriano Garsia, which we began in 1991 and have never finished. Adriano taught me a lot of mathematics and even more about how to do mathematics. In 1992 Adriano introduced me to Claudio Procesi, and we told him about our discoveries. The insight that the geometry of the Hilbert scheme should contain the solution to our problems was his. In those days my knowledge of algebraic geometry was rudimentary, and I had never heard of Hilbert schemes. I have Claudio's impetus to thank for whatever more I might know about those subjects today.

432 NOTICES OF THE AMS VOLUME 51, NUMBER 4 2004 Conant Prize

The 2004 Levi L. Conant Prize was awarded at the 24-dim ensionallattice known as the 11 Oth Annual Meeting of the AMS in Phoenix in Leech lattice. Elkies keeps the reader's January 2004. attention throughout, judging well The Conant Prize is awarded annually to recog­ which points to expand upon and nize an outstanding expository paper published in which points to skip over. It is hard either the Notices of the AMS or the Bulletin of the to put the article down as it takes un­ AMS in the preceding five years. Established in expected turns and weaves together 2001, the prize honors the memory of Levi L. different areas of pure mathematics. Conant (1857- 1916), who was a mathematician at In Part II, Elkies avoids the temp­ Worcester Polytechnic University. The prize carries tation to expand this development a cash award of $1,000. further, in a way that might tire the The Conant Prize is awarded by the AMS Coun­ reader, and instead he develops an cil acting on the recommendation of a selection ostensibly different topic, stress­ committee. For the 2004 prize the members of ing a detailed analogy with the ma­ the selection committee were: Anthony W. Knapp, terial in Part I. Here the topic is one Brian]. Parshall (chair), and Carl Pomerance. in applied mathematics, specifically Previous recipients of the Conant Prize are: Carl that of linear error-correcting codes. Pomerance (2001), Elliott Lieb and Jakob Yngvason He introduces "Hamming space" as Noam D. Elkies (2002), and Nicholas Katz and Peter Sarnak (2003). the space of ordered n-tuples from The 2004 Conant Prize was awarded to NoAM D. a finite alphabet, especially from the elements of ELKIES. The text that follows presents the commit­ a finite field. Error-correcting codes become the ana­ tee's citation, a brief biographical sketch, and the log of sphere packings. Linear error-correcting awardee's response upon receiving the prize. codes become the analog of sphere packings with centers at the points of a lattice. "Weight enumer­ Citation ators" play the role of theta functions, and he pur­ The Levi L. Conant Prize in 2004 is granted to sues the topic through the same kinds of twists and Noam D. Elkies for his enlightening two-part turns as in Part I. Eventually he arrives again at the constructs the lat­ article "Lattices, Linear Codes, and Invariants", Leech lattice, and this time he e proper­ Notices of the AMS 4 7, nos. 10- 11 (2000): Part I, tice and examines some of its remarkabl 1238-45; Part II, 1382- 91. ties. The article leaves the reader with a good feeling Part I, which is of prize-winning quality by itself, about the unity of mathematics, and its underlying begins with the problem of finding the densest pack­ beauty. It is a masterful exposition. ing of 24-dimensional marbles whose centers are placed at the points of a lattice. It carries the reader effortlessly along a journey through the space of Biographical Sketch lattices, through the subject of theta functions and Noam D. Elkies is a number theorist whose work modular forms, through classical number-theoretic mostly concerns Diophantine geometry, computa­ identities, through sporadic finite simple groups, and tional number theory, and connections with other finally to some hints of an exceptional fields such as sphere packing and error-correcting

APRIL2004 NOTICES OF THE AMS 433 • Read . codes. He also publishes occasionally in enumera­ • tive combinatorics and combinatorial games. He twice represented the United States at the Inter­ Something .• • national Mathematical Olympiad, winning gold Different . medals both times, and was a Putnam Fellow in each • of the three years he took the Putnam examination. He has been at Harvard since coming there as a graduate student in 1985; after earning his Ph.D. Meet the mathematician there under Barry Mazur and Benedict Gross, he • at our website. was a junior fellow, then associate professor, and was granted tenure in 1993 at age twenty-six, the youngest in Harvard's history. His work has also : Radon Transforms and the Rigidity of been recognized by awards such as a Packard Fel­ : the Grassmannians lowship and the Prix Peccot of the College de France. : jacques Gasqui and Hubert Goldschmidt Elkies's main interest outside mathematics is . --~----~------: This book provides the first unified examination of the rela- music, mainly classical piano and composition. : tionship between Radon transforms on symmetric spaces Recently performed works include a full-length : of compact type and the infinitesimal versions of two opera, Yossele Solovey; Brandenburg Concerto : fundamental rigidity problems in Riemannian geometry. no. 7, commissioned and performed by the Meta­ : Annals of Mathematics Studies morphosen Chamber Orchestra; and several other : Paper $45.00 Cloth $75.00 orchestral compositions, one of which had Elkies . playing the solo piano part in Boston's Symphony : The Birth of Model Theory Hall. He still has some time for chess, where he • Uiwenheim's Theorem in the Frame of the Theory of Relatives specializes in composing and solving : Calixto Badesa problems; . ------he won the world championship for solving chess : Calixto Badesa provides both the first sustained, book- problems in 1996 and earned the Solving Grand­ : length analysis of Li:iwenheim's proof and a detailed des- master title in 2001. : cription of the theoretical framework- and, in particular, of • the algebraic tradition-that made his theorem possible. Response : Cloth $49.95 . I am honored and grateful to receive the 2004 L. L. Conant Prize : for my article "Lattices, Linear The Structure of Spherical Buildings Codes, and Invariants" in the Notices of the AMS. • Richard M. Weiss I . ------also thank Tony Knapp for soliciting the article and : This book provides a clear and authoritative introduction for working with me on the mathematical writing. : to the theory of buildings, a topic of central importance It was already gratifying to have the opportunity : to mathematicians interested in the geometric aspects of to introduce the Notices readership to a beautiful : group theory. Buildings are described throughout in the circle of mathematical ideas whose continuing : language of graph theory. vitality is exemplified • Cloth $45.00 by such recent work as the . unified treatment by Nebe, Rains, and Sloane of . various generalizations of the MacWilliams identity : Across the Board and Gleason's theorems, • The Mathematics of Chessboard Problems and the proof by Cohn and : john j. Watkins • Kumar that the Leech lattice yields the densest . ------lattice packing of spheres in dimension 24. I am : "Watkins has a friendly writing style, and the reader is delighted that my exposition was selected for the : brought along nicely from simple concepts to slightly more Conant Prize and hope that this additional : complicated ones." -Ron Graham, expo­ President, sure will entice more : Mathematical Association of America mathematicians to learn about • Cloth $24 the invariants associated with lattices and codes. • .95 • : Behind Deep Blue : Building the Computer That Defeated the World Chess Champion : Feng-hsiung Hsu

: New in paperback $19.95 • .• • • : PRINCETON 800-777-4726 . READ EXCERPTS ONLINE : : University Press MATH.PUPRESS. PRINCETON.EDU

434 NOTICES OF THE AMS VOT.TIMF. :\1_ NTTMRFR 4 2004Awardfor Distinguished Public Service

The 2004 Award for Distinguished Public Service founded and leads, such as the Rice University Cen­ was presented at the 11 Oth Annual Meeting of the ter for Excellence and Equity in Education, represent AMS in Phoenix in January 2004. a continuing tribute to his energy and perseverance. The Award for Distinguished Public Service is pre­ More than half of Richard's Ph.D. students have been sented every two years to a research mathematician women, and more than a third have been underrep­ who has made a distinguished contribution to the resented minorities. In addition, his life has been mathematics profession during the preceding five filled with many other forms of public service: years. The purpose of the award is to encourage and he was a member of the National Science Board recognize those individuals who contribute their from 1996-2002; in 1996 he was one of the first time to public service activities in support of math­ recipients of a Presidential Award for Excellence in ematics. The award carries a cash prize of $4,000. Science, Mathematics, and Engineering Mentoring; and he is a founding member of the Society for The Award for Distinguished Public Service is the Advancement of Chicanos and Native Americans made by the AMS Council, acting on the recommen­ in Science (SACNAS). dation of a selection committee. For the 2004 award the members of the selection committee were: D.]. Biographical Sketch Lewis (chair), William James Lewis, Calvin C. Moore, Richard Tapia is a mathematician and professor William Y. Velez, and Margaret H. Wright. in the Department of Computational and Applied Previous recipients of the award are: Kenneth M. Mathematics at Rice University in Houston, Texas. Hoffman (1990), Harvey B. Keynes (1992), I. M. He is internationally known for his research in Singer (1993), D.]. Lewis (1995), Kenneth C. Millett the computational and mathematical sciences and (1998), Paul J. Sally Jr. (2000), and Margaret H. is a national leader in education and outreach Wright (2002). programs. The 2004 Award for Distinguished Public Service Tapia's current Rice positions are Noah Harding was presented to RicHARDT APIA. The text that follows Professor of Computational and Applied Mathe­ presents the selection committee's citation, a brief matics; associate director of Graduate Studies, Office biographical sketch, and the recipient's response of Research and Graduate Studies; and director of upon receiving the award. the Center for Excellence and Equity in Education. Tapia was born in Los Angeles to parents who Citation separately immigrated from Mexico as teenagers in The award for Distinguished Public Service is given search of educational opportunities for themselves to Richard A. Tapia for inspiring and teaching thou­ and for future generations. Tapia was the first in sands of people (from elementary school students to his family to attend college. He received B.A., M.A., senior citizens) to study and appreciate the mathe­ and Ph.D. degrees in mathematics from the Uni­ matical sciences. His dedication to opening doors versity of California, Los Angeles. In 196 7 he joined for underrepresented minorities and women is the Department of Mathematics at UCLA and then legendary, as is his determination to reach students spent two years on the faculty at the University of who would otherwise be discouraged or overlooked. Wisconsin. In 1970 he moved to Rice University, Educational and outreach programs that he has where he was promoted to associate professor in

APRIL 2004 NOTICES OF THE AMS 435 19 72 and full professor in symposium entitled "The Richard Tapia Celebration 1976. He chaired the depart­ of Diversity in Computing". It is the first in a series ment from 1978 to 1983. He is of events designed to celebrate the technical con­ currently an adjunct faculty tributions and career interests of diverse people in member of Baylor College of computing fields. The symposium, sponsored by the Medicine and the University of Association for Computing Machinery and IEEE­ Houston. Computer Society, took place in Houston, Texas. Tapia has authored or coau­ • In May 2000 Cornell University established a thored two books and over lecture series to honor Tapia and David Blackwell, pro­ eighty mathematical research fessor at the University of California, Berkeley. The papers. He has delivered nu­ lecture series provides a forum for the research of merous invited addresses at African-American, Latino, and American Indian sci­ national and international entists working in the fields of mathematical and mathematical conferences and statistical sciences. serves on several national ad­ • In September Tapia received a 2000 Peace visory boards. Award for Education from the Spiritual Assembly Due to Tapia's efforts, Rice of the Baha'is of Houston. With unity of humanity Richard Tapia has received national recogni- as a guiding principle, the Baha'is of Houston pre­ tion for its educational outreach sent three awards-for education, for humanitar­ programs, and the Rice Computational and Applied ianism, and for peace-each year to individuals or Mathematics Department has become a national leader in producing women and underrepresented organizations for their work in serving the com­ minority Ph.D. recipients in the mathematical sci­ munity and breaking down barriers of culture, race, ences. Thirty-five mathematics students have re­ class, and creed. The awards are presented in as­ ceived, or are currently working on, the Ph.D. degree sociation with the International Day of Peace, a under his direction or co direction. Of these thirty-five day designated by the United Nations "to com­ students, fifteen have been women and eight have memorating and strengthening the ideas of peace been underrepresented minorities. both within and among all nations and peoples." As associate director of Graduate Studies at Rice • The Society for the Advancement of Chicanos University, Tapia supervises a group of graduate and Native Americans in Science (SACNAS) hon­ students from all areas. He meets with the group ored Tapia with the 2000 SACNAS Distinguished regularly to monitor their progress, and many of Scientist Award at their annual national meeting these students are involved in community and in Atlanta, Georgia, on October 14, 2000. Tapia educational outreach. was selected for his ongoing commitment to edu­ Under Tapia's direction Rice's Alliances for Grad­ cational opportunities for women and minority uate Education and the Professoriate (AGEP) Program, students and in honor of a lifetime of achievement funded by the National Science Foundation, provides in his field and for dedication to the future of opportunities for undergraduate and graduate stu­ young scientists. dents in science, mathematics, and engineering to • In 1999 Tapia was awarded the Giants in Sci­ participate in university activities and work for the ence Award by the Quality Education for Minorities summer under the guidance of researchers at Rice. (QEM) Network. Over the years Tapia has impacted hundreds of teachers through two summer programs: the Math­ ·Tapia received the 1997 Lifetime Mentor Award ematiCal and Computational Sciences Awareness from the American Association for the Advance­ workshop and GirlTECH. ment of Science. ·In 1997 Tapia was inducted into the Hispanic Among his many honors: Engineer National Achievement Awards Conference • The National Atomic Museum Foundation of Hall of Fame. Hispanics in Science and Engineering named Tapia Exhibit Honoree in Albuquerque, New Mexico, in • President Clinton appointed Tapia to the October 2003. National Science Board (NSB), the governing body • In January 2002 Tapia was inducted into the of the National Science Foundation in 1996. Tapia Texas Science Hall of Fame. The Texas Science Hall also received the 1996 Presidential Award for of Fame is a tribute to the "giants" who shape the Excellence in Science, Mathematics, and Engineering world through their innovative use of science. Mentoring. Later that year he was named the ·In October 2001 Tapia was honored with the Hispanic Engineer of the Year by Hispanic Engineer Reginald H. Jones Distinguished Service Award by magazine, the first academic to receive this honor. NACME, Inc., in Baltimore, Maryland. • Tapia was awarded the inaugural A. Nico Haber­ • Tapia's work at improving the representation mann Award by the Computer Research Association of underrepresented groups was celebrated with a in 1994 for outstanding contributions in aiding

436 NOTICES OF THE AMS VOLUME 51, NUMBER 4 members of underrepresented groups within the mathematics in a broader community, and other computing research community. public service activities are both valuable and • In the same year Tapia was selected Professor necessary for the scientific health of our nation. of the Year by the Association of Hispanic School My own development benefitted enormously Administrators of the Houston Independent School from the guidance and support of others. My mother District. and my father came separately from Mexico to • In 1992 Tapia was elected to the National the United States as young teenagers in search of Academy of Engineering, the first native-born educational opportunities. Times were difficult Hispanic to receive this honor. when they arrived. My parents were not able to • Students at Rice University voted Tapia the achieve their own educational goals, but their 1991 winner of the George R. Brown Award for dreams were realized for their five children, each superior teaching. of whom graduated from college. • Tapia was given the College Level Educator of My siblings and I were born and raised in Los the Year Award by Hispanic Engineer magazine Angeles. I am a product of public education, from and named one of the twenty most influential my primary education in the Los Angeles public leaders in minority math education by the National schools through my doctoral degree from UCLA. Research Council in 1990. I strongly believe that quality public education is • Tapia was asked to serve as chair of the National essential to the educational health and scientific Research Council's Board on Higher Education and competitiveness of our nation. Workforce, as cochair of all educational outreach As a graduate student at UCLA, I was greatly and training activities for both the University of influenced by my professors. In particular, David Illinois Supercomputer Center (NCSA) and the Sanchez gave me direction at a time when I greatly San Diego Supercomputer Center, and as cochair of needed guidance, while Magnus Hestenes shaped the Research Board for Building Engineering and how I think about mathematics. Science Talent (BEST). My first faculty position was at the Mathematics Research Center at the University of Wisconsin. My Response experiences at MRC, where I learned so much in so It is a great honor to be recognized by the American many ways from so many people, were crucial to Mathematical Society. No recognition can be more my professional development. I particularly thank cherished than recognition conferred by one's peers Michael Golomb, Barkley Rosser, I. ]. Schoenberg, and colleagues. I thank the selection committee and Hans Weinberger for mentoring me and for for choosing me for this prestigious award. Even showing me that excellence and graciousness need more, I thank the AMS for establishing this award, not be mutually exclusive. which formally recognizes the importance of out­ Rice University has been my home for more than reach and public service. three decades. The students I have taught and Throughout my formative years my parents in­ known, and from whom I have learned, have played stilled in me the value of education, community, and an essential role in shaping my vision of what is outreach to others. As a result, in my professional important. The Rice administration, especially Ken life I have valued not only academic scholarship Kennedy as director of the Center on Parallel but also teaching and mentoring, public service, and Computation and current Rice president Malcolm outreach to the general community. I never thought Gillis, have strongly supported that vision and that these activities detracted from each other-I allowed me to pursue it. grew up thinking that they went hand in hand, Recently, my six years on the National Science each influencing and supporting the others. For Board further expanded my horizons, allowing me example, credibility in scholarly research facili­ to discern critical national needs in science and tates credibility in public service, while outreach mathematics, including representation by all mem­ activities broaden one's perspective, revealing that bers of our society. I thank the National Science different people learn and understand mathemat­ Foundation for this extraordinary opportunity to ics in different ways. learn and to serve. This award is especially satisfying, because Finally, I thank my family. My wife, Jean, has formal recognition by prestigious organizations been a wise advisor and an enthusiastic supporter validates the importance of public service and out­ of my activities. And I have learned so much from reach activities. In turn, this validation promotes my children: my daughter, Becky; my son, Richard; public service within the mathematics community. and my late daughter, Circee, to whose memory I I want young mathematicians to see that there dedicate this award. are many dimensions to mathematical scholarship. In addition to scholarly research, the activities of teaching and mentoring, expository writing, increasing awareness and understanding of

APRIL 2004 NOTICES OF THE AMS 437 2003 Morgan Prize

The 2003 AMS-MAA-SIAM and the awardee's response upon receiving the Frank and Brennie Morgan prize. The same information is provided for the Prize for Outstanding Research honorable mention. in Math ematics b y an Undergraduate Student was Melanie Wood awarded at the Joint Citation Mathematics Meetings in The winner of the 2003 Morgan Prize for Out­ Phoenix in January 2004. standing Research by an Undergraduate is Melanie Morgan Prize is The Wood. The award is based on research on two dif­ awarded annually for out­ ferent topics: Belyi-extending maps and P-orderings. standing research in mathe­ with finite coverings matics by an undergraduate The first topic is concerned student (or students having of the projective line that are ramified only at three submitted joint work). Stu­ points of the projective line. The absolute Galois dents in Canada, Mexico, or group of the field of rational numbers acts on (that Grothendieck Melanie Wood the United States or its pos­ these coverings and on diagrams sessions are eligible for con­ named dessins d'enfants) associated with the cov­ sideration for the prize. Es­ erings. Melanie Wood's research gives a way to tablished in 1995, the prize was endowed by Mrs. generate genuinely new Galois invariants of des sins Frank Morgan and carries the name of her late hus­ from old ones. Her work yields important insights band. The prize is given jointly by the AMS, the into the actions of the Galois group on funda­ Mathematical Association of America (MAA), and mental groups. This research has attracted the the Society for Industrial and Applied Mathemat­ attention and admiration of the specialists work­ ics (SIAM) and carries a cash award of $1,000. ing in this field. The paper has been submitted for Recipients of the Morgan Prize are chosen by a publication. joint AMS-MAA-SIAM selection committee. For the In a separate project, Melanie Wood studies 2003 prize the members of the selection committee P-orderings in Dedekind rings. These P-orderings were: Kelly]. Black, Fan Chung Graham, Thomas C. were introduced by Bhargava in 199 5 to generalize Hales (chair), Svetlana R. Katok, Kris Stewart, and the usual factorial function. It is well known that a Philippe M. Tondeur. polynomial with rational coefficients takes integer Previous recipients of the Morgan Prize are: Kan­ values at the integers if and only if it is an integer nan Soundararajan (1995), Manjul Bhargava (1996), linear combination of binomial coefficient polyno­ Jade Vinson (1997), Daniel Biss (1998), Sean mials xCk . One of her results in this area implies McLaughlin (1999), jacob Lurie (2000), Ciprian that, in imaginary quadratic fields, the integer-valued Manolescu (2001), and Joshua Greene (2002). polynomials cannot possess a basis of this same The 2003 Morgan Prize was awarded to MELANIE general form. Melanie began this work during the Wooo. Receiving an honorable mention was KAREN 2000 Duluth Summer Research Program (directed YEATS. The text that follows presents the selection by Joseph Gallian), and her paper on P-orderings has committee's citation, a brief biographical sketch, recently appeared in the journal ofN umber Theory.

438 N OTICES OF THE AMS VOLUME 51, NUMBER 4 Richard Hairr (with help from Makoto Matsumoto) has proved a multiplicative version for Dirichlet mentored her work at Duke. series of a classical estimate of Schur on the size of Melanie Wood's research has been described in the coefficients of a product of two power series. glowing terms by her mentors and by other experts In her second paper Yeats determines bounds in her field. The work is deep and original. The com­ on the size of values of a character, expressed as a mittee commends her for the mature mathemati­ function of the degree of the character, for excep­ cal perspective in her writings. The AMS, the MAA, tional compact Lie groups. This research completes and SIAM are pleased to award the 2003 Frank and the work of other researchers, who had previously Brennie Morgan Prize to Melanie Wood. obtained results for classical compact Lie groups. Biographical Sketch In a third paper she makes a model-theoretic Melanie Wood graduated from Duke University in investigation of exotic identities of the positive May 2003 with highest distinction in mathematics. integers. An exotic identity is one involving addition, Her math competition honors include top place multiplication, and exponentiation that is not a finishes in the USA Mathematical Olympiad and the consequence of eleven basic arithmetic identities, Asian Pacific Mathematical Olympiad, and the articulated by Dedekind in 1888. designation of Putnam Fellow. She won both a The committee was impressed by the quality of the Gates Cambridge Scholarship and a Fulbright to papers, the enthusiastic letters from her mentors, study at the University of Cambridge, where she is and the speed and independence of her research. currently doing a one-year math program. This fall The committee is proud to honor Karen Yeats with she will enter the math Ph.D. program at Princeton this award. on a National Science Foundation Graduate Fel­ Biographical Sketch lowship. Her current research interests are in Karen Yeats is a native of Halifax, NS, Canada. She algebraic number theory and arithmetic algebraic began enjoying mathematics through regional, geometry. Melanie also enjoys acting, especially national (Canadian), and foreign contests. She classical acting and voice work; directing; danc­ entered the University of Waterloo in September ing; and philosophy. 1998 and graduated with an honors BMath in Pure Response Math and a Governor General's Silver Medal in I am extremely honored to be awarded this prize. 2003. During that time she had the opportunity to My experiences doing math research have been spend three summers as an NSERC (Natural Sci­ tremendously rewarding and the critical factor in ences and Engineering Research Council of Canada) my decision to continue on to graduate work in undergraduate research assistant and benefited mathematics. That I had these experiences at all is greatly from the strong faculty and program in due to two institutions that enable and encourage pure mathematics at Waterloo. She is now pursu­ undergraduate math research: Duke University and ing a Ph.D. in mathematics at Boston University. the REU [Research Experiences for Undergraduates] Karen is an accomplished recorder player and also at the University of Minnesota, Duluth. At Duke, I enjoys playing clarinet and singing in choirs, as well wish to thank Richard Hairr, who supervised my as the occasional foray into making teddy animals research on the absolute Galois group, and Robert and working on free software. Bryant, who was available for many helpful con­ Response versations. I wish to thank Makoto Matsumoto for I am truly honored to have been named honorable quick and helpful responses to technical questions. mention for this year's Morgan Prize. Great thanks I also wish to thank Joe Gallian, director of the to the creators and organizers to whom the prize Duluth REU, for his support of my research, and all owes its existence. I also owe great thanks to NSERC, those affiliated with the Duluth REU who gave me Kathryn Hare, Frank Zorzitto, and especially Stan feedback on my P-orderings paper. Burris for my summer research terms, which have made all this possible. At the University of Water­ Honorable Mention: Karen Yeats loo I also want to thank everyone in Math and Pure Citation Math for making it clear to me that I was in the right The Morgan Prize Committee is pleased to award place from the very beginning, and in Halifax to honorable mention for the 2003 Morgan Prize for everyone who encouraged me on the contests. Undergraduate Research to Karen Yeats for a series of outstanding contributions on topics ranging from asymptotics and number theory to mathe­ matical logic. A few examples indicate the broad versatility of her research. One of Karen Yeats's research projects is moti­ vated by a precise analogy between results in addi­ tive number theory and results in multiplicative number theory. Based on this analogy, Karen Yeats

APRIL 2004 NOTICES OF THE AMS 439 Mathematics People

of Mathematicians, and The Fields Institute's Distinguished Dawson Awarded CRM-Fields Lecture Series in the Statistical Sciences. His numerous Prize editorial contributions include serving as co-editor-in-chief of the Canadian journal ofMathematics. He has served his DoNALD DAwsoN, professor emeritus at Carleton University, profession through numerous committees of the CMS and has been awarded the 2004 CRM-Fields Prize. The prize, of the National Science and Engineering Research Council. awarded annually by the Cen­ He is currently president-elect of the Bernoulli Society for tre de Recherches Mathema­ Mathematical Statistics and Probability. tiques in Montreal and The Previous recipients of the CRM-Fields Prize are Fields Institute for Research H. S.M. (Donald) Coxeter, George A. Elliot, James G. Arthur, in Mathematical Sciences in Robert V. Moody, Stephen A. Cook, Israel Michael Sigal, Toronto, recognizes excep­ William T. Tutte, John B. Friedlander, John McKay, and tional contributions by a math­ Edwin Perkins. ematician working in Canada. The prize carries a cash award -From a CRM announcement of 5,000 Canadian dollars (ap­ proximately US$3,850), and the recipient is expected to present a lecture at the CRM AWM Essay Contest Winners and at The Fields Institute. Dawson was recognized for Announced exceptional achievement in the The Association for Women in Mathematics (AWM) has an­ Donald Dawson field of probability. He has nounced the winners of its 2003 essay contest, "Biographies made seminal contributions to of Contemporary Women in Mathematics". The Grand Prize the study of spatially distributed stochastic processes and went to EsTHER FELDBLUM, Maimonides School, Sharon, infinite-dimensional branching systems, including the Massachusetts, for her essay "Dr. Harpreet Chowdhary: Dawson-Watanabe superprocess. The Mathematician As Executive". This essay won first place Dawson received his B.Sc. from McGill in 1958 and his in the grade 9-12 category, and as grand prizewinner it doctorate from the Massachusetts Institute of Technology will be published in the AWM Newsletter. The first-place in 1963. He taught at both McGill University and Carleton winner in the college category was JESSICA JoHN of Willamette University, where he is now professor emeritus. His lead­ University, Salem, Oregon, for "Elizabeth Stanhope: Over­ ership within the Canadian mathematical community coming Silent Barriers". In the middle school category includes a term as director of The Fields Institute from 1996 the first-place winner was SERGEI SHUBIN of Joaquin Miller to 2000. Middle School, San Jose, California, for "I Seek an Answer Dawson is a fellow of the Royal Society of Canada, the to the Question 'Why?'-Dr. Helen Moore". A complete list International Statistical Institute, and the Institute of Math­ of the winners, as well as copies of their essays, may be ematical Statistics. Other honors include the 1991 Gold found on the AWM website at http: I /www. awm-math. Medal Lecture of the Statistical Society of Canada, the 1994 org/biographies/contest/2003.html. Jeffery-Williams lecture of the Canadian Mathematical Society (CMS), an invited lecture at the 1994 International Congress -From an A WM announcement

440 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Mathematics Opportunities

Director, The Fields Institute, 222 College Street, Toronto, New NSF Funcling Ontario M5T 3]1, Canada.

Opportunities -From a Fields Institute announcement Two new funding opportunities, both with a deadline date of April 7, 2004, have been announced in the National Sci­ ence Foundation (NSF) program solicitation "Mathematical AP Calculus Readers Sought Sciences: Innovations at the Interface with the Sciences and Engineering" (NSF 04-538). This solicitation is available on The Educational Testing Service and the College Board in- . the Web at http: I lwww. nsf. govlpubsyslodslgetpub. vite interested college faculty to apply to be readers for cfm?nsf04538. the Advanced Placement Calculus Exam. The AP Calculus The two new activities are New Mathematical and exams (AB and BC) were taken by approximately 200,000 Statistical Tools for Understanding Complex Systems in the high school students last year. The six free-response prob­ Environment (MSPA-CSE) and Interactions between the lems on the exam are graded during seven days in June Mathematical Sciences and Computer Science (MSPA-MCS). by more than 650 high school and college mathematics The solicitation NSF 04-538 also contains information teachers at Colorado State University in Ft. Collins, about other activities related to the NSF's Mathematical Colorado. This is an excellent opportunity for teachers, Sciences Priority Area (MSPA). These include activities especially those just starting their professional careers, to enhance their knowledge of the AP Calculus Program and covered by existing target dates or deadlines previously an­ of teaching and to meet with other faculty from around nounced for fiscal year 2004, activities covered by separate the country. To learn more about this opportunity or to solicitations, and competitions whose announcement is apply for a position as a reader, see the website http: I I expected in the next few months. See the full text of NSF 04- apcent ra l . co ll egeboard. com and click on the link to 538 and the separate solicitations for detailed information. "Faculty Involvement" under the dropdown menu for "Colleges & Universities", or send email to apreader@ -From an NSF announcement ets. org. Questions about the reading may be sent to Caren Diefenderfer, chief reader for the AP Calculus Call for Nominations Program, at cdi efenderfer@ho ll ins. edu. for CRM-Fields Prize -Caren L. Diefenderfer, Hollins University The Centre de Recherches Mathematiques (CRM) and The Fields Institute for Research in Mathematical Sciences News from the Clay invite nominations for the joint CRM-Fields Prize, which recognizes exceptional achievement in the mathematical Mathematics Institute sciences. The candidate's research should have been The Clay Mathematics Institute is organizing a summer conducted primarily in Canada or in affiliation with a school on Floer homology, gauge theory, and low­ Canadian university. The prize carries a cash award of dimensional topology June 5-26, 2004. The school will 5,000 Canadian dollars (approximately US$3,850). The be held at the Alfred Renyi Institute of Mathematics in recipient is expected to present a lecture at the CRM and Budapest, Hungary, and will be aimed at graduate stu­ at The Fields Institute. dents and mathematicians within five years of the Ph.D. The deadline for nominations is October 1, 2004. For The school will consist of two weeks of foundational more information, see http: I lwww . fields. uto ron to. courses and one week of minicourses focusing on more calproposalslcrm-fields_prize.html,orcontactthe advanced topics and recent developments. These courses

APRIL 2004 NOTICES OF THE AMS 441 Mathematics Opportunities will concentrate on recent activity at the crossroads of mathematical sciences and to better prepare undergrad­ mathematical disciplines around low-dimensional topol­ uate biology or mathematics students to pursue graduate ogy-the theory of holomorphic curves, gauge theory, study and careers in fields that integrate the mathemati­ knot theory, smooth four-manifold topology, and contact cal and biological sciences. geometry. The aim of this summer school is to provide a The core of the activity is long-term research experiences comprehensive introduction to these exciting areas through for interdisciplinarily balanced cohorts of at least four week-long courses in Heegaard Floer theory of three- and undergraduates. Projects should focus on research at the four-manifolds, Seiberg-Witten theory, contact topology, intersection of the mathematical and biological sciences. and knot theory. The third week of advanced courses will Projects should provide students exposure to contempo­ focus on the frontiers of research in these areas. Lectur­ rary mathematics and biology, addressed with modern ers will include John Etnyre, Ronald Fintushel, Cameron research tools and methods. That is, projects must be Gordon, Peter Ozsvath, Andras Stipsicz, and Zoltan Szabo. genuine research experiences rather than rehearsals of While there are no formal prerequisites, preference research methods. Projects must involve students from will be given to applicants with some prior knowledge of both areas in collaborative research experiences and topology or symplectic geometry. Funding is available include joint mentorship by faculty in both fields. In for graduate students and postdoctoral fellows (within addition, it is expected that projects will strengthen five years of the Ph.D.). Standard support amounts will the research and education capacity, infrastructure, and include funds for local expenses and accommodations culture of the participating institutions. To this end, plus economy travel. projects should create models for education in the The application deadline is March 31, 2004. For more mathematical and biological sciences and influence the information and an application form, please visit direction of academic programs for a broad range of http://www.claymath.org/summerschool or email: students. [email protected]. UBM is a joint effort of the Education and Human Resources, Biological Sciences, and Mathematical and -CMI announcement Physical Sciences directorates at the NSF. The program announcement is available on the web page http://www.nsf.gov/pubsys/ods/getpub. cfm?nsf04546. The next proposal deadline is April 26, News from the International 2004. Centre for Mathematical Sdences -NSF announcement The International Centre for Mathematical Sciences, Edin­ burgh, United Kingdom, announces a workshop on the Mathematical Sodal and moonshine conjectures and vertex algebras, to be held July 5-15, 2004. Part 1 of the workshop will consist of Behavioral Sdences expository sessions on such areas as Borcherds's proof The National Science Foundation (NSF) has established a of the Conway-Norton conjecture, construction of the program called Mathematical Social and Behavioral Sciences Monster, vertex (operator) algebras, modular moonshine, (MSBS). Research proposals are invited for projects that BKM algebras and automorphic forms, FLM's construction, advance the mathematical or statistical foundations of and proof of the McKay-Thompson conjecture. Part 2 will research in the social, behavioral, or economic sciences. consist of invited talks on current research. The scientific The resulting research is expected both to further under­ committee consists of A. Baker, Glasgow; A. Ivanov, Imper­ standing of social and/or behavioral science phenomena ial College; J. Lepowsky, Rutgers University; J. McKay, and to address a topic of interest to the mathematical Concordia; V. Nikulin, Liverpool; and M. Tuite, Galway. sciences. Proposals for workshops or symposia that fos­ For further information and application forms, see the ter the interaction of social, behavioral, and/or economic website http://www.ma.hw.ac.uk/icms/meetings/ scientists with mathematicians and/or statisticians also 2004/moonshi ne. are welcome. The program announcement is available on the web -ICMS announcement at http://www.nsf.gov/pubsys/ods/getpub. cfm?nsf04548. The proposal deadline is April 30, 2004. Training in Biological and -NSF announcement Mathematical Sdences The National Science Foundation (NSF) has begun a pro­ gram called Undergraduate Biology and Mathematics (UBM). The goal of UBM is to enhance undergraduate education and training at the intersection of the biological and

442 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Inside the AMS

establishing accounts with a few news services that sum­ A Summer at Voice of America marized the latest science news, I was ready to go! Each year the AMS sponsors a fellow to participate in the Since David was often busy working on his own stories, Mass Media Fellowship program of the American Associa­ I generally had both the luxury and the challenge of set­ tion for the Advancement of Science. This program places ting my own agenda. This started with pitching stories, science and mathematics graduate students in summer which had to be relevant to our global audience. The first internships at media outlets. In the piece below, the 2003 story I pitched that was accepted focused on the results AMS fellow, Claudia A. Clark of Northeastern University, of a recent study on wind power. After a few weeks read­ describes her experiences during her fellowship at Voice of ing the study, researching wind power, and interviewing America. For information about applying for the fellowship, one of the study's authors-which might take my experi­ see the October 2003 issue of the Notices, page 1102, or visit enced colleagues a few hours-! thought I was ready to the website http: I /ehrweb. aaas. org/massmedi a. htm. write the piece. At this point I got my first lesson in jour­ nalistic objectivity: I needed to interview someone whose It was already warm when I entered the building in conclusions differed from those of this researcher. This Dupont Circle to attend the 9:00 a.m. press conference at was one of the many valuable lessons impressed upon the Earth Policy Institute. After all, it was late June in me by more than one colleague at VOA: no one listening Washington, DC. But when I left the building over an hour to your report should be able to tell what you, the writer, later, I hardly noticed the increased heat and humidity. I'd believe. Contrasting points of view are more than a means just recorded a press conference for the first time on my to add drama. own and had asked the questions I'd come to ask. I was After interviewing a spokesperson from a trade asso­ heading back to my office at Voice of America (VOA), ciation of U.S. electric power suppliers, I began to write. where I would use this and subsequent interviews to pro­ This was the most frustrating, creative, and rewarding duce my first radio piece: a four-minute "backgrounder" (in radio terminology) on wind power. And I couldn't have part of the process for me. First of all, I had to tell a com­ been happier. pelling story in a very short period of time (although four More than six months earlier I had read about the ten-week minutes is considered to be a very long time inradio).Yes, AMS-AAAS Mass Media Science and Engineering Fellowship I was writing about science, but the audience responds in an AMS Member Newsletter. I was intrigued by the idea of to what it can relate to: the humans involved and their being a bridge between the world of science and the public, emotions, such as excitement or disappointment. Since I as well as by the opportunity to work in the media. This seemed found that people generally love talking about their work, like a natural extension of my interest in mathematics and radio is a wonderful medium for science reporting: you science, my enjoyment of writing and teaching, my love of can convey not only the interviewee's thoughts but also learning, and a deep curiosity aboutwhatmakes people tick. his or her feelings. To my delight, I was offered a fellowship in radio-my first However, it must be done effectively. I was warned not choice-at VOA in Washington, DC. to automatically use quotes like "This was a very exciting The summer began with a three-day orientation at result!" but to use the interviewee's words to summarize the headquarters of the American Association for the or emphasize the important points. It was my job to Advancement of Science (AAAS). I attended workshops on explain the details in concise, accurate, "visual" and pitching story ideas, interviewing people, and writing news understandable language, because the listener has only one pieces; and toured the nearby National Public Radio offices chance to hear it: in David's words, I had to "write for the and studios. Besides meeting the other fellows, I met ear." After numerous rewrites and final editorial approval, VOA's science editor and my mentor for the summer, the piece was ready to be "voiced" by David: read and David McAlary, as well as Sam Rankin of the AMS Wash­ recorded. I produced the piece, putting the recorded text, ington, DC, office. A few days later the fellowship began. the "sound bites", and any sound effects together to make David trained me to use all of the hardware and soft­ the finished product, "Winds of Change". ware I would need to record interviews or press conferences The second piece that I wrote, at David's suggestion, and to produce news stories for broadcast. After attend­ was the most demanding because of the highly technical ing a few press conferences together with David and nature of the material. This "backgrounder" focused on a

APRIL 2004 NOTICES OF THE AMS 443 Inside the AMS

recently published paper about a technique of using very Group IV produced 239 new doctorates, of short, low-energy laser pulses and the fluorescent proper­ which 98 (41%) are females, compared to all ties of certain chemicals in the body to perform noninvasive other groups combined, where 206 (26%) are biopsies. Itwas a challenge to get the researchers to explain females. In group IV 109 (46%) of the new the technique in simple terms so that I could understand doctoral recipients are U.S. citizens (while in it and explain it to the audience using the researchers' own the other groups 49% are U.S. citizens). words. After much editing and a few suggestions from one Table 3 on page 220 should have reported total full-time of the paper's authors, I produced the piece "Bloodless graduate students for 2002 as 9,972 (the figure 997 Biopsies". I was rewarded with compliments from an appeared incorrectly in the printed table). editor, who said it was one of the best medical pieces she'd seen in a long time, and from one of the researchers, who said I'd done a great job making this very complicated Correction: vonNeumann material understandable. My third and final story was on a paper that used a math­ Awards ematical model to determine the most effective way to halt The January 2004 issue of the Notices, page 52, carried the spread of SARS. Titled "Soap Fights SARS", this piece a report about the AMS participation in the Budapest was my most serious; I was talking about something that celebration of the lOOth anniversary of the birth of John killed people. However, writing and producing this story von Neumann. The report gave an incomplete list of the took me the least amount of time: the subject matter was recipients of the first John von Neumann Awards. In more familiar to me (mathematics), and I was getting more addition to Marina von Neumann Whitman and Charles experienced with the process. Simonyi, George Dyson received one of the awards for his In addition to completing these three pieces, I worked historical research on von Neumann and the computer. on other projects, including recording a press conference Dyson is the author of Darwin among the Machines: The on the improved condition of the ozone layer and dis­ Evolution of Global Intelligence. The Notices thanks John cussing solar sailing with NASA scientists. Ball, who attended the celebration as president of the This summer was an inspirational experience. I got to International Mathematical Union, for pointing out this know some of the other newsroom correspondents, who omission. were fascinating and helpful. I was constantly pushed to try new things, take risks, and learn from the mistakes I -Allyn jackson made. I experienced the responsibility and the opportu­ nity that journalists have to tell the stories and to get them right and came away deciding to pursue science journalism as a career. Deaths of AMS Members The hunger for science news is there: it came from other JosE F. EscoBAR, of Cornell University, died on January 3, "language services" at VOA, who wanted more science news 2004. Born on December 20, 1954, he was a member of to translate and broadcast to their own countries, and the the Society for 20 years. South African journalists I interviewed at AAAS who were ABRAHAM P. HILLMAN, professor emeritus, University of training to be science journalists. The need for science news New Mexico, died on January 19, 2004. Born on December 18, is there. As California Congressman Bob Filner, who has a 1918,hewas amemberofthe Society for 63 years. Ph.D. in the history of science, said when I interviewed him at an AMS Congressional Briefing in Washington: "We live in RicHARD A. LEIBLER, retired, from Reston, VA, died on a society which is dominated by science and technology and October 25, 2003. Born on March 18, 1914, he was a yet 99 percent of the population and 99.5 percent of the pol­ member of the Society for 66 years. icymakers know nothing about science and technology ....We DAN LORENZ, of Technion-Israel Institute of Technology, cannot make intelligent decisions unless we have a broader died on January 25, 2001. Born on June 29, 1931, he was sense of what's going on." I'm there. a member of the Society for 42 years. MEIR REICHAW, professor emeritus, Technion-Israel In­ -Claudia A. Clark stitute of Technology, died on February 28, 2000. Born on December 20, 1923, he was a member of the Society for 38 years. Corrections to the 2003 Annual NINA M. RoY, professor emeritus, from Rosemont, PA, died on December 25, 2003. Born on November 5, 1931, Survey of the Mathematical she was a member of the Society for 32 years. WILLIAM STRANGE, retired, U. S. Naval Academy, died on Sciences (First Report) October 13, 2003. Born on June 19, 1918, he was a mem­ An incorrect percentage was quoted in the "Highlights" on ber of the Society for 62 years. page 219 of the February 2004 issue of the Notices of the JACK WILLIAMSON, retired, University of Hawaii, died on AMS. The correct percentage (49%) appears in the last line October 15, 2003. Born on November 21, 1940, he was a of the quote below: member of the Society for 36 years.

444 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Reference and Book List

The Reference section of the Notices Upcoming Deadlines school. See "Mathematics Opportuni­ is intended to provide the reader with March 31, 2004: Nominations for ties" in this issue. frequently sought information in Third World Academy of Sciences April1, 2004: Applications for New an easily accessible manner. New prizes. See http: I /www. i ctp. Directions program at the Institute information is printed as it becomes trieste.it/~twas/twas_prizes. for Mathematics and its Applications. available and is referenced after the html. See http://www.ima.umn.edu/ new-directions/. first printing. As soon as information March 31, 2004: Nominations for the Prize for Achievement in Infor­ April 7, 2004: Two new NSF is updated or otherwise changed, it mation-Based Complexity. Contact programs: New Mathematical and will be noted in this section. Joseph F. Traub, email: traub@cs. Statistical Tools for Understanding columbia. edu. Complex Systems in the Environment, Contacting the Notices March 31, 2004: Applications for and Interactions between the Mathe­ The preferred method for contacting Clay Mathematics Institute summer matical Sciences and Computer 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. Articles include feature articles, AMS Bylaws-November 2003, p. 1283 memorial articles, communications, AMS Email Addresses-November 2003, p. 1266 opinion pieces, and book reviews. The AMS Ethical Guidelines-june/july 2002, p. 706 editor is also the person to whom to send news of unusual interest about AMS Officers 2002 and 2003 (Council, Executive Committee, Publications Committees, Board of Trustees)-May 2003, p. 594 other people's mathematics research. AMS Officers and Committee Members-October 2003, p. 1115 The managing editor is the person to whom to send items for "Mathe­ Backlog of Mathematics Research journals-September 2003, p. 961 matics People", "Mathematics Op­ Conference Board of the Mathematical Sciences-September 2003, p. 945 portunities", "For Your Information", Information for Notices Authors-june/july 2003, p. 706 "Reference and Book list", and "Math­ Mathematics Research Institutes Contact Information-August 2003, ematics Calendar". Requests for p. 821 permissions, as well as all other National Science Board-january 2004, p. 54 inquiries, go to the managing editor. New journals for 2002-june/july 2003, p. 708 The electronic-mail addresses are NRC Board on Mathematical Sciences and Their Applications-March 2004, noti ces@math. ou. edu in the case of p. 350 the editor and noti ces@ams. org in NRC Mathematical Sciences Education Board-April 2004, p. 446 the case of the managing editor. The NSF Mathematical and Physical Sciences Advisory Committee-February fax numbers are 405-325-7484 for 2004,p.242 the editor and 401-331-3842 for the Program Officers for Federal Funding Agencies-October 2003, p. 1107 (DoD, DoE); December 2003, p. 1429 (DMS Program Officers); December managing editor. Postal addresses 2003, p. 1430 (NSF Education Program Officers) may be found in the masthead.

APRIL 2004 NOTICES OF THE AMS 445 Reference and Book List

Science. See "Mathematics Opportu­ telephone 202-334-2760; fax 202-334- Hung Hsi Wu, University of Cali­ nities" in this issue. 2759; email: [email protected]. fornia, Berkeley April 16, 2004: Applications for September 30, 2004: Nominations Project NExT. See http: I larchi ves. for Information-Based Complexity MSEBStaff math.utk.edulprojnextl. Young Researcher Award. Contact Carole Lacampagne, Director April 26, 2004: NSF Undergraduate Joseph F. Traub at traub@cs. Vicki Stohl, Research Associate Biology and Mathematics program. columbia. edu. Dianna Williams, Senior Project See "Mathematics Opportunities" in October 1, 2004: Nominations for Assistant this issue. CRM-Fields Prize. See "Mathematics April 30, 2004: NSF Mathematical Opportunities" in this issue. The contact information is: Mathe­ Social and Behavioral Sciences pro­ January 1, 2005: Entries for Cryp­ matical Sciences Education Board, gram. See "Mathematics Opportuni­ tologia undergraduate paper compe­ National Academy of Sciences, 500 Fifth ties" in this issue. titions. See http: I lwww. dean. usma. Street, NW, 11 thFloor, Washington, DC May 1, 2004: Applications for edulmathlpubslcryptologial or 20001; telephone 202-334-3294; fax National Research Council Research contact Cryptologia, Department of 202-344-1294; email: mseb@nas. edu; Associateships. See http: I lwww4. Mathematical Sciences, United States World Wide Web http: I lwww7. nationalacademies.orglpgalrap. Military Academy, West Point, NY nationalacademies.orglmsebl nsf, or contact Research Associate­ 10996; email: Crypto l ogi a@usma. index. html. ship Programs, Keck Center of the edu. National Academies, 500 Fifth Street, NW, GR322A, Washington, DC 20001; Mathematical Sciences telephone 202-334-2760; fax 202-334- Education Board, National Book List 2759; email: rap@nas. edu. Research Council The Book List highlights books that May 1, 2004: Applications for AWM Thomas Banchoff, Brown Uni­ have mathematical themes and are Travel Grants. See http: I lwww. versity aimed at a broad audience potentially awm-math.orgltravelgrants.html, ]ere Confrey(vice chair), Washing­ including mathematicians, students, or contact Association for Women in ton University in St. Louis and the general public. When a book Mathematics, 4114 Computer and ]an de Lange, Freudenthal Insti­ has been reviewed in the Notices, a Space Sciences Building, University of tute, The Netherlands reference is given to the review. Gen­ Maryland, College Park, MD 20742- Louis Gomez, Northwestern Uni­ erally the list will contain only books 2461; telephone 301-405-7892; email: versity published within the last two years, [email protected]. Douglas A. Grouws, University of though exceptions may be made in June 1, 2004: Applications for the Iowa cases where current events (e.g., the fall program of the Christine Mirzayan Arthur Jaffe, Harvard University death of a prominent mathematician, Science and Technology Policy In­ Eric jolly, Education Development coverage of a certain piece of mathe­ ternship Program of the National Center matics in the news) warrant drawing Academies. See http: I lwww7. joan Leitzel (chair), University of readers' attention to older books. nationalacademies.orgl New Hampshire Suggestions for books to include on i nternshi pli ndex. html or contact jim Lewis, University of Nebraska, the list may be sent to notices­ The National Academies Christine Lincoln [email protected]. Mirzayan Science and Technology Pol­ George McShan, National School ''Added to Book List since the list's icy Internship Program, 500 Fifth Boards Association last appearance. Street, NW, Room 508, Washington, Karen Michalowicz, The Langley DC 20001; telephone: 202-334-2455; School, McLean, VA 1089 and All That: A journey into fax: 202-334-1667. · judith Mumme, WestEd, Camarillo, Mathematics, by David Acheson. June 30, 2004: Proposals .for CA Oxford University Press, July 2002. DMS/NIGMS Program in Mathematical Casilda Pardo, Valle Vista Elemen­ ISBN 0-19-851623-1. Biology. See http: I lwww. nsf . gov I tary School, Albuquerque, NM Abel's Proof: An Essay on the pubsl2002lnsf02125lnsf02125. Sue Parsons, Cerritos College Sources and Meaning ofMathematical htm. Marge Petit, National Center for Unsolvability, by Peter Pesic. MIT Press, August 1, 2004: Applications for the Improvement of Educational May 2003. ISBN 0-262-16216-4. National Research Council Research Assessment (Reviewed March 2004.) Associateships. See http: I lwww4. Donald Saari, University of Cali­ After Math, by Miriam Webster. nationalacademies.orglpgalrap. fornia, Irvine Zinka Press, June 1997. ISBN 0-9647- nsf, or contact Research Associate­ Richard Scheaffer, University of 1711-5. (Reviewed October 2003.) ship Programs, Keck Center of the Florida All the Mathematics You Missed (But National Academies, 500 Fifth Street, Francis Sullivan, Center for Com­ Need to Know for Graduate School), by NW, GR322A, Washington, DC 20001; puting Sciences Thomas A. Garrity. Cambridge

446 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Reference and Book List

University Press, December 2001. ISBN Wallace. W. W. Norton & Co., October John Wiley & Sons, January 2003. ISBN 0-521-79707-1. 2003. ISBN 0-393-00338-8. 0-4 71-08601-0. Beyond the Limit: The Dream of Four Colors Suffice: How the Map Linked: The New Science of Net­ Sofya Kovalevskaya, by Joan Spicci. Problem Was Solved, by Robin Wilson. works, by Albert-Laszlo Barabasi. Forge, August 2002. ISBN 0-765- Princeton University Press, March Perseus Publishing, May 2002. ISBN 30233-0. (Reviewed January 2004.) 2003. ISBN 0-691-11533-8. (Reviewed 0-738-20667-9. (Reviewed February The Book of My Life, by Girolamo February 2004.) 2004.) Cardano. New York Review of Books The Fractal Murders, by Mark Math through the Ages: A Gentle Classics Series/ Granta. ISBN 1-590- Cohen. Muddy Gap Press, May 2002. History for Teachers and Others, by 17016-4. 0-9718986-0-X. (Reviewed October William P. Berlinghoff and Fernando Q. Calculated Risks: How to Know 2003.) Gouvea. Oxton House, 2002. ISBN When Numbers Deceive You, by Gerd Gamma: Exploring Euler's Constant, 1-881929-21-3. Gigerenzer. Simon & Schuster, March by Julian Havil. Princeton University Mathematical Constants, by Steven R. 2003. ISBN 0-743-25423-6. Press, May 2003. ISBN 0-691-09983-9. Finch. Cambridge University Press, California Dreaming: Reforming Geometrical Landscapes: The August 2003. ISBN 0-521-81805-2. Mathematics Education, by Suzanne M. Voyages of Discovery and the Trans­ A Mathematician's Survival Guide: Wilson. Yale University Press, January formation of Mathematical Practice, Graduate School and Early Career 2003. ISBN 0-300-09432-9. (Reviewed by Amir R. Alexander. Stanford Uni­ Development, by Steven G. Krantz. November 2003.) versity Press, September 2002. ISBN AMS, August 2003. ISBN 0-821-83455- * The Changing Shape of Geometry: 0-804-73260-4. X. (Reviewed in this issue.) Celebrating a Century of Geometry Geometry: Our Cultural History, by Mathematicians under the Nazis, by and Geometry Teaching, edited by Audun Holme. Springer, April 2002. Sanford L. Segal. Princeton University Chris Pritchard. Cambridge Univer­ ISBN 3-540-41949-7. Press, July 2003. ISBN 0-691-00451-X. sity Press, January 2003. ISBN 0-521- Godel's Proof, by Ernest Nagel and Mathematics: A very short intro­ 53162-4. James R. Newman. New York Univer­ duction, by Timothy Gowers. Oxford Codebreakers: Arne Beurling and sity Press, revised edition, February University Press, October 2002. ISBN the Swedish Crypto Program during 2002. ISBN 0-8147-5816-9. (Reviewed 0-192-85361-9. World War II, by Bengt Beckman, trans­ March 2004.) Mathematics and Culture I, edited lated by Kjell-Ove Widman. AMS, The Golden Ratio: The Story of Phi, by Michele Emmer. Springer, January February 2003. ISBN 0-8218-2889-4. the World's Most Astonishing Number, 2004. ISBN 3-540-01770-4. (Reviewed September 2003.) by Mario Livio. Broadway Books, Mathematics and the Roots ofPost­ The Constants of Nature: From October 2002. ISBN 0-767-90815-5. modern Thought, by Vladimir TasiC. Alpha to Omega-The Numbers That How Economics Became a Mathe­ Oxford University Press, 2001. ISBN Encode the Deepest Secrets of the Uni­ matical Science, by E. Roy Weintraub. 0-195-13967-4. (Reviewed August verse, by John D. Barrow. Jonathan Duke University Press, June 2002. 2003.) Cape, September 2002. Pantheon ISBN 0-822-32856-9. Mathematics, Art, Technology, and Books, January 2003. ISBN 0-375- Imagining Numbers (particularly the Cinema, edited by Michele Emmer and 42221-8. square root of minus fifteen), by Barry Mirella Manaresi. Springer, 2003. ISBN Correspondance Grothendieck­ Mazur. Farrar, Straus and Giroux, Feb­ 3-540-00601-X. Serre, Pierre Colmez and Jean-Pierre ruary 2003. ISBN 0-374-17469-5. (Re­ Mathematics by Experiment: Plau­ Serre, editors. Societe Mathematique viewed November 2003.) sible Reasoning in the 21st Century, by de France, 2001. ISBN 2-85629-104-X. Infinity: The Quest to Think the Un­ David Bailey and Jonathan Borwein. (Reviewed October 2003.) thinkable, by Brian Clegg. Carroll & Graf, A K Peters, September 2003. ISBN Doing Mathematics: Convention, December 2003. ISBN 0-786-71285-6. 1-568-81136-5. Subject, Calculation, Analogy, by Mar­ Information: The New Language of Mathematics Elsewhere: An Explo­ tin H. Krieger. World Scientific, April Science, by Hans Christian vonBaeyer. ration of Ideas across Cultures, by 2003. ISBN 9-812-38200-3. Weidenfeld&Nicolson, October 2003. Marcia Ascher. Princeton University Einstein's Clocks, Poincare's Maps: ISBN 0-297-60725-1 (hardcover), 0-753- Press, September 2002. ISBN 0-691- Empires of Time, by Peter Galison. 81782-9 (softcover). 07020-2. (Reviewed May 2003.) W. W. Norton & Co., August 2003. Isaac Newton, by James Gleick. Pan­ Mathematics for the Imagination, ISBN 0-393-02001-0. theon Books, May 2003. ISBN 0-3 75- by Peter M. Higgins. Oxford University Emergence of the Theory of Lie 42233-1. (Reviewed December 2003.) Press, November 2002. ISBN 0-198- Groups. An Essay in the History of ]a nos Bolyai, Euclid, and the Nature 60460-2. Mathematics, 1869-1926, by Thomas ofSpace, by Jeremy J. Gray. MIT Press, The Mathematics of Juggling, by Hawkins. Springer-Verlag, 2000. ISBN May 2003. ISBN 0-262-5 7174-9. Burkard Polster. Springer, November 0-387-98963-3. (Reviewed June/ July Kepler's Conjecture: How Some of 2002. ISBN 0-387-95513-5. (Reviewed 2003.) the Greatest Minds in History Helped January 2004.) Everything and More: A Compact Solve One of the Oldest Math Prob­ Memoirs of a Proof Theorist: Godel History of Infinity, by David Foster lems in the World, by George G. Szpiro. and Other Logicians, by Gaisi Takeuti,

APRIL 2004 NOTICES OF THE AMS 447 Reference and Book List

translated by Mariko Yasugi and Six Degrees: The Science of a Con­ Nicholas Passell. World Scientific, Feb­ nected Age, by Duncan]. Watts. W. W. ruary 2003. ISBN 981-238-279-8. Norton & Co., February 2003. ISBN 0- The Millennium Problems: The 393-04142-5. (Reviewed February 2004.) Seven Greatest Unsolved Mathemati­ Sync: The Emerging Science ofSpon­ cal Puzzles of Our Time, by Keith J. taneous Order, by Steven Strogatz. Devlin. Basic Books, October 2002. Hyperion, February 2003. ISBN 0-786- ISBN 0-465-01729-0. (Reviewed Sep­ 86844-9. (Reviewed March 2004.) tember 2003.) Travels in Four Dimensions: The Enig­ More Mathematical Astronomy mas of Space and Time, by Robin Le Morsels, by Jean Meeus. Willmann-Bell Poidevin. Oxford University Press, Feb­ Inc., 2002. ISBN 0-943396-743. ruary 2003. ISBN 0-19-875254-7. The Music of the Primes: Searching What the Numbers Say: A Field Guide to Solve the Greatest Mystery in to Mastering Our Numerical World, by Mathematics, by Marcus Du Sautoy. Derrick Niederman and David Boyum. HarperCollins, April2003. ISBN 0-066- Broadway Books, April 2003. ISBN 0- 21070-4. 767-90998-4. Newton's Apple: Isaac Newton and the English Scientific Renaissance, by Peter Aughton. Weidenfeld & Nicolson, October 2003. ISBN 0-297-84321-4. On the Nature ofHuman Romantic Interaction, by Karl Iagnemma. Dial Press, April2003. ISBN 0-385-3 3 593-8. The One True Platonic Heaven: A Sci­ entific Fiction of the Limits of Knowl­ edge, by John L. Casti. Joseph Henry ·~----·----~ Press, May 2003. ISBN 0-309-08547-0...... ~ <( Origami3 , edited by Thomas Hull. AK ~-- ~ Peters, July 2002. ISBN 1-568-81181-0. Predicting Presidential Elections and Other Things, by Ray C. Fair. Stanford University Press, August 2002. ISBN 0- 804-74509-9. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem, by John Derbyshire. Joseph Henry Press, March 2003. ISBN0-309-08549-7. Proofs from the Book, by Martin Aigner and Giinter M. Ziegler. Springer­ Verlag, third edition, December 2003. ISBN 3-540-40460-0. Remarkable Mathematicians, by loan James. Cambridge University Press, February 2003. ISBN 0-521- 52094-0. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathe­ matics, by Karl Sabbagh. Farrar Straus & Giroux, April 2003. ISBN 0-374- 25007-3. Science in the Looking Glass, by E. Brian Davies. Oxford University Press, August 2003. ISBN 0-19-852543-5. The Search for Certainty: A Philo­ sophical Account of Foundations of Mathematics, by Marcus Giaquinto. Oxford University Press, October 2002. ISBN 0-198-75244-X.

448 NOTICES OF THE AMS VOLUME 51, NUMBER 4 AMERICAN MATHEMATICAL SOCIETY

CALL FO

The selection committees for these prizes request nominations for consideration for the 2005 awards, which will be presented at the Joint Mathematics Meetings in Atlanta, GA, in January 2005. Information about these prizes may be found in the November 2003 Notices, pp. 1289-1299 (also available at http:/ /www.ams.org/prizes-awards).

BOCHER MEMORIAL PRIZE

The Bacher Memorial Prize, founded in 1923 in honor of Maxime Bacher, is awarded every three years for a notable paper in analysis published during the preceding six years. To be eligible, the nominee should be a member of the American Mathematical Society or the paper should have been published in a recognized North American journal. LEVI L. CONANT PRIZE

The Levi L. Conant Prize, first awarded in January 2001, is presented annually for an outstanding expos­ itory paper published in either the Notices or the Bulletin of the American Mathematical Society during the preceding five years. FRANK NELSON COLE PRIZE IN NUMBER THEORY

The Frank Nelson Cole Prizes in Algebra and Number Theory are now presented at three year intervals. The award in January 2005 will be for a notable paper in number theory. The nominee should be a member of the American Mathematical Society or the paper should have been published in a recognized North American journal. RUTH LYTTLE SATTER PRIZE

The Ruth Lyttle Satter Prize is presented every two years in recognition of an outstanding contribution to mathematics research by a woman during the previous six years. ALBERT LEON WHITEMAN MEMORIAL PRIZE

The Albert Leon Whiteman Memorial Prize is awarded every four years, for notable exposition on the history of mathematics. The ideas expressed and understandings embodied in that exposition should reflect exceptional mathematical scholarship.

Nominations should be submitted to the Secretary, Robert ]. Daverman, American Mathematical Society, 3 1 2D Ayres Hall, University ofTennessee, Knoxville TN 37996-1330. Include a short description of the work that is the basis of the nomina­ tion, with complete bibliographic citations when appropriate. A brief curriculum vita of the nominee also should be includ­ ed. The nominations will be forwarded by the Secretary to the appropriate prize selection committee, which, as in the past, will make final decisions on the awarding of these prizes. ci)AMS Deadline for nominations is June 30, 2004. ~ AMERICAN MATHEMATICAL SOCIETY Books may be nominate(:! by members of the Society, by members of the . . . selection committee, ·.by . members of AMS Editorial Committees or .by publishers. For each nominated '} '* "'' "' "' g< • book, the nominator should include a short descriptior,-t . of the work, with. comple~e bibliographic citations .. when appropriate. The no~ination must also include a ~ letter that articulates the .ways in_which the book satisfiesth,e stated purpose of the prize.The ,letter , should ,not .ex.ceed two pages. · Published submitting nominations must provide a single copy of each b~ok they nominate. Puolishers may nominate at most three books.

y -'.r- ~ Nominations should be submitted to the · AMS Secretary, Robert J. Daverman, American ' Mathematical SG>ciety, 3 I1D Ayres Hall, University of Tennessee, Knoxville TN 37996-1330. 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 June I, 2004. . rrhe pri'ze is awarded each year: to an ... · · .undergraduate student··(crr ' studen~s · having submit'ted join.t work) for out- standing . r~search in mathematics. Any stu­ .dent who is an undergraduate in ,a college or university i.n ttie United States or its. posses­ sions or Canada or Mexico, is eligible to .be J ~ . ' ... considered for this prize. he recipients of the prize are to be selected by a standing joint committee not . T h·e.' prize reciei,ent's 'researci:J ,ne~d of the AMS, MAA, and SIAM. The deci­ be confined .to a single paper; 1t may 4 sions of this committee are final. The 2004 be contained in several papers. prize will be awarded for papers submitted However, the paper (or papers) to be consid­ for consideration no later than june 30, ered for the prize must be submitted whil~. 2004, by (or on behalf of) students who were ·the student is an undergraduate; they cail- . undergraduates in December 2003. not be submitted. aft~~ the student's gradua- tion. The research pap·er (or pap~rs) may be submitted for consideration by the student or a nominator. AU ~ submissions f or the · p.rize must include at least one letter of sup­ port from a person, usually a faculty mem­ ber,. familiar: with the student's research. P1:1blication of researc.h is not required. Add this Cover Sheet to all of your Academic Job Applications

How to use this form The purpose of the cover The AMS suggests that employers. Also, reference l. Using the facing page form is to aid department applicants and employers writers can submit their or a photocopy, staff in tracking and visit the Job Application letters online. A paperless (or visit the AMS web site responding to each appli­ Database for application process is for a choice of electronic versions at www.ams.org/ cation for employment. Mathematicians possible with this system, coversheet/), fill in the Mathematics departments (www. mathjobs. org), a however; employers can answers which apply to all in Bachelor's-, Master's-, new electronic resource choose to use any portion of your academic and Doctorate-granting being offered by the AMS of the service. There will applications. Make photocopies. institutions are expecting (in partnership with Duke be annual employer fees to receive the form from University) for the second beginning this year. This 2. As you mail each each applicant, along with year in 2002-03. The system was developed at application, fill in the the other application mate­ system provides a way for the Duke University remaining questions neatly on one cover rials they require. applicants to produce Department of sheet and include it printed coversheet forms, Mathematics. on top of your applica­ apply for jobs, or publicize tion materials. themselves in the "Job Please direct all questions Wanted" list. Employers and comments to: emp- can post a job listing, and i nfo@ams. org. once applications are made, search and sort among their applicants. Note-taking, rating, e-mail, data downloading and customizable EOE func­ tions are available to Academic Employment in Mathematics AMS STANDARD COVER SHEET Last Name First Name Middle Names Address through next june ______Home Phone

e-mail Address

Current Institutional Affiliation Work Phone

Highest Degree Held or Expected This form is provided Granting Institution ______Date (optional) ______courtesy of the American Ph.D. Advisor Mathematical Society. Ph.D. Thesis Title (optional) ______Indicate the mathematical subject area(s) in which you have done research using the Mathematics Subject This cover sheet Classification printed on the back of this form or on the AMS website. Use the two-digit classification which best is provided as an aid fits your interests in the Primary Interest line and additional two-digit numbers in the Secondary Interest line. to departments Primary Interest ~ in processing job applications. Secondary Interests optional ~ ~ It should be included with your application Give a brief synopsis of your current research interests (e.g. finite group actions on four-manifolds). material. Avoid special mathematical symbols and please do not write outside of the boxed area.

Please print or type. Do not send this form to the AMS. Most recent, if any, position held post Ph.D. University or Company Position Title Indicate the position for which you are applying and position posting code, if applicable

If applying for a position which requires U.S. citizenship or U.S. permanent residency, indicate your eligibility D Yes D No If unsuccessful for this position, would you like to be considered for a temporary position? D Yes D No If yes, please check the appropriate boxes. D Postdoctoral Position D 2+ Year Position D 1 Year Position List the names, affiliations, and e-mail addresses of up to four individuals who will provide letters of recom­ mendation if asked. Mark the box provided for each individual whom you have already asked to send a letter. D ______D ______D ______D ______2000 Mathematics Subject Classification

00 General 51 Geometry 01 History and biography 52 Convex and discrete geometry 03 Mathematical logic and foundations 53 Differential geometry 05 Combinatorics 54 General topology 06 Order, lattices, ordered algebraic structures 55 Algebraic topology 08 General algebraic systems 57 Manifolds and cell complexes 11 Number theory 58 Global analysis, analysis on manifolds 12 Field theory and polynomials 60 Probability theory and stochastic processes 13 Commutative rings and algebras 62 Statistics 14 Algebraic geometry 65 Numerical analysis 15 Unear and multilinear algebra, matrix theory 68 Computer science 16 Associative rings and algebras 70 Mechanics of particles and systems 17 Nonassociative rings and algebras 74 Mechanics of deformable solids 18 Category theory, homological algebra 76 Fluid mechanics 19 K-theory 78 Optics, electromagnetic theory 20 Group theory and generalizations 80 Classical thermodynamics, heat transfer 22 Topological groups, Lie groups 81 Quantum theory 26 Real functions 82 Statistical mechanics, structure of matter 28 Measure and integration 83 Relativity and gravitational theory 30 Functions of a complex variable 85 Astronomy and astrophysics 31 Potential theory 86 Geophysics 32 Several complex variables and analytic spaces 90 Operations research, mathematical programming 33 Special functions 91 Game theory, economics, social and behavioral 34 Ordinary differential equations sciences 35 Partial differential equations 92 Biology and other natural sciences 37 Dynamical systems and ergodic theory 93 Systems theory, control 39 Difference and functional equations 94 Information and communication, circuits 40 Sequences, series, summability 97 Mathematics education 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus 45 Integral equations 46 Functional analysis 47 Operator theory 49 Calculus of variations and optimal control, optimization Mathematics Calendar

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

April 2004 Bhargava (Clay Institute and Princeton), number theory; A. Borodin (Clay Institute and CalTech), integrable systems; T. Colding (Courant *5-6 The Fifth Annual Colloquiumfest, Institut Henri Poincare, Institute, NYU), differential geometry; A. Huber (Leipzig), arithmetic Paris, France. geometry; E. Miller (Minnesota), combinatorics; M. Mustata (Clay Topics: Valuation Theory in Algebraic Geometry and Model Theory. Institute), algebraic geometry; G. Staffilani (MIT), partial differential Organizers: Z. Chatzidakis (Equipe de Logique Mathematique, equations. Univ. Paris 7), and F.-V. Kuhlmann (Research Unit "Algebra and Information: http: I /www. math. duke. edu/ conferences/ dmj 04/. Logic", Mathematical Sciences Group, Univ. of Saskatchewan, cur­ rently visiting the Equipe de Geometrie et Dynamique, Institut de May 2004 Mathematiques de Jussieu). Invited Speakers: P. Roquette (Heidelberg), A. Prestel (Konstanz), *5-9 Workshop on the Representation Theory of p-Adic Groups, B. Teissier (Inst. Math. de ]ussieu), M. Spivakovsky (Toulouse), Ottawa University, Ottawa, Ontario, Canada. V. Cossart (Versailles-Saint Quentin), 0. Piltant (Versailles-Saint Information: http: I /www. fields. utoronto . ca/programs/ Quentin),]. Koenigsmann (Basel/Freiburg), H. Knaf (Kaiserslautern), scientific/03-04/p-adic/. R. Cluckers (ENS Ulm, Paris). Information: http: I /math. us ask. ca/fvk/Mb5. htm. '' 7-8 Interdisciplinary Seminar (I)-law, Probability and Risk, ICMS, Edinburgh, UK. * 16-17 Eighth Annual Informal UK Meeting on Integrable Models, Information: http: I /www. rna. hw . ac. uk/icms/meetings/. Conformal Field Theory and Related Topics, ICMS, Edinburgh, UK. * 10-14 PIMS-MITACS Graduate Industrial Mathematics Modelling Information: http: I /www. rna. hw. ac . uk/icms/meetings/. Camp, University of Victoria, Victoria, British Columbia, Ontario, Canada. '' 2 3-24 Workshop on Mathematical Oncology, The Fields Institute, Deadline: Apply: March 15, 2004. Toronto, Ontario, Canada. Information: http: I /www. pims. math . ca/ industrial/2004/ Information: http : I /www. fields . utoronto . cal programs/ gimmc/; email: gimmc©pims. math. ca. scientific/03-04/mathoncology/. * 12-14 Workshop: Shape Optimization and Applications, Univer­ '' 23-25 The Third Duke Mathematical journal Conference, Duke sity of Ottawa, Ontario, Canada. University, Durham, North Carolina. Information: http: I /www. fields. utoronto . ca/programs/ Description: There will be eight talks by prominent "younger" scientific/03-04/shape_theory/. mathematicians representing a cross-section of areas frequently appearing in the journal: D. Auroux (MIT), symplectic geometry; M. * 1 2-1 5 Workshop on large Scale Nonlinear and Semidefinite

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

APRIL 2004 NOTICES OF THE AMS 455 Mathematics Calendar

Programming, University of Waterloo, Waterloo, Ontario, Canada. Information: Further information, including details of the program Information: http : I I orion. math. uwaterloo. ca/\%7Ehwolkowi/ and information on accommodations, will be posted at http: I I w04workshop.d/readme.html. math. northwestern. edu/Strings as it becomes available.

'' 12-16 The Coxeter Legacy: Reflections and Projections, The *25-June 25 PIMS Summer School in Probability, University of Fields Institute, Toronto, Ontario, Canada. British Columbia, Vancouver, BC, Canada. Description: Conference in honour of H.S.M. Coxeter. We intend Program: This summer school will consist of two advanced grad­ to reflect on Coxeter's legacy, emphasizing his main achievements, uate courses, given by M. T. Barlow (UBC) and G. F. Lawler (Duke demonstrating the impact of Coxeter's ideas on current research, University). and exploring future directions. During the conference there will Information: http://www.pims.math.ca/science/2004/ be talks by invited principal speakers and sessions for contributed ssprob/. talks. Information: http : I /www. fields. utoronto. calprograms/ '' 26-27 Workshop on Electronic Voting-Theory and Practice, scientific/03-04/coxeterlegacy/. DIMACS Center, Rutgers University, Piscataway, New Jersey. Description: To many technologists, electronic voting represents '' 17-21 PIMS-MITACS Industrial Problem Solving Workshop, Uni­ a seemingly simple exercise in system design. In reality, the many versity of British Columbia, Vancouver, BC, Canada. requirements it imposes with regard to correctness, anonymity, and Information: http: I /www. pims . math. ca/industrial/2004/ availability pose an unusually thorny collection of problems, and the ipsw/; email: ipsw@pims .math. ca. security risks associated with electronic voting, especially remotely over the Internet, are numerous and complex, posing major * 17- 21 Workshop on Integrable and Near-Integrable Hamiltonian technological challenges for computer scientists. The problems PDE, The Fields Institute, Toronto, Ontario, Canada. range from the threat of denial-of-service-attacks to the need for Sponsors: Joint with the CRM. careful selection of techniques to enforce private, and correct, Organizing Committee: W. Craig (chair), P. Deift, S. Kuksin, P. tallying of ballots. Other possible requirements for electronic Olver,]. Toth, P. Winternitz. voting schemes are resistance to vote buying; defenses against Information: email: abrand@fields. utoronto. ca. malfunctioning software, viruses; and related problems; audit ability; and the development of user-friendly and universally * 19-23 Canadian Operator Symposium (COSy) 2004, University accessible interfaces. of Waterloo, Waterloo, Ontario, Canada. Goal: To bring together and foster an interplay of ideas among Topics: Operator theory, operator algebras, abstract harmonic researchers and practitioners in different areas of relevance to analysis. voting. For example, the workshop will investigate prevention of Organizers: K. R. Davidson, L. W. Marcoux, A. M. Nica, N. Spronk. penetration attacks that involve the use of a delivery mechanism to Speakers: D. V. Voiculescu (main speaker) (Univ. Calif., Berkeley); transport a malicious payload to the target host. This could be in K. J. Dykema (Texas A&M); G. Elliott (Univ. Toronto); A. T. M. the form of a "Trojan horse" or remote control program. It will also Lau (Univ. Alberta); S. C. Power (Lancaster Univ.); I. Putnam (Univ. investigate vulnerabilities of the communication path between the Victoria); Z. ]. Ruan (Univ. Illinois-Urbana); R. Speicher (Queen's voting client (the devices where a voter votes) and the server (where Univ.). votes are tallied). Especially in the case of remote voting, the path Information: http : I /www. math. uwaterloo. car cosy. html; must be "trusted", and a challenge is to maintain an authenticated email: cosy©math. uwaterloo. ca. communications linkage. Although not specifically a security issue, reliability issues * 20-22 PIMS Seminar on Stochastic Processes 2004, University of British Columbia, Vancouver, BC, Canada. are closely related and will also be considered. The workshop will consider issues dealingwithrandomhardware and software failures Invited Speakers: R. Carmona (Princeton), R. Dalang (EPF Lausanne), (as opposed to deliberate, intelligent attack). A key difference A. Guionnet (Ecole Normale Superieure de Lyon), Y. Le Jan (Orsay), between voting and electronic commerce is that in the former, B. Virag (Univ. of Toronto). one wants to irreversibly sever the link between the ballot and the Information: http: I /www. pims. math. ca/science/2004/ ssp/. voter. The workshop will discuss audit trails as a way of ensuring this. The workshop will also investigate methods for minimizing * 23-27 Geometry and Topology of String Theory, Northwestern coercion and fraud, e.g., schemes to allow a voter to vote more University, Evanston, Illinois. than once and having only the last vote count. Description: During the 2003-04 academic year, the Department of Mathematics of Northwestern University is hosting a special year Organizers: M. Jakobsson, RSA Laboratories, email: mj akobsson@ in the geometry and topology of string theory. As a culmination rsasecuri ty. com; A. Juels, RSA Laboratories, email: ajuels@ of this activity, we have organized a conference bringing together rsasecuri t y. com. mathematicians and physicists working in different areas related Local Arrangements: M. Mercado, DIMACS Center, mercado@ to string theory. dimacs. rutgers. edu, 732-445-5928. Organizers: E. Getzler and E. Zaslow, Dept. of Math., Northwestern Information: http: I /dimacs. rutgers . edu/Workshops/Voting/. Univ. Speakers: M. Aganagic (Washington), A. Bondal (IUM), T. Bridgeland * 31-June 11 1Oth European Intensive Course on "Complex Anal­ (Edinburgh), R. Bryant (Duke), T. Coates (Harvard), E. Diaconescu ysis and Its Generalizations (with applications to partial differ­ (Rutgers), K. Fukaya (Kyoto), A. Givental (Berkeley), S. Gukov (Har­ ential equations)", University of Coimbra, Coimbra, Portugal. vard), K. Hori (Toronto), A. Kapustin (CalTech), M. Liu (Harvard), M. Description: This intensive course will have a total of 40 hours Marino (CERN), N. Nekrasov (IHES), Y.-G. Oh (Wisconsin), R. Pand­ of lectures and is at postgraduate level. It is organized by the haripande (Princeton), A. Polishchuk (Oregon), P. Seidel (Chicago), Universities of Coimbra and Aveiro with the same goals as the R. Thomas (Imperial College) ones organized under the programme Socrates and is open to Workshop: In the two weeks before the conference, we will all young mathematicians interested in complex analysis and its organize a workshop built around minicourses on background applications. material relevant to recent developments in string theory. These Lecturers: W. VanAssche (Katholieke Univ. Leuven, Belgium); A. Mar­ will be suitable for graduate students as well as faculty in other tinez (Univ. de Almeria, Spain); S. Bernstein (Univ. of Weimar/Univ. fields of mathematics. of Freiberg, Germany); M. Mackey (Univ. College, Dublin, Ireland).

456 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Mathematics Calendar

Organizers: H. Malonek, A. Branquinho, ]. Carvalho e Silva, P. '' 7-26 Probability Models and Statistical Analyses for Ranking Cerejeiras. Data, University of Ottawa, Ontario, Canada. Information: Email: ajplb@mat. uc .pt, hrmalon@mat. ua.pt, Information: http: I /www. fields . utoronto. ca/programs/ jaimecs@mat . uc .pt,pceres©mat.ua. pt. scientific/03-04/shape_theory/.

June 2004 '' 8-9 DIMACS Workshop on Genomic Instability in Cancer: Biolog­ ical and Mathematical Approaches, Institute for Advanced Study, * 5-26 Clay Mathematics Institute Summer School: Floer Homol­ Princeton, New Jersey. ogy, Gauge Theory, and Low Dimensional Topology, Alfred Renyi Description: Many cancers are characterized by genomic instabil­ Institute of Mathematics, Budapest, Hungary. ity. This often involves gross chromosomal abnormalities such as Overview: The Clay Mathematics Institute is organizing a summer deletion and duplication of chromosomes or chromosome parts, school, Floer Homology, Gauge Theory, and Low Dimensional chromosomal rearrangements and mitotic recombinations, gener­ Topology in June 2004. The school will be held at the Alfred ally termed as "chromosomal instability". Other types of genomic Renyi Institute of Mathematics and will be aimed at graduate instability are characterized by an increased rate of small-scale students and mathematicians within five years of their Ph.D. The genetic changes (such as microsatellite instability). The role of school will consist of two weeks of foundational courses and one genomic instability (GI) for cancer progression is a very important week of minicourses focusing on more advanced topics and recent yet unresolved question. developments. These courses will concentrate on recent activity at Sponsors: DIMACS and Institute for Advanced Study. the crossroads of mathematical disciplines around low-dimensional topology, the theory of holomorphic curves, gauge theory, knot Information: http : I I dimacs. rutgers . edu/Workshops/ theory, smooth four-manifold topology, and contact geometry. Genomic/. The aim of this summer school is to provide a comprehensive '' 14-1 8 (REVISED) Conference on Surface Water Waves, The Fields introduction to these exciting areas through weeklong courses in Institute, Toronto, Ontario, Canada. Heegaard Floer theory of three- and four-manifolds, Seiberg-Witten theory, contact topology, and knot theory. The third week of Organizing Committee: J. Carter (Colorado), W. Craig (McMaster), advanced courses will focus on the frontiers of research in these B. Deconinck (Colorado State),]. Hammack (Pennsylvania State), D. areas. Henderson (Pennsylvania State), D. Nicholls (Minnesota), H. Segur (Colorado), C. Sulem (Toronto). Partial List of Participants: Lecturers will include ]. Etnyre, R. Fintushel, C. Gordon, P. Ozsvath, A. Stipsicz, and Z. Szabo. Information: email: abrand©fields. utoronto. ca. Organizing Committee: D. Ellwood, P. Ozsvath, A. Stipsicz, Z. * 1 5-1 7 DIMACS Workshop on Mobile and Wireless Security, Szabo. DIMACS Center, Rutgers University, Piscataway, New Jersey. Mathematical Preparation: While there are no formal prerequisites, preference will be given to applicants with some prior knowledge Description: This workshop will focus on addressing the many of topology or symplectic geometry. outstanding issues that remain in wireless cellular and WLAN networking such as (but not limited to): Management and monitor­ Application Forms: The application form is available online at ing, ad hoc trust establishment, secure roaming between overlay http: I /www . claymath. org/ summerschool. Completed application networks, availability and denial of service mitigation, and network forms will be accepted by mail or fax. The application deadline is and link layer security protocols. March 31, 2004. Graduate and Postdoctoral Funding: Funding is available for Information: http://dimacs.rutgers.edu/Workshops/ graduate students and postdoctoral fellows (within 5 years of MobileWireless/. their Ph.D.). Standard support amounts will include funds for local '' 1 7-1 8 Analysis, Probability, and Logic: A Conference in Honor expenses and accommodations plus economy travel. of Edward Nelson, University of British Columbia, Vancouver, BC, Information: http: I /www. claymath. org/ summerschool; email: Canada. [email protected]. Background: Edward Nelson, professor at Princeton University, * 7-11 Computational and Statistical Aspects of Microarray Anal­ has done beautiful and influential work in probability, functional ysis, Bressanone, Italy. analysis, mathematical physics, nonstandard analysis, stochastic mechanics, and logic. This conference will be an occasion where Description: This five-day, highly interactive advanced course his students, colleagues, and friends can meet for an overall look (halfway to a workshop) is aimed at scientists that are already at his work and its influence on current research. The experience involved in analyzing genomic data. is intended to be serious, illuminating, and enjoyable. Focus: The focus will be on analyzing and interpreting DNA microarray data, but other topics will also be covered. Among Program: The mathematics department at the University of British these: visualization, machine learning, and exploratory analyses Columbia, in conjunction with the Pacific Institute of Mathematical will be emphasized. Methods for quality control and for making use Sciences, will host the meeting. The program will include review of biological metadata in a structured way will be demonstrated. talks on Nelson's research and a few invited talks on current Attendees should bring with them data or specific methodological directions in the areas of his interest. innovations that they want to work on in collaboration with the Organizing Committee: D. Brydges (UBC), E. Carlen (Georgia Tech), organizers and other attendees. Participants should have sufficient B. Faris (Arizona), and G. Lawler (Cornell). minimal background on biological, statistical, and computational Plenary Speakers: S. Buss (UCSD), E. Carlen (Georgia Tech), L. Gross aspects ofmicroarray data and preferably be used to the R statistical (Cornell), G. Lawler (Cornell), B. Simon (Ca!Tech), C. Villani (ENS de environment. The maximum number of participants is 30. Lyon). Organizers: R. Gentleman, Dept. of Biostatistics, Harvard School Contact: email: nelson04©math . gatech. edu. of Public Health, Boston, Massachusetts, USA (member of the Information: http : I /www. pims. math. ca/ science/2004/ Bioconductor research project) and W. Huber, Dept. for Molecular nelson04/. Genome Analysis, DKFZ (German Cancer Research Center), Hei­ delberg, Germany (also a member of the Bioconductor research * 17-19 Conference in Honour of Dale Brownawell, University of project). Waterloo, Waterloo, Ontario, Canada. Information: email: stefano . iacus©unimi. it; http: I /www . Information: http : I /www. fields. utoronto . cal programs/ economia.unimi . it/marray/. scientific/03-04/brownawell/.

APRIL 2004 NOTICES OF THE AMS 457 Mathematics Calendar

* 19-24 Symmetries and Integrability of Difference Equations­ tion and proof of the McKay-Thompson conjecture. The second EuroConference on Analytic Difference Equations, Special part of the meeting will consist of invited talks on current research. Functions and Quantum Models on the Lattice, Helsinki, Finland. Scientific Organizing Committee: A. Baker (Glasgow), A. Ivanov Description: This meeting is the second in a series of two devoted (Imperial College), ]. Lepowsky (Rutgers), ]. McKay (Concordia), V. to discrete systems and their integrability and symmetries (the first Nikulin (Liverpool), M. Tuite (Galway). took place in Giens, France, in 2002). This second conference will Speakers: Expository talks will be given by the following(* awaiting emphasize linear and nonlinear special functions, and associated confirmation);]. Bruinier (Cologne)", C. Dong(Santa Cruz), T. Gannon quantum problems and geometry. The topics covered in this (Alberta)*, R. Griess (Michigan), Y.-Z. Huang (Rutgers), E. Jurisich meeting will include: analytic difference equations and spectral (Charleston), H. Li (Rutgers), A. Meurman (Lund), M. Miyamoto theory; difference bispectral problems; difference Galois theory; (Tsukuba), A. Ryba (CUNY). Q-hypergeometric and elliptic modular functions; representation Other Invited Talks: C. Cummins (Concordia), G. Mason (Santa theory and orthogonal polynomials; algebraic curves and addition Cruz), S. Norton (Cambridge), R. Wilson (Rutgers). formulae of Abelian functions; discrete and quantum geometry; Information: http: I /www. rna . hw. ac . uk/icms/meetings/2004/ quantum models on the lattice; quantum mappings. moonshine/index .html. Chair: J. Hietarinta (Turku Univ., Fl); Vice-Chair: F. W. Nijhoff (Leeds Univ., UK). '' 7-20 MSRI-PIMS Summer Graduate Programme: Knots and 3- Invited Speakers: K. Aomoto (Nagoya, JP); R. Askey (Wisconsin­ Manifolds, University of British Columbia, Vancouver, BC, Canada. Madison, US); A. Bobenko (TU Berlin, DE); A. Doliwa (Warsaw, PL); Information: http: I /www. pims. math. ca/ science/2004/ S. Elaydi (Trinity, San Antonio, Texas, US); V. Enolskii (Heriot­ KT3Mgrad/. Watt, Edinburgh, UK); L. Faddeev (Steklov, St. Petersburg, RU); A. Grunbaum (Berkeley, US); L. Haine (Louvain, BE); M. Ismail (Univ. '' 11-1 7 XI Meeting on Real Analysis and Measure Theory South Florida, US); T. Koornwinder (Amsterdam, NL); I. Krichever (CARTEMI), Hotel Continental Terme, Ischia, Naples, Italy. (Columbia, US); I. Laine (Joensuu, FI); F. Marcellan (Univ. Carlos Topics: Mathematical disciplines such as measure theory, real III, Madrid, ES); M. Noumi (Kobe, JP); 0. Ragnisco (Roma Tre, IT); analysis, lattice theory, taking into consideration their role in J.-P. Ramis (Univ. Paul Sabatier, FR); V. Roubtsov (Angers, FR); S. applications to decision theory, economics, mathematical finance, Ruijsenaars (CWI, NL); P. Santini (Roma La Sapienza, IT); M. van der and theoretical physics. Put (Groningen, NL); J. Felipe van Diejen (Talca, CL); P. Vanhaecke Invited Speakers: C. D. Aliprantis (Univ. of Purdue, USA), N. Fusco (Poitiers, FR); A. Zhedanov (Donetsk, UA). (Univ. Federico II, Naples, Italy), M. G. Graziano (Univ. of Calabria, Deadline: March 19, 2004. Italy), D. Koelzow (Univ. of Erlangen, Germany), P. Mattila (Univ. of Helsinki, Finland), A. Olevskii (Univ. of Tel Aviv, Israel), W. Pfeffer Information: Full program and application form available on the (Univ. of California, Davis), J.D.M. Wright (Univ. of Reading, UK). Euresco website at http: I /www. esf. org/ euresco/04/pc04185. Information: http: I /www. dma. unina . it;-cartemi. * 24-26 Statistical Mechanics: A Conference in Honour of the '' 12-16 School on Atmospheric Sciences and Climate Dynamics, 75th Birthday of Oliver Penrose, ICMS, Edinburgh, UK. IST, Lisbon, Portugal. Description: The main goal of the meeting is to present a program Speakers: M. Allen (Univ. of Oxford, UK), ]. Ildefonso Diaz (Univ. of first-class lectures in statistical mechanics and related fields for Complutense de Madrid, Spain), M. Ghil (ENS-Paris, France, and the benefit of British mathematicians. Most of the talks will be UCLA, USA), P. Haynes (Univ. of Cambridge, UK), E. G. Tabak related to some of the many topics in statistical mechanics where (Courant lnst., NYU, USA). Prof. Penrose has contributed greatly. For example, John Ball's talk Organizers: ]. Videman (Inst. Superior Tecnico, Portugal), ]. M. will be related to his joint work with Prof. Penrose on applying Urbano (Univ. de Coimbra, Portugal), D. Bresch (Univ. Joseph­ ideas from statistical mechanics to some problems in continuum Fourier, LMC-CNRS, France). mechanics. Other topics to be covered during the meeting include Information: http: I /www. mat . uc . pt;-tt2004/ atmosphere/. nonequilibrium models, spin models, phase separation and Bose­ Einstein condensation. * 16-20 Algebraic Topology in Computer Science Workshop, Program: The meeting will run for at least two full days, starting University of Western Ontario, London, Canada. on Thursday, 24 June. Talks may continue into Saturday morning. Information: http : I /www .math. uwo. ca/\%7Ej ardine/at-csii . Speakers: ]. Ball (Oxford), ]. Bricmont (Louvain), P. Fratzl (Berlin), html. ]. Lebowitz (Rutgers), M. Penrose (Bath), R. Penrose (Oxford), H. Spohn (Munich), B. Toth (Budapest). '' 18-23 International Conference on Differential Equations and Participation: Number of participants is strictly limited. Anyone Applications in Mathematical Biology, Malaspina University Col­ interested in attending should contact T. Dart at ICMS email: lege, Nanaimo, British Columbia, Canada. tracey©maths . ed. ac . uk, who will forward the request to the Organizers: University of Calgary, University of Alberta, and organizers. Official invitations will be issued in February. Malaspina University-College Nanaimo. Information: http: I /www. rna . hw. ac. uk/ icms/meetings/2004/ Organizing Committee: E. Braverman (coordinator), M. Li, L. Idels, penrose/index.html. L. Berezansky, A. Ponosov, E. Litsyn. Scientific Committee: R. Agarwal, E. Beretta, F. Brauer, C. Castillo­ July 2004 Chavez, K. L. Cooke, C. Corduneanu,]. M. Cushing, 0. Diekmann, K. Gopalsamy, I. Gyori, J. K. Hale, I. T. Kiguradze, V. B. Kolmanovskii, * 5-l 5 Moonshine-the First Quarter Century and Beyond: A Y. Kuang, M. Lewis, A. D. Myshkis, S. Reich, I. P. Stavroulakis,]. Wu. Workshop on the Moonshine Conjectures and Vertex Algebras, Information: http: I /web . mala . be. ca/math/ conference/. International Centre for Mathematical Sciences, Edinburgh, UK. Aims and Topics: Moonshine and related topics have been active * 19 or 20 The Second International Workshop on Declarative research areas since the late 1970s. The aim of this workshop is Agent Languages and Technologies (DAL T-2004), The Third to review the impact of this research area on mathematics and International Joint Conference on Autonomous Agents & Multi­ theoretical physics and to highlight possible new directions. Agent Systems (AAMAS 2004), New York, New York. The first part of the meeting will be expository, including Overview: Building multiagent systems still calls for models such areas as Borcherds's proof of the Conway-Norton conjecture, and technologies that ensure system predictability, enable feature construction of the Monster, vertex (operator) algebras, modular discovery and verification, and accommodate flexibility. Declarative moonshine, BKM algebras and automorphic forms, FLM's constuc- approaches offer to satisfy precisely these properties of large-scale

458 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Mathematics Calendar

multiagent systems. Recent advances in the area of computational Information: http:llwww.pims.math.calsciencel2004l logics provide a strong foundation for declarative languages and KT3Mwksp/. technologies. Equipped with such a strong foundation, declarative approaches can enable agents to reason about their interactions * 19-23 Quantum Information and Quantum Control Conference, and their environment and hence not only establish the required The Fields Institute, Toronto, Ontario, Canada. tasks but also handle exceptions that arise in many systems. Information: http : I lwww. fields . utoronto . cal programs I Topics: (include but are not limited to): Declarative agent commu­ scientificl04-05lquantumiCI. nication and coordination languages, Declarative approaches to the engineering of agent systems, Experimental studies of declarative '' 1 9-24 School and Workshop on Oceanography, Lakes and Rivers, technologies, Industrial and commercial experiences with declar­ IST, Lisbon, Portugal. ative agent technologies, Formal methods for the specification Speakers (School): P. Constantin (Univ. of Chicago, USA); B. and verification of agent systems, Computational logics in multi­ Cushman-Raisin (Dartmouth College, USA); E. Grenier (ENS-Lyon, agent systems, Argumentation and dialectical systems, Declarative France); J. Pedlosky (Woods Hole Oceanographic Inst., USA); B. description of contracts and negotiation issues, Lessons learned Perthame (ENS-Paris, France). from the design and implementation of agent systems, Declar­ Workshop: B. Desjardins (Ecole Polytech/CEA, France); R. Farwig ative paradigms for the combination of heterogeneous agents, (TU Darmstadt, Germany); F. Guillen-Gonzalez (Univ. de Sevilha, Constraints and agent systems, Declarative policies and security Spain); D. Marshall (Univ. of Reading, UK); J. Teixeira (UCAR/NRL, in MAS, Knowledge-based and knowledge-intensive MAS, Modeling USA); S. Wang (Indiana Univ., USA). of agent rationality. Organizers: ]. Videman (Inst. Superior Tecnico, Portugal); ]. M. Information: http: I /centria. di. fct. unl. pt;-jleite/dalt04/ Urbano (Univ. de Coimbra, Portugal); D. Bresch (Univ. Joseph­ index.htm. Fourier, LMC-CNRS, France). Information: http: I /www. mat. uc. ptrtt2004/ ocean/. * 19-22 11th Conference of the International Linear Algebra Society, University of Coimbra, Coimbra, Portugal. August2004 Description: The conference is dedicated to Richard Brualdi in honor of his 65th birthday and his numerous contributions to linear * 2-5 CSIR-Sponsored Seminar: Recent Advances in Combinatorics, algebra, ILAS, and mathematics. It includes invited speakers, special Graphs and Codes and Their Applications, Eluru, India. topic sessions (minisymposia), and presentations of submitted Director: R.N. Mohan (NSC2004), Sir CRR College, Eluru-534007, papers. A special issue of linear algebra and its applications will AP, India. publish the conference proceedings (these proceedings will include Keynote Address: Graph Theory and Applications, B. D. Acharya, research papers based on the talks given in the conference). Dept. of Sci. and Tech., New Delhi. Plenary Speakers: R. Bhatia (Indian Statistical Institute New Delhi); Principal Speakers: B. K. Sinha, Indian Stat. Inst., Kolkata; I. H. H. Caswell (Woods Hole Oceanographic Institution); G. Cybenko N. Rao, Gayarthri Vidhyapith, Visakhapatnam; G. C. Rao, Andhra (Dartmouth College); E, Elrnroth (Ume6 Univ.); S. Friedland (Univ. Univ., Visakhapatnam; N. K. Mandal, Calcutta Univ.; M. Acharya, of Illinois Chicago); P. Gritzmann (Technical Univ. Munich); R. Delhi College of Engineering. Guralnick (Univ. of Southern California); U. Helmke (Wlirzburg Information: email: vjwrnmohan@sancharnet. in. Univ.); W. Helton (Univ. of California San Diego); C. Krattenthaler (Claude Bernard Univ., Lyon); M. Omladic (Univ. of Ljubljana); X. * 5-6 Workshop on Missing Data Problems, The Fields Institute, Puerta (Polytechnic Univ. Catalonia); A. Ram (Univ. of Wisconsin Toronto, Ontario, Canada. Madison); J. Rosenthal (Notre Dame Univ.); S. Rump (Technical Univ. Information: http: I lwww. fields. utoronto. calprograms I Hamburg-Harburg); F. Silva (Lisbon Univ.). scientificl04-05lmissing-datal. Special Lectures: P. Lancaster (Univ. of Calgary), H. Schneider, Prize speaker; B. Meini (Univ. of Pisa), SlAG/LA speaker; J. Mora (Carlos * 6-11 17th International Conference on Multiple Criteria Decision III Univ. Madrid), SlAG/LA speaker; P. Semrl (Univ. of Ljubljana), Analysis, Whistler, British Columbia, Canada. Taussky-Todd speaker; A. J. Hoffman (Watson Res. Center, IBM), Description: This conference brings together researchers and banquet speaker. practitioners who address "Multiple Criteria Decision Making". Minisymposia: Group representations, organized by A. P. Santana Theme: "New Paradigms for New Decisions" emphasizes the and C. Andre; Combinatorial Matrix Theory, organized by B. Shader; advancements that occur in both the development and practice of Markov methods for search engines, organized by I. Ipsen and S. decision making. Participants are particularly welcome to submit Kirkland; Nonnegative matrices, organized by T. Laffey. Additional work relevant to this theme or to any other topic related to decision suggestions for minisymposia will be discussed by the organizing making with multiple criteria. This includes measurement and committee, taking into account scheduling constraints. mathematical formulations. In addition to the conference sessions, Deadlines: Submission of contributed papers: April 30, 2004. The there are social activities and numerous networking opportunities preregistration deadline is May 31, 2004. The fees before the May 31 with colleagues. deadline are 130 euros for non-ILAS members, 120 euros for ILAS Information: email: wedley@sfu. ca. members, and 90 euros for graduate students. After the deadline they are 150-150-120. '' 8-11 Euler 2004-Third Annual Meeting of the Euler Society, Organizing Committee: D. Hershkowitz (ILAS president), H. Schnei­ Roger Williams University Conference Center, Portsmouth, Rhode der, T. Laffey, R. Loewy, I. Zaballa, B. Shader, G. de Oliveira, J. Dias Island. da Silva, E. Marques de Sa, J. F. Queir6 (chair). Topics: Papers are invited on all topics related to Euler, his work Local Organizing Committee: A. P. Santana, A. L. Duarte, C. and his times, emphasizing but not limited to the 1750s. Caldeira, J. C. Gallardo, 0. Azenhas, J. F. Queir6. Keynote Address: R. Thiele, Univ. Leipzig. Information: http : I lwww. mat . uc . ptl ilas2004l; email: ilas2004@ Workshops: On reading Euler from original sources in French and mat.uc.pt. Latin. Deadline: Titles and abstracts due to the organizers by June 15, * 19-23 Knots in Vancouver: Workshop in Knot Theory and 3- 2004. Earlier is better. Manifolds, PIMS, University of British Columbia, Vancouver, BC, Organizer: E. Sandifer, sandifere@wcsu. edu. Canada. Information: http: I lwww. EulerSociety. org.

APRIL 2004 NOTICES OF THE AMS 459 Mathematics Calendar

* 1 7-20 Degenerate PDEs and Singular Geometries, University of example involves use of auctions in corporate procurement and in Potsdam, Potsdam, Germany. government deregulation efforts. When many items with interre­ Workshop Topics: Recent developments in singular analysis and lated values are being sold, economic efficiency can be increased by geometry. allowing bids on combinations of items. Procedures for auctioning Scientific Advisory Board: S. Albeverio (Bonn), B. Fedosov (Moscow), combinations of items have inherent computational problems to G. Grubb (Copenhagen), G. Mendoza (Philadelphia), B.-W. Schulze overcome, and the emergence of these issues has sparked consid­ (Potsdam), M. Shubin (Boston), N. Teleman (Ancona). erable research activity in the computer science and combinatorial Organizing Committee: ]. Gil (Altoona), T. Krainer (Potsdam), P. optimization communities. The most prominent example is com­ Popivanov (Sofia), L. Rodino (Torino), I. Witt (Freiberg). binatorial auctions in which multiple goods are auctioned and Information: T. Krainer, Inst. of Math., Univ. of Potsdam, P.O. bidders have and wish to express different valuations on which Box 60 15 53, D-14415 Potsdam, Germany; email: dpsgillmath. uni­ goods complement each other and which goods substitute for potsdam.de;http:lldpsg.math.uni-potsdam.de. each other. Allowing bidders to submit "aU-or-nothing" bids for combinations of goods yields NP-complete allocation problems * 23-September 2 International Conference-School on Geometry that need to be solved efficiently when proper care is given to and Analysis Dedicated to the 75th Anniversary of Academician designing an auction. Furthermore, bidders face computational Yu. G. Reshetnyak, Sobolev Institute of Mathematics, Novosibirsk, and communication problems in combinatorial auctions since they Russia. might not be feasibly able to express all possible preferences for all Goal: The main goal of the meeting is to provide students and subsets of goods. Another area of auction design that has been de­ young mathematicians with a possibility to attend lectures of veloping rapidly in research and in practice is short-term electricity internationally recognized researchers in geometry, quasiconformal auctions in which allowing bidders to make bids that reflect their analysis, nonlinear potential theory, and variational problems and nonconvex costs requires solving large mixed integer programming to present results of their own research. problems and finding prices that support decentralized generation Topics: Riemannian geometry in the large, Quasiconformal analysis, and transmission operations. Nonlinear potential theory and Sobolevspaces, Variational problems Organizers:J.Kalagnanam,IBMWatsonLab, email: j ayantillus. ibm . and related equations. com; E. Maskin, Inst. for Advanced Study, email: maskin@ias. edu; D. Scientific Program: Minicourses in geometry and analysis intended Parkes, Harvard Univ., email: parkes@eecs. harvard. edu; A. Pekec, for students and young mathematicians (up to five 90-min.lectures), Duke Univ., email: pekec@duke . edu; M. Rothkopf, Rutgers Univ., plenary lectures (90 min.), and short communications (20 min.). email: rothkopfillrutcor. rutgers. edu. Program Committee: M. Agranovskii (Israel),]. Ball (Great Britain), local Arrangements: M. Mercado, DIMACS Center, mercadoill A. A. Borisenko (Ukraine), V. N. Berestovskii (Russia), Yu. D. Burago dimacs.rutgers.edu, 732-445-5928. (Russia), V. I. Burenkov (Great Britain), V. N. Dubinin (Russia), Information: http : I I dimacs . rutgers. eduiWorkshopsl F. Gehring (USA), V. Gol'dshtein (Israel), ]. Heinonen (USA), T. AuctionDesignl. Iwaniec (USA), S. S. Kutateladze (Russia), V. I. Kuz'minov (Russia), 0. Martio (Finland), V. Maz'ya (Sweden), V. M. Miklyukov (Russia), I. * 1 3-1 6 Conference on Automorphic Forms and the Trace Formula, Nikolaev (Russia & USA) , M. Shubin (USA), V. D. Stepanov (Russia), in Honour of James Arthur on the Occasion of His 60th Birthday, I. A. Taimanov (Russia), V. A. Toponogov (Russia), A. M. Vershik The Fields Institute, Toronto, Canada. (Russia), V. V. Vershinin (Russia). Information: http: I lwww. fields. utoronto. calprogramsl Organizers: Sobolev Institute of Mathematics and Novosibirsk scientificl04-05larthurconfl. State University. * 14-1 5 DIMACS Workshop on Cryptography: Theory Meets Prac­ Deadlines: May 1 for registration; July 15 for submission of an tice, DIMACS Center, Rutgers University, Piscataway, New Jersey. abstract. Description: Recent advances in information technology and its Information: ContactS. Vodopyanov (chairman of the Organizing rapid acceptance by the business community have allowed for Committee), email: angeom@math. nsc. ru; http : I lwww . math. nsc. expediting complex business transactions. The most prominent rulconferencelag2004lindengl.htm. example involves use of auctions in corporate procurement and in government deregulation efforts. September 2004 In addition to the research community, the combinatorial * 7-11 2004 Workshop on Algebraic Geometry and Physics, and optimization problems that are involved with auction design Instituto Superior Tecnico (IST), Lisbon, Portugal. and general micro economic considerations have generated interest Information: http: I lwww. math. ist . utl . ptlgalgiWAGP041. from IT businesses such as IBM, industrial users of combinatorial procurement auctions such as Mars, Inc., and government agencies * 27-0ctober 2 Workshop on Elliptic Cohomology and Its Relation such as the FCC and the FERC -regulated electricity system operators to the Geometry of Loop Spaces, The Fields Institute, Toronto, PJM andNYISO (see http: I lwww .pjm. com and http: I lwww .nyiso. Canada. com). This workshop will bring together researchers in computer Organizing Committee: M. Ando (UIUC), H. Miller (MIT),]. Morava science, optimization, operations research, and economics who are (Johns Hopkins). working on computational aspects of auction design. The aim is Information: email: abrand@fields . utoronto . ca. to discuss the most prominent issues in auction design and try to design implementable and efficient auction procedures that allow * 28-0ctober 1 48th Annual Conference of the Australian Mathe­ for a large preference space while maintaining several desirable matical Society, RMIT University, Melbourne, VIC 3001, Australia. properties such as fairness, failure-freeness, and computational Director: K. Horadam; email: kathy. horadam@ems . rmi t. edu. au. feasibility for all participants. Information: http : I lwww .ma.rmit. edu. aulaustms04. Organizer: D. Boneh, Stanford, dabo@cs. stanford. edu. local Arrangements: M. Mercado, DIMACS Center, mercado@ October 2004 dimacs.rutgers.edu, 732-445-5928. Information: http : I I dimacs. rutgers. eduiWorkshopsl * 7-8 DIMACS Workshop on Computational Issues in Auction Practice/. Design, DIMACS Center, Rutgers University, Piscataway, New Jersey. Description: Recent advances in information technology and its November 2004 rapid acceptance by the business community have allowed for expediting complex business transactions. The most prominent * 11-12 DIMACS Workshop on Markets as Predictive Devices

460 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Mathematics Calendar

(Information Markets), DIMACS Center, Rutgers University, Pis­ safety, security and logistics with respect to their applications. The cataway, New Jersey. idea of this symposium is to assemble researchers and practitioners Description: For decades economists have studied an astonishing from universities, institutions, industries and government working "side effect" of financial and wagering markets: their ability to in these fields all over the world. Common methods and models used serve as highly accurate forecasting devices. This workshop aims in reliability, safety, security and logistics will be considered from to explore the use of markets as a substitute for, or complement to, a general point of view. Theoretical, modeling, computational and more traditional forecasting tools. We will examine how information case study contributions will range from academic considerations flows from traders to the market and back again, how market to industrial applications. There will be invited talks, plenary mechanisms process information, how market prices communicate sessions, parallel sessions, posters, and exhibitions. The symposium information and forecasts, and what mechanisms best foster will pose an opportunity to Ph.D. students to participate and accurate and statistically testable predictions. The workshop will present their works. The talks will be selected by the Scientific bring together researchers and practitioners from a variety of Program Committee and will be included in the symposium relevant fields, including economics, finance, computer science, proceedings. Selected papers after review and revision will be and statistics, in both academia and industry, to discuss the state published in special issues of International journal of Reliability, of the art today and the challenges and prospects for tomorrow. Quality and Safety Engineering, journal of Air Transportation, As part of the workshop, one or more tutorials are planned Transport and Telecommunication, Computer Modeling and New for the benefit of students and other newcomers to the field; little Technologies, Technological and Economic Development ofEconomy, or no background knowledge will be assumed. and Communications in Dependability and Quality Management. Organizers: R. Hanson, George Mason Univ., email: rhanson@ Information: I. B. Frenkel, Industrial Engineering and Manage­ gmu. edu; J. Ledyard, Calif. Inst. of Tech., email: j ledyard© ment Department, Negev Academic College of Engineering (NACE), hss . caltech . edu; D. Pennock, Overture Services, email: David. Bialik/Bazel Sts., P.O. Box 45, Beer Sheva, 84100, Israel; tel: +972- [email protected]. 8-6475642; fax: +972-8-64 75643; email: SMRSSL05@nace. ac . il; Local Arrangements: M. Mercado, DIMACS Center, email: mercado© http://www.nace.ac.il/extra/SMRSSL05/. dimacs. rutgers. edu, 732-445-5928. Information: http : I I dimacs. rutgers. edu/Workshops/ March 2005 Markets/. '' 21-2 5 Workshop on N= 1 Compactifications, The Fields Institute, Toronto, Ontario, Canada. '' l 5-l 7 Coxeter Lecture Series, The Fields Institute, Toronto, Organizers: M. Douglas, K. Hori, S. Sethi. Ontario, Canada. Information: email: abrand©fields. utoronto. ca. Organizer: N. Hitchin, Mathematical Institute, Oxford. Information: email: abrand©fields . utoronto . ca. * 28-April l Workshop on String Phenomenology, The Perimeter Institute, Waterloo, Ontario, Canada. "19-23 Workshop on Mirror Symmetry, The Perimeter Institute, Organizers: J. Louis, R. Myers, G. Shiu. Waterloo, Canada. Information: email: abrand©fields . utoronto. ca. Organizing Committee: D. Auroux, M. Gross, K. Hori, N. Yui. Information: email: abrand©fields. utoronto . ca.

December 2004 The following new announcements will not be repeated until the criteria in the next to the last paragraph at the bottom of "l 3-l 7 The 9th Asian Technology Conference in Mathematics the first page of this section are met. (ATCM2004), National Institute of Education, Singapore. Program: This annual conference will cover a broad range of April2005 topics on the application and use of technology in mathematics * 1-July 8 Special Semester on "Modern Methods of Time­ research and teaching. Researchers, mathematicians, educators, Frequency Analysis", Erwin Schroedinger Institute (ESI) for Math­ and teachers are invited to share their knowledge in the area of ematical Physics, Vienna, Austria. using technology to engage learners and empower teachers of Description: The special semester will bring together 100 sci­ mathematics or to enable research in any field of mathematics. entists from mathematics, engineering, and physics to explore Organizers: National Institute of Education, Singa.pore; ATCM Inc. new directions in time-frequency analysis. The ESI offers an ideal Deadlines: Submission of abstracts: June 15, 2004. Submission of environment for research and interaction. In addition, the program full papers: July 15, 2004. Early-bird registration: October 15, 2004. will contain four specialized workshops and a big conference on Information: http: I /math.nie. edu. sg/atcm or http: I /www. "Progress in Time-Frequency Analysis" (May 23-28, 2005). atcminc. com; email: atcm2004©nie. edu. sg or wyang©radford. Main Topics: (a) Non-orthogonal expansions and representation edu. theory, (b) Combined phase space methods: Between Gabor and wavelets, (c) Non-linear approximation theory and computational january 2005 harmonic analysis, (d) Time-frequency methods and pseudodiffer­ ential operators. '' l 0-14 Workshop on Topological Strings, The Fields Institute, Organizers: H. G. Feichtinger (Univ. of Vienna), K. Groechenig (GSF Toronto, Ontario, Canada. Research Center, Munich), J. J. Benedetto (Univ. of Maryland). Topics: Emphasis on Gromov-Witten invariants and open-closed Information: http: I /www. univie. ac. at/NuHAG/ESI05, http : I I duality. www.esi.ac.at; email: hans . feichtinger©univie.ac.at, email: Organizing Committee: E. Getzler, K. Hori, S. Katz. karlheinz.groechenig@gsf .de. Information: email: abrand©fields . utoronto. ca. May 2005 February 2005 * 2-6 Workshop on Gravitational Aspects of String Theory, The * l 5-l 7 International Symposium on Stochastic Models in Relia­ Fields Institute, Toronto, Ontario, Canada. bility, Safety, Security and Log is tics (SM RSSL'OS), Negev Academic Organizers: C. Johnson, P. Kraus, D. Marolf, A. Peet. College of Engineering (NACE), Beer Sheva, Israel. Information: email: abrand@fields. utoronto. ca. Description: The SMRSSL'05 will serve as a forum for discussing different issues of stochastic models and methods in reliability,

APRIL 2004 NOTICES OF THE AMS 461 New Publications Offered by the AMS

Algebra and Algebraic The Maximal Geometry Subgroups of The Maximal Subgroups Positive Dimension of Positive Dimension in Exceptional in Exceptional Chiral Algebras Algebraic Groups Martin w. Llebeck Algebraic Groups Alexander Beilinson and Gary M. Seltz Vladimir Drinfeld, University Martin W. Liebeck, Imperial Chiral Algebras College, London, and of Chicago, IL I Am c~ l Soc>c! y Equivariant "quantum" counterpart of the space of the global solutions of a differential equation. Ergodic Theory Diffeomorphisms: of Equivariant The book is intended for researchers working in algebraic Diffeomorphisms: Markov Partitions geometry and its applications to mathematical physics and Markov Partitions representation theory. and Stable Ergodicity and Stable Ergodicity Michael Field Contents: Introduction; Axiomatic patterns; Geometry of Matthew Nicol Michael Field, University of D -schemes; Local theory: Chiral basics; Global theory: Chiral Houston, TX, and Matthew homology; Bibliography; Index and notation. Nicol, University of Surrey, Colloquium Publications, Volume 5 1 Guildford, England May 2004, approximately 352 pages, Hardcover, ISBN 0-8218- Contents: Introduction; Equivariant geometry and dynamics; 3528-9, LC 2003063872, 2000 Mathematics Subject Technical preliminaries; Part 1. Markov Partitions: Markov AMS members $55, List $69, Classification: 17Bxx; 14Fxx, All partitions for finite group actions; Transversally hyperbolic Order code COLL/ SlN

462 NOTICES OF TilE AMS VOLUME 51, NUMBER 4 New Publications Offered by the AMS sets; Markov partitions for basic sets; Part 2. Stable Ergodicity: Gabor systems and Newton's method; G. Olafsson and Preliminaries; Livsic regularity and ergodic components; Stable D. Speegle, Wavelets, wavelet sets, and linear actions on IJil.»; ergodicity; Appendix A. On the absolute continuity of v; A. M. Powell, Orthonormalized coherent states; Q. Sun, Local­ Appendix. Bibliography. ization of stability and p-frames in the Fourier domain; Memoirs of the American. Mathematical Society, Volume 169, J. Yang, L. Shen, M. Papadakis, I. Kakadiaris, D. J. Kouri, and Number 803 D. K. Hoffman, Orthonormal wavelets arising from HDAFs. May 2004, 100 pages, Softcover, ISBN 0-8218-3599-8, Contemporary Mathematics, Volume 345 LC 2003070916, 2000 Mathematics Subject Classification: April 2004, 342 pages, Softcover, ISBN 0-8218-3380-4, 37A25, 37B10, 37C80, 37D30; 37C70, Individual member $32, LC 2004041027, 2000 Mathematics Subject Classification: List $54, Institutional member $43, Order code 20C20,41A17,42C15,42C40,43A85,46C05,46C99,46E25, MEM0/ 169/ 803N 47C05, 65T60, All AMS members $71, List $89, Order code CONM/ 345N Applications Differential Equations Wavelets, Frames and Operator Theory Well-Posedness for General2 x 2 Wavelets, Frames Christopher Heil, Georgia and Operator Theory Institute of Technology, Well-Posedness Systems of Christopher Heil Atlanta, Palle E.T. Jorgensen, for General Palle E.T. Jorgensen 2 x 2 Systems of Conservation Laws David R. La rson University of Iowa, Iowa City, Conservation Laws Editors Fabio Ancona, Bologna, Italy, and David R. Larson, Texas Fabio Ancona Andrea Marson and Andrea Marson, A&M University, College University of Padova, Italy Station, Editors

Amorocnn ~lnthomallcoiSoc•oty Contents: Introduction; Preliminaries; In the past two decades, wavelets and Outline of the proof; The algorithm; frames have emerged as significant tools in mathematics and Basic interaction estimates; Bounds on technology. They interact with harmonic analysis, operator the total variation and on the interaction potential; Estimates theory, and a host of other applications. on the number of discontinuities; Estimates on shift differen­ This book grew out of a special session on Wavelets, Frames tials; Completion of the proof; Conclusion; Bibliography. and Operator Theory held at the Joint Mathematics Meetings Memoirs of the American Mathematical Society, Volume 169, in Baltimore and a National Science Foundation-sponsored Number 801 workshop held at the University of Maryland. Both events were associated with the NSF Focused Research Group. The volume May 2004, 170 pages, Softcover, ISBN 0-8218-3435-5, includes both theoretical and applied papers highlighting the LC 2003070915, 2000 Mathematics Subject Classification: many facets of these interconnected topics. It is suitable for 35L65, Individual member $37, List $61, Institutional graduate students and researchers interested in wavelets and member $49, Order code MEM0/ 169/ 801N their applications. This item will also be of interest to those working in analysis. Contents: A. Aldroubi, C. Cabrelli, and U. M. Molter, How to construct wavelet frames on irregular grids and arbitrary dila­ tions in llll.d; L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, An analogue of Bratt eli-Jorgensen loop group actions for GMRA's; R. L. Benedetto, Examples of wavelets for local fields; M. Bownik and Z. Rzeszotnik, The spectral func­ tion of shift-invariant spaces on general lattices; P. G. Casazza, Custom building finite frames; P. G. Casazza and G. Kutyniok, Frames of subspaces; D. E. Dutkay, The local trace function for super-wavelets; H. Feichtinger and I. Pesenson, Recovery of band-limited functions on manifolds by an iterative algorithm; J. E. Gilbert and J. D. Lakey, On a characterization of the local Hardy space by Gabor frames; A. L. Gonzalez and R. A. Zalik, Riesz bases, multiresolution analyses, and perturbation; D. Han and Y. Wang, The exis­ tence of Gabor bases and frames; B. D. Johnson, Co-affine systems in IJil.d; K. A. Kornelson and D. R. Larson, Rank-one decomposition of operators and construction of frames; D. Labate, G. Weiss, and E. Wilson, An approach to the study of wave packet systems; M. C. Lammers, Convolution for

APRIL 2004 NOTICES OF THE AMS 463 New Publications Offered by the AMS

The volume is suitable for graduate students and research Geometry and Topology mathematicians interested in various aspects of the theory of quadratic forms. This item will also be of interest to those working in algebra ME1Yl9IRS Equivariant, and algebraic geometry. AmerlcanMothernotkaiSoctely Almost-Arborescent Contents: R. Aravire and B. Jacob, The Milnor sequence for Wq.J"(x) in characteristic 2 when .r is perfect; Equivariant, Representations of Almost-Arborescent L. Arenas-Carmona, Spinor norm for local skew-Hermitian Representations of Open Simply-Connected 3-Manifolds; Open Simply­ forms; P. Balmer, An introduction to triangular Witt groups A Finiteness Result and a survey of applications; W. K. Chan, A. G. Earnest, and V. Po~naru Connected B.-K. Oh, Regularity properties of positive definite integral C. Tanas i quadratic forms; W. K. Chan and M. I. Icaza, Effective i/1!1!< 3-Manifolds; A results \~: on representations of quadratic forms; W. K. Chan and Finiteness Result M. Peters, Quaternary quadratic forms and Hilbert modular Am~rlcan M o th~mstl oo l Soct<\ 1 surfaces; V. Poenaru, Universite Paris M. Dickmann and A. Petrovich, Real semigroups and abstract real spectra. I; M. Gaulter, The role of characteristic Sud, Orsay, and C. Tanasi, vectors in the neighbour lattice process; L. J. Gerstein, On Palermo, Italy representation by quadratic IFq [x]-lattices; D. W. Hoffmann, Diagonal forms of degree p in characteristic p; D. G. James, Contents: Introduction; The case V3 = 3 M of Theorem I and Local densities and the representations of an integer by a defi­ Theorem II; The accumulation pattern of the double point nite quadratic form; M. Karoubi, Periodicity of Hermitian M2 (j) c X 2; Arbitrary open simply-connected 3-manifold; K-theory and Milnor's K-groups; N. A. Karpenko, Third proof Bibliography. of second gap in dimensions of quadratic forms from In; Memoirs of the American Mathematical Society, Volume 169, M.-H. Kim, Recent developments on universal forms; Number 800 W. Kohnen, Special Siegel modular forms and singular series May 2004, 89 pages, Softcover, ISBN 0-8218-3460-6, polynomials of quadratic forms; A. Laghribi, Quasi-hyperbol­ icity of LC 2003070918, 2000 Mathematics Subject Classification: totally singular quadratic forms; D. B. Leep and 57M30, 37M10, 57M40, Individual member $32, List $54, T. L. Smith, Witt kernels of triquadratic extensions; Institutional member $43, Order code MEM0/169/800N D. W. Lewis, Anti-automorphisms of the second kind; J. Martinet, Reduction modulo 2 and 3 of Euclidean lattices, II; M. Marshall, The elementary type conjecture in quadratic form theory; A. Pfister, On Hilbert's theorem about ternary quartics; Number Theory R. Schulze-Pillot, Representation by integral quadratic forms-A survey; R. Schulze-Pillot and F. Xu, Representations by spinor genera of ternary quadratic forms; T. Watanabe, A Algebraic and survey and a complement of fundamental Hermite constants. Arithmetic Theory of Contemporary Mathematics, Volume 344 April2004, 350 pages, Softcover, ISBN 0-8218-3441-X, Algebraic and Quadratic Forms 2000 Mathematics Subject Arithmetic Theory Classification: llExx, 11Fxx, llHxx, of Quadratic Forms Ricardo Baeza, Universidad de All AMS members $71, List $89, Order code CONM/ 344N Talca, Chile, John S. Hsia, Ohio State University, Columbus, Bill Jacob, University of California, Santa Barbara, and Alexander Prestel, University of Konstanz, Germany, Editors This proceedings volume contains papers presented at the International Conference on the algebraic and arithmetic theory of quadratic forms held in Talca (Chile). The modern theory of quadratic forms has connections with a broad spectrum of mathematical areas including number theory, geometry, and K-theory. This volume contains survey and research articles covering the range of connections among these topics. The survey articles bring readers up-to-date on research and open problems in representation theory of integral quadratic forms, the algebraic theory of finite s quare class fields, and developments in the theory of Witt groups of triangulated categories. The specialized articles present important develop­ ments in both the algebraic and arithmetic theory of quadratic forms, as well as connections to geometry and K-theory.

464 NOTICES OF THE AMS VOLUME 51, NUMBER 4 New Publications Offered by the AMS

Distributional comparisons with Pe; Comparisons with Pe: More Publications Available point probabilities; Proofs; Technical complements; Refer­ ences; Notation index; Author index; Subject index. from the AMS EMS Monographs in Mathematics December 2003, 352 pages, Hardcover, ISBN 3-03719-000-0, New Series! 2000 Mathematics Subject Classification: 60C05, All AMS members $63, List $79, Order code EMSMONO/ lN The AMS is now distributing books from the European Mathe­ matical Society Publishing House. Selections will include a broad range of publications within the following series: EMS Series of Lectures in Mathematics EMS Monographs in Mathematics This book series publishes polished notes arising from semi­ nars or lecture series in all fields of pure and applied This book series publishes research monographs and graduate mathematics. Individual volumes give rapid and accessible level textbooks from all fields of mathematics. These books introductions and guide readers to topics of current research give comprehensive, self-contained accounts of the particular and to more advanced and specialized literature. subjects covered. Volumes present new or previously unpub­ lished mathematical results. Uhlenbeck Logarithmic Compactness Combinatorial Katrin Wehrheim, Princeton Structures: University, N] This book gives a detailed account of A Probabilistic the analytic foundations of gauge Approach theory-Uhlenbeck's compactness theorems for general connections and Richard Arratia, University of for Yang-Mills connections. It guides Southern California, University graduate students into the analysis of Park, Los Angeles, Yang-Mills theory as well as serves as a A. D. Barbour, University of reference for researchers in the field. Zurich, Switzerland, and The volume is largely self contained. It contains a number of Simon Tavare, University of Southern California, appendices (e.g., on Sobolev spaces of maps between mani­ University Park, Los Angeles folds) and an introductory part covering the LP-regularity theory for the inhomogenous Neumann problem. The two The elements of many classical combinatorial structures can main parts contain the full proofs of Uhlenbeck's weak and be naturally decomposed into components. Permutations can strong compactness theorems on closed manifolds as well as be decomposed into cycles, polynomials over a finite field into their generalizations to manifolds with boundary and noncom­ irreducible factors, mappings into connected components. In pact manifolds. These parts include a number of useful all of these examples, and in many more, there are strong analytic tools such as general patching constructions and local similarities between the numbers of components of different slice theorems. sizes that are found in the decompositions of "typical" The book is suitable for graduate students and research math­ elements of large size. For instance, the total number of ematicians interested in differential geometry, global analysis, components grows logarithmically with the size of the and analysis on manifolds. element, and the size of the largest component is an appre­ ed within the Americas by the American Mathematical Society. ciable fraction of the whole. Distribut Contents: The Neumann problem; Weak compactness; Strong This book explains the similarities in asymptotic behavior as compactness; Appendix; List of symbols; Index; Bibliography. the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition. The EMS Series of Lectures in Mathematics discussion is conducted in the language of probability, January 2004, 250 pages, Softcover, ISBN 3-03719-004-3 , 2000 enabling the theory to be developed under rather general and Mathematics Subject Classification: 53C07; 58El5, All AMS explicit conditions; for the finer conclusions, Stein's method members $36, List $45, Order code EMSSERLEC/ lN emerges as the key ingredient. The book is thus of particular interest to graduate students and researchers in both combi­ natorics and probability theory. This item will also be of interest to those working in probability. Distributed within the Americas by the American Mathematical Society. Contents: Permutations and primes; Decomposable combina­ torial structures; Probabilistic preliminaries; The Ewens sampling formula: methods; The Ewens sampling formula: asymptotics; Logarithmic combinatorial structures; General setting; Consequences; A Stein equation; Point probabilities;

APRIL 2004 NOTICES OF THE AMS 465 New Publications Offered by the AMS

EMS Textbooks in Mathematics ESI Lectures in Mathematics and Physics This book series is aimed at students or professional mathe­ The International Erwin Schrodinger Institute (ESI) is a meeting maticians seeking an introduction into a particular field. place that nurtures the development and exchange of ideas Individual volumes provide relevant techniques, results and among leading experts in mathematics and mathematical applications and afford insight into the motivations and ideas physics. In particular, the ESI encourages intellectual exchange behind the theories. between scientists from Eastern Europe and the rest of the world. Books in this series arise from the Institute's research Course in programs. Error-Correcting Lectures on Real Codes Semisimple Lie Jmn Justesen and Tom Algebras and Their Heholdt, Technical University of Denmark, Lyngby, Denmark Representations Arkady Onishchik, Yaroslavl This book is written as a text for a course aimed at advanced undergradu­ State University, Russia ates. Only some familiarity with In 1914, E. Cartan posed the problem elementary linear algebra and proba­ to find all irreducible real linear Lie bility is directly assumed, but some maturity is required. The algebras. An updated exposition of his students may specialize in discrete mathematics, computer work was given by Iwahori (1959). This science, or communication engineering. The book is also a theory reduces the classification of irreducible real representa­ suitable introduction to coding theory for researchers from tions of a real Lie algebra to a description of the so-called related fields or for professionals who want to supplement self-conjugate irreducible complex representations of this their theoretical basis. It gives the coding basics for working algebra and to the calculation of an invariant of such a repre­ on projects in any of the above areas, but material specific to sentation (with values + 1 or -1) which is called the index. one of these fields has not been included. Moreover, these two problems were reduced to the case when Chapters cover the codes and decoding methods that are the Lie algebra is simple and the highest weight of its irre­ currently of most interest in research, development, and appli­ ducible complex representation is fundamental. A complete cation. They give a relatively brief presentation of the essential case-by-case classification for all simple real Lie algebras was results, emphasizing the interrelations between different given (without proof) in the tables of Tits (1967). But actually methods and proofs of all important results. A sequence of a general solution of these problems is contained in a paper of problems at the end of each chapter serves to review the Karpelevich (1955) that was written in Russian and not widely results and give the student an appreciation of the concepts. known, where inclusions between real forms induced by a In addition, some problems and suggestions for projects indi­ complex representation were studied. cate direction for further work. The presentation encourages The author begins with a simplified (and somewhat extended the use of programming tools for studying codes, imple­ and corrected) exposition of the main part of this paper and menting decoding methods, and simulating performance. relates it to the theory of Cartan-Iwahori. He concludes with Specific examples of programming exercise are provided on some tables, where an involution of the Dynkin diagram that the book's home page. allows for finding self-conjugate representations is described This item will also be of interest to those working in applica­ and explicit formulas for the index are given. In a short tions. addendum, written by ]. V. Silhan, this involution is inter­ Distributed within the Americas by the American Mathematical Society. preted in terms of the Satake diagram. EMS Textbooks in Mathematics The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where January 2004, 192 pages, Hardcover, ISBN 3-03719-001-9, these theories are used. Readers should know the classical 2000 Mathematics Subject Classification: 94Bxx, All AMS theory of complex semisimple Lie algebras and their finite­ members $36, List $45, Order code EMSTEXT/ 1N dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the corre­ spondence between real forms and involutive automorphisms, the Cartan decompositions and the conjugacy of maximal compact subgroups of the automorphism group. Distributed within the Americas by the American Mathematical Society. ESI Lectures in Mathematics and Physics December 2003, 100 pages, Softcover, ISBN 3-03719-002-7, 2000 Mathematics Subject Classification: 17B20, All AMS members $22, List $28, Order code EMSESILEC/ 1N

466 NOTICES OF THE AMS VOLUME 51, NUMBER 4 New Publications Offered by the AMS

Zurich Lectures in Advanced Mathematics Contents: M. Vergne, Cohomologie equivariante et theoreme Mathematics in Zurich has a long and distinguished tradition. de Stokes; F. Rouviere, Espaces de Damek-Ricci, geometrie et The writing of lecture notes volumes and research monographs analyse; J. Faraut, Espaces hilbertiens invariants de fonctions is a prominent part of this tradition. This series aims to make holomorphes; Seminaire. some of these publications better known to a wider audience. Seminaires et Congres, Number 7 Moderately priced, concise and lively in style, the volumes will November 2003, 178 pages, Softcover, ISBN 2-85629-142-2, appeal to researchers and students alike, who seek an 2000 Mathematics Subject Classification: 19110, 22£25, 32M05, informed introduction to important areas of current research. 43A80,43A90, 53B20, 53C22, 53C35, 53D50, 55N91, Lectures on Partial Individual member $36, List $40, Order code SEC0/ 7N Hyperbolicity and Spectral Analysis Stable Ergodicity and Its Applications: Yakov Pesin, Pennsylvania State University, University Park Ion Colojoara Anniversary Volume This book is an introduction to the modern theory of partial hyperbolicity Aurelian Gheondea, Bilkent with applications to stable ergodicity University, Turkey, and theory of smooth dynamical systems. Mihai Sabac, University of It provides a systematic treatment of the theory and describes all the basic concepts and major Bucharest, Romania, Editors results obtained in the area since its creation in the early This book is dedicated to Ion Colojoara, 1970s. It can be used as a textbook for a graduate student one of the main contributors to the course and is also of interest to professional mathematicians development of spectral theory in Banach spaces. It contains working in the field of dynamical systems and their applica­ survey papers and a careful selection of research papers by tions. Colojoara's colleagues, former students, and prominent Distributed within the Americas by the American Mathematical Society. researchers in the field. Zurich Lectures in Advanced Mathematics Topics include: January 2004, 144 pages, Softcover, ISBN 3-03719-003-5, · growth properties of resolvents 2000 Mathematics Subject Classification: 37D30; 37Cxx, . positive definite operator measures and operator-valued All AMS members $26, List $32.50, Order code EMSZLEC/1N functions · spectral spaces of bounded operators . realization techniques for analytic functions of several vari- Analyse sur les ables Seminaires . perturbation theory and spectral resolutions in Krein spaces groupes de Lie et . orthogonal polynomials in several non-commuting variables & Congres . idempotent linear relations and generalized projection on Col l ect ion SMF theorie des Hilbert spaces .~.·­ representations The book is suitable for graduate students and researchers ~""'ANALYSE SURLES interested in operator theory. GROUPES DE LIE Jacques Faraut, Universite ETTHEoRIE A publication of the Theta Foundation. Distributed worldwide, except DES REPREsENTATIONS Pierre et Marie Curie, Paris, in Romania, by the AMS. J. FARAUT. F. ROUV1£RE. M. VERGNE Franc;:ois Rouviere, Universite Contents: E. Albrecht and W. Ricker, Local spectral theory for de Nice, France, and Michele operators with thin spectrum; C.-G. Ambrozie, A remark on Vergne, Ecole Polytechnique, operator-valued positive-definite functions; T. Ya. Azizov and Palaiseau, France v. A. Straus, Spectral decompositions for special classes of self-adjoint and normal operators on Krein spaces; This book, Analysis on Lie Groups and Representation Theory, T. Constantinescu, Orthogonal polynomials in several non­ is based on the Summer School held at Kenitra University commuting variables. 1.; ]. Eschmeier and M. Putinar, On (Morocco). Michele Vergne presented equivariant cohomology bounded analytic extensions in en; P. Jonas, On locally defi­ in the case of the circle acting on a manifold due to Paradan's nite operators in Krein spaces; J.-Ph. Labrousse, Idempotent localization formula. Franc;ois Rouviere lectured on Damek­ linear relations; V. Lyance and G. Chuiko, Rings of projectors Ricci spaces, which provide examples of nonsymmetric and operator-valued measures; M. Sabac, Commutators, inter­ harmonic Riemannian manifolds. jacques Faraut discussed twining operators and invariant subspaces; F.-H. Vasilescu, analysis, using Choquet's theory, of the invariant Hilbert Spectral measures and moment problems. spaces of holomorphic functions, with an application to the Bargmann-Segal transform on a compact symmetric space. International Book Series of Mathematical Texts The volume is suitable for graduate students and researchers October 2003, 216 pages, Hardcover, ISBN 973-85432-3-1, interested in Lie groups. 2000 Mathematics Subject Classification: 47-06; 47All, 47B40, 47A56,46C20,47B50,46G25, 32A70,47A48,47A60,47A05, A publication of the Societe Mathematique de France, Marseilles (SMF), 47B15, 47B20, 47B25, All AMS members $27, List $34, Order distributed by the AMS in the U.S ., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive code THETA/5N a 30% discount from list.

APRIL 2004 NOTICES OF THE AMS 467 New Publications Offered by the AMS

Operator Algebras Stochastic Analysis and Applications on Large Scale Hideki Kosaki, Kyushu Interacting Systems University, Fukuoka, Japan, Tadahisa Funaki, University of Editor Tokyo, and Hirofumi Osada, This volume reflects the proceedings of Nagoya University, japan, the U.S.-Japan seminar on Operator Editors Algebras and Applications held at Kyushu University Qapan). Contributors This volume is a collection of 15 to the volume are leading researchers in research and survey papers written by the theory of operator algebras. the speakers from two international conferences held in Japan, The 11th Mathematical Society of Readers will find a variety of current research topics reflecting Japan International Research Institute's Stochastic Analysis on the state of the art of the subject. Topics covered include the Large Scale Interacting Systems and Stochastic Analysis and classification of C* -algebras by K-theoretical data and related Statistical Mechanics. topics, a detailed study on various classes of C* -algebras, subfactor analysis and related topics, free probability theory, Topics discussed in the volume cover the hydrodynamic limit, and more. The material is suitable for graduate students and fluctuations, large deviations, spectral gap (Poincare inequality), researchers interested in operator algebras and their applica­ logarithmic Sobolev inequality, Ornstein-Zernike asymptotics, tions. random environments, determinantal expressions for systems Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, including exclusion processes (stochastic lattice gas, Kawasaki and distributed worldwide, except in Japan, by the AMS. dynamics), zero range processes, interacting Brownian parti­ cles, random walks, self-avoiding walks, Ginzburg-Landau Contents: C* -Algebras: B. Blackadar, Semiprojectivity in model, interface models, Ising model, Widom-Rowlinson model, simple C* -algebras; N. Brown, On quasidiagonal C* -algebras; directed polymers, random matrices, Dyson's model, and more. N. Brown and M. Dadarlat, Extensions of quasidiagonal C* -algebras and K-theory; J. H. Hong, The ideal structure of The material is suitable for graduate students and researchers graph algebras; ]. A. Jeong, Stable rank and real rank of graph interested in probability theory, stochastic processes, and C* -algebras; Q. Lin and N. C. Phillips, Direct limit decompos­ statistical mechanics. tion for C* -algebras of minimal diffeomorphisms; M. Nagisa, This item will also be of interest to those working in differential Single generation and rank of C* -algebras; H. Nakamura, equations and mathematical physics. Aperiodic automorphisms of certain simple C* -algebras; Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, C.-G. Park, C* -algebras over sphere with fibres non-commuta­ and distributed worldwide, except in Japan, by the AMS. tive tori; M. Rerdam, Stable C* -algebras; von Neumann Algebras: M. Izumi, Non-commutative Markov operators Contents: S. R. S. Varadhan, Large deviations for the asym­ arising from subfactors; Y. Kawahigashi, Braiding and nets of metric simple exclusion process; M. Campanino, D. Ioffe, factors on the circle; N. Sato, A notion of Morita equivalence andY. Velenik, Random path representation and sharp corre- between subfactors; Y. Ueda, Amalgamated free product over lations asymptotics at high temperatures; P. Caputo, Spectral Cartan subalgebra, II. Supplementary results and examples; gap inequalities in product spaces with conservation laws; Miscellaneous Topics: T. Digernes, Finite approximations and M.-F. Chen, Ten explicit criteria of one-dimensional processes; physics over unconventional fields; F. Hiai and H. Kosaki, F. Comets, T. Shiga, and N. Yoshida, Probabilistic analysis of Operator means and their norms; T. Matsui, Quantum spin directed polymers in a random environment a review; ]. Fritz, chain and Popescu systems; K. Matsumoto, Topological conju­ Entropy pairs and compensated compactness for weakly asym­ gacy invariants of symbolic dynamics arising from C* -algebra metric systems; T. Funaki and H. Sakagawa, Large deviations K-theory; Y. Watatani, Relative positions of four subspaces in for \7 c:p interface model and derivation of free boundary prob­ a Hilbert space and subfactors. lems; M.-H. Giga and Y. Giga, A PDE approach for motion of phase-boundaries by a singular interfacial energy; Y. Higuchi, Advanced Studies in Pure Mathematics, Volume 38 J. Murai, and J. Wang, The Dobrushin-Hryniv theory for the January 2004, 328 pages, Hardcover, ISBN 4-931469-23-X, two-dimensional lattice Widom-Rowlinson model; M. Katori, 2000 Mathematics Subject Classification: 46Lxx; 11Rxx, T. Nagao, and H. Tanemura, lnfinite systems of non-colliding All AMS members $74, List $92, Order code ASPM/38N Brownian particles; C. Landim, S. Olla, and S. R. S. Varadhan, Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes; H. Osada, Non-collision and collision properties of Dyson's model in infinite dimension and other stochastic dynamics whose equilibrium states are deter­ minantal random point fields; T. Shirai andY. Takahashi, Random point fields associated with Fermion, Boson and other statistics; H. Tanaka, Levy processes conditioned to stay posi­ tive and diffusions in random environments; K. Uchiyama, Zero-range-exclusion particle systems. Advanced Studies in Pure Mathematics, Volume 39 January 2004, 406 pages, Hardcover, ISBN 4-931469-24-8, 2000 Mathematics Subject Classification: 60-XX; 82-XX, All AMS members $77, List $96, Order code ASPM/39N

468 NOTICES OF THE AMS VOLUME 51, NUMBER 4 New Publications Offered by the AMS

Guide to 16\'Il:X: Learning 16\'Il:X Fourth Edition David F. Griffiths, University Helmut Kopka and of Dundee, Scotland, and Patrick W. Daly, Max Planck Desmond J. Higham, Institut fiir Aeronomie, University of Strathclyde, Katlenburg-Lindau, Germany Glasgow, Scotland This book shows readers how to begin A clear, simple, up-to-date, sometimes amusing introduction to ~TJ!(-brief, using ~TEX to create high-quality docu­ ments. It also serves as a reference for siam. yet surprisingly comprehensive. Covers '------' basics, graphics, bibliographies, all ~Tpl: users. In this completely indexes, slides, electronic resources, revised edition, the authors cover the ~Tpc2, standard and offer more details, examples, exercises, tips, and tricks. They differences between "old" and "new" ~T£(, etc. Rich with brief go beyond the core installation to describe the key contributed but pertinent examples. -The American Mathematical Monthly packages that have become essential to ~Tpl: processing. In the book, readers will find: Most beginners in ~TJ}( do not wish to read through a large comprehensive manual. They want a brief account which · Complete coverage of ~Tpc fundamentals, including how to input text, symbols, and mathematics; how to produce lists covers the essential elements of the subject. This book by and tables; how to include graphics and color; and how to Griffiths and Higham is the answer. organize and customize documents -George Phillips, Mathematical Institute, · Discussion of more advanced concepts such as bibliograph­ University of St. Andrews, Scotland ical databases and BIBTEX, math extensions with .A.JvtS-~TEX, This book is playful, witty, intelligent, and extremely easy to drawing, slides, and letters read .. . tells you what you need to know, with examples, in · Helpful appendices on installation, error messages, creating sensible, clear, entertaining language. packages, using ~Tpc with HTML and XML, and fonts -Robert M. Corless, University of Western Ontario · An extensive alphabetized listing of commands and their This book offers a brief introduction to the k\Tpl: system for uses typesetting documents. It covers the essential material while New to this edition: avoiding unnecessary detail. The contents are aimed squarely · More emphasis on k\Tpl: as a markup language that separates at k\Tpl: beginners who wish to learn the basics with a content and forms minimum of fuss. It is suitable for students needing to · Detailed discussions of contributed packages alongside rele­ produce a first-time report or thesis and for more experienced vant standard topics users who are using older typesetting systems. A large part of · In-depth information on PDF output, including extensive the book consists of "before and after" illustrations showing coverage of how to use the hyperref package to create links, the effect of k\Tpl: commands. bookmarks, and active buttons. Published by the Society for Industriai and Applied Mathematics. This Guide to ~T£(, Fourth Edition will prove indispensible to Contents: Preamble; Basic k\TEX; Typesetting mathematics; anyone wishing to gain the benefits of k\Tpl:. Further essential k\Tpl:; More about ~Tpc; Old ~Tpl: versus The accompanying CD-ROM is part of the Tpl: Live set. It k\Tp1:2,; A sample article; A sample report; Slides; Internet contains a full ~Tpc installation for Windows, Mac OSX, and resources; Bibliography; Index. Linux, as well as many extensions, including those discussed December 1996, 84 pages, Softcover, ISBN 0-89871-383-8, in the book 2000 Mathematics Subject Classification: 00-XX, 68N15, Published by Addison-Wesley. Individual member $16, List $23, Order code LLTEXN Contents: Basics: Introduction; Text, symbols, and commands; Document layout and organization; Displaying text; Text in boxes; Tables; Mathematical formulas; Graphics inclusion and color; Floating tables and figures; User custornizations; Beyond the Basics: Document management; Bibliographic databases and BIBTEX; PostScript and PDF; Multilingual ~Tpc; Math exten­ sions with .A.JvtS-k\Tpl:; Drawing with k\Tpl:; Presentation material; Letters; Apendices: The new font selection scheme; Installing and maintaining ~TEX; Error messages; ~TEX programming; k\Tpl: and the world wide web; Obsolete k\Tpl:; Command summary; Bibliography; Index. November 2003, 597 pages, Softcover, ISBN 0-321-17385-6, 2000 Mathematics Subject Classification: 00-XX, 68N15, Individual member $45, List $49.99, Order code GLTEXN

APRIL 2004 NOTICES OF THE AMS 469 New Publications Offered by the AMS

Handbook of Writing for the Mathematical Sciences Nicholas J. Higham, University of Manchester, England Any professional mathematician (whether in the corporate world or in academe) who writes, and therefore puts math into print, should have this handbook on the reference shelf ... The information needed by writers, editors, and their assistants to prepare clear, accurate, and understandable mathematical material is contained in this book. -Barbara A. Simmons, Technical Communication Higham's handbook is to the technical writer what The Elements of Style by Strunk and White is to the liberal arts writer. I've reached for The Elements of Style many times and expect to reach for Handbook of Writing for the Mathematical Sciences even more frequently. -Keith Parris, Alcatel Network Systems The subject of mathematical writing is infused with life once again by Nick Higham. He follows up his successful Handbook of Mathematical Writing with this much-anticipated second edition. As is Higham's style, the material is enlivened by anecdotes, unusual paper titles, and humorous quotations. This volume provides even more information on the issues you will face when writing a technical paper or talk, from choosing the right journal in which to publish to handling your references. Its overview of the entire publication process is invaluable for anyone hoping to publish in a technical journal. This second edition is completely revised, making use of feed­ back from readers as well as Higham's own large file of ideas based on his experiences in reading, writing, editing, exam­ ining, and supervising theses. Published by the Society for Industrial and Applied Mathematics. Contents: General principles; Writer's tools and recommended reading; Mathematical writing; English usage; When English is a foreign language; Writing a paper; Revising a draft; Publishing a paper; Writing and defending a thesis; Writing a talk; Giving a talk; Preparing a poster; TJ:X and IbTJ:X; Aids and resources for writing and research; The Greek alphabet; Summary of TJ:X and IbTJ:X symbols; GNU emacs commands; Mathematical and other organizations; Prizes for expository writing; Glossary; Bibliography; Name index; Subject index. October 1998, 302 pages, Softcover, ISBN 0-89871-420-6, 2000 Mathematics Subject Classification: 00-XX, 68N15, Individual member $31, List $44.50, Order code HBWMSN

470 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Classified Advertisements Positions available, items for sale, services available, and more

St. John's University is an Equal Opportu­ CONNECTICUT MICHIGAN nity Employer and encourages applications YALE UNIVERSITY MICHIGAN STATE UNIVERSITY from women and minorities. Geometry and Topology East Lansing, Ml 48824 proMSc Program in The Department of Mathematics of Yale Industrial Mathematics University invites applications for a posi­ TENNESSEE tion as a tenured associate or full profes­ Direct your students toward one of the sor in the general area of geometry and professional M.Sc. programs. Industry VANDERBILT UNIVERSITY topology and especially in the fields of needs business-savvy mathematicians. See Department of Mathematics symplectic geometry, gauge theory, and http://www.sciencemasters.com/. 4-dimensional topology, and Riemannian We invite applications for a nontenure­ and large-scale geometry. We seek schol­ track assistant professor position in uni­ ars with a record of outstanding achieve­ NEW YORK versal algebra, algebraic logic, and related ment in research who are accomplished areas beginning fall 2004. The position is teachers at both the undergraduate and ST. JOHN'S UNIVERSITY for an initial two-year term, with the possi­ graduate level. We are interested in can­ Department of Mathematics & bility of a one-year renewal. It is intended didates with breadth of expertise in the Computer Science for a recent Ph.D. recipient with demon­ strated research potential and a strong above-mentioned area. 2 Tenure-Track Positions Please send curriculum vitae, descrip­ commitment to excellence in teaching. Du­ tion of research interests, and at least three St. John's University, one of the largest ties include a very active research program letters of recommendation by April 30, Catholic universities in the nation, cur­ and a 2-2 teaching load. 2004, to: rently has two tenure-track positions avail­ Submit your application and supporting Department of Mathematics able in the Department of Mathematics & material in a single mailing to the address Yale University Computer Science. below, inclusive of an email address, a fax P.O. Box 208283 These positions are at the assistant or number, an AMS Standard Cover Sheet, New Haven, CT 06520-8283 associate professor rank in mathematics a research summary and research plan, Attn: Search Committee Geometry on the Queens campus for September and the names and addresses of three and Topology 2004. Applicants should possess a Ph.D. referees who have agreed to write letters Yale University is an Affirmative Ac­ in mathematics and a commitment to both of recommendation for you. Evaluation tion/Equal Opportunity Employer. Women teaching and research. of the applications will commence on and members of minority groups are urged To apply please send a cover letter, January 26, 2004, and will continue until to apply. curriculum vitae, and the names of three the position is filled. references to: Dept. of Mathematics & Algebra and Logic Search Committee Computer Science SJH 334, St. John's Department of Mathematics University, 8000 Utopia Pkwy., Jamaica, Vanderbilt University NY 11439. You may also send email to: 1326 Stevenson Center email:trainac@stjohns . edu. Nashville, TN 37240

Suggested uses for classified advertising are positions available, books 2004 issue-May 28, 2004; September 2004 issue-June 28, 2004; October or lecture notes for sale, books being sought, exchange or rental of 2004 issue-July 28, 2004; November 2004 issue-August 27, 2004. houses, and typing services. U.S. laws prohibit discrimination in employment on the basis of color, The 2004 rate is $100 per inch or fraction thereof on a single column age, sex, race, religion, or national origin. "Positions Available" advertise­ (one-inch minimum), calculated from top of headline. Any fractional text ments from institutions outside the U.S . cannot be published unless they of l/ 2 inch or more will be charged at the next inch rate. No discounts for are accompanied by a statement that the institution does not discriminate multiple ads or the same ad in consecutive issues. For an additional $10 on these grounds whether or not it is subject to U.S. laws. Details and charge, announcements can be placed anonymously. Correspondence will specific wording may be found on page 1373 (vol. 44). be forwarded. Situations wanted advertisements from involuntarily unemployed Advertisements in the "Positions Available" classified section will be set mathematicians are accepted under certain conditions for free publi­ with a minimum one-line headline, consisting of the institution name cation. Call toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or above body copy, unless additional headline copy is specified by the 401-455-4084 worldwide for further information. advertiser. Headlines will be centered in boldface at no extra charge. Ads Submission: Promotions Department, AMS, P. 0. Box 6248, Providence, will appear in the language in which they are submitted. Rhode Island 02940; or via fax: 401-331-3842; or send email to class ads@ There are no member discounts for classified ads. Dictation over the ams. or g. AMS location for express delivery packages is 201 Charles telephone will not be accepted for classified advertising. Street, Providence, Rhode Island 02904. Advertisers will be billed upon Upcoming deadlines for classified advertising are as follows: May 2004 publication. issue-February 27, 2004; June/July 2004 issue-April 29, 2004; August

APRIL 2004 NOTICES OF THE AMS 471 Classified Advertisements

Prospective applicants are invited to visit rank starting fall 2004. We are seeking the Vanderbilt University Deparnnent of applicants from all areas of mathemat­ Mathematics website: http: I lwww. math. ics and applied mathematics, including vanderbilt. edu/. Vanderbilt University is numerical optimization, computational bi­ an Affirmative Action/Equal Opportunity ology, and dynamical systems. Successful Employer. candidates must have a Ph.D. degree in mathematics, applied mathematics, or in a closely related field and will be expected to CYPRUS teach, perform research, and provide ser­ vices to the university. Information about UNIVERSITY OF CYPRUS the university can be found at http: II Department of Mathematics and www .lau. edu.lb/. Statistics Interested applicants should send a let­ / ter of application, a curriculum vitae, three l The department invites applications for letters of recommendation, and official one position in algebra-geometry at the transcripts to: Dr. Abdallah Sfeir, V.P. for rank of lecturer or assistant professor. Academic Affairs, Lebanese American Uni­ The official languages of the university versity, 475 Riverside Drive #1846, New are Greek and/Turkish. The deadline for York, NY 10115; email: [email protected]. applications is May 27, 2004. For more in­ The Lebanese American University is an formation, see http: 1/vrww. mas . ucy. ac. Equal Opportunity Employer operating un­ cy /positions. html. der a charter from the Regents of the State of New York. ENGLAND

OXFORD UNIVERSITY PAKISTAN St Peter's College and the Mathematical Institute LAHORE UNIVERSITY OF Pure Mathematics MANAGEMENT SCIENCES (LUMS) Department of Mathematics We are seeking to appoint, with effect from 1st October 2004, a pure mathema­ The Department of Mathematics invites tician who will be Fellow and Tutor at St. applications for faculty positions at all Peter's College and Faculty Lecturer in the levels to begin in September 2004. The Mathematical Institute of the University of deparnnent has recently started a Ph.D. Oxford, and will teach both in the college program. Candidates are eXPected to have and the institute. The preferred research demonstrated excellence in both teach­ area is in algebraic geometry, broadly in­ ing and research after the Ph.D. LUMS, terpreted, but outstanding candidates in a premier institute of higher learning in other areas of pure mathematics should Pakistan, offers an excellent academic en­ not be deterred from applying. The com­ vironment, as well as growth and develop­ bined college and university salary will ment prospects with a competitive remu­ be according to age on a scale up to neration package. To apply, send a letter $42,900 at age 45 (under review). College of application and a copy of your resume allowances will additionally be payable. with three names of references to lsmat Further particulars of the post and Beg (email: ibeg@lums. edu. pk). For more the application procedure are available at information, visit the university website: http://www.maths.ox.ac . uk/notices/ http://www .lums . edu. pk. The university vacancies. The closing date for applica­ is an Equal OpportunityI Affirmative Ac­ tions is 2 April 2004. Potential applicants tion Employer. are welcome to make informal contact with Dr. lionel Mason, College Fellow and Tutor in Applied Mathematics, by tele­ WANTED phone (+ 44 (0)1865 278913) or email: (lmason@maths. ox. ac. uk). MATH REVIEWS Wanted Math Reviews a,b,c,d from '98 volume and whole volumes '99 and '00; LEBANON email: adam@alpha .lehman. cuny. edu.

LEBANESE AMERICAN UNIVERSITY Division of Computer Science and Mathematics The Division of Computer Science and Mathematics at the Lebanese American University, Lebanon, invites applications for two tenure-track faculty positions (one in mathematics and the other in applied mathematics) at the assistant professor

472 NOTICES OF THE AMS VOLUME 51, NUMBER 4 CAMBRIDGE

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Fold here Meetings & Conferences oftheAMS

IMPORTANTINFORMATIONREGARDING MEETINGSPROGRAMS:AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information withlinks totheabstractforeachtalkcanbefoundon theAMSwebsite. See http: I /www. ams. org/meeti ngs/.Programs and abstracts will continue to be displayed on the AMS website in the Meetings and Conferences section until about three weeks after the meeting is over. Final programs for Sectional Meetings will be archived on the AMS website in an electronic issue of the Notices as noted below for each meeting.

Special Sessions Athens, Ohio Algebraic Coding Theory, Marcus Greferath, San Diego Ohio University State University, and Sergio R. L6pez-Permouth, Ohio Uni­ versity. March 26-27,2004 Differential Equations and Control Theory, Sergiu Aizicovici Friday - Saturday and Nicolai Pavel, Ohio University. Meeting #995 Discrete Structures and Complex Dynamics, Mario Bonk and Central Section Lukas Geyer, University of Michigan. Associate secretary: Susan ]. Friedlander Dynamical Systems, Patrick D. McSwiggen, University of Announcement issue of Notices: January 2004 Cincinnati, and Todd Young, Ohio University. Program first available on AMS website: February 12, 2004 Fast Algorithms in Numerical Analysis, George Farm, Oak Program issue of electronic Notices: March 2004 Ridge National Laboratory, and Martin J. Mohlenkamp, Issue of Abstracts: Volume 25, Issue 2 Ohio University. Deadlines Groups, Representations, and Characters, Mark Lewis, For organizers: Expired Kent State University, and Thomas R. Wolf, Ohio Univer­ For consideration of contributed papers in Special Sessions: sity. Expired Integrable Systems in Mathematics and Physics, Michael For abstracts: Expired Gekhtman, University of Notre Dame, and Luen Chau U, Pennsylvania State University. Invited Addresses Linear Algebra and Its Applications, S. K. Jain, Ohio Uni­ Mario Bonk, University of Michigan, Dynamics on fractal versity, and Michael Neumann, University of Connecticut. spheres. Probabilistic and Asymptotic Aspects of Group Theory, Irene Gamba, University of Texas at Austin, On the Boltz­ Rostislav Grigorchuk, Texas A&M University, Mark Sapir, mann equation for inelastic interactions. Vanderbilt University, and Zoran Sunik, Texas A&M Rostislav Ivanovich Grigorchuk, Texas A&M University, University. Kolmogorov complexity, growth and algorithmic problems. Recent Trends in Infinite-Dimensional Banach Space Theory, Eric G. Zaslow, Northwestern University, Mirror symme­ Beata Randrianantoanina and Narcisse Randrianantoanina, try: An orientation. Miami University.

APRIL 2004 NOTICES OF THE AMS 479 Meetings & Conferences

Statistics and Probability, Maria Rizzo and Vladimir Vino­ Fluid Problems and Related Questions, Maria Elena Schon­ gradov, Ohio University. bek, University of California Santa Cruz, and Yuxi Zheng, Theory of Rings and Modules, Nguyen Viet Dung, Franco Pennsylvania State University. Guerriero, Dinh Van Huynh, and Pramod Kanwar, Ohio Modern Problems of Integration: Theory and Applications, University. Mark Burgin, University of California Los Angeles. Wavelets, Other Multiscale Methods and Their Applications, Noncommutative Algebra and Algebraic Geometry, En-Bing lin, University of Toledo, and Xiaoping Annie Lance W. Small, University of California San Diego, and Paul Shen, Ohio University. Smith, University of Washington. Nonlinear and Harmonic Analysis, Rowan Killip and Christoph Martin Thiele, University of California Los Los Angeles, Angeles. Partial Differential Equations, Igor Kukavica, University of California Southern California, and Qi S. Zhang, University of California University of Southern California Riverside. Recent Advances in the Mathematical Analysis of Geophysical April3-4, 2004 and Hydrodynamical Models, Mohammed Ziane, University Saturday - Sunday of Southern California. Smooth Ergodic Theory and Related Topics, Nicolai Haydn, Meeting #996 University of Southern California, and Huyi Hu, Michigan Western Section State University. Associate secretary: Michel L. Lapidus Announcement issue of Notices: February 2004 Registration and Meeting Information (Updated) Program first available on AMS website: February 19, 2004 The registration desk will be open on Saturday, April 3, Program issue of electronic Notices: April 2004 from 7:30 a.m. until 4:00 p.m. and Sunday, April 4, from Issue of Abstracts: Volume 25, Issue 2 8:00 a.m. until noon. Deadlines For organizers: Expired For consideration of contributed papers in Special Sessions: Lawrenceville, Expired For abstracts: Expired New Jersey Rider University Invited Addresses Dan Boneh, Stanford University, Applications of bilinear Aprill 7-18,2004 maps to cryptography. Saturday - Sunday Maria Elena Schonbek, University of California Santa Cruz, Meeting #997 Fluid equations and their asymptotic behavior. Eastern Section Paul Smith, University of Washington, Noncommutative Associate secretary: Lesley M. Sibner algebraic geometry. Announcement issue of Notices: February 2004 Christoph Martin Tirlele, University of California Los Angeles, Program first available on AMS website: March 4, 2004 Some basic questions in nonlinear Fourier analysis. Program issue of electronic Notices: April 2004 Issue of Abstracts: Volume 25, Issue 3 Special Sessions Deadlines Arithmetic Geometry and K- Theory, Thomas Geisser and Wayne Raskind, University of Southern California. For organizers: Expired For consideration of contributed papers in Special Sessions: Complex and Hyperbolic Geometry, Francis Bonahon and Expired Dragomir Saric, University of Southern California. For abstracts: Expired Contact and Symplectic Geometry, Dragomir Dragnev, Ko Honda, and Sang Seon Kim, University of Southern Invited Addresses California. Sylvia Serfaty, New York University-Courant Institute, Dynamic Equations on Time Scales: Theory and Applications, Analysis of vortices in the magnetic Ginzburg-£andau John M. Davis and Johnny Henderson, Baylor University, m~d I and Qin Sheng, University of Dayton. Dennis P. Sullivan, Graduate Center of CUNY and SUNY Financial Mathematics, Jaksa Cvitanic and Janfeng Zhang, at Stony Brook, Poincare duality, free loop space, and University of Southern California. strings.

480 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Meetings & Conferences

Wim F. Sweldens, Bell Laboratories, Title to be announced. Variational Methods in Classical Mechanics, John N. Mather, Gaoyong Zhang, Polytechnic University, Affine isoperi­ Princeton University, and Vadim Y. Kaloshin, Institute for metric inequalities. Advanced Study.

Special Sessions Algebraic Geometry and Mirror Symmetry, Ciprian Borcea, Houston, Texas Rider University. Hyatt Regency Houston Analytic Convex Geometry, Alina Stancu, Polytechnic May 1 3-1 5, 2004 University, and Elisabeth Werner, Case Western Reserve University. Thursday - Saturday Automorphic Forms and Analytic Number Theory, Stephen Meeting #998 Miller, Rutgers University, and Ramin Takloo-Bighash, Sixth International ]oint Meeting of the AMS and the Princeton University. Sociedad Matematica Mexicana (SMM). CR Geometry and Singularities, Joseph J. Kohn, Princeton Associate secretary: John L. Bryant University, John P. D'Angelo, University of Illinois, Xiaojun Announcement issue of Notices: February 2004 Huang, Rutgers University, and Andreea Nicoara, Harvard Program first available on AMS website: March 11, 2004 University. Program issue of electronic Notices: April 2004 Issue of Abstracts: Volume 25, Issue 3 Commutative Algebra and Algebraic Geometry, Alberto Corso, University of Kentucky, Claudia Polini, University Deadlines of Notre Dame, and Wolmer V. Vasconcelos, Rutgers For organizers: Expired University. For consideration of contributed papers in Special Sessions: Convergence ofRiemannian Manifolds, Christina Sormani, Expired Herbert H. Lehman College (CUNY), XiaochunRong, Rutgers For abstracts: Expired State University, and Guofang Wei, University of California Santa Barbara. Invited Addresses Elliptic Surfaces and Elliptic Fibrations, William L. Hoyt, Luchezar Avramov, University of Nebraska, Homomor­ Rutgers University, Joseph H. Silverman, Brown University, phisms of commutative rings. and Charles F. Schwartz, Rider University. Persi W. Diaconis, Stanford University, Title to be Geometry and Arithmetic of Lattices, John H. Conway, announced. Princeton University, and Derek A. Smith, Lafayette College. Samuel Gitler, CINVESTAV, Title to be announced. Geometry ofProteinModelling, lleana Streinu, Smith College, Adolfo Sanchez-Valenzuela, Centro de Investigacion en and Jack Snoeyink, University of North Carolina at Chapel Matematicas, Title to be announced. Hill. Jose Seade-Kuri, UNAM, Knots, fibered links, and singu­ Group Cohomology and Related Topics (in Honor of William larities. Browder's 70th Birthday), Alejandro Adem, University of Bernd Sturmfels, University of California Berkeley, Trop­ Wisconsin, and Jonathan Pakianathan, University of icalgeometry (Erdos Memorial Lecture). Rochester. Special Sessions Homotopical Physics, James Stasheff, University of North Carolina, Thomas J. Lada, North Carolina State University, Algebraic Geometry, Pedro Luis Del Angel R., CIMAT, and Alexander A. Voronov, University of Minnesota. Emma Previato, Boston University, Frank Sottile, Univer­ sity of Massachusetts, Amherst, and Isidro Nieto, Univer­ Homotopy Theory (in Honor of William Browder's 70th sidad de San Nicolas Hidalgo. Birthday), Martin Bendersky, Hunter College, and Donald Davis, Lehigh University. Algebraic Topology, Miguel A. Xicotencatl, CINVESTAV, and Frederick R. Cohen, University of Rochester. Strings and Branes, Thomas P. Branson, University of Iowa, and S. James Gates, University of Maryland. Associative Rings, jose Rios Montes, UNAM, Maria-Jose Arroyo, UAM-Iztapalapa, and Sergio R. L6pez-Permouth, Surgery (in Honor of William Browder's 70th Birthday), Ohio University. Frank S. Quinn, Virginia Polytechnic Institute & State Coding Theory and Cryptography, Horacio Tapia-Recillas, University. UAM-Iztapalapa, and Neal I. Koblitz, University of Wash­ Theory of Initial- and Boundary-Value Problems for Some ington. Nonlinear Partial Differential Equations, Laihan Luo, Complex Analysis and Operator Theory, Enrique Ramirez Richard Stockton College of New Jersey. de Arellano, CINVESTAV, John E. Fornaess, University of Tomography and Integral Geometry, Andrew Markoe, Michigan, Ann Arbor, and Norberto Salinas, University of Rider University, and Eric Todd Quinto, Tufts University. Kansas.

APRIL 2004 NOTICES OF THE AMS 481 Meetings & Conferences

Continua Theory and General Topology (in Honor of David Bellamy's 60th Birthday), Janusz J. Charatonik, UNAM, Nashville, Tennessee Charles L. Hagopian, California State University, Sacra­ Vanderbilt University mento, and Sergio Macias, UNAM. Continuous Distributed Parameter Models in Mathematical October 16-1 7, 2004 Biology, William E. Fitzgibbon, University of Houston, and Saturday - Sunday Jorge X. Velasco Hernandez, Instituto Mexicana del Meeting #999 Petroleo. Southeastern Section Curvature and Geodesics, David D. W. Bao, University of Associate secretary: John L. Bryant Houston, and Lilia Del Riego, Universidad Aut6noma San Announcement issue of Notices: August 2004 Luis Potosi. Program first available on AMS website: September 2, 2004 Designing Frames and Wavelets: From Theory to Digitiza­ Program issue of electronic Notices: October 2004 tion, Peter R. Massopust, Tuboscope Pipeline Services, Issue of Abstracts: Volume 25, Issue 4 and Manos I. Papadakis, University of Houston. Deadlines Differential Geometry, Raul Quiroga Barranco, CINVESTAV, For organizers: March 16, 2004 and Alberto Candel, California State University, Northridge. For consideration of contributed papers in Special Sessions: Dynamical Systems, Renato Iturriaga, CIMAT, and Rafael June 29, 2004 de la Uave, University of Texas at Austin. For abstracts: August 24, 2004 Geometric Variational Problems, Clara Garza-Hume, IIMAS, Invited Addresses UNAM, and Michael Wolf and Robert M. Hardt, Rice University. Ruth M. Charney, Brandeis University, Title to be announced. Graph Theory and Combinatorics, Gelacio Salazar, IICO, Peter S. Ozsvath, Columbia University, Title to be UASLP, Isidoro Gitler, CINVESTAV, and Nathaniel Dean, announced. Texas Southern University. Sorin T. Popa, University of California Los Angeles, Title Harmonic and Functional Analysis, Salvador Perez-Esteva, to be announced. UNAM -Cuernavaca, Carlos Bosh-Giral, ITAM, and Josefina Rudi Weikard, University of Alabama at Birmingham, Title Alvarez, New Mexico State University. to be announced. Homological Algebra of Commutative Rings, Srikanth Iyen­ gar, University of Nebraska, and Graham J. Leuschke, University of Toronto. Albuquerque, Low Dimensional Topology, Victor Nunez, CIMAT, and Luis G. Valdez, University of Texas, El Paso. New Mexico Mathematical Physics, Carlos Villegas-Blas, UNAM, and University of New Mexico Alejandro Uribe, University of Michigan, Ann Arbor. October 16-1 7, 2004 Mathematical Problems in the Analysis of Synchronous Saturday - Sunday States in Networks, Kresimir Josie, University of Houston, and Valentin Afraimovich, IICO-UASLP. Meeting #1 000 Nonlinear Analysis, Monica Clapp, UNAM, and Alfonso Western Section Castro, University of Texas at San Antonio. Associate secretary: Michel L. Lapidus Probability and Stochastic Processes, Daniel Hernandez Announcement issue of Notices: August 2004 Program first available on AMS website: September 3, 2004 Hernandez, CIMAT, and Christian Houdre, Georgia Insti­ Program issue of electronic Notices: October 2004 tute of Technology. Issue of Abstracts: Volume 25, Issue 4 Problems and Issues in Electronic Publishing, Klaus Kaiser, University of Houston, Bernd Wegner, Technische Univer­ Deadlines sWit Berlin, and Enrique Ramirez de Arellano, CINVESTAV. For organizers: March 16, 2004 For consideration of contributed papers in Special Sessions: Representations of Algebras, Rita Esther Zuazua Vega, June 29, 2004 UNAM, and Gordana G. Todorov, Northeastern University. For abstracts: August 24, 2004 Space and Time Decomposition Methods in Computational and Applied Mathematics, Roland Glowinski and Tsorng­ Invited Addresses Whay Pan, University of Houston, and L. Hector Sara C. Billey, University of Washington, Seattle, Title to Juarez V., UAM-Iztapalapa. be announced.

482 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Meetings & Conferences

Peter Ebenfelt, University of California San Diego, Title to Program first available on AMS website: September 9, 2004 be announced. Program issue of electronic Notices: October 2004 Theodore Stanford, New Mexico State University, Title to Issue of Abstracts: Volume 2S, Issue 4 be announced. Deadlines Craig A. Tracy, University of California Davis, Title to be For organizers: March 23, 2004 announced. For consideration of contributed papers in Special Sessions: Special Sessions July 7, 2004 For abstracts: August 31, 2004 Algebraic Geometry (Code: SS 3A), Hirotachi Abo and Chris Peterson, Colorado State University. Invited Addresses Arithmetic Geometry (Code: SS 6A), Alexandru Buium and Ian Agol, University of Illinois at Chicago, Title to be Michael J. Nakamaye, University of New Mexico. announced. Categories and Operads in Topology, Geometry, Physics and Robert W. Ghrist, University of Illinois, Title to be Other Applications (Code: SS SA), Hanna Ewa Makaruk and announced. Robert Michal Owczarek, Los Alamos National Laboratory, Yuri Manin, Northwestern University, Title to be announced. and Zbigniew Oziewicz, Universidad Nacional Aut6noma de Mexico. Paul Seidel, Imperial College-London and University of Chicago, Title to be announced. Financial Mathematics:The Mathematics of Derivative Securities (Code: SS 4A), Maria Cristina Mariani, New Mexico Special Sessions State University, and Osvaldo Mendez, University of Texas at El Paso. Algebraic Topology: Interactions with Representation Theory and Algebraic Geometry (Code: SS 13A), Paul G. Goerss, Interactions in Riemannian Geometry (Code: SS 8A), Northwestern University, and Jesper Kragh Grodal, Charles P. Boyer and Krzysztof Galicki, University of University of Chicago. New Mexico. Applications ofMotives (Code: SS 18A), Eric M. Friedlander, Mathematical Methods in Turbulence (Code: SS 9A), Monika Northwestern University, Alexander Goncharov, Brown Nitsche and Vachtang Poutkaradze, University of New University, Mikhail Kapranov, Yale University, and Yuri Mexico. Manin, Max Planck Institute for Mathematics. Multiscale Methods and Sampling in Time-Frequency Analy­ Codes and Applications (Code: SS SA), William C. Huffman, sis (Code: SS lOA), Jeffrey Andrew Hogan, University Loyola University of Chicago, and VeraS. Pless, Univer­ of Arkansas, and Joseph D. Lakey, New Mexico State sity of Illinois at Chicago. University. Computability Theory and Applications (Code: SS 8A), Nonlinear Partial Differential Equations Applied to Materials Robert I. Soare and Denis R. Hirschfeldt, University of Science (Code: SS llA), Patricia Bauman, Purdue University, Chicago. and Tiziana Giorgi, New Mexico State University. Differential Geometry (Code: SS lOA), Anders Ingemar Random Matrix Theory and Growth Processes (Code: SS lA), Linner and Hongyou Wu, Northern Illinois University. Craig A. Tracy, University of California Davis. Extremal Combinatorics (Code: SS 2A), Dhruv Mubayi and Several Complex Variables and CR Geometry (Code: SS Yi Zhao, University of Illinois at Chicago. 2A), Peter Ebenfelt, University of California San Diego, and Marshall A. Whittlesey, California State University San Fluid Dynamics, Diffusion and Reaction (Code: SS 4A), Marcos. Peter S. Constantin and Leonid V. Ryzhik, University of Chicago. Spectral Geometry (Code: SS 7A), Ivan G. Avramidi, New Mexico Institute of Mining and Technology, and Thomas Geometric Aspects of the Langlands Program (Code: SS Patrick Branson, University of Iowa. 17A), Edward Frenkel, University of California Berkeley, Mark Goresky, Institute for Advanced Study, and Kari Vilonen, Northwestern University. Geometric Partial Differential Equations (Code: SS 7A), Gui­ Evanston, Illinois Qiang Chen and Jared Wunsch, Northwestern University. Northwestern University Index Theory, Morse Theory, and the Witten Deformation October 23-24, 2004 Method (Code: SS 3A), Igor Prokhorenkov and Ken Richard­ son, Texas Christian University. Saturday - Sunday Iterated Function Systems and Analysis on Fractals (Code: Meeting #1 001 SS 12A), Ka-Sing Lau, Chinese University of Hong Kong, Central Section and Stephen S.-T. Yau, University of Illinois at Chicago. Associate secretary: Susan]. Friedlander Mathematical Problems in Robotics (Code: SS lSA), Announcement issue of Notices: August 2004 Robert W. Ghrist, University of Illinois at Urbana-Champaign.

APRIL 2004 NOTICES OF 1HE AMS 483 Meetings & Conferences

Modern Schubert Calculus (Code: SS 1A), Ezra Miller, Invariants of Knots and 3-Manifolds (Code: SS 1A), University of Minnesota, and Frank Sottile, University of Marta M. Asaeda, University of Maryland, jozef H. Przy­ Massachusetts. tycki, George Washington University, and AdamS. Sikora, Nonlinear Partial Differential Equations and Applications SUNY at Buffalo. (Code: SS 6A), Gui-Qiang Chen, Northwestern University, and Mikhail Feldman, University of Wisconsin at Madison. Nonlinear Waves (Code: SS 14A), jerry L. Bona, University Atlanta, Georgia of Illinois at Chicago, Shuming Sun, Virginia Polytechnic Atlanta Marriott Marquis and Institute and State University, and Bingyu Zhang, Univer­ sity of Cincinnati. Hyatt Regency Atlanta Solving Polynomial Systems (Code: SS 9A), Anton Leykin January 5-8, 2005 and Jan Verschelde, University of Illinois at Chicago. Wednesday - Saturday Spectral Problems ofDifferential Operators (Code: SS 16A), Qingkai Kong, Hongyou Wu, and Anton Zettl, Northern Meeting #1 003 Illinois University. ]oint Mathematics Meetings, including the 111 th Annual Stability Issues in Fluid Dynamics (Code: SS 11A), Susan]. Meeting of the AMS, 88th Annual Meeting of the Mathe­ Friedlander and Roman Shvydkoy, University of Illinois matical Association ofAmerica (.MAA), annual meetings of at Chicago. the Association of Women in Mathematics (A WM) and the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic (ASL). Pittsburgh, Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2004 Program first available on AMS website: To be announced Pennsylvania Program issue of electronic Notices: January 2005 University of Pittsburgh Issue of Abstracts: To be announced

November 6-7,2004 Deadlines Saturday - Sunday For organizers: April 5, 2004 For consideration of contributed papers in Special Sessions: Meeting #1 002 To be announced Eastern Section For abstracts: To be announced Associate secretary: Lesley M. Sibner For summaries of papers to MAA organizers: To be Announcement issue of Notices: September 2004 announced Program first available on AMS website: September 23, 2004 Program issue of electronic Notices: November 2004 Issue of Abstracts: Volume 25, Issue 4 Bowling Green, Deadlines Kentucky For organizers: April 7, 2004 Western Kentucky University For consideration of contributed papers in Special Sessions: July 20, 2004 March 1 8-1 9, 2004 For abstracts: September 14, 2004 Friday - Saturday

Invited Addresses Meeting #1 003 Jeffrey F. Brock, Brown University, Title to be announced. Southeastern Section Der-Chen Chang, Georgetown University, Title to be Associate secretary: John L. Bryant announced. Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Robert Schapire, Princeton University, Title to be Program issue of electronic Notices: To be announced announced. Issue of Abstracts: To be announced Ofer Zeitouni, University of Minnesota, Minneapolis, Title to be announced. Deadlines For organizers: July 19, 2004 Special Sessions For consideration of contributed papers in Special Sessions: Convexity and Combinatorics (Code: SS 2A), James F. To be announced Lawrence and Valeriu Soltan, George Mason University. For abstracts: To be announced

484 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Meetings & Conferences

Associate secretary: Michel L. Lapidus Newark, Delaware Announcement issue of Notices: To be announced Program first available on AMS website: To be announced University of Delaware Program issue of electronic Notices: To be announced April2-3, 2005 Issue of Abstracts: To be announced Saturday - Sunday Deadlines Meeting #1 005 For organizers: To be announced For consideration of contributed papers in Special Sessions: Eastern Section To be announced Associate secretary: Lesley M. Sibner For abstracts: To be announced Announcement issue of Notices: 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 Dynamical Systems in Neuroscience (Code: SS lA), Eugene M. Izhikevich, The Neurosciences Institute. Deadlines For organizers: September 2, 2004 For consideration of contributed papers in Special Sessions: Mainz, Germany To be announced For abstracts: To be announced June 16-19,2005 Thursday - Sunday Meeting #l 008 Lubbock, Texas Second ]oint International Meeting with the Deutsche Texas Tech University Mathematiker-Vereinigung (DMV) and the Oesterrichische Mathematische Gesellschaft (OMG) April8-1 0, 2005 Associate secretary: Susan]. Friedlander Friday - Sunday Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Meeting #1 006 Program issue of electronic Notices: To be announced Central Section Issue of Abstracts: To be announced Associate secretary: Susan ]. Friedlander Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be announced For organizers: To be announced Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Sessions: Issue of Abstracts: To be announced To be announced For abstracts: To be announced Deadlines For organizers: To be announced Invited Addresses For consideration of contributed papers in Special Sessions: Helene Esnault, University of Essen, Title to be announced. To be announced For abstracts: To be announced Richard Hamilton, Columbia University, Title to be announced. Special Sessions Michael J. Hopkins, Massachusetts Institute of Technology, Topology of Continua (Code: SS lA), Wayne Lewis, Texas Title to be announced. Tech University. Christian Krattenthaler, University of Lyon, Title to be announced. Frank Natterer, University of Muenster, Title to be Santa Barbara, announced. Horng-Tzer Yau, New York University and Stanford California - University, Title to be announced. University of California Santa Barbara Special Sessions April16-1 7, 2005 Algebraic Geometry, Yuri Tschinkel, Georg-August­ Saturday - Sunday Universitat Gottingen, and Brendan E. Hassett, Rice University. Meeting #1 007 Discrete Geometry, Jacob Eli Goodman, The City College Western Section of New York, CUNY, Emo Welzl, Eidgen Technische

APruL2004 NOTICES OF THE AMS 485 Meetings & Conferences

Hochschule, and Gunter M. Ziegler, Technical University of Berlin. San Antonio, Texas Functional Analytic and Complex Analytic Methods in Linear Henry B. Gonzalez Convention Center Partial Differential Equations, R. Meise, University of Dusseldorf, B. A. Taylor, University of Michigan, and January 1 2-1 5, 2006 Dietmar Vogt, University of Wuppertal. Thursday - Sunday Mathematics Education, Gunter Torner, Universitat joint Mathematics Meetings, including the 112th Annual Duisburg-Essen. Meeting of the AMS, 89th Annual Meeting of the Mathematical Association of America, annual meetings of Nonlinear Waves, Herbert Koch, University of Dortmund, the Association for Women in Mathematics (A WM) and the and Daniel I. Tataru, University of California Berkeley. National Association of Mathematicians (NAM), and the Stochastic Analysis on Metric Spaces, Laurent Saloff-Coste, winter meeting of the Association for Symbolic Logic (ASL}. Cornell University, Karl-Theodor Sturm, University of Associate secretary: John L. Bryant Bonn, and Wolfgang Woess, Graz Technical University. Announcement issue of Notices: October 2005 Program first available on AMS website: To be announced Program issue of electronic Notices: January 2006 Johnson City, Issue of Abstracts: To be announced Tennessee Deadlines For organizers: April12, 2005 East Tennessee State University For consideration of contributed papers in Special Sessions: October 1 5-16, 2005 To be announced For abstracts: To be announced Saturday - Sunday For summaries of papers to MAA organizers: To be Southeastern Section announced Associate secretary: John L. Bryant Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced New Orleans, Issue of Abstracts: To be announced Louisiana Deadlines New Orleans Marriott and For organizers: March 15, 2005 For consideration of contributed papers in Special Sessions: Sheraton New Orleans Hotel To be announced January 4-7, 2007 For abstracts: To be announced Thursday - Sunday joint Mathematics Meetings, including the 113th Annual Meeting of the AMS, 90th Annual Meeting of the Mathe­ Lincoln, Nebraska matical Association ofAmerica (MAA), annual meetings of University of Nebraska in Lincoln the Association for Women in Mathematics (A WM) and the National Association of Mathematicians (NAM), and the October 21-22, 2005 winter meeting of the Association for Symbolic Logic (ASL). Friday - Saturday Associate secretary: Susan]. Friedlander Central Section Announcement issue of Notices: October 2006 Associate secretary: Susan]. Friedlander Program first available on AMS website: To be announced Announcement issue of Notices: August 2005 Program issue of electronic Notices: January 2007 Program first available on AMS website: To be announced Issue of Abstracts: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines For organizers: April 4, 2006 Deadlines For consideration of contributed papers in Special Sessions: For organizers: To be announced To be announced For consideration of contributed papers in Special Sessions: For abstracts: To be announced To be announced For summaries of papers to MAA organizers: To be For abstracts: To be announced announced

486 NOTICES OF THE AMS VOLUME 51, NUMBER 4 Meetings & Conferences Photo Index to Pages 400, 401 San Diego, California 2004 Joint Mathematics San Diego Convention Center 1 Meetings inPhoenix 16 January 6-9, 2008 2 17 Sunday - Wednesday 7 I ]oint Mathematics Meetings, including the 114th Annual ~ Meeting of the AMS, 91st Annual Meeting of the Mathe­ 3 18 matical Association ofAmerica (MAA), annual meetings of 8 the Association for Women in Mathematics (A VVM) and the 4 19 National Association of Mathematicians (NAM), and the - winter meeting of the Associaiton for Symbolic Logic (ASL). 5 9 20 Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2007 10 Program first available on AMS website: November 1, 2007 6 21 Program issue of electronic Notices: January 2008 Page 401 Issue of Abstracts: Volume 29, Issue 1 Page400

Deadlines Page 400 For organizers: April 6, 2007 For consideration of contributed papers in Special Sessions: 1. Entrance to Exhibits Hall To be announced 2. Ribbon cutting-opening of exhibits For abstracts: To be announced 3. Registration desk For summaries of papers to MAA organizers: To be 4. AMS author Steven Krantz (left) at book signing announced 5. AMS booth in exhibits area 6. Exhibition area Washington, District 7. Entrance to Phoenix Civic Plaza 8. Historic St. Mary's Basilica of Columbia 9. Exhibits Hall from above Marriott Wardman Park Hotel 10. 2004 Steele Prize winner Cathleen Morawetz and Omni Shoreham Hotel January 7-1 0, 2009 Page 401 Wednesday - Saturday 11. AMS Executive Director John Ewing with retiring ]oint Mathematics Meetings, including the 115th Annual MR Executive Editor Jane Kister Meeting of the AMS, 92nd Annual Meeting of the Mathe­ 12. 2004 Distinguished Public Service Awardee Richard matical Association ofAmerica (MAA), annual meetings of Tapia the Association for Women in Mathematics (A VVM) and the 13. Woven baskets, Mathematics and the Arts exhibit National Association of Mathematicians (NAM), and the 13. Invited Address speaker Stephen Wolfram winter meeting of the Association for Symbolic Logic (ASL). Associate secretary: Lesley M. Sibner 14. Left to right AMS President David Eisenbud, 2003 Announcement issue of Notices: October 2008 Morgan Prize winner Melanie Wood, MAA President Program first available on AMS website: November 1, 2008 Ron Graham, SIAM President James M. Hyman Program issue of electronic Notices: January 2009 15. Who Wants To Be A Mathematician contestants Issue of Abstracts: Volume 30, Issue 1 with host Mike Breen 16. Game contestant Jonathan Lo Deadlines For organizers: April 7, 2008 17. David Eisenbud, 2004 Norbert Wiener Prize winner For consideration of contributed papers in Special Sessions: James Sethian, James Hyman To be announced 18. Lobby, Civic Plaza For abstracts: To be announced 19. Emaillab For summaries of papers to MAA organizers: To be 20. Employment Center announced

APRIL 2004 NOTICES OF THE AMS 487 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; 92521-013 5; e-mail: l a pi dus@math. uc r. edu; telephone: 909- e-mail: lsi bner@duke . poly. edu; telephone: 718-260-3505. 787-3113. Southeastern Section: John L. Bryant, Department of Math­ Central Section: Susan J, Friedlander, Department of Math­ ematics, Florida State University, Tallahassee, FL 32306-4510; ematics, University of Illinois at Chicago, 851 S. Morgan (M/ C e-mail: bryant@math. fsu. edu; telephone: 850-644-5805. 249), Chicago, lL 60607-7045; e-mail: susan@math. nwu. edu; tele­ phone: 312-996-3041.

The Meetings and Conferences section of the Notices January 4-7 New Orleans, Louisiana p.486 gives information on all AMS meetings and conferences Annual Meeting approved by press time for this issue. Please refer to the page 2008 numbers cited in the table of contents on this page for more January 6-9 San Diego, California p.487 detailed information on each event. Invited Speakers and Annual Meeting Special Sessions are listed as soon as they are approved by 2009 the cognizant program committee; the codes listed are needed January 7-10 Washington, DC p.487 for electronic abstract submission. For some meetings the list Annual Meeting may be incomplete. Information in this issue may be dated. Up-to-date meeting and conference information can be found at www. ams. org/meeti ngs/. Important Information regarding AMS Meetings Potential organizers, speakers, and hosts should refer to Meetings: page 84 in the January 2004 issue of the Notices for general 2004 information regarding participation in AMS meetings and March 26-27 Athens, Ohio p.479 conferences. April3-4 Los Angeles, California p.480 Aprill7-18 Lawrenceville, New Jersey p.480 Abstracts May 13-15 Houston, Texas p.481 Several options are available for speakers submitting October 16-17 Nashville, Tennessee p.482 abstracts, including an easy-to-use interactive Web form. No October 16-17 Albuquerque, New Mexico p.482 knowledge of mpc is necessary to submit an electronic form, October 23-24 Evanston, Illinois p.483 although those who use mpc may submit abstracts with such coding, and all math displays and similarily coded ma­ November 6-7 Pittsburgh, Pennsylvania p.484 terial (such as accent marks in text) must be typeset in mpc. To see descriptions of the forms available, visit http: I I 2005 www. ams. org/abstracts/i nstructi ons. html, or send mail January 5-8 Atlanta, Georgia p.484 to abs-submi t@ams. org, typing help as the subject line; de­ Annual Meeting scriptions and instructions on how to get the template of your March 18-19 Bowling Green, Kentucky p.484 choice will be e-mailed to you. April2-3 Newark, Delaware p.485 Completed email abstracts should be sent to abs-submi t@ April8-10 Lubbock, Texas p.485 ams. org, typing submission as the subject line. Questions Aprill6-17 Santa Barbara, California p.485 about abstracts may be sent to abs-i nfo@ams. org. June 16-19 Mainz, Germany p.485 Paper abstract forms may be sent to Meetings & Confer­ October 15-16 Johnson City, Tennessee p.486 ences Department, AMS, P.O. Box 6887, Providence, RI 02940. October 21-22 Lincoln, Nebraska p.486 There is a $20 processing fee for each paper abstract. There is no charge for electronic abstracts. Note that all abstract dead­ 2006 lines are strictly enforced. January 12-15 San Antonio, Texas p.486 Close attention should be paid to specified deadlines in this Annual Meeting issue. Unfortunately, late abstracts cannot be accommodated. 2007

Conferences: (See http://www.ams.org/meetings/ for the most up-to-date information on these conferences.) June 5-July 24, 2004: Joint Summer Research Conferences in the Mathematical Sciences, Snowbird, Utah. (See November 2003 Notices, page 1363.)

488 NOTICES OF THE AMS VOLUME 51, NUMBER 4 AMERICAN MATHEMATICAL SOCIETY

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