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KOX, Universiteit van Amsterdam, The Netherlands (Eds.) Contributors: A. Aldroubi, L. Baggett, G. Benke, C.A. d'Heres, France and SHRAWAN KUMAR, University of North This volume from the Einstein Series is based largely on Cabrelli, P.G. Casazza, 0. Christensen, W. Czaja, C. Davis, Carolina, Chapel Hill papers presented at the Sixth International Conference H. G. Feichtinger, M. Fickus, J.-P. Gabardo, K. Groechenig, The theory of Frobenius splittings has made a signifi­ on the History of General Relativity. These contributions K. Guo, E. Hayashi, C. Heil, H.P. Heinig, J. Hogan, P.E.T. cant impact in the study of the geometry of flag varieties from notable experts offer both new and historical Jorgensen, K. Kornelson, J. Kovacevic, D. Labate, J.D. and representation theory. This work systematically insights on gravitation, general relativity, cosmology, Lakey, D.R. Larson, M.T. Leon, S. Li, W.-Q. Lim, A. Linder, develops the theory of Frobenius splittings and covers unified field theory, and the history of science. U.M. Molter, E. Schulz, T. Sorrels, D. Speegle, K.F. Taylor, all its major developments. The concise, efficient J.C. Tremain, D.F. Walnut, Y. Wang, G. Weiss, E. Wilson Contributors include: K. Brading, G. Gale, H.F.M. exposition unfolds from basic introductory material on 2005/APPROX. 320 PP./HARDCOVER Goenner, J. Goldberg, S. Katzir, D. Kennefick, H. Kragh, Frobenius splittings- definitions, properties, and ISBN 0-8176-3778-8/$89.95 (TENT.) C. Lehner, U. Majer, J. Mattingly, E.T. Newman, J.D. examples- to cutting-edge research. Many examples, APPLIED AND NUMERICAL HARMONIC ANALYSIS Norton, J. Renn, R. Rynasiewicz, J.M. Sanchez-Ron, T. exercises, and open problems suggested throughout. Sauer, C. Smeenk, J. Stachel, M. Wazeck, D. WUnsch 2005/246 PP./HARDCOVER/ISBN 0-8176-4191-2/$69.95 PROGRESS IN MATHEMATICS, VOL. 231 2005/APPROX. 510 PP./HARDCOVER ISBN 0-8176-4380-X/$99.95 !TENT.) EINSTEIN STUDIES, VOL. 11

CALL: 1-800-777-4643 • FAX: (201) 348-4505 E-MAIL: [email protected] • www.birkhauser.com Please mention promotion #Y9698 when ordering. Prices are valid in the Americas only Birkhiiuser and are subject to change without notice. For price and ordering information outside the Americas, Boston · Basel · Berlin please contact Birkhauser Verlag AG by E-mail: [email protected] 2/05 Promotion #Y9698 Notices March 2005 of the American Mathematical Society

Feature Articles 324 Nonholonomic Dynamics Anthony M. Bloch, jerrold E. Marsden, Dmitry V. Zenkov Nonholonomic dynamical systems are mechanical systems, like a falling rolling disk, whose velocities have constraints that can't be derived from their position constraints. The authors review the history and some recent progress in the study of such systems.

334 A 1940 Letter of Andre Weil on Analogy in Mathematics Translated by Martin H. Krieger Martin H. Krieger translates Andre Weil's 1940 letter to his sister. Weil comments, from prison, and to a nonexpert, on the history of number theory and on analogy in mathematics research. Weil's writing is of interest both historically and for his insights into doing mathematical research.

Communications Commentary

342 WHAT IS ... a Dimer? 321 Opinion Richard Kenyon and Andrei Okounkov 322 Letters to the Editor 348 A Tribute to Deane Montgomery 344 The Golden Ratio-A Book Review Ronald Fintushel Reviewed by George Markowsky 3 50 Presidential Views: Interview with James Arthur Allyn jackson Departments

Notices ...... 353 of the American Mathematical Society Mathematics People ...... TWAS Pr izes Announced, Lemieux a nd Dick Win Infor mation- Professor of the EDITOR: Andy Magid Based Complexity Yo ung Researcher Awar d, Year Awards Announced, Young Mathematician Honored in ASSOCIATE EDITORS: Susanne C. Brenner, Bill Casselman (Graphics Editor), Sie mens Westinghouse Competition. Robert ]. Daverman, Nathaniel Dean, Rick Durrett, Susan Friedlander, Robion Kirby, Steven G. Krantz, Mathematics Opportunities ...... 354 Elliott H. Lieb, Mark Saul, Karen E. Smith, Audrey Project NExT: New Terras, Lisa Traynor Collaboration in Mathematica l Geosciences, Experiences in T eaching, Inter national Mathematics Competition SENIOR WRITER and DEPUTY EDITOR: Allyn jackson for Un iversit y Stude nts, Summer Program for Women MANAGING EDITOR: Sandra Frost Undergraduates, European Mathematical Society Announces for Lucien CONTRIBUTING WRITER: Elaine Kehoe New Article Competitions, Call for Nominations Godea ux Prize, Call for Nominations for Infor mation-Based PRODUCTION ASSISTANT: Muriel Toupin Complexity Pr ize, NSF Co mputing Equipmen t and PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Murphy, Lori Nero, Karen Ouellette, Donna Instrumentation Pr ograms, News f rom MSR I. Salter, Deborah Smith, Peter Sykes . .... 357 ADVERTISING SALES: Anne Newcomb For Your Information ...... Everett Pitcher Lectures, /ClAM Establishes Su Buchin Prize, SUBSCRIPTION INFORMATION: Subscription prices OMS Employmen t O ppor tunities. for Volume 52 (2005) are $4 17 list; $334 institutional member; $250 individual member. (The subscription Inside the AMS ...... 358 price for members is included in the annual dues.) A late charge of 10% of the subscription price will be AMS Sponsors NExT Fellows, Trjitzinsky Me morial Awards imposed upon orders received from nonmembers Presente d, Deaths of AMS Members. after january 1 of the subscription year. Add for postage: Surface delivery outside the United States Reference and Book list ...... 360 and lndia-$20; in India- $40; expedited delivery to destinations in North America-$35; elsewhere-$87. Subscriptions and orders for AMS publications should Mathematics Calendar ...... 365 be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. All New Publications Offered by the AMS ...... 373 orders must be prepaid. ADVERTISING: Notices publishes situations wanted Classified Advertisements ...... 377 and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be faxed to Meetings and Conferences Table of Contents ...... 392 401-331-3842 (indicate "Notices advertising" on fax cover sheet). SUBMISSIONS: Articles and letters may be sent to the editor by email at noti ces@math. ou . edu, b y fax at 405-325-5765, or by postal mail at Department of Mathematics, 601 Elm, PHSC 423, University of Okla­ homa, Norman, OK 73019-0001. Email is preferred. Correspondence with the managing editor may be sent to noti ce s@ams. or g. For more information, see the section "Reference and Book List". From the NOTICES ON THE AMS WEBSITE: Most of this publi­ cation is available electronically through the AMS web­ site, the Society's resource for delivering electronic AMS Secretary products and services. Use the URL http : I / WNW . ams . org/ noti ces/ to access the Notices on the website. Call for Nominations for AMS Award for [Notices o{the American Mathematica l Society is published Exemplary Programs ...... 318 monthly except bimonthly in june/ july by the American Mathematical Society at 201 Charles Street, Providence, Rl 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals Call for Nominations for Leroy P. Steele Prizes ...... 364 postage paid at Providence, Rl, and additional mailing offices. POSTMASTER: Send address change notices to Notices of che American Mathematical Society, P.O. Box 6248, Providence, Rl 02940-6248 USA.] Publication here of the Society's street address and the other information in brackets above is a tech­ nical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: noti ces@ams . org. ©Copyright 2005 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and fa lls within the guidelines established to ensure permanence and durability. Opinions expressed in signed Notices arti cles are those of the authors and do not necessarily rellect opinions of the editors or policies of the American Ma themati cal Society. Mathematics from Hindawi

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

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

The Prize Selection Committee requests nominations for the initial award, which will be presented at the Joint Mathematics Meetings in San Antonio, Texas, in January 2006. Letters of nomination may be submitted by one or more individuals. Nomination of the writer's own institution is per­ mitted. The letter should describe the specific program(s) for which the department is being nominated as well as the achievements that make the program( s) an outstanding success, and may include any ancillary documents which support the success of the program(s). The letter should not exceed two pages, with supporting documentation not to exceed an additional three pages.

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

Deadline for nominations is Aprill, 2005. CAMBRIDGE NEW AND NOTEWORTHY

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Letter from the Ed itor

a library or bookstore (Kreiger). Much like a research ar­ Breaking News ticle published in an obscure but refereed and archival journal, once something is in the literature, a resourceful As our readers know, the Notices serves both as the jour­ use nal of record of the American Mathematical Society and reader who needs the information can track it down, as a leading periodical of mathematical news, exposition, it, and, if need be, cite it. The Notices, a leading source for and comment. In the second role, we publish the articles information about the mathematical world, will from time that appear under our editorial classifications of "Feature to time bring to its large readership gems like the Fintushel Articles", "Communications", and "Opinion". These pieces and Kreiger articles which we feel deserve the exposure are put into production the first of month N for the issue and which will inform and entertain our readers. dated month N + 3 (so, for example, the articles in this This is not the first time the Notices has carried articles March issue went into production December 1). "Put into that have previously been available from other sources or production" means typesetting of the final version of au­ in other languages. For example, last month's Notices, like thors' manuscripts, or rather the electronic version thereof, the February 2004 Notices, carried an interview with the by Notices staff based at AMS headquarters in Providence. Abel Prize winners that originally appeared in the Euro­ And although the final manuscripts are themselves com­ pean Mathematical Society Newsletter. We have carried puter files, and hence their transmission-from the author, translations of mathematical articles that previously ap­ to the editor, to the production staff-allows for adjust­ peared elsewhere recently as well. The Notices welcomes ments up to almost the last minute, in fact most of the suggestions for pieces that may have appeared previously have a long gestation. The typical history of a pieces in outlets or in languages from which republication in the Notices article begins with a suggestion-from an editor, Notices would be rewarding to our readers. from the a uthor, or from a third party- which then be­ Returning to my previous theme of timeliness, I note comes an invitation to the author. Then there is an initial submission, usually one or more rounds of revisions based that while the Notices may not be a good medium for rapid on content reviews, and then some final rounds of revi­ news, it can be the subject thereof. To take a couple of re­ sions based on style reviews. Although this process can cent examples, in December, National Public Radio's Week­ sometimes be months, it can also take years. (Our main end Edition program interviewed Ed Saff about his work mathematical feature this month, an exposition of the with Doug Hardin on discretizing manifolds. The inter­ history and current developments in nonholonomic dy­ viewer, Scott Simon, closed by advising listeners that they namical systems, is an example of months.) In short, and could read more by consulting Saff and Hardin's paper in although there are exceptions, the Notices can't really the November issue of the Notices of the American Math­ rapidly deliver mathematical news. ematical Society. And, a few weeks earlier, the daily email Notices readers are of course aware of this limitation newsletter of the Chronicle of Higher Education, which and turn to its pages for explanation, exposition, com­ notes a few articles of general scholarly interest to its mentary, and, I hope, entertainment while using other recipients, recommended Martin Krieger's article "Some of sources for quick news updates. A couple of articles in this What Mathematicians Do", also from the November Notices. issue, however, also illustrate another function of the Just as the Notices is pleased when it can bring a piece from publishing the latest Notices that is almost antipodal to a less common source to the attention of its audience, we news. I refer to Ronald Fintushel's tribute memorializing are pleased when the general media can draw the atten­ Deane Montgomery and Martin Kreiger's translation of tion of its audience to articles in the Notices. Andre Weil's letter to Simone Weil. Fintushel's memorial article was written over a decade ago and has been ap­ pearing on his webpage. Weil's letter was written more than - A ndy Magid half a century ago and appears (in French) in Weil's col­ lected works. (The translation here is updated from the one Kreiger did in his 2003 book published by World Scientific.) There is, of course, no way to consider either document as breaking news. That these two pieces are well worth reading will be obvious. But I would like to comment too about why they are also well worth publishing here in the Notices despite their appearance and availability elsewhere. "Availability" is not always equivalent to "accessibility", but anyone who knew about the pieces could presumably seek out the rel­ evant webpage on the Internet (Fintushel) or the book in

MARCH 2005 NOTICES OF THE AMS 321 letters to the Editor [non-governmental organization]. I Keynes, and me as editors of the MET chaired it from 1982 to 1989. At that report. The MET report is the work of time it was not involved in electoral many. This is more accurately repre­ Post Papers Online to Benefit politics. My personal involvements in sented by listing an entity like CBMS Mathematicians in Poor elections and ecology parties were as its author, so the suggested citation Countries few and far between. for the Mathematical Education of Teachers is: In the article "The Elephant in the - Pierre Samuel Conference Board of the Mathemat­ Internet" ("Notices of the AMS", Bourg-la-Reine, France ical Sciences (2001), The Mathematical November 2004), D. Biss discusses Education of Teachers, Washington, the fact that Internet postings have an (Received November 25, 2004) DC, and Providence, RI: Mathematical overall negative effect on the quality Association of America and American of mathematical writing and thus dam­ Mathematical Society. age the aesthetic side of mathematics. A World without Mathematics ... He says, "I, for one, am hesitant to On page 1334 in the December 2004 -Cathy Kessel post my papers online; it always feels issue of the Notices, a map represent­ [email protected] a little like leaving my infant in a ing the "world without mathematics" dumpster." caught my attention. As it turns out, (Received November 29, 2004) While I fully join the author in em­ this is a map of Iceland published phasizing importance of beauty in in 1590 by a Dutch mapmaker. It is mathematical writing, I do not feel stunningly detailed for that time, as More on Underrepresentation that the quality of writing by serious anyone can see who compares it to The central argument in David Gale's mathematicians has deteriorated modern maps of Iceland. The map was letter on underrepresentation (De­ since they started to put papers on the originally created by an Icelandic cember 2004, page 1318) is highly arXiv. In any case, for me there is an scholar, Gudbrandur Thorlaksson questionable at best, and the letter as issue here that is much more signifi­ (1541-1627). Gudbrandur was an ex­ a whole is unfortunately regressive. cant: in many countries, including, cellent mathematician and astronomer. I applaud David Gale for realizing for example, Russia and China, many He studied at the University of Copen­ that" .. .it is desirable to provide condi­ libraries have no money to buy even hagen at the same time Tycho Brahe tions that might increase the doctoral "cheap" journals. So for many math­ was there, and later he became the rec­ percentages of. .. underrepresented ematicians in these countries, the tor of the school at Skalholt in Iceland groups." However, I strongly disagree arXiv and other websites are the only before becoming bishop at Bolar. His with his explanation of the roots of this way to gain access to literature. This many activities included measuring the underrepresentation. is why I think that mathematicians correct geodetic latitude of Bolar as 65 In his letter he states, "The fact is, should put their papers online and try degrees and 44 minutes. He is known some groups in our society-whether to protect the aesthetic side of math­ to have studied books by astronomers · cultural, ethnic, or even religious­ ematics just by being demanding of and mathematicians including Georg have put more emphasis on mathe­ themselves. Peurbach, Erasmus Reinhold, Peter matical achievement than others." Apian, and Oronce Fine. My great-great­ This statement is vague and difficult -Pavel Etingof (repeat a total of 11 times) grandfa­ to argue. (What exactly does David MIT ther's map makes for a nice display in Gale think he knows about, for [email protected] the Notices but is maybe not the best instance, contemporary black culture representation of a "world without that would support this?) More im­ (Received November 18, 2004) mathematics". Rather, its accuracy is a portantly, this is merely a symptom wonderful example of how mathe­ of the deeper causes of underrepre­ matics was used to create a map of one sentation. Grothendieck's Environmental country. The verifiable fact is that women Inspiration and minorities face significant ob­ In the second of the wonderful arti­ -Agust Egilsson structions to education in the United cles about A. Grothendieck, it is said University of California, Berkeley States. There is a terrific amount of that I was "one of the founders of the [email protected] literature on this subject; for an in­ French Green Party" (vol. 51, no. 10, troduction see Savage Inequalities by p. 1200). This is formally inaccurate. (Received November 29, 2004) Jonathan Kozol. A lack of access to What is true, and more important, education not only explains the dis­ is that my friend Grothendieck's crepancies in participation but also influence made me become an Citing "The Mathematical explains why some groups may environmental activist. I worked Education of Teachers" appear to be less interested in math­ mostly at "Les Amis de la Terre", In their December Notices "Opinion" ematics. How can one be interested in the French branch of "Friends of piece, Peter Ewell and Bernard Madi­ something that one has never been the Earth International", an N.G.O. son list Judith Epstein, Michael exposed to?

322 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Letters to the Editor

A similar argument based on Gale Responds to Karp About the Cover culture and religion was used in The letter to which Dagan Karp is re­ the mid- to late nineteenth century sponding criticized a very specific From dimers to random surfaces against providing literacy training to project that stipulated a target date This month's cover accompa­ former slaves. See, for example, A at which time math Ph.D's earned by nies the short article "WHAT IS ... a People's History of the United States by various ethnic groups should be Dimer?" in this issue, by Richard Howard Zinn for more details: roughly proportional to their respec­ Kenyon and Andrei Okounkov. The In retrospect it is clear that the ob­ tive populations. I won't repeat the article discusses the figure. struction to interest in reading was argument here. My letter never took Kenyon adds, "The tiling of the due to lack of access to resources and up the admittedly important subject octagonal region on the cover with not the culture and religion of slaves. of the causes and cures of the present 60° rhombi can be seen as the pro­ It does not take a professional math­ unequal representation (aside from a jection of a three-dimensional, ematician to recognize the pattern of brief sentence in support of increas­ piecewise linear surface. In the social inequities facing women and mi­ ing opportunities for minorities in limit of small lattice spacing, the norities in the United States. Women the mathematical community), yet surface arising from a randomly and minorities are significantly un­ Mr. Karp attacks me for what he chosen tiling has an almost-sure derrepresented in positions of power thinks are my views in these matters. shape (with facets delimited by the throughout society. It would be im­ I would welcome debate on the yellow cardioid)." possible to argue that for cultural or population proportional issue at Other similar images can be religious reasons women and minori­ any time, but it makes no sense for seen on Kenyon's gallery page ties simply do not desire positions of Mr. Karp to take me to task for views http://www.math.ubc.ca/ power. Professional mathematicians he attributes to me which I have never -kenyon/gallery/gallery. and the universities they work for are expressed and which he therefore html. part of, and not separate from, the cannot know. -Bill Casselman, greater society. Women and minorities Graphics Editor have been historically excluded from -David Gale ([email protected]) higher education in general, including University of California mathematics, and the lack of access to [email protected] education they face today amounts to continued exclusion. (Received December 17, 2004) David Gale also makes the paren­ thetical remark "(Unfortunately, we can't be like the students in the schools of Lake Woebegone who are all above average)." Given the subject matter of his letter, this is an irre­ sponsible comment, and I hope the implication that some ethnic groups are naturally more talented in math­ ematics than others was not intended. It is unfortunate that the culture of mathematics is so behind the times. I look forward to the day when all people, including gays, lesbians, bisexuals, visible minorities, biracial individuals, transgender individuals, and women, will be welcomed into the mathematics community without the discouragement of members within the community who claim pub­ licly that members of "those groups" are somehow less inclined or able to do mathematics.

-Dagan Karp University of British Columbia [email protected]

(Received December 12, 2004)

MARcH 2005 NOTICES OF THE AMS 323 Nonholonomic Dynamics

Anthony M. Bloch, jerrold E. Marsden, Dmitry V. Zenkov

Introduction theorem); nonholonomic systems are almost Pois­ Nonholonomic systems are, roughly speaking, me­ son but not Poisson (i.e., there is a bracket that to­ chanical systems with constraints on their veloc­ gether with the energy on the phase space defines ity that are not derivable from position constraints. the motion, but the bracket generally does not sat­ They arise, for instance, in mechanical systems isfy the Jacobi identity); and finally, unlike the that have rolling contact (for example, the rolling Hamiltonian setting, volume may not be preserved of wheels without slipping) or certain kinds of slid­ in the phase space, leading to interesting asymp­ ing contact (such as the sliding of skates). They are totic stability in some cases, despite energy con­ servation. a remarkable generalization of classical Lagrangian The purpose of this article is to engage the reader's and Hamiltonian systems in which one allows po­ interest by highlighting some of these differences along with some current research in the sition constraints only. area. There has been some confusion in the liter­ There are some fascinating differences between ature for quite some time over issues such as the nonholonomic systems and classical Hamiltonian variational character of nonholonomic systems, so or Lagrangian systems. Among other things: non­ it is appropriate that we begin with a brief review holonomic systems are nonvariational-they arise of the history of the subject. from the Lagrange-d'Alembert principle and not Some History from Hamilton's principle; while energy is pre­ The term "nonholonomic system" was served for nonholonomic systems, momentum is coined by Hertz (1894). The oldest publication that addresses not always preserved for systems with symmetry the dynamics of a rolling rigid body known to the (i.e., there is nontrivial dynamics associated with authors is Euler (1734), in which small oscillations the nonholonomic generalization of Noether's of a rigid body moving without slipping on a hor­ izontal plane were studied. Later, the dynamics of Anthony M. Bloch is professor of mathematics at the Uni­ a rigid body rolling on a surface was studied in versity of Michigan, Ann Arbor. His email address is Routh (1860), ab l och@umi ch. edu. Research partially supported by NSF Slesser (1861), Vierkandt (1892), and grants DMS 0103895 and 0305837. Walker (1896). The derivation of the equations of motion of a jerrold E. Marsden is professor of engineering and control and dynamical systems at the California Institute of Tech­ nonholonomic system in the form of the Euler­ nology. His email address is marsden@cds. cal tech. edu. Lagrange equations corrected by some additional Research partially supported by NSF grant DMS-0204474. terms to take into account the constraints (but Dmitry V. Zenkov is assistant professor of mathematics at without Lagrange multipliers), was outlined by Fer­ North Carolina State University. His email address is rers (1872). The formal derivation of this form of dvzenkov@uni ty. ncsu. edu. Research partially supported equations was performed in Voronetz (1901). In the by NSF grant DMS-0306017. case in which some of the configuration variables

324 NOTICES OF THE AMS VOLUME 52, NUMBER 3 are cyclic, such equations (now called Chaplygin equations) were obtained by Chaplygin in 1895 (and published two years later). This result of Chap­ lygin eventually gave rise to the modern technique of nonholonomic reduction. Chaplygin also was first to realize the importance of an invariant mea­ sure in nonholonomic dynamics. One of the more interesting historical events was the paper of Korteweg (1899). Up to that point (and even persisting until recently) there was some confusion in the literature between nonholonomic mechanical systems and variational nonholonomic systems (also called "vakonomic" systems). The latter are appropriate for optimal control prob­ lems. One of the purposes of Korteweg's paper was to straighten out this confusion, and in doing so, he pointed out a number of errors in papers up to that point. We refer the reader to Cendra, Mars­ den, and Ratiu [2001] for an elaboration on some of these points and a more comprehensive histor­ ical review. Figure 1. Wolfgang Pauli and Niels Bohr­ Classic books in mechanics, as well as their examining "tippe top" inversion (at the Institute modern counterparts, have discussed in detail the of Physics at Lund, Sweden in 1955). geometry of Hamiltonian and Lagrangian systems; on the other hand, there has not been much work The lagrange-d'Aiembert Principle until recently on the geometry of nonholonomic sys­ We now describe the equations of motion for a tems. The geometry and reduction of such sys­ nonholonomic system. We confine our attention to tems is discussed in the recent book by Bloch nonholonomic constraints that are homogeneous [2003], in which a fairly comprehensive survey is in the velocity. Accordingly, we consider a me­ given together with a discussion of the natural chanical system with a configuration manifold Q, connections to control theory. A comprehensive set whose local coordinates are denoted qi, and an of references to the literature may be found in this (n- p)-dimensional nonintegrable constraint dis­ reference together with many other topics not tribution 'D c TQ. The distribution 'D can be de­ touched on here. In the last section of the present scribed locally by equations of the form paper, we do, however, give a brief discussion of interesting topics that still await further investi­ (1) .sa +A~(r,s)r"' = 0, a = 1, ... ,p, gation, such as integrability. Toys and Warnings where q = (r, s) E ~n-p x ~Pare appropriately cho­ Figure 1 shows the famous physicists Wolfgang sen local coordinates in Q, which we write as Pauli and Niels Bohr examining the "tippe top" toy qi = (r"',sa), 1 :o; a :o; n - p and 1 :o; a :o; p. Note undergoing its interesting inversion. It is simply a that we only consider here constraints that are lin­ half-sphere, with a cylindrical stem mounted on the ear in the velocities. These linear constraints cover flat part of the half-sphere used to spin the toy. If essentially all physical systems of interest. Non­ one spins it fast enough, then it undergoes a 180- linear constraints are of interest, however-a dis­ degree flip of its axis of rotation. There are simi­ cussion and history may be found, for example, in lar toys, such as the rattleback which we discuss Marle [1 998]. below, that also undergo rather nonintuitive mo­ Consider, in addition to the constraint distrib­ tions. ution, a given Lagrangian L: TQ ~ ~. As in halo­ However, one has to be quite careful about how nomic mechanics, the Lagrangian for many systems one models such systems. For example, while it is the kinetic energy minus the potential energy. might seem quite appealing to model the initial moe The equations of motion are then given by the fol­ tion of the tippe top as a sphere rolling on a flat lowing Lagrange-d' Alembert principle. surface, in this and some similar situations (such as the "rising egg") it turns out that sliding friction Definition 1. The Lagrange-d'Alembert equations (which would mean using a holonomic mechanical of motion for the system with the Lagrangian L model) plays a very important role, and so model­ and constraint distribution 'D are those determined ing it as a nonholonomic system is too simplistic by a view. For further discussion and simulations, see Bou-Rabee, Marsden, and Romero [2004].

MARCH 2005 NOTICES OF THE AMS 325 where we choose variations 8q(t) of the curve q(t) where Bt13 is defined by that satisfy the constraints for each t E [a, b] and vanish at the endpoints, i.e., 8q(a) = 8q(b) = 0. aAb aAb aAb aAb) Bb = ( __a ___{3+Aa __{3_Aa __e< This principle is supplemented by the condition IX{3 arf3 are< IX asa {3 asa • that the curve itself satisfies the constraints; that is, we require that q E 'D. There is a beautiful geometric interpretation of these equations: the constraints define an Ehres­ It is also of interest to consider the role of Dirac mann connection on the tangent bundle TQ and structures in nonholonornic mechanics. Interest­ B is the curvature of the connection which vanishes ingly, this point of view enables one to formulate precisely when the constraints are integrable; that the variation of the Lagrangian and constraint as is, are holonornic. one condition (see Yoshimura and Marsden [2004] and references therein). Note carefully that in the above definition, we take the variation before imposing the constraints; that is, we do not impose the constraints on the (} family of curves defining the variation. These op­ erations do not commute, and this fact is a central y reason that nonholonornic mechanics is nonvari­ ational in the usual sense of the word. This dis­ tinction, already remarked on in our historical in­ troduction, is well known to be important for obtaining the correct mechanical equations (see Bloch, Krishnaprasad, Marsden, and Murray [1996] and Bloch [2003] for a discussion and references). Figure 2. The geometry for the rolling disk. The usual arguments in the calculus of variations show that the Lagrange-d'Alembert principle is The Falling Rolling Disk equivalent to the equations The falling rolling disk is a simple but instructive example to consider. We consider a disk (such as d ar ar) ; (2) -fJL = ( dt aqi - aqi fJq = o a coin) that rolls without slipping on a horizontal plane and that can "tilt" as it rolls. As Figure 2 indicates, we denote the coordi­ for all variations [Jqi = (8ra, fJsa) satisfying the nates of contact of the disk with the xy-plane by constraints at each point of the underlying curve (x, y) and let e, cp, and l/J denote the angle between q(t), i.e., such that fJsa + A~fJra = 0. Substituting the plane of the disk and the vertical axis, the variations of this type, with bra arbitrary, into (2) "heading angle" of the disk, and the "self-rotation" gives angle of the disk, respectively. 1 While the equations of motion are straightfor­ ward to develop, they are somewhat complicated. (3) One can show that this example is, in an appropriate sense, an integrable system and that it conserves for all oc = 1, ... , n- p. One can equivalently write volume in the phase space and, in addition, it ex­ these equations in terms of Lagrange multipliers. hibits stability but not asymptotic stability. See Equations (3), combined with the constraint Zenkov, Bloch, and Marsden [1998] and Bloch [2003] equations (1), give the complete equations of mo­ for more details. tion of the system. This system demonstrates unusual conserva­ A useful way of reformulating equations (3) is tion laws, but ones that are typical for nonholo­ l/J to define a constrained Lagrangian by substituting nornic systems. One can check that while cp and are cyclic variables (that is, they do not appear ex­ the constraints (1) into the Lagrangian: plicitly in the constrained Lagrangian), their asso­ ciated momenta

The equations of motion can be written in terms 1 A classical reference for the rolling disk is Vierkandt of the constrained Lagrangian in the following way, [1892], who showed something very interesting: On an ap­ as a direct coordinate calculation shows: propriate symmetry-reduced space, namely, the con­ strained velocity phase space modulo the action of the group of Euclidean motions of the plane, all orbits of the system are periodic. ·

326 NOTICES OF THE AMS VOLUME 52, NUMBER 3 oLe d oLe that the equations of motion have the form (4). If P1 = ocp an Pz = otj; these equations had a cyclic variable, say r 1, then all the quantities L, Le, and B~tl would be inde­ are not conserved. pendent of r 1. This is equivalent to saying that However, there exist two independent vector there is a translational symmetry in the r 1 direc­ fields r]l(8) and TJ 2 (8) such that the momentum tion. Let us also suppose, as is often the case, that components along these fields are preserved by the the s variables are also cyclic. Then the equation dynamics. We emphasize that the vector fields for the momentum p 1 = oLeI or 1 becomes '11 (8) and TJz{e) do not equal the fields o I ocp and oI ot.jJ. See Zenkov [2003] and the references therein · _ -~Bb ·tl for details. P1 - osb 1/lr .

Momentum Equation This fails to be a conservation law in general since Note that the Assume there is a Lie group G (with Lie algebra de­ the right-hand side need not vanish. in and the equation noted g) that acts freely and properly on the con­ right-hand side is linear r, r 1 a very special figuration space Q. A Lagrangian system is called does not depend on itself. This is G -invariant if its Lagrangian L is invariant under case of what is called the momentum equation. For group, the induced action of G on TQ. Recall the defini­ systems with a noncommutative symmetry tion of the momentum map for an unconstrained such as the Chaplygin sleigh discussed below, the Lagrangian system with symmetry: The momentum above analysis for cyclic variables, while giving the to capture the full story. map J: TQ ~ gQ_ is the bundle map taking TQ to right idea, fails the bundle gQ_ whose fiber over the point q is the dual Thus the nonholonomic momentum is a dynam­ Lie algebra g* that is defined by ically evolving quantity. The momentum dynamics is specified in Theorem 3 (see Bloch, Krishnaprasad, (5) (J(vq),~) = (IFL(vq).~Q) := :~i(~QY, Marsden, and Murray [1996]). Let gv be the bundle over Q whose fiber at the point q is given by gG.

where~ E g, Vq E TQ, and where ~Q E TQ is the Theorem 3. Assume that the Lagrangian is invariant generator associated with the Lie algebra element under the group action and that ~q is a section of ~. the bundle gv. Then a solution q(t) of the Lagrange­ d'Alembert equations for a nonholonomic system A nonholonomic system is called G -invariant if must satisfy the momentum equation both the Lagrangian L and the constraint distrib­ ution 'D are invariant under the induced action of G on TQ. Let 'Dq denote the fiber of the constraint distribution 'D at q E Q. nonholonomic Noether Definition 2. The nonholonomic momentum map We thus have the following 1nhe is defined as the collection of the components theorem: of the ordinary momentum map J that are consis­ Corollary 4. If~ is a horizontal symmetry, i.e., if tent with the constraints, i.e., the Lie algebra ele­ ~Q(q) E 'Dq for all q E Q, then the following con­ ments ~ in equation (5) are chosen from the sub­ servation law holds: space gG of Lie algebra elements in g whose infinitesimal generators evaluated at q lie in the in­ (7) tersection 'Dq n Tq(Orb(q)). Unlike Hamiltonian systems, G-invariant non­ A somewhat restricted version of the momen­ holonomic systems often do not have associated tum equation was given by Kozlov and Kolesnikov momentum conservation laws. Besides the rolling [1978], and the corollary was given by Arnold, penny, the rattleback and the snakeboard are well­ Kozlov, and Neishtadt [1988], page 82. known examples (see Bloch, Krishnaprasad, Mars­ den, and Murray [1996] and Zenkov, Bloch, and The Poisson Geometry of Nonholonomic Marsden [1998]). The rattleback is discussed fur­ Systems ther below. So far we have adopted the philosophy of La­ It is easy to see why the momentum quantities grangian mechanics; now in this section, we con­ are generally not conserved from the Lagrange­ sider the Hamiltonian description of nonholonomic d'Alembert equations of motion. The simplest sit­ systems. Because of the necessary replacement of uation would be the case in which the Lagrangian conservation laws with the momentum equation, and the constraint have a cyclic variable (more gen­ it is natural to let the value of the momentum be eral definitions of cyclic symmetry that apply to a variable, and for this reason it is natural to take problems like the falling disk are possible). Recall a Poisson viewpoint. Some of this theory was

MARcH 2005 NOTICES OF THE AMS 327 initiated in van der Schaft and Maschke [1994]. First, a constraint phase space :M = IFL(1J) c What follows builds on their work, further devel­ T * Q is defined in the same way as in Bates and Sni­ ops the theory of nonholonomic Poisson reduction, atycki [1993], so that the constraints on the Hamil­ and ties this theory to other work in the area. See tonian side are given by p E :M.. In local coordinates, also Koon and Marsden [1997]. The following two complications make this ef­ :M = {(q,p) E T *Q I wf~: = o}. fort especially interesting. First of all, as we have mentioned, symmetry need not lead to conserva­ Let {XIX} be a local basis for the constraint distri­ tion laws but rather to a momentum equation. Sec­ ond, the natural Poisson bracket fails to satisfy bution 'D and let {wa} be a local basis for the an­ the Jacobi identity. In fact, the so-called nihilator 1J 0 • Let {Wa} span the complementary ]acobiator (the cyclic sum that vanishes when the subspace to 'D SUCh that (Wa, Wb) = 0/:, where Ob Jacobi identity holds) or, equivalently, the Schouten is the usual Kronecker delta. Here a = 1, ... , k and bracket, is an interesting expression involving the ()( = 1, ... , n - k. Define a coordinate transforma­ curvature of the underlying distribution describ­ tion (q, p) ._. (q, piX, Pa) by ing the nonholonomic constraints. Thus in the non­ holonomic setting we have an almost Poisson struc­ (8) ture. Poisson Formulation It is shown in van der Schaft and Maschke [1994] The approach of van der Schaft and Maschke that in the new (generally not canonical) coordinates [1994] starts on the Lagrangian side with a con­ (q, piX, Pa), the Poisson tensor becomes figuration space Q and a Lagrangian L (possibly of the form kinetic energy minus potential energy, i.e., (9) L(q, q) = ~ ((q, q)) - V(q), Let (piX , Pa) satisfy the constraint equations where (( · , · )) is a metric on Q defining the kinetic ::a (q, p) = 0. Since energy and V is a potential energy function). As above, our nonholonomic constraints are :M = {(q,p

(;~) E :M.. As in the Lagrangian setting it is desirable to model the Hamiltonian equations without the Lagrange multipliers by a vector field on a submanifold of Here]:M is the (2n - k) x (2n - k) truncated ma­ T *Q. In van der Schaft and Maschke [1994] it is trix of J restricted to :M and is expressed in the done through a clever change of coordinates. In induced coordinates. Bloch [2003] we recall how they do this. Here we The matrix ] :M defines a bracket { · , · }:M on the just present the results. constraint submanifold :M as follows:

328 NOTICES OF THE AMS VOLUME 52, NUMBER 3 'DIG under appropriate conditions. The Lagrangian induces a well-defined function, the constrained re­ duced Lagrangian

lc: 'DIG- IRI., for any two smooth functions F:M, G:M on the con­ straint submanifold .Jvl. Clearly, this bracket sat­ on this phase space. We do not discuss this here isfies the first two defining properties of a Poisson for reason of space but refer the reader to Bloch, bracket, namely, skew symmetry and the Leibniz Krishnaprasad, Marsden, and Murray [1996), Bloch rule, and one can show that it satisfies the Jacobi [2003), and Cendra, Marsden, and Ratiu [2001) for identity if and only if the constraints are halo­ both the Lagrangian and Hamiltonian analysis of nomic. Furthermore, the constrained Hamiltonian the equations of motion and the reduced equa­ H:M is an integral of motion for the constrained dy­ tions of motion. The last paper cited, in particular, namics on .Jv1 due to the skew symmetry of the gives an intrinsic, coordinate-free formulation that bracket. also gives a very neat interpretation to the mo­ mentum equation in terms of parallel transport on A Formula for the Constrained Hamilton the appropriate bundle. In particular, the coordi­ Equations nate form of the reduced equations is quite com­ In holonomic mechanics, it is well known that plicated, whereas the intrinsic formulation reveals the Poisson and the Lagrangian formulations are their structure more clearly. equivalent via a Legendre transform. And it is nat­ ural to ask whether the same relation holds for the Measure-Preserving Systems on lie Groups nonholonomic mechanics as developed in van der and Asymptotic Dynamics Schaft and Maschke [1994) and in Bloch, Krish­ In this section we demonstrate that nonholonomic naprasad, Marsden, and Murray [1996). dynamics is not necessarily measure-preserving. We can use the general procedures of van der This is in contrast to the volume-preserving nature Schaft and Maschke [1994) to write down a com­ of Hamiltonian systems and follows from the fact pact formula for the nonholonomic equations of that nonholonomic systems are only almost Pois­ motion. son. Energy, however, is preserved. This illustrates Theorem 5. Let qi = (r"', sa) be the local coordi­ the very special nature of Hamiltonian systems in nates in which wa has the form which both energy and volume are preserved. The existence of an invariant measure as a nec­ essary condition for integrability of a nonholo­ nomic system was pointed out by Kozlov. The pro­ A~(r, s) is coordinate expression of the where the cedure of integration of a measure-preserving Then the nonholonomic Ehresmann connection. dynamical system goes back to jacobi (1866). constrained Hamilton equations of motion on .Jv1 can be written as Euler-Poincare-Suslov Equations An important special case of the (reduced) non­ ·a - - Aa oH:M holonomic equations is the dynamics of a con­ s - " ofJ" , strained generalized rigid body. ·a oH:M The configuration space for a generalized rigid r = ofJa ' body is a Lie group G. The Lagrangian L: TG - IRI. ~ _ oH:M b oH:M b oH.111 is a left-invariant metric on G, i.e., L(g, g)= l(g- 1g), Pa - ---;-;;-+A"'~- PbBa/3-;;:;::-- • ur uS upp where I : g - IRI. is the reduced Lagrangian defined by the formula /(Q) = ~ Iab oao b , Q = (Q1 , ... ,on) lie in a Lie algebra g and lab are the components where Btil are the coefficients of the curvature of of the positive-definite inertia tensor 0 : g ,_. g*. The the Ehresmann connection. Here Pb should be un­ ed dynamics of the generalized rigid body are derstood as Pb restricted to .Jv1 and, more precisely, reduc should be denoted by (pb):M. governed by the Euler-Poincare equations One can show that the equations in this theorem (12) Pb = c~b l ad P c P d = c~bP c O a , ar e equivalent to those in the Lagrange­ d'Alembert formulation (see Bloch [2003)). where Pb = IabQb are the components of the mo­ We remark that the theory of reduction for non­ mentum and C~b are the structure constants of the holonomic systems is elegant and interesting­ Lie algebra g. The system (1 2) is Hamiltonian. one can formulate the equations in intrinsic fash­ Nonetheless, it can fail the phase volume preser­ ion on the constrained reduced velocity phase space vation property, as the following theorem states.

MARCH 2005 NOTICES OF THE AMS 329 Theorem 6 (Kozlov [1988]). The Euler-Poincare (17) w-1a, a], a) = (Jla, a) equations (12) have an invariant measure if and only and, since the left-hand side is zero, J1 must be zero. if the group G is unimodular.2 Thus in this case only constraint vectors a that com­ The constrained generalized rigid body is the dy­ mute with D- 1a allow the measure to be preserved. namical system (12) subject to the left-invariant This means that a and D- 1 a must lie in the same nonholonomic constraint maximal commuting subalgebra. In particular, if a (13) is an eigenstate of the inertia tensor, measure is pre­ served. When the maximal commuting subalgebra where a is a fixed element of the dual Lie algebra is one-dimensional, this is a necessary condition. g* and ( · , ·) denotes the natural pairing between This is the case for groups such as 50(3) (see the Lie algebra and its dual (multiple constraints below). may be imposed as well). The two classical exam­ Theorem 7 can be restated as the following sym­ ples of such systems are the Chaplygin sleigh (Chap­ metry requirement imposed on the constraints: lygin, 1911) and the Suslov problem (Suslov, 1902) discussed below. Theorem 8. A compact Euler-Poincare-Suslov sys­ The reduced dynamics of the constrained gen­ tem is measure-preserving if the constraint vectors eralized rigid body is governed by the Euler­ a are eigenvectors of the inertia tensor, or if the con­ Poincare-Suslov equations strained system is 71. 2 symmetric about all principal axes. If the maximal commuting subalgebra is one­ i\ab ~ bP c D.a + i\ab (14) Pb = c~biadP c Pd + = c dimensional, this condition is necessary. together with the constraint (13). If the Lagrange The Euler-Poincare-Suslov Problem on S0(3). multiplier i\ is eliminated, (14) becomes the mo­ As an illustration, consider the classical Suslov mentum equation. problem, which can be formulated as the standard existence Next, we formulate a condition for the Euler top dynamics subject to the constraint of an invariant measure of the Euler-Poincare­ 1 2 3 Suslov equations: (18) (a, D. ) = a1D. + azD. + a 3 D. = 0,

Theorem 7. Equations (14) have an invariant mea­ where D.= (D. 1,D.2,D. 3) E so(3) is the angular ve­ sure if and only if locity of the top. Constraint (18) forces the projection of the an­ where gular velocity along the direction a = (a1 , a 2 , a3 ) rel­ to vanish. The reduced (15) ative to the body frame K=1 / (a,D - 1a ) and J1E ill!. . nonholonomic equations of motion are then given by (14) with This result was proved by Kozlov [1988] for com­ pact algebras and by Jovanovic [1998] for arbi­ Cf2 = C:h = C~ 1 = - Ci1 = -Ch = -Ci3 = 1 theorem, one eliminates trary algebras. To prove the and ct = 0 otherwise. the multiplier i\ and obtains a system of differen­ tial equations with quadratic right-hand sides. Ac­ As (17) implies, the momentum dynamics is cording to Kozlov [1988], a system of differential measure (phase volume) preserving if and only if equations with homogeneous polynomial right­ the constraint direction a is an eigenvector of the hand sides is measure preserving if and only if it inertia tensor D. is divergence free. The condition (15) is then ob­ An alternative way to obtain this conclusion is tained by setting the divergence of the right-hand to compute the eigenvalues of the linearized mo­ side of (14) equal to zero. mentum flow at the equilibria. If az = a 3 = 0 (a According to the definition, for a unimodular constraint that is an eigenstate of the moment of group, CJk vanishes. In particular, if the group is inertia operator) one gets zero eigenvalues while compact or semisimple, it is unimodular, and we in general one gets a real non-zero eigenvalue and can identify g* with g and rewrite condition (15) two zero eigenvalues, which is incompatible with as measure preservation. (16) [D- 1 a, a] = Jla, J1 E Ill!. . The Chaplygin Sleigh One of the simplest mechanical systems that il­ Pairing a with itself (via the Killing form or a mul­ lustrates the possible "dissipative nature" of non­ tiple of the trace) we have holonomic systems, even though they are energy preserving, is the Chaplygin sleigh. This system con­ 2 Recall that a Lie group is called unimodular if the struc­ rigid body sliding on a plane. The body ture constants satisfy the equations C&c = 0. A standard sists of a fact is that a unimodular group has a bilaterally invari­ is supported at three points, two of which slide ant measure. freely without friction while the third is a knife edge,

330 NOTICES OF THE AMS VOLUME 52, NUMBER 3 z a constraint that allows no motion orthogonal to this edge. To analyze the system, one can use a coordinate system 0 xy fixed in the plane and a coordinate sys­ tem A~ 17 fixed in the body with its origin at the point of support of the knife edge and the axis A~ through the center of mass C of the rigid body. The configuration of the body is described by the co­ a ordinates x, y of the contact point and the angle e between the moving and fixed sets of axes, i.e., the configuration space is the group S£(2). Let m be X the mass and I the moment of inertia of the body about the center of mass. Let a be the distance from A to C (see Figure 3). The nonholonomic momen­ Figure 3. The Chaplygin sleigh is a rigid body moving on two tum has two components: p1, the angular mo­ sliding posts and one knife edge. mentum of the system relative to the contact point, and p 2 , the projection of the linear momentum of the system on the ~ axis. The momentum equations written relative to P2 the body frame become ~~ ?I d:l .. • 2 . "" ~ ~~ ?J ~ . . ... ~ . mapf ~.?=J ~~ p V' oP . . ~ (19) P2 = (I + ma2)2. l? ' ~ ~ 1 p p ...... "!. z p ~ l\S This dynamics has a family of equilibria (i.e., [t 17' 1} . ' ~ ~ ~ points at which the right-hand sides vanish) given 1F 17" . . "It "'U" IF i1 PI by {(p1. P2l I P1 = 0, P2 = const} . ·1. 0 1 .lf "'U" "It • 17" 1r Assuming a > 0 and linearizing about any of 1f 1} these equilibria, one finds a zero eigenvalue, and ~ ~ ~ ' . 17' [t "!...... p p a negative eigenvalue if p2 > 0 or a positive eigen­ l\S ~ ·1 .. p value if P2 < 0. ~ ~ ~ ' . . ,p V' p l? Thus the volume in the momentum plane is pre­ ~~ ~ ... . . ~ ?J .?=J~ served if and only if a = 0, which is equivalent to ~~ ~ . - 2 .. d:l ?I ?1~ (15).3 In fact, the solution curves are ellipses in the "" P1P2-plane with the positive P2-axis attracting all solutions (see Figure 4). If a = 0, the dynamics is integrable, and in par­ Figure 4. Chaplygin sleigh phase portrait. ticular, the body momentum relative to the group S£(2) is preserved. Recall that a free rigid body on remarkable demonstration of the nontriviality of the plane conserves the spatial momentum. This the momentum equation. Moreover, the stable spin illustrates how different momentum conservation direction is in fact asymptotically stable. laws are in the case of nonhorizontal symmetry. We adopt the ideal model (with no energy dis­ The Rattleback sipation and no sliding) and within that context no approximations are made. In particular, the rat­ We end with a brief discussion of one of the most tieback's shape need not be ellipsoidal. Walker did fascinating nonholonomic systems- the rattleback top or Celtic stone. A rattleback is a convex asym­ some initial stability and instability investigations metric rigid body rolling without sliding on a hor­ by computing the spectrum, while Bondi extended izontal plane (see Figure 5). It is known for its abil­ this analysis and also used what we now recognize ity to spin in one direction and to resist spinning as the momentum equation. See Bloch [2003] and in the opposite direction for some parameter val­ Zenkov, Bloch, and Marsden [1 998] for the explicit ues, and for other values to exhibit multiple re­ form of the momentum for the rattleback. A dis­ versals in clear violation of conservation of angu­ cussion of the momentum equation for the rattle­ lar momentum or of damped angular momentum. back may also be found in Burdick, Goodwine, and In fact, this phenomenon m ay be viewed as a Ostrowski [1994]. Karapetyan carried out a stabil­ ity analysis of the relative equilibria, while Mar­ 3 This ca,Iculation is perfo rmed f or the nonholonomic mo­ keev's and Pascal's main contributions were to the mentum, whereas in the Suslov problem example the full study of spin reversals using small-parameter and momentum is used. averaging techniques. Energy methods were used

MARCH 2005 NOTICES OF THE AMS 331 those discussed for the rolling penny earlier in this paper. An important topic is the control of nonholonornic systems. This is dis­ cussed in detail in Bloch [2003), where many references are given. There is a natural link between nonlinear control systems and nonholonornic distribu­ tions: the control vector fields in a control system provide controllabil­ Figure 5. The rattleback. ity precisely when the distribution they span is nonintegrable, thus giv- ing rise to new to analyze the problem in Zenkov, Bloch, and Mars­ directions of motion. den [1998). A key example is the "nonholonornic integrator", a system with two There are many other remarkable nonholonornic controls defined on the Heisen­ berg group introduced and studied in systems and for these we refer the reader to Bloch Brockett [1981). The role of sub-Riemannian geometry [2003), the references therein, and in the many other optimal control papers. of the nonholonornic integrator is discussed in Bloch [2003). For more on sub­ Further Topics Riemannian geometry see Montgomery [2002). In sub-Riemannian geometry one has an evolution of This review has touched on just a few of the fas­ a variational or Hamiltonian system subject to a cinating aspects of nonholonornic mechanics. There nonholonornic constraint-this should not be con­ are many other topics of interest, and we conclude fused with nonholonornic mechanical systems. The by mentioning some of these. differences are very interesting and are exposed in One question of interest is when a nonholo­ detail in Bloch [2003). nornic system is integrable. There is no known ana­ Another topic of interest is numerical integra­ logue of the Liouville-Arnold theorem well-known tion of nonholonornic systems. The idea is to pre­ from holonornic mechanics. One can show that serve key mechanical quantities of interest such as. nonholonornic systems are integrable if the di­ momentum conservation laws. For a survey of mension of the phase space of the system is n and these ideas in the Hamiltonian case see Marsden if there exist (n - 2) integrals of motion and an in­ and West [2001). variant measure (see, e.g., Arnold, Kozlov, and Remark. A complete set of references for this Neishtadt [1998)). Examples of integrable nonho­ paper may be found at http: //www .cds. lonornic systems include the rolling disk discussed caltech.edu/mechanics_and_control / . above, Routh's problem of a homogeneous sphere rolling on a surface of revolution (see, e.g., Zenkov [1995)), and Chaplygin's sphere-a balanced inho­ References mogeneous sphere rolling on a plane (see, e.g., V. I. ARNOLD, V. V. KOZLOV, and A. I. NEISHTADT, Arnold, Dynamical Kozlov, and Neishtadt [1988)). One of the systems III, Encyclopedia of Mathematics, vol. 3, interesting aspects of such systems is that one can Springer, Berlin, 1988. obtain invariant tori as in the Hamiltonian case, but L. BATES and J. SNIATYCKJ, Nonholonomic reduction, Reports the dynamics on these tori may be nonuniform. It on Math. Phys. 32 (1993), 99- 115. is possible in some such cases to make the system A.M. BLOCH, with J. Baillieul, P. E. Crouch and J. E. Marsden, Hamiltonian by a trajectory-dependent time repa­ Nonholonomic Mechanics and Control, Springer, Berlin, 2003. rameterization. This is discussed in the work of Ko­ A.M. BLOCH, P. S. KRISHNAPRASAD, J. E. MARSDEN, zlov and more recently in Ehlers, Koiller, and R. MURRAY, Mont­ Nonholonomic mechanical systems with symmetry, gomery, and Rios [2004), who denote the process Arch. Rat. Mech. An. 136 (1996), 21-99. "Harniltonization". ·(This process of Harniltoniza­ N. M. Bou-RABEE, J. E. MARSDEN, and L. N. ROMERO, Tippe top tion need not necessarily be applied to the inte­ inversion as a dissipation induced instability, SIAM]. grable case.) These systems conserve a measure, but on App/. Dyn. Systems 3 (2004), 352- 377. other systems such as the Chaplygin sleigh do not R. W. BROCKETT, Control theory and singular Riemannian and are still solvable. geometry, New Directions in Applied Mathematics (P. J. Hilton and Analysis of the stability of nonholonornic mo­ G. S. Young, eds.), Springer-Verlag, 1981, 11-27. tion is also of interest, and there is a natural gen­ J. BURDICK, B. GOODWINE, and J. P. 0STROWSKl, The Ratt/eback eralization of the energy-momentum method of Revisited, preprint, 1994. Arnold and of Marsden and collaborators. This is H. CENDRA, J. E. MARSDEN, and T. S. RATIU, Geometric mechan­ discussed in Zenkov, Bloch, and Marsden [1 998). ics, Lagrangian reduction and nonholonomic systems, This method makes use of integrals similar to Mathematics Ui:limited: 2001 and Beyond (B . Enguist

332 NOTICES OF THE AMS VOLUME 52, N UMBER 3 and W. Schmid, eds.), Springer-Verlag, New York, 2001, 221-273. K. EHLERS, ]. KOILLER, R. MONTGOMERY, and P. M. R.!os, Non­ ECOLE POLYTECHNIQVE holonomic Mechanics via Moving Frames: Cartan's FEDERALE DE LAUSANNE Equivalence and Hamiltonizable Chaplygin Systems, in "The Breadth of Symplectic and Poisson Geometry: Festschrift for Alan Weinstein", preprint, 2004. Faculty Positions in B. JovANOVIC, Nonholonomic geodesic flows on Lie groups and the integrable Suslov problem on 50(4),]. Phys. Mathematics A: Math. Gen. 31 (1998), 1415-1422. W. S. KOON and ]. E. MARSDE N, The Hamiltonian and at Ecole Polytechnique Federale de Lagrangian approaches to the dynamics of nonholo­ Lausanne (EPFL) nomic systems, Reports onMathPhys. 40 (1997), 21-62. V. V. KozLOv, Invariant measures of the Euler-Poincare equations on Lie algebras, Functional An. Appl. 22 (1998), 69-70. V. V. KO ZLOV and N. N. KOLESNIKOV, On theorems of dynamics,]. Appl. Math. Mech. 42 (1978), 28-33. A. LEWIS,]. P. 0 STROWSKJ, R. M. MURRAY, and]. BURDICK, Non­ holonomic mechanics and locomotion: The snake­ board example, IEEE Intern. Conf. on Robotics and Au­ tomation (1994). C.-M. MARLE, Various approaches to conservative and non­ conservative nonholonomic systems, Reports on Math­ ematical Physics 42 (1998), 211-229. ]. E. MARSDEN and T. S. RA nu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, vol. 17; first Over the next few years EPFL intends to make numer­ edition 1994, second edition, 1999. ous faculty appointments of exceptional individuals ]. E. MARSDEN and M. WEST, Discrete mechanics and varia­ across the range of mathematics. We seek mathemati­ tional integrators, Acta Numerica 10 (2001), 357-514. cal scientists with outstanding accomplishments in any R. MONTGOMERY, A Tour of Sub-Riemannian Geometries, their Geodesics and Applications, Mathematical Surveys domain of pure and applied mathematics, including and Monographs, vol. 91, Amer. Math. Soc. 2002. statistics. Preference will be given at the assistant and A.]. VAN DER SCHAFT and B. M. MASCHKE, On the Hamilton­ associate professor levels, but top senior candidates ian formulation of nonholonomic mechanical sys­ will also be considered. tems, Rep. on Math. Phys. 34 (1994), 225-233 . A. VI ERKANDT, Ober gleitende und rollende Bewegung, Successful candidates will establish and lead vigorous Monatshefte der Math. und Phys. lll, (1982) 31-54. H. YOSHIM URA and ]. E. MARSDEN, Variational principles, independent research programs, interact with existing Dirac structures, and implicit Lagrangian systems, projects, and be committed to excellence in teaching. Preprint, 2004. Significant start-up resources and research infrastruc­ D. V. ZENKOV, The geometry of the Routh problem,]. Non­ ture will be available. linear Sci. 5 (1995), 503-519. __ , Linear conservation laws of nonholonomic systems with symmetry, Discrete and Continuous Dynamical Applications should be made through http://sma.epfl. Systems (supplementary volume), 2003, 963-972. ch/search. Candidates will be required to submit cur­ D. V. ZENKOV and A. M. BLOC H, Dynamics of the n-dirnen­ riculum vitae, concise statement of research and teach­ sional Suslov problem, journal of Geometry and Physics ing interests, and the names and addresses (including 34 (2000), 121-136. email) of five referees as a single PDF file (at most 20 D. V. ZENKOV, A. M. BLOCH, and]. E. MARSDEN, The energy­ sides of A4, plus list of publications). A printed version momentum method for the stability of nonholonomic systems, Dyn. Stab. of Systems 13 ( 1 ~98), 12 3-166. of this file should be sent to: ·

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MARCH 2005 NOTICES OF THE AMS 333 A 1940 Letter of Andre Weil on Analogy in Mathematics Translated by Martin H. Krieger

This article is excerpted from the book Doing Mathematics (2003) by Martin H. Krieger. It is reprinted with permission from World Scientific Publishing.

or Andre Weil, "having a disagreement argument of that book, including what I called the with the French authorities on the subject Dedekind-Weil analogy. of [his] military 'obligations' was the rea­ The Weilletter is a gem, of wider interest to the Fson [he] spent February through May [of mathematical and philosophical community, con­ 1940] in a military prison." When he was cerned both with the actual mathematics and with released, he went into the service. Weil wrote this how mathematicians describe their work. I pro­ fourteen-page letter to Simone Weil, his sister, from vided a translation from the French in the book's Bonne-Nouvelle Prison in Rouen in March 1940, appendix. I am grateful to the editor of the Notices; sixty-five years ago this month. (Keep in mind that publication herein will allow for an even wider au­ the letter was not written for a mathematician, dience. even though Simone could not understand most of The letter is from Andre Weil, Oeuvres Scien­ it.) tifiques, Collected Papers, volume 1 (New York: I first heard of the letter from a small passage Springer, 1979), pp. 244-255. The translation aims translated in a book by D. Reed (Figures of Thought, to be reasonably faithful, not only to the meaning London: Routledge, 1995). At the time I was trying but also to sentence structure. Brackets are in the to understand the range of solutions to the Ising Oeuvres Scientifiques text. Braces indicate foot­ model in mathematical physics, and in going to notes therein. My editorial insertions are indicated Well's letter I found poignant his exposition of a by braces-and-brackets, {[ ]}. It is slightly revised, threefold analogy out of Riemann and Dedekind, as taken from Martin H. Krieger, Doing Mathe­ one that proves to organize a great deal of disparate matics: Convention, Subject, Calculation, Analogy material. Moreover, I had just begun to appreciate (Singapore: World Scientific, 2003), pp. 293-305. In the significance of the Langlands Program for my the notes to the Oeuvres Scientifiques, Weil indicates problem. [See the "Notes Added in Proof" to Mar­ that he was wrong then about the influence of the tin H. Krieger, Constitutions of Matter: Mathemati­ theory of quadratic forms in more than two vari­ cally Modeling the Most Everyday of Physical Phe­ ables and that Hilbert is explicit about the analogy nomena (Chicago: University of Chicago Press, in his account of the Twelfth Problem (for which 1996), pp. 311-312.] Eventually, in chapter 5 of see David Hilbert, "Mathematical problems", Bul­ Doing Mathematics, I worked out the analogy and letin of the American Mathematical Society 37, provided an exposition of the Weilletter. A recent 2000, 407-436). Notices article ("Some of what mathematicians do", While this article was in proof, Philip Horowitz November 2004, pp. 1226- 1230) summarizes the sent me his unpublished translation of the letter, which I had not known of before. I am grateful to Martin H. Krieger is professor of planning at the Univer· Horowitz for allowing me to use his translation to · sity of Southern California. His e mail address is improve mine in a number of places. kri eger@usc . ed u. - Martin H. Krieger

334 NOTICES OF THE AMS VOLUME 52, NUMBER 3 March 26, 1940 radicals, on the general processes whereby, after having searched for a long time and in vain to Some thoughts I have had of late, concerning my solve this problem by a foreordained procedure, arithmetic-algebraic work, might pass for a re­ mathematicians inverted the question and began sponse to one of your letters, where you asked me to develop adequate methods). The problem had what is of interest to me in my work. So, I decided been solved subsequently for all second-degree to write them down, even if for the most part they equations which had solutions in negative numbers; are incomprehensible to you. when the equation had no solution, the usual for­ The thoughts that follow are of two sorts. The mula having led to the imaginaries, about which first concerns the history of the theory of numbers; there remained many doubts (and it was thus until you may be able to understand the beginning; you Gauss and his contemporaries); just because of will understand nothing of what follows that. The the suspicion of these imaginaries, the so-called other concerns the role of analogy in mathemati­ Cardan and Tartaglia formula for the solution of cal discovery, examining a particular example, and the equation of the 3rd degree in radicals pro­ perhaps you will be able to profit from it. I advise duced some discomfort. Be that as it may, when you that all that concerns the history of mathe­ Gauss began the Disquisitiones with the notion of matics in what follows is based on insufficient congruences for building up his systematic expo­ scholarship, and is derived from an a priori re­ sition, it was also natural to solve congruences of construction, and even if things ought to have the second degree, after having solved those of the happened this way (which is not proven here), I first degree (a congruence is a relationship among cannot say that they did happen this way. In integers a, b, m, which is written a= b modulo m, mathematics, moreover, as much as in any other abbreviated a =b (mod m) or a =b(m), meaning field, the line of history has many turning points. that a and b have the same remainder in division With these precautions out of the way, let us start by m, or a - b is a multiple of m; a congruence of with the history of the theory of numbers. It is dom­ the first degree is ax+ b =O(m), of the second de­ inated by the law of reciprocity. This is Gauss's the­ gree is ax2 + bx + c =O(m), etc.); the latter lead (by ore rna aureum (?I need to refresh my memory of the same procedure through which one reduces an this point: Gauss very much liked names of this sort, ordinary second degree equation to an extraction he had as well a theorema egregium, and I no of roots) to x2 =a (mod m); if the latter has a so­ longer know which is which), published by him in lution, one says that a is a quadratic residue of m, his Disquisitiones in 1801, which was only begin­ if otherwise, a is a non-residue (1 and -1 are ning to be read and understood toward 1820 by residues of 5, 2 and -2 are non-residues). If these Abel, Jacobi, and Dirichlet, and which remained notions were around for some time before Gauss, as the bible of the number theorist for almost it was not necessary that they be associated with a century. But in order to say what this law is, a notion of congruence; for the notions presented whose statement was already known to Euler and themselves in diophantine problems (solutions of Legendre [Euler had found it empirically, as did Le­ equations in integers or rationals) which were the gendre; Legendre claimed more in giving a proof object of Fermat's most important work; the first in his Arithmetic, which apparently supposed the degree diophantine equations, ax+ by= c, are truth of something which was approximately as dif­ equivalent to first degree congruences ficult as the theorem; but he complained bitterly ax= c (mod b); the second degree equations of of the "theft" committed by Gauss, who, without the type studied by Fermat (decompositions in knowing Legendre, found, empirically as well, the terms of squares, x2 + y 2 =a, and equations statement of the theorem, and gave two very beau­ x2 + ay2 = b, etc.) are not equivalent to congru­ tiful proofs in his Disquisitiones, and later up to 4 ences, but congruences and the distinction be­ or 5 others, all based on different principles.]: it is tween residues and non-residues play a large role necessary to backtrack a bit in order to explain the in his work, in truth they did not appear explicitly law of reciprocity. in Fermat's work (it is true that we do not possess Algebra began with the task of finding, for given his proofs, but he seems to have employed other equations, solutions within a given domain, which principles about which we can make some ap­ might be the positive numbers, or the reals, or proximate inferences), but which, as far as I know later the complex numbers. One had not yet con­ (based on second-hand evidence) were already well ceived of the ingenious idea, characteristic of mod­ in evidence in Euler. ern algebra, of starting with an equation and then The law of reciprocity permits us to know, given constructing ad hoc a domain in which it has a so­ two prime numbers p, q, whether q is or is not a lution (I have a fair amount to say about this idea, (quadratic) residue of p, if one knows already which has shown itself to be extremely productive; whether, (a) pis or is not a residue of q; (b) if p and moreover, Poincare has somewhere or other some q are respectively congruent to 1 or -1 modulo 4 beautiful thoughts, a propos of the solution by (or for q = 2, if p is congruent to 1, 2, 5, or 7,

MARCH 2005 NOTICES OF THE AMS 335 modulo 8). For example, 53 = 5 = 1 (mod 4), and Gauss recognized as well, and even thought (there 53 is not a residue of 5, therefore 5 is not a residue is a trace of this in his notes) of studying the do­ of 53. Since the problem for non-primes leads nat­ main of the n-th roots of unity, at the same time urally to the problem for primes, this law gives an thinking to try to provide a proof of "Fermat's the­ easy means of determining if a is or is not a residue orem" (xn + yn = zn is impossible), which he sus­ of b as soon as one knows their prime factoriza­ pected would be a simple application (that is what tion. But this "practical" application is insignificant. he said) of such a theory. But then he encountered What is crucial is there be laws. It is obvious that the fact that there was no longer a unique prime the residues of m form an arithmetic progression decomposition (except fori and j, as 4th and 3rd of increment m, for if a is a residue, it is the same roots of unity, and I believe also for the 5th roots). for all mx +a; however it is beautiful and surpris­ There are many separate threads; it would take ing that the prime numbers p for which m is a 12 5 years to unravel them and assemble them residue are precisely those which belong to certain anew into a new skein. The great names here are arithmetic progressions of increment 4m; for the Dirichlet (who introduced the zeta functions or others m is a non-residue; and what is even more L-functions into the theory of quadratic forms, amazing, if one recalls on the other hand that the through which he proved among other things that distribution of prime numbers in any given arith­ every arithmetic progression contains an infinity metic progression Ax + B (which one knows from of primes; but above all, since that time we have Dirichlet will have an infinity of primes as long as only needed to follow his model in order to apply A and B are relatively prime) does not follow any these functions to the theory of numbers), Kum­ other known law other than a statistical one (the mer (who elucidated the fields generated by roots approximate number of primes which are ~ T, of unity by inventing "ideal" factors, and went far which, for a given A, is the same for any B prime enough in the theory of these fields in order to ob­ to A) and appears, for each concrete case that one tain some results on Fermat's theorem), Dedekind, examines numerically, to be as "random" as a list Kronecker, Hilbert, Artin. Here is a sketch of the of numbers generated by a roulette wheel. picture that results from their efforts. The rest of the Disquisitiones contains above all: I cannot say anything without using the notion 1. the definitive theory of quadratic forms in 2 of a field, which according to its definition, if one variables, ax2 + bxy + cy2 , having among other limits oneself to its definition, is simple (it is a set consequences the complete resolution of the prob­ where one has in effect the usual "four elementary lem which gave birth to the theory: to know if {[arithmetic]} operations," these having the usual ax2 + bxy + cy2 = m has solutions in integers. properties of commutativity, associativity, dis­ 2. the study of the n-th roots of unity, and, as tributivity); the algebraic extension of a field k (it we would say, the Galois theory of the fields given is a field k', containing k, of which all elements are by these roots and their sub fields (all without using roots of an algebraic equation O

336 NOTICES OF THE AMS VOLUME 52, NUMBER 3 abelian extension of the field of rationals), nor simple and it generalizes the law of reciprocity in what connections they had with the work of a straightforward and direct manner. For the arith­ Lagrange, with Abel's own work on elliptic functions metic progression in which the prime numbers are (where the division takes place, from Abel's point found, with residue a, one substitutes ideal classes of view, in the abelian equation [the roots gener­ {[des classes d'ideaux]}, the definition of which is ating abelian extensions], results which were already simple enough. The classes of quadratic forms in known to Gauss, but not published, at the very two variables, studied by Gauss, correspond to a least for the particular case of the so-called lem­ particular case of these classes of ideals, as was rec­ niscate) and abelian functions, or with Jacobi's ognized by Dedekind; Dirichlet's analytic methods work on the same subject (the same Jacobi who in­ (using zeta or L-functions) for studying quadratic vented "abelian functions" in the modern sense and forms, is translated readily to the more general gave them that name, see his memoir "De tran­ classes of ideals that had been considered in this scendentibus quibusdam abelianis"), nor with theory; for example, for the theorem on arithmetic Galois's work (which was only understood little by progressions there corresponds the following re­ little, and much later; there is no trace in Riemann sult: in each of these ideal classes ink, there is an that he had learned from it, although (this is most infinity of prime ideals, therefore an infinity of remarkable) Dedekind, Privatdozent in Gottingen ideals of k which may be factored in a given fash­ and close friend of Riemann, had since 1855 or 6, ion in k'. Finally, the decomposition of ideals of k when Riemann was at the height of his powers, into classes determines k' in a unique way: and, by given a course on abstract groups and Galois the­ the theorem called the law ofArt in reciprocity (be­ ory). cause it implicitly contains Gauss's law and all To know if a (not a multiple of p) is a residue known generalizations), there is a correspondence of p (prime), is to know whether x2 - a = py has (an "isomorphism") of the Galois group of k' with solutions; in passing to the field extension of fo, respect to k, and the "group" of ideal classes ink. one gets (x - .jii)(x + .jii) = py, so in this field pis Thus, once one knows what happens ink, one has not prime to x- fo, which, nevertheless, it does complete knowledge of abelian extensions of k. not divide. In the language of ideals, that is as This does not mean there is nothing more to do much to say that in this field p is not prime, but about abelian extensions (for example, one can may be decomposed into two prime ideal factors. generate these by the numbers exp(- 2rri 1n) if k Thus one is presented with a problem: k being a is the field of rationals, thus by means of the ex­ field (here the field of rationals), k' (here, k' is k ponential function; if k is the field generated by adjoined by fo) an algebraic extension of k, to .;=a., a a positive integer, one knows how to gen­ know if a prime ideal (here, a number) in k re­ erate these extensions by means of elliptic func­ mains prime in k' or if it decomposes into prime tions or their close relatives; but one knows noth­ ideals, and how: a being gi ven, the law of reci­ ing for all other k). But these questions are well procity points to those p for which a is the residue, understood and one can say that everything that and so resolves the problem for this particular has been done in arithmetic since Gauss up to re­ case. Here and in all of what follows, k, k', etc. are cent years consists in variations on the law of rec­ fields of algebraic numbers (roots of algebraic iprocity: beginning with Gauss's law; and ending equations with rational coefficients). with and crowning the work of Kummer, Dedekind, When it is a question of biquadratic residues, one Hilbert, is Artin's law, it is all the same law. This is works with a field generated by -0fa; but such a field beautiful, but a bit vexing. We know a little more is not in general an abelian extension of the "base than Gauss, without doubt; but what we know more field" k unless the adjunction of a 4th root of a (or a bit more) is just that we do not know more. brings along at the same time three others (namely, This explains why, for some time, mathemati­ if lX is one of them, the others are - IX, i lX, and cians have focused on the problem of the non­ - ilX), this requires that k contains i = J=I; one abelian decomposition laws (problems concerning would have nothing so simple if one takes as the k, k', when k' is any nonabelian extension of k; we base field the rationals, but all goes well if one takes remain still within the realm of a field of algebraic (as did Gauss) the field of "complex rationals" numbers). What we know amounts to very little; and r + si (r, s rational). The same is the case for cubic that little bit was found by Artin. To each field is residues. In these cases, one studies the decom­ attached a zeta function, discovered by Dedekind; position, in the field k' obtained by the adjunction if k' is an extension of k, the zeta function at­ of a 4th (or, respectively, 3rd) root, starting with a tached to k' decomposes into factors; Artin dis­ base field k containing i (respectively, j), of an covered this decomposition; when k' is an abelian ideal (here, a number) prime in k . extension of k, these factors are identical to Dirich­ So, this problem of the decomposition in k' of let's L-functions, or rather to their generalization ideals of k is completely resolved when k' is an for fields k and classes of ideals ink, and the iden­ abelian extension of k, and the solution is very tity between these factors and these functions is

MARCH 2005 NOTICES OF THE AMS 337 (in other words) Artin's reciprocity law; and this is pp. 47-49.]} Of course, I am not foolish enough to the way Artin first arrived at this law as a bold con­ compare myself to Riemann; but to add a little bit, jecture (it seems that Landau made fun of him), whatever it is, to Riemann, that would already be, some time before being able to prove it (a curious as they say in Greek, to do something {[faire quelque fact, his proof is a simple translation of another re­ chose]}, even if in order to do it you have the silent sult by Tchebotareff that had just been published, help of Galois, Poincare and Artin. which he cited; however it is Artin, justly having it Be that as it may, in the time (1875 to 1890) when bear his name, who had the glory of discovering Dedekind created his theory of ideals in the field it). In other words, the law of reciprocity is noth­ of algebraic numbers (in his famous "XI Supple­ ing other than the rule for forming the coefficients ments": Dedekind published four editions of Dirich­ of the series that represents the Artin factors (which let's Lectures on the theory of numbers, given at are called "Artin L-functions"). As the decomposi­ Gbttingen during the last years of Dirichlet's life, tion into factors remains valid if k' is a non-abelian and admirably edited by Dedekind; among the ap­ extension, it is these factors, for these "non-abelian pendices or "Supplements" of these lectures, which L-functions", that it is natural to tackle in order to contain nothing indicating they are Dedekind's discover the law of formation of their coefficients. original work, and which indeed they are only in It is worth noting that, in the abelian case, it is part, beginning with the 2nd edition there are three known that the Dirichlet L-functions, and conse­ entirely different expositions of the theory of ideals, quently the Artin L-functions, which scarcely dif­ one for each edition), he discovered that an anal­ fer from them, are entire functions. One knows ogous principle permitted one to establish, by nothing of this sort for the general case: it is there, purely algebraic means, the principal results, called as already indicated by Artin, that one might find "elementary", of the theory of algebraic functions an opening for an attack (please excuse the of one variable, which were obtained by Riemann metaphor): since the methods known from arith­ by transcendental {[analytic]} means; he published metic do not appear to permit us to show that the with Weber an account of the consequences of this Artin functions are entire functions, one could principle. Until then, when the topic of algebraic hope that in proving it one could open a breach functions arose, it concerned a function y of a which would permit one to enter this fort (please variable x, defined by anequationP(x, y) = 0 where excuse the straining of the metaphor). Pis a polynomial with complex coefficients. This lat­ Since the opening is well defended (it had de­ ter point was essential in order to apply Riemann's fied Artin), it is necessary to inspect the available methods; with those of Dedekind, in contrast, those artillery and the means of tunneling under the fort coefficients could come from an arbitrary field (please excuse, etc.). {The reader who has the pa­ (called "the field of constants"), since the argu­ tience to get to the end will see that as artillery, I ments were purely algebraic. This point will be im­ make use of a trilingual inscription, dictionaries, portant shortly. adultery, and a bridge which is a turntable {[or a The analogies that Dedekind demonstrated were turnbridge]}, not to speak of God and the devil, who easy to understand. For integers one substituted also play a role in this comedy.} And here is where polynomials in x, to the divisibility of integers cor­ the analogy that has been referred to since the be­ responded the divisibility of polynomials (it is well ginning finally makes its entrance, like Tartuffe ap­ known, and it is taught even in high schools, that pearing only in the third act. there are other such analogies, such as for the de­ It is widely believed that there is nothing more rivation of the greatest common divisor), to the ra­ to do about algebraic functions of one variable, be­ tionals correspond the rational fractions {[?of poly­ cause Riemann, who had discovered just about all nomials, or the rational functions]}, and to algebraic that we know about them (excepting the work on numbers correspond the algebraic functions. At uniformization by Poincare and Klein, and that of first glance, the analogy seems superficial; to the Hurwitz and Severi on correspondences), left us no most profound problems of the theory of numbers indication that there might be major problems that (such as the decomposition into prime ideals) there concern them. I am surely one of the most knowl­ would seem to be nothing corresponding in alge­ edgeable persons about this subject; mainly because braic functions, and inversely. Hilbert went fur­ I had the good fortune (in 19 2 3) to learn it directly ther in figuring out these matters; he saw that, for from Riemann's memoir, which is one of the great­ example, the Riemann-Roch theorem corresponds est pieces of mathematics that has ever been writ­ to Dedekind's work in arithmetic on the ideal called ten; there is not a single word in it that is not of "the different"; Hilbert's insight was only published consequence. The story is not closed, however; for by him in an obscure review (Ostrowski pointed me example, see my memoir in the Liouville Journal to it), but it was already transmitted orally, much (see the introduction to this paper). {["Generalisa­ as other of his ideas on this subject. The unwrit­ tion des fonctions abeliennes," Journal de Mathe­ ten laws of modern mathematics forbid writing matiques Pures et Appliquees IX 17 (1938): 47-87, down such views if they cannot be stated precisely

338 NOTICES OF THE AMS VOLUME 52, NUMBER 3 nor, all the more, proven. To tell the truth, if this fact that this point plays exactly the same role as were not the case, one would be overwhelmed by all the others is essential. Let R(x) = work that is even more stupid and if not more use­ a(x - 011) ... (x - Oim) l (x- /h) ... (x- /3n) be a ra­ less compared to work that is now published in the tional fraction, with its decomposition into fac­ journals. But one would love it if Hilbert had writ­ tors as indicated; it will have zeros 011, ... , Oim, the ten down all that he had in mind. poles /31, ... , f3n, and the point at infinity, which is Let us examine this analogy more closely. Once zero if n > m, and is infinite if n < m . In the do­ it is possible to translate any particular proof from main of rational numbers, one always has a de­ one theory to another, then the analogy has ceased composition into prime factors, r =Pl ... to be productive for this purpose; it would cease Pm I q1 ... qn , each prime factor corresponding to to be at all productive if at one point we had a mean­ a binomial factor (x - 01); but nothing apparently ingful and natural way of deriving both theories corresponds to the point at infinity. If one models from a single one. In this sense, around 1820, the theory of functions on the theory of algebraic mathematicians (Gauss, Abel, Galois, Jacobi) per­ numbers, one is forced to give a special role, in the mitted themselves, with anguish and delight, to be proofs, to the point at infinity, sweeping the prob­ guided by the analogy between the division of the lem into a corner, if we are to have a definitive state­ circle (Gauss's problem) and the division of ellip­ ment of the result: this is just what Dedekind­ tic functions. Today, we can easily show that both Weber did, this is just what was done by all who problems have a place in the theory of abelian have written in algebraic terms about algebraic equations; we have the theory (I am speaking of a functions of one variable, until now, I was the first, purely algebraic theory, so it is not a matter of num­ two years ago, to give (in Crelle's Journal {["Zur al­ ber theory in this case) of abelian extensions. Gone gebraischen Theorie der algebraischen Funktio­ is the analogy: gone are the two theories, their con­ nen", 179 (1938), pp. 129-133]}) a purely algebraic flicts and their delicious reciprocal reflections, proof of the main theorems of this theory, which their furtive caresses, their inexplicable quarrels; is as birationally invariant (that is to say, not at­ alas, all is just one theory, whose majestic beauty tributing a special role to any point) as were Rie­ can no longer excite us. Nothing is more fecund mann's proofs; and that is of more than method­ than these slightly adulterous relationships; noth­ ological importance. {Actually, I was not quite the ing gives greater pleasure to the connoisseur, first. The proofs, to be sure very roundabout, of the whether he participates in it, or even if he is an his­ Italian school (Severi above all) are, in principle, of torian contemplating it retrospectively, accompa­ the same sort, although drafted in classical lan­ nied, nevertheless, by a touch of melancholy. The guage.} However fine it is to have these results for pleasure comes from the illusion and the far from the function field, it seems that one has lost sight clear meaning; once the illusion is dissipated, and of the analogy. In order to reestablish the analogy, knowledge obtained, one becomes indifferent at the it is necessary to introduce, into the theory of al­ same time; at least in the Gita there is a slew of gebraic numbers, something that corresponds to prayers (slokas) on the subject, each one more the point at infinity in the theory of functions. final than the previous ones. But let us return to That is what one achieves, and in a very satisfac­ our algebraic functions. tory manner, too, in the theory of "valuations". Whether it is due to the Hilbert tradition or to This theory, which is not difficult but I cannot ex­ the attraction of this subject, the analogies be­ plain here, depends on Hensel's theory of p-adic tween algebraic functions and numbers have been fields: to define a prime ideal in a field (a field given on the minds of all the great number theorists of abstractly) is to represent the field "isomorphi­ our time; abelian extensions and abelian functions, cally'' in a p-adic field: to represent it in the same classes of ideals and classes of divisors, there is way in the field of real or complex numbers, is (in material enough for many seductive mind-games, this theory) to define a "prime ideal at infinity". This some of which are likely to be deceptive (thus the latter notion is due to Hasse (who was a student appearance of theta functions in one or another of Hensel), or perhaps Artin, or to both of them. If theory). But to make something of this, two more one follows it in all of its consequences, the theory recent technical contrivances were necessary. On alone permits us to reestablish the analogy at many the one hand, the theory of algebraic functions, points where it once seemed defective: it even per­ that of Riemann, depends essentially on the idea mits us to discover in the number field simple and of birational invariance; for example, if we are elementary facts which however were not yet seen concerned with the field of rational functions of (see my 1939 article in la Revue Rose which con­ one variable x, one introduces (initially, I take tains some of the details {["Sur l'analogie entre les the field of constants to be the complex numbers) corps de nombres algebriques et les corps de fonc­ as the points corresponding to the various complex tions algebriques," Revue Scientifique 77 (1939) values of x, including the point at infinity, denoted 104-106, and the comments in the Oeuvres Scien­ symbolically by x = oo , and defined by 1 I x = 0; the tifiques, volume 1, pp. 542-543]}). It is not so much

MARCH 2005 NOTICES OF THE AMS 339 this point of view that has been used up to now for still polynomials: that is just what occupies me at giving satisfactory statements of the principal the moment, all of this permits me to believe that results of the theory of abelian extensions (I for­ all results for these fields could inversely, if one got to say that this theory is most often called could formulate them appropriately, be translated "class field theory"). An important point is that the to the number fields. p-adic field, or respectively the real or complex field, On the other hand, between the function fields corresponding to a prime ideal, plays exactly the and the "Riemannian" fields, the distance is not so role, in arithmetic, that the field of power series in large that a patient study would not teach us the the neighborhood of a point plays in the theory of art of passing from one to the other, and to profit functions: that is why one calls it a local field. in the study of the first from knowledge acquired With aU of this, we have made great progress; about the second, and of the extremely powerful but it is not enough. The purely algebraic theory means offered to us, in the study of the latter, of algebraic functions in any arbitrary field of con­ from the integral calculus and the theory of ana­ stants is not rich enough so that one might draw lytic functions. That is not to say that at best all useful lessons from it. The "classical" theory (that will be easy; but one ends up by learning to see is, Riemannian) of algebraic functions over the something there, although it is still somewhat con­ field of constants of the complex numbers is infi­ fused. Intuition makes much of it; I mean by this nitely richer; but on the one hand it is too much the faculty of seeing a connection between things so, and in the mass of facts some real analogies be­ that in appearance are completely different; it does come lost; and above all, it is too far from the the­ not fail to lead us astray quite often. Be that as it ory of numbers. One would be totally obstructed may, my work consists in deciphering a trilingual if there were not a bridge between the two. text {[cf. the Rosetta Stone]}; of each of the three And just as God defeats the devil: this bridge ex­ columns I have only disparate fragments; I have ists; it is the theory of the field of algebraic func­ some ideas about each of the three languages: but tions over a finite field of constants (that is to say, I know as well there are great differences in mean­ a finite number of elements: also said to be a Ga­ ing from one column to another, for which noth­ lois field ,, or earlier "Galois imaginaries" because ing has prepared me in advance. In the several Galois first defined them and studied them; they years I have worked at it, I have found little pieces are the algebraic extensions of a field with p ele­ of the dictionary. Sometimes I worked on one col­ ments formed by the numbers 0, 1, 2, .. . , p- 1 umn, sometimes under another. My large study where one calculates with, them modulo p, p = that appeared in the Liouville journal made nice ad­ prime number). They appear already in Dedekind. vances in the "Riemannian" column; unhappily, a A young student in Gottingen, killed in 1914 or large part of the deciphered text surely does not 1915, studied, in his dissertation that appeared in have a translation in the other two languages: but 1919 (work done entirely on his own, says his one part remains that is very useful to me. At this teacher Landau), zeta functions for certain of these moment, I am working on the middle column. All fields, and showed that the ordinary methods of of this is amusing enough. However, do not imag­ the theory of algebraic numbers applied to them. ine that this work on several columns is a frequent Artin, in 1921 or 1922, took up the question again, occasion in mathematics; in such a pure form, this again from the point of view of the zeta function; is almost a unique case. This sort of work suits me F. K. Schmidt made the bridge between these re­ particularly; it is unbelievable at this point that sults and those of Dedekind-Weber, in the process distinguished people such as Hasse and his stu­ of providling a definition of the zeta function that dents, who have made this subject the matter of was birationally invariant. In the last few years, their most serious thoughts over the years, have, these fields were a favorite subject of Hasse and not only neglected, but disdained to take the Rie­ his school; Hasse made a number of beautiful con­ mannian point of view: at this point they no longer tributions. know how to read work written in Riemannian (one I spoke of a bridge; it would be more correct to day, Siegel made fun of Hasse, who had declared speak of a turntable {[?turnbridge]}. On one hand the himself incapable of reading my Liouville paper), analogy with number fields is so strict and obvious and that they have rediscovered sometimes with a that there is neither an argument nor a result in arith­ great deal of effort, in their dialect, important re­ metic that cannot be translated almost word for sults that were already known, much as the ideas word to the function fields. In particular, it is so for of Severi on the ring of correspondences were re­ all that concerns zeta functions and Artin functions; discovered by Deming. But the role of what I call and there is more: Artin functions in the abelian case analogies, even if they are not always so clear, is are polynomials, which one can express by saying that nonetheless important. It would be of great inter­ these fields furnish a simplified model of what hap­ est to study these things for a period for which we pens in number fields; here, there is thus room to are well provided with texts; the choice would be conjecture that the non-abelian Artin functions are delicate.

340 NOTICES OF THE AMS VOLUME 52, NUMBER 3 P.S. I send this to you without rereacling .. .I fear attention to each other and so waste their time, ... having made more of my research than I intended; some like the Hebrews in the desert, others like Han­ that is, in order to explain (following your request) nibal at Capua {[where the troops were said to have how one develops one's research, I have been fo­ been entranced by the place)}. The current organi­ cusing on the locks I wish to open. In speaking of zation of science does not take into account (un­ analogies between numbers and functions, I do happily, for the experimental sciences; in mathe­ not want to give the impression of being the only matics the damage is much less great) the fact that one who understands them: Artin has thought pro­ very few persons are capable of grasping the en­ foundly about them as well, and that is to say a tire forefront of science, of seizing not only the great deal. It is curious to note that one work weak points of resistance, but also the part that is (signed by a student of Artin who is not otherwise most important to take on, the art of massing the known, which without proof to the contrary, allows troops, of making each sector work toward the one to presume that Artin is the real source) ap­ success of the others, etc. Of course, when I speak peared 2 or 3 years ago which gives perhaps the of troops the term (for the mathematician, at least) only example of a result from the classical theory, is essentially metaphoric, each mathematician being obtained by a double translation, starting with an himself his own troops. If, under the leadership arithmetic result (on abelian zeta functions), and given by certain teachers, certain "schools" have no­ which is novel and interesting. And Hasse, whose table success, the role of the individual in mathe­ combination of patience and talent make him a kind matics remains preponderant. Moreover, it is be­ of genius, has had very interesting ideas on this sub­ coming impossible to apply a view of this sort to ject. Moreover (a characteristic trait, and which science as a whole; it is not possible to have some­ would be sympathetic to you, of the school of mod­ one who can master enough of both mathematics ern algebra) all of this is spread by an oral and epis­ and physics at the same time to control their de­ tolary tradition more than by orthodox publications, velopment alternatively or simultaneously; all at­ so it is difficult to make a history of all of it in de­ tempts at "planning" become grotesque and it is nec­ tail. essary to leave it to chance and to the specialists. You doubt and with good reason that modern axiomatics will work on difficult material. When I invented (I say invented, and not discovered) uni­ form spaces, I did not have the impression of work­ ing with resistant material, but rather the impres­ sion that a professional sculptor must have when he plays by making a snowman. It is hard for you to appreciate that modern mathematics has be­ come so extensive and so complex that it is es­ sential, ifmathematics is to stay as a whole and not become a pile of little bits of research, to provide a unification, which absorbs in some simple and general theories all the common substrata of the diverse branches of the science, suppressing what is not so useful and necessary, and leaving intact what is truly the specific detail of each big prob­ lem. This is the good one can achieve with ax­ iomatics (and this is no small achievement). This is what Bourbaki is up to. It will not have escaped you (to take up the military metaphor again) that there is within all of this great problems of strat­ egy. And it is as common to know tactics as it is rare (and beautiful, as Gandhi would say) to plan strat­ egy. I will compare (despite the incoherence of the metaphor) the great axiomatic edifices to commu­ nication at the rear of the front: there is not much glory in the Commissariat and logistics and trans­ port, but what would happen if these brave folks did not consecrate themselves to secondary work (where, moreover, they readily earn their subsis­ tence)? The danger is only too great that various fronts end up, not by starving (the Council for Re­ search is there for that), but by paying insufficient

MARCH 2005 NOTICES OF THE AMS 341 W H A T s aDimer? Richard Kenyon and Andrei Okounkov

A dimeris a polymer with only two atoms. A dimer alistic physical model, the dimer model has in­ covering of a graph (j is a collection of edges that trinsic interest as an exactly solvable model that covers all the vertices exactly once, that is, each ver­ exhibits certain types of phase transitions. Since tex is the endpoint of a unique edge. One can think phase transitions are quite complicated phenom­ of vertices of (j as univalent atoms, each bonding ena in nature, any approximate model whose phase to exactly one neighbor. Dimer coverings are also transitions can be studied analytically is valuable. called perfect matchings. The dimer model is the Kasteleyn, contemporaneously with Temperley study of natural measures ("Gibbs measures") on and Fisher, showed how to count the number of the set of dimer coverings of a graph, usually ape­ dimer coverings of an m x n square grid, and later · riodic planar graph such as z2. While not a very re- on any planar graph. While Kasteleyn's result holds for any planar graph, the statement is particularly simple when (j is a subgraph of the honeycomb graph H (the graph of the regular tiling of the plane by hexagons) bounded by a simple polygon (a "simply connected" subgraph). Then the num­ ber of coverings Z is the square root of the deter­ minant of the adjacency matrix of (j. That is, Z = .JdetK where K is the matrix indexed by the vertices of (j defined by Kv,v' = 1 or 0 according to whether v, v' are adjacent or not. A similar state­ ment (but with extra signs inK) holds for any pla­ nar graph. One can compute, for example, the num­ ber of dimer coverings of an m x n grid to be

m n I -1! k 11 /2 n n 2 cos__!!______+ 2i cos_!!______P=l k=l m + 1 n + 1

For example, the number of domino tilings _of a checkerboard (which are just dimers ori an 8 x 8 grid) is 12988816 = 24 17253 2 tilings. Dimer coverings of bipartite graphs such as the honeycomb graph or square grid can be viewed as

Richard Kenyon is professor of mathematics at the Uni­ v ersity of British Columbia. Hi s email address is [email protected]. Andrei Okounkov is professor of mathematics at Prince­ ton University . .flis e mail address is okounkov@ Figure 1. math.princeton.edu.

342 NOTICES OF THE AMS VOLUME 52, NUMBER 3 (random) surfaces in JR3 . Concretely, dimer cover­ Partial results are available in recent work by the ings of the honeycomb or any of its simply con­ authors. nected subgraphs can be represented as tilings of Similarly, it is expected that the limiting prob­ the plane with 60° rhombi: in this case each atom ability of observing any given finite local pattern is a triangle, and dimers are obtained by gluing ad­ of tiles at a given point is a function that depends jacent triangles along an edge. See Figure 1 for an only on the slope of the limit shape at that point. example. Clearly any such rhombus tiling can be That is, if two limit shapes have the same slopes viewed as the projection of a three-dimensional at corresponding points, then they have the same piecewise-linear surface. The function on a tiling local measures (defined by the densities of local pat­ that gives the third coordinate of this projection terns) there. All these probabilities can be computed is called the height function. A similar kind of explicitly. For example, if s is the y-slope of the height function also exists for domino tilings, or height function, then the density of horizontal dimers on any bipartite planar graph, though the rhombi is s + l I 3, and the covariance between two definition is a little more complicated. horizontal rhombi, one of which is k steps directly sin2 srrk The tiling in Figure 1 is perfectly random, and above the other, equals - (rrk)2 . One derives yet one can see that there is something regular nearly identical formulas for correlations of eigen­ about its height function. In fact, as we take tilings values of random matrices. This is one indication of a fixed region (e.g., a polygon) with smaller and that dimers and random matrices have very much smaller rhombi, the height functions of a typical in common. Indeed, in a certain natural sense a tiling will converge to some nonrandom surface random dimer covering is a discretization of the called the limit shape. What is this limit shape? Fig­ time evolution of the eigenvalues of a random Her­ ure 1 shows it has some linear pieces-those occur mitian matrix. in the frozen regions near the corners where all tiles Instead of the honeycomb graph, one can make there are lined up the same way. How large are the similar calculations for dimers on any periodic bi­ fluctuations around the limit shape ? What is the partite planar graph. In fact, it is natural to allow the probability of occurrence of some fixed local pat­ edges of the graph to be weighted, in which case the tern at a given point ? weight of a dimer covering is proportional to the The answers to these and other questions de­ product of the weights of its edges. Again the sta­ pend on the asymptotics of the inverse Kasteleyn tistics can be worked out exactly using Kasteleyn's matrix K-1 as the size of the graph goes to infin­ method, and one obtains more complicated PDEs ity. Since K can be viewed (using work of the first for the limit shapes. Curiously, these PDEs can all author) as a discretization of the a-operator, it is be solved, and one can find the limit shapes via not so surprising that complex analysis and alge­ analytic data, in a way resembling Weierstrass's braic geometry enter the scene. In the nonfrozen parametrization of minimal surfaces in JR 3 via region, the limit shape satisfies a nonlinear ellip­ analytic data. These more general models enjoy a tic PDE closely related to the complex Burgers equa­ further property that their limit shapes can have tion 1:>x + 1:>1:>r = 0. To be precise, the arguments facets in many other rational directions. The inter­ of 1:>, 1 - 1/1:> and 1 I ( 1 - 1:>) are proportional to the play between analyticity and facet formation makes densities of tiles of the three orientations. These for a fascinating study in real algebraic geometry. densities extend continuously to constants on the frozen regions. For polygonal domains like the one Further Reading in Figure 1, the solution to this free boundary prob­ [1] P. KAsTELEYN, Graph theory and crystal physics, Graph lem is algebraic. In particular, it turns out that the Theory and Theoretical Physics, Academic Press, London, 1967,pp.43-110. boundary of the frozen region in the figure is an [2] H. COHN, R. KENYON,]. PROPP, A variational principle for inscribed cardioid (a degree 4, genus 0 curve with domino tilings. ]. Amer. Math. Soc. 14 (2001), no. 2, 1 real and 2 complex cusps-the yellow curve in 297-346. the figure). [3] R. KENYoN, The planar dimer model with boundary: A From the complex Burgers equation, the non­ survey, Directions in mathematical quasicrystals, CRM frozen part of the limit shape comes equipped Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, with a natural complex and, hence, conformal struc­ 2000, pp. 307- 328. ture, which is different from the conformal struc­ ture induced from the plane. It is expected that in this new conformal structure the fluctuations The "WHAT IS ... ?" column carries short (one- or two-page) nontechnical articles aimed at graduate stu­ around the limit shape are described by a confor­ dents. Each article focuses on a single mathematical mally invariant process called the Gaussian free object rather than a whole theory. The Notices wel­ field, a two-dimensional cousin of Brownian motion. comes feedback and suggestions for topics. Messages It is still an open problem (even for the honey­ may be sent to noti ces-whati s@ams. or g. comb graph) to prove this in complete generality.

MARCH 2005 NOTICES OF THE AMS 343 Book Review

The Golden Ratio

Reviewed by George Markowsky

The Golden Ratio were being made about MarioLivio the golden ratio. Broadway Books, 2003 The results of my re­ Paperback, 304 pages, $14.95 search were published in ISBN 0-7679-0816-3 "Misconceptions about the Golden Ratio" (The The number (1 + JS)/ 2 = 1.618 ... is widely College Mathematics known as the golden ratio, ¢ and phi. Phi appears journal, Vol. 23, No . 1, in many different equations and formulas and has Jan. 1992, 2-19). This many interesting properties. Many people have paper debunks many of heard marvelous tales about phi and how it per­ the more prominent meates art and nature. My first exposure to phi was claims about phi and in a comic book entitled Donald in Mathmagic documents their perva­ Land, which later became an animated cartoon sive presence in the seen by millions of people. As I grew up I kept see­ mathematical literature. For example, the name ing the same "facts" repeated in many differ~nt places, including popular books on mathematiCS, "golden ratio" is a nineteenth-century creation and various mathematics textbooks, newspapers, and is not an ancient name for phi. Furthermore, it does not appear that phi was used to design either even in scholarly papers. It seemed as if every­ the Great Pyramid or the Parthenon. For example, body knew these basic "facts" about phi. the Parthenon is 228 feet and 1/8 inch long, 101 Around 1990 I decided to give a talk to the Uni­ feet and 3.7 5 inches wide, and 45 feet and 1 inch versity of Maine Classics Club and thought that the high. Taking the obvious ratios of length/~dt.h golden ratio would be a fascinating topic for this and width/height yields the number 2.25, which1s audience. During the preparation of the talk I col­ quite far from phi, which is 1.618 .... The number lected all of the usual stories about the golden 2.2 5 = 9/4 is the ratio of two squares, and further ratio being used to design the Great Pyramid and study indicated that the gate to the Acropolis was the Parthenon, as well as about its aesthetic prop­ built using this same ratio. erties and its use by painters. I found the references It also does not appear that Leonardo da Vinci to be quite vague, and in the process of trying to used phi, nor is phi present in the proportions of make my talk more precise, I actually began to the United Nations building in New York. look up measurements of buildings. Much to my Furthermore, in a large number of informal audi­ surprise, the results did not support the claims that ence participation events, I have found that peo­ George Markowsky is professor of computer science at the ple do not pick golden rectangles more frequently University of Maine. His email address is markov@ than others (in fact, they are often picked less fre­ maine. edu. quently than others), so that the statements about

344 NOTICES OF THE AMS VOLUME 52, NUMBER 3 the aesthetic superiority of phi do not stand up to analyzes them in the same manner. Later in the empirical tests. These claims and others are de­ chapter he reproduces a diagram from my paper molished in my paper in some detail. (he attributes the diagram to me, but does not give Since publishing my paper, I have tried to get a reference) that shows forty-eight rectangles of dif­ people, in particular mathematicians, to tell the ferent proportions that I have used a number of truth about phi. Phi has many interesting mathe­ times to ask people which rectangle they find most matical properties that deserve to be brought to the pleasing. public's attention. It is, however, a disservice to On p. 183 of his book, Livio states: mathematics to mix the interesting properties of phi with dubious claims about its importance in art, You can test yourself (or your friends) architecture, human anatomy, and aesthetics. on the question of which rectangle you Mario Livio's book, The Golden Ratio, is a broad prefer best. Figure 84 shows a collection survey of the properties of phi. The book is some of forty-eight rectangles, all having the 270 pages long, counting ten appendices, and same height, but with their widths rang­ bounces along describing various mathematical ing from 0.4 to 2.5 times their height. properties of phi, while at the same time trying to University of Maine mathematician astonish the reader. It is the constant desire to as­ George Markowsky used this collection tonish the reader that gets Livio into trouble and in his own informal experiments. that is undoubtedly the source of the subtitle: The Interestingly, Livio does not reference my paper Story of Phi, the World's Most Astonishing Number. and does not quote my conclusion: When I first heard about this book, I was hopeful that it would finally put many of the bogus stories In the experiments I have conducted so about phi to rest, but unfortunately this book does far, the most commonly selected rec­ not quite do so. tangle is one with a ratio of 1.83. For example, in his discussion of the Parthenon, Livio quotes from my paper and gives proper at­ Also, Livio does not point out that there are ac­ tribution. However, I believe that he waffles on the tually two golden rectangles in the diagram-one issue of whether phi was used in the design of the is oriented with the long dimension horizontal and Parthenon. On p. 74 he states: the other with the long dimension vertical. Livio could have performed a valuable service to So, was the Golden Ratio used in the the mathematical community had he written an ac­ Parthenon's design? It is difficult to say curate book about phi that treated it in a balanced for sure. While most of the mathemat­ manner and that consistently and thoroughly de­ ical theorems concerning the Golden bunked the various misconceptions about phi that Ratio (or "extreme and mean ratio") ap­ continue to circulate. Throughout this book, Livio pear to have been formulated after the struggles with the problem of wanting to "amaze" Parthenon had been constructed, con­ the public without going too far and losing re­ siderable knowledge existed among the spectability, but unfortunately he does not suc­ Pythagoreans prior to that. ceed in solving it. I take strong issue with his conclusion that it is He deserves credit for surveying a wide range difficult to say for sure whether phi was used in of sources about phi, but in my opinion he is very the construction of the Parthenon. It seems to me inconsistent in how he uses them. In some cases that to even entertain the notion there has to be he does an effective job of debunking nonsense, but some reason to believe that it was true. It is clear in others his debunking is halfhearted. In some that the Greeks were not as enamored of phi as peo­ cases he omits data that would be harmful to es­ ple became once it received the name golden ratio tablishing phi as the "most astonishing" number. in the nineteenth century. Calling phi division into Unfortunately, he also seems interested in spawn­ mean and extreme ratio does not generate great ex­ ing some new myths. citement on the part of artists and architects. I For example, on p. 9 he discusses Salvador Dali's have found no credible evidence that phi was ever "Sacrament of the Last Supper". The first "fact" that used by Greek artists and architects for any pro­ we are presented with is that the canvas measures ject at all. approximately 105.5 inches by 65.75 inches, which In Chapter 7 of his book Livio discusses the "are in a Golden Ratio to each other." The ratio possible presence of phi in various paintings and 105.5/65.75 is approximately 1.605, which is close its role in aesthetics. Again, he closely parallels my to, but not equal to phi. If it was important for the paper but does not cite the paper either in the text painting to have phi as the ratio of its width to or in the notes to the text. In his discussion of height, why not use a canvas of size approximately Leonardo da Vinci, Livio reproduces exactly the 106 inches by 66 inches, which has a ratio of 1.606, painting and drawing discussed in my paper and which is even closer to phi? We are next told that:

MARCH 2005 NOTICES OF THE A.MS 345 Perhaps more important, part of a huge the recurrence relation and not on the starting dodecahedron(. .. ) is seen floating above points. In particular, pick any two positive integers the table and engulfing it. ... As we shall and use the formula Fn = Fn -1 + 2Fn-2· One will see in Chapter 4, regular solids (like the find that eventually the ratio of consecutive terms cube) that can be precisely enclosed by will approach 2. In this regard, phi is no more a sphere with all their corners resting amazing than just about any other number. on the sphere, and the dodecahedron in Livio devotes a fair amount of space to dis­ particular, are intimately related to the cussing Luca Pacioli and his work on the "Divine Golden Ratio. Proportion". Livio notes that Pacioli ends up rec­ ommending a system of proportions for art not This paragraph is odd for a number of reasons. based on phi, even after he spends a lot of time dis­ First, it seems to suggest that somehow the cube cussing phi. A favorite gambit of Livio is to ask is related to phi. Fortunately, when one reads chap­ rhetorical questions such as the one on p. 178: ter 4, one learns that "The Golden Ratio, ¢,plays "Short of intellectual curiosity, for what reason a crucial role in the dimensions and symmetry would so many artists even consider employing the properties of some Platonic solids." As one might Golden Ratio in their works?" The placement of this expect, the cube is not one of these solids. Another question is interesting because it follows a long sec­ oddity is that while phi is present in the various tion generally showing that artists have not been proportions of the dodecahedron, it is interesting using phi in their work in any significant way. Of to note that the dodecahedron in the painting is dis­ course, part of the answer to the question is that torted by the perspective that Dali used. Thus the people keep writing books and papers extolling proportions that we see in the painting itself are the aesthetic virtues of phi. With so much being not those of the dodecahedron. Livio makes no at­ written about phi by "experts", many artists feel tempt to actually measure any of the dimensions strong pressure to at least look at phi. or to relate what we see to phi. He then proceeds At times the book appeals to mysticism. It talks to ask: "Why did Dali choose to exhibit the Golden about the "mystical" properties of integers andre­ Ratio so prominently in this painting?" This is as­ peats a lot of nonsense about 666, the number of tonishing because he did not give any evidence the beast; there is even a ridiculous formula relat­ that phi is present in any significant way or that ing 666 and phi. In particular, we are expected to Dali had any interest in displaying phi in his paint­ be amazed (p. 23) that sin666° + cos(6 x 6 x 6) 0 is ings. Since Dali wrote about his paintings, one a "good approximation" of the negative of phi. would expect that he would have mentioned his use Doing some "research" of this type, I was amazed of phi if that was of importance to him. to find that tan 666° + tan 666° (about -2 . 75276) is Another way of expanding what it means to "sort of" a good approximation of - e. In his dis­ "use phi" is to take all applications of Fibonacci cussion of pyramidology, Martin Gardner shows numbers as applications of phi. Of course, one can how in the absence of any rules one can torture express the Fibonacci numbers in terms of powers numbers to come up with just about any result one of phi, but Livio, like most authors writing to as­ wants. tonish people, neglects to mention that repre­ Livio describes the rectangle construction that senting the Fibonacci numbers in terms of phi phi enthusiasts are so fond of (pp. 85-86). The would make it much harder to "use" them in many fact that one gets a spiral of rectangles is consid­ applications. In particular, one can use the Fi­ ered amazing. Of course, one can do the same bonacci numbers happily without ever knowing thing with any rectangle by dividing it into two about phi. Most properties of the Fibonacci num­ pieces: a smaller rectangle similar to the original bers are best derived from the recurrence relation rectangle, and another rectangle that always has Fn = Fn - 1 +Fn- 2, rather than by using phi. The fact some fixed proportion. One can then create a spi­ that Fibonacci numbers can be written in terms of ral of smaller rectangles that converges to a point. phi is a special case of the much more general re­ Why the spiral derived from phi should be called sults available as part of the theory of linear re­ "the Eye of God" is not explained. It is also not men­ currence equations with constant coefficients. tioned that the rectangle having dimensions 2 by On p. 86 Livio notes that if one takes any two .J2 is even more amazing, since if one divides the positive integers and forms a series in which each long side in half one gets two rectangles similar to new term is the sum of the preceding two terms, the original rectangle instead of just one rectangle then eventually the ratio of a term to the preced­ and a square, as one does with phi. ing term converges to phi. He holds this out as an In addition to its tendency to exaggerate the amazing fact but does not mention that in general "uses" of phi, the book contains outright errors. For if one picks any linear recurrence to generate terms example, on p . 19 we are told that "we could even in such a s equence, one will find that consecutive argue theoretically that the fact that 13 is a prime terms converge to some ratio that depends only on number, divisible only by I and itself, gives it an

346 NOTICES OF THE AMS VOLUME 52, NUMBER 3 advantage over 10, because most fractions would to find the golden ratio somewhere can be irreducible in such a system." Given that divis­ alter two numbers by± 1% and alter their ibility properties are independent of the base, this ratio by roughly ± 2%. statement makes no sense. I was surprised to find the following discussion On p. 116 the book says that "Jacques Bernoulli's in Livio's book (p. 47) without attribution: association with the Golden Ratio comes through another famous curve." The curve referred to is the The second point that is often ignored logarithmic spiral. However, the definition of the by the too-passionate Golden Ratio afi­ logarithmic spiral does not depend on phi, as can cionados is that any measurements of be seen from its equation r = aece in polar coor­ lengths involve errors or inaccuracies. It dinates, where a > 0 and c > 0. This curve spirals is important to realize that any inaccu­ infinitely often in both directions if - 00 < e < 00 . racy in length measurements leads to a If one permits c = 0 then one gets a circle. One can yet larger inaccuracy in the calculated certainly use phi as a parameter, but clearly one can ratio. For example, imagine that two also use any other positive number as a parame­ lengths, of 10 inches, each, are mea­ ter. It is seriously misleading to claim that the sured with a precision of 1 percent. This properties of the logarithmic spiral somehow de­ means that the result of the measure­ pend on phi. Even though the book has ten math­ ment of each length could be anywhere ematical appendices that contain formulas, between 9.9 and 10.1 inches. The ratio nowhere in the book does the formula for the log­ of these measured lengths could be as arithmic spiral appear. Of course, the formula for bad as 9.9/ 10.1 = 0.98, which repre­ the logarithmic spiral would reveal that the curve sents a 2 percent inaccuracy-double has no special dependence on phi. The claims about that of the individual measurements. the logarithmic spiral being related to phi are re­ Therefore, an overzealous Golden Num­ peated at several places in the book. berist could change two measurements Chapter 8 has some interesting material about by only 1 percent, thereby affecting the tilings and quasi-crystals. It is a shame that this ma­ obtained ratio by 2 percent. terial is not developed with more technical details. Even though Livio is aware of my paper and Chapter 9, the final chapter, contains a long dis­ quotes it in various places, it is not even men­ cussion about the unreasonable effectiveness of tioned in the notes for the chapter where the pre­ mathematics. The breezy way in which discussions ceding paragraph appears. This chapter also dis­ of phi, the Fibonacci numbers, god, relativity, and cusses the Great Pyramid and seems to follow the string theory all roll into one another serves to outlines of the discussion in my paper, again with­ glorify the role of phi. out any attribution. For example, compare p. 6 of The book also suffers from sloppy scholarship. my article with p. 56 of Livio's book. As in my In several places it follows my paper closely with­ paper he includes the link to Martin Gardner's dis­ out giving any attribution. For example, a key point cussion of pyramidology, which is the crank dis­ that I addressed in my paper (p. 5) was to develop cipline of predicting the future by playing around some way of determining whether a measurement with various measurements from the Great Pyra­ can actually be an indicator of the presence of phi. mid. In my paper I pointed out that some of the In my paper I proposed that people use a ±2% "facts" that Martin Gardner used in his classic range around phi to at least treat the claim of the book, Fads and Fallacies in the Name of Science, to presence of phi as being worthy of consideration. debunk the pyramidologists are actually based on I gave the rationale for this as follows: their work. Again, no citations are given to my Another point overlooked by many work. To his credit Livio concludes that the Golden golden ratio enthusiasts is the fact that Ratio was most likely not consciously incorporated in the design of the Great Pyramid. measurements of real objects can only I think that Livio lost a great opportunity. If he be approximations. Surfaces of real ob­ had focused on the mathematics of phi and spent jects are not perfectly flat. Furthermore, less time on trying to astonish people with dubi­ it is necessary to specify the precision of ous claims, he would have done the mathematical any measurements and to realize that in­ community a great service. Given his ability to accuracies in measurements lead to write, he would have also produced a much more greater inaccuracies in ratios. For interesting book. example, a ±1% variation in the mea­ surement of two lengths can lead to a roughly ±2% variation (0.99 / 1.01 >::; 0.98 to 1.01 /0.99 >::; 1.02) in the ratio that is computed. Thus someone eager

MARcH 2005 NOTICES OF THE AMS 347 A Tribute to Deane Montgomery

Ronald Fintushel

One afternoon during my meetings there.) We talked for a while about the third year as a graduate stu­ graduate program at Binghamton and then moved dent at S.U.N.Y. Binghamton, to the blackboard at the front of the office. I ex­ my adviser, Louis McAuley, plained my thesis work, and Deane encouraged popped into my office and me and offered some ideas for future research. I asked if I would like to talk returned to Binghamton invigorated and excited with Deane Montgomery about mathematics. Within a year, due in a large about my thesis. "Certainly", part to Deane's help, I had a postion at Tulane Uni­ I answered, tongue-in-cheek, versity. There are dozens of mathematicians who thinking that he must be jok­ have told me similar stories about their relation­ ing. How could Deane Mont­ ship with Deane Montgomery. He was justly fa­ gomery, one of the founders mous for his efforts in helping young topologists of the theory of transforma­ and, more generally, making all visitors feel wel­ Deane Montgomery tion groups, possibly be in- come at the Institute. He was especially ardent at terested in hearing about the searching for students like myself from smaller, less work of a student at S.U.N.Y. Binghamton? Half an prestigious graduate programs and encouraging hour later my adviser appeared once more, an­ their careers. nouncing that, sure enough, we were driving to In 1979-1981, I was fortunate to serve as Deane Princeton early the next morning. Montgomery's assistant. Throughout his career at That next day was a turning point in my career in mathematics. I appeared at Deane Montgomery's the Institute, Deane had twenty-one assistants. Al­ office with more than a little trepidation, but though most of them were mainly interested in Deane's warmth and sincerity put me at ease. His some aspect of transformation groups, this was not office was expansive, with a beautiful view of the always the case. For example, M. Kuranishi, E. Moise, Institute grounds. It had a large library and was and C. D. Papakyriakopoulos were his assistants. arranged with a dual purpose. The rear was fur­ And again, many were chosen from outside "pres­ nished comfortably in order to facilitate conver­ tige" departments. In order to occupy that won­ sation. (Later I would learn that the Institute's derful office in the corner of Fuld Hall adjacent to Faculty of Mathematics would often hold their Deane Montgomery's, his assistants assumed two duties-to meet with him once a week in a private seminar to study a topic of the assistant's choice Ronald Fintushel is University Distinguished Professor of Mathematics at Michigan State University. His email and to allow Deane to buy him a cup of coffee af­ address is ronfi nt@math. msu. edu. He wrote this memo­ terward. rial tribute for Deane Montgomery's memorial service at During our weekly sessions I learned much about the Institute for Advanced Study in 1992. this remarkable man. Deane often spoke of his ca-

348 NOTICES OF THE AMS VOLUME 52, NUMBER 3 reer and of the history of the Institute. I have ex­ tiful structure theory for such actions on homotopy coriated myself several times since for not keep­ 7-spheres, and they showed that one can find ex­ ing a journal, for most of the details have drifted amples with arbitrarily many exceptional orbits. away with time. However, there are certain basic When I first came to the Institute, I was interested aspects of these conversations that I will never in the same question for pseudofree circle actions forget-Deane's love of mathematics and his joy at on the 5-sphere, and my discussions with Deane en­ the success of others, his gentleness and personal couraged me further. It was on this problem that humility, his abhorrence of pretense in any form, Ron Stern and I first began our collaboration, and his pride in the Institute and conviction to uphold although the problem itself remains unsolved, it has its standards. Most of all, he absolutely never gave been a major motivation for most of our work false praise. His midwestern upbringing and math­ since then. It has served as a testing ground for our ematical training at the University of Iowa gave knowledge of Kirby calculus, of the theory of sin­ him a point of view that often served as a refreshing gular spaces, and finally of gauge theory, without foil to the intense sophistication all around him. I ever revealing all its secrets. Yet in turn it has remember one Thursday-morning topology semi­ taught us much about 4-dimensional topology. In nar whose topic was not au courant and whose pre­ recent years there has been a resurgence of ex­ sentation was, to put it kindly, rough at the edges. citement among young researchers in calculating Afterward, when asked for my opinion, I mumbled gauge-theoretic invariants of Seifert fiber spaces, noncommittedly. Deane's response was quite dif­ and many facets of their interest can be traced ferent. "I thoroughly enjoyed that," he said, and back to the papers of Montgomery andYang via this called the speaker "salt of the earth", one of his route. highest forms of praise. Deane had the knack of The admiration of the mathematical commu­ making people feel comfortable and important. He nity for Deane was universal. There was a large con­ knew that I was an avid runner, so we often talked ference held in 1983 in honor of his seventy-fifth about our exercise schedules. Deane was himself birthday at the University of Colorado and also a an early-morning exerciser who enjoyed going for conference at the University of North Carolina in a long walk before coming to his office. Now, I am honor of Deane's eightieth birthday in 1989. At a person who likes to work in the early morning, each of these conferences many of the mathe­ but in my two years at the Institute, I never arrived maticians spoke extemporaneously about the ways at Fuld Hall before Deane. Usually his door was that their lives and careers had been touched by closed, and I could hear the muffled sound of con­ their friendship and mathematical association with versation with an early guest. Many mathemati­ Deane Montgomery. I found the story of one of the cians who came to the Institute began their visit by participants to be particularly moving. He recounted calling on Deane. how, early in his career at an East Coast university, Because of his humility and personal distaste for his desire to be a mathematician was nearly over­ self-promotion, mathematicians whose work does whelmed by anti-Semitism. It was Deane Mont­ not involve the study of transformation groups gomery who helped him gain the resolve to fight are often unaware of his many contributions to the bigotry and to persevere in his work. After I left the Institute, Deane topology. Others have spoken and written about his and I kept up a steady correspondence. He had developed an interest in solution of Hilbert's fifth problem, but perhaps not enough is said about his later work, especially gauge theory, and this was often a topic of dis­ cussion. I could always count on his letters for his joint work with C. T. Yang. In a long series of support and advice. In many ways I felt that Deane papers written in the late 1960s and early 1970s, Montgomery was my mathematical father. In this they used the study of group actions on homotopy sense, I have many siblings. His mathematics and ?-spheres to showcase and test the growing new his unwavering character have inspired all of techniques of differential topology, especially index us. By his actions, he has shown us how to conduct theory and surgery theory. At a time when much our relationships with our own colleagues and work in topology consisted in building these ma­ students. Although we all miss him terribly, if we chines, their papers demonstrated the beauty of ap­ follow his example, his spirit will never die. plying this theory to unfurl complexities of sym­ metry and structure. As a part of this series, Montgomery and Yang studied pseudofree circle ac­ tions, those that have no points fixed by the entire circle group but that have isolated circles that are pointwise fixed by finite cyclic subgroups. Since a linear action of this sort on a 2n - 1 sphere can have at most n such exceptional orbits, it was nat­ ural to ask whether such a restriction existed for smooth actions. In their papers they found a beau-

MARCH 2005 NOTICES OF THE AMS 349 Presidential Views: Interview with james Arthur

Every other year, when a new AMS president takes office, the Notices publishes inter­ views with the incoming and outgoing president. What follows is an edited version of an interview with James Arthur, whose two-year term as president begins on Febru­ ary 1, 2005. The interview was conducted in fall2004 by Notices senior writer and deputy editor Allyn Jackson. Arthur is a University Professor of Mathematics at the Univer­ sity of Toronto. An interview withAMS president David Eisenbud appeared in the February 2005 issue of the Notices.

Notices: When you found out that the Nominat­ Notices: The way mathematics is funded by the ing Committee wanted you to run for president, federal government in Canada is very different why did you say yes? from how it's done in the United States. Can you ex­ Arthur: I actually was quite surprised, not being plain the difference? an American and not working in the United States. Arthur: Yes. I don't know the figures about total I was a little concerned about whether it would be funding per mathematician. I imagine they might a reasonable thing to do. So I made inquiries to both be comparable, or maybe a little higher in Canada. AMS people and also to a number of my colleagues, I think the main difference, as far as mathemati­ and I was convinced that it would be viable. I don't cians are concerned, is that there is no summer believe there has been an AMS president who has money in Canada. The size of the grants is variable, not been actually living in the United States. There and the money is used for postdoctoral fellows and have been other Canadians, such as Cathleen graduate students. Morawetz and Irving Kaplansky. But I cut my math­ Notices: Do most Canadian mathematicians have ematical teeth in the United States, and I worked grants? there for twelve years, during graduate school and Arthur: I think most active ones have a grant. afterwards. As it happens I am the only member That suggests there are more grants in Canada per of my family who is not an American citizen. My capita than in the United States. wife is American, and my two sons are American Notices: Are the grants perceived the same way citizens and are both working at this time in the as they are in the US? Here an NSF grant is a big United States. I am very proud to be a member of validation, and mathematics departments take it se­ the Canadian mathematical community. I came riously in tenure decisions. Is that also true in back to Canada in 1979, and I am proud to see the Canada? progress that Canadian mathematics has made Arthur: Yes. Of course, the size of the grant is since then. relevant also, because there are some small grants Mathematics is an international enterprise, and in Canada. But it's taken very seriously. There are the AMS has something like one-third of its mem­ of course good sides and bad sides to that. If a per­ bers outside the United States. The AMS has some son receives a disappointing decision from the purely American mathematical concerns, but it is granting agency, it can be harmful psychologically. a major force in international mathematics. It is the On the other hand, the work of granting commit­ most influential mathematical organization in the tees in both countries is very careful, and I think world and reaches well beyond the boundaries of we would not have that kind of careful, informed the United States. I don't have a congressperson I analysis of a person's work and of what a person can write to! But I suppose that one letter from a proposes to do if the analysis were done by uni­ president to his local congressperson does not versity administrators who are not mathemati­ make that much of a difference. I expect to be able cians. So in that sense I think the granting process to play a significant role in discussions with peo­ is on the whole helpful to how mathematics oper­ ple in Washington on mathematical questions. ates and how it is organized in universities. The

350 NOTICES OF THE AMS VOLUME 52, NUMBER 3 problem is that in the United States, relatively few At the undergraduate level, mathematicians have people are funded. been criticized for not being good teachers. The cal­ Notices: Do you have thoughts on the NSF-funded culus sequence has of course been one of the prime VIGRE program? examples. But we have to remember that a greater Arthur: The idea is very good. It really helped proportion of students are taking calculus now build up graduate programs at some universities. than have done so in the past. In teaching calcu­ It must be a little bit capricious as to which de­ lus we are being asked to be the gatekeepers for partment gets a VIGRE grant and which one doesn't. other academic programs. That makes for an ad­ And there is always the danger that a department versarial relationship. Probably we as mathemati­ might be worse off in the long run if it becomes cians are better teachers than we give ourselves dependent on a program that is not of infinite du­ credit for being. We need to remember that, and ration. On the other hand, I think VIGRE probably we need to tell people has been very helpful to departments in attracting in our universities. good graduate students. It's really hard to get good Notices: People say graduate students. This is something that I would that in recent years hope the AMS tries to affect in one way or another. math departments and We want the people who are best at mathematics mathematicians are as undergraduate or high school students to think much more concerned very seriously about going into a career in mathe­ about the quality of their matics. These are talented people, and they might teaching than they were well be good at other things. If society does not im­ ten or fifteen years ago. plicitly send them the message that mathematics Is that your sense? is important, for example by providing good fund­ Arthur: Oh, ab­ ing for graduate school, they might well think of solutely. Part of it is that other things to do. young mathematicians Notices: In your election statement you noted simply cannot survive if that mathematics has been the beneficiary of much they are not competent greater interest on the part of the public in recent teachers. That's taken years. Why did that happen? very seriously by search Arthur: Well, it is quite remarkable. I am not sure, committees. So most James Arthur but you can point to some obvious things. Andrew young mathematicians Wiles's proof of Fermat's Last Theorem, in a way who are hired by universities are pretty good teach­ that we would not have expected, caught people's ers. That puts pressure on everybody to become bet­ imagination. Books like the one on John Nash, A ter teachers-if you are an older mathematician, you Beautiful Mind, have also brought a good deal of don't want to be shown up by the instructor or as­ attention to mathematics. And of course in movies, sistant professor you have just hired! mathematics has been chic in the last five or ten Notices: What about minority students in math­ years. There is a sympathetic curiosity about math­ ematics? There are so few, for example, black peo­ ematics. People don't know what mathematicians ple getting Ph.D.'s in mathematics. What can be do, but they somehow have the feeling that math­ done about that? It's a difficult question. ematics has mystery, power, and beauty. I think peo­ Arthur: It is, and I don't think there is any magic ple who are not mathematicians have a sense of way to do it. It's the question of role models. As a that. We should stoke their curiosity. We already white male, when I was beginning in mathematics, do so, but I think we can do more. I certainly did not understand the importance that This is probably the most important aspect of role models played for me. When I thinl< back now, strengthening mathematics in the United States it was just something I took for granted. There and around the world. To have public sympathy and were people who were like me but twenty years interest in mathematics would help just about older than me, and they were mathematicians. It's everything that concerns us as mathematicians. It hard to imagine what it would have been like if no would persuade more talented young people to go such people existed. For black students, there are into mathematics, because mathematics would be few role models as mathematicians. These are ob­ regarded with interest, and maybe even awe, by vious things to say, but the AMS should do what it their parents and their friends, and would seem to can to increase the visibility and the prominence be a worthy cause to spend one's life pursuing. It of mathematicians who belong to minorities. One would help funding for mathematics, because vot­ obvious way is through membership on its com­ ers would have a sympathy for it. Also, public ap­ mittees. The AMS committees do wonderful work preciation of the beauty of mathematics would supporting the many different objectives of the So­ help encourage good people to become high school ciety. They also lend a certain prominence to the teachers. people who are on the committees. I think we

MARCH 2005 NOTICES OF THE AMS 351 already work very hard on this, and when I am session, in which the speaker is given friendly ad­ president, I hope we can find overlooked minority vice: "No, this is not going to work, you will have mathematicians and women mathematicians and to polish up on that." I don't know if we mathe­ make them more visible as role models. My pre­ maticians would stand for it! decessor David Eisenbud has done a very good job In September I went to a Congressional briefing on this. organized by the AMS Washington Office. Fred It's a little sobering to be taking over this pres­ Roberts [director of DIMACS] spoke on how math­ idency! I look back at the past four or five presi­ ematics can help with emergency preparedness, dents, and while they perhaps had different inter­ disaster prevention, and related security matters. ests, they were all very impressive in the things they I don't think Fred would mind me saying that he did. The AMS, in its professional operations, does didn't speak about anything particularly deep in many different things. From the executive director mathematics on this occasion. It was a very charm­ down, it's an extraordinary organization. The AMS ing talk, and there were all kinds of questions at publication operation is really very important. I the end. One person, who appeared to be a Con­ don't want to get into trouble with commercial gressional aide, asked "What's it like to be a math­ publishers, but they are making it difficult for ematician? What does a mathematician's typical mathematics and other subjects because the fees day consist of?" It was quite amazing, they were that they charge are enormous. Just look at what really curious. There was something about the way the AMS does. We have the journal of the AMS, and Fred talked that invited this-you felt that he was it's one of the best three or four journals in math­ conversing with you. ematics. I doubt that other professional societies In calculus courses, we have the potential to in­ have such outstanding journals. Mathematicians fluence a huge number of university students, look forward with anticipation to the monthly No­ some of whom are going to be the leaders of to­ tices of the AMS. We also have MathSciNet, with its morrow. And we somehow forget that. We can't unequaled mathematical database. make every calculus student happy, but neverthe­ One thing we have to be concerned about is the less we can try to communicate to them that math­ lack of AMS membership among younger mathe­ ematics is interesting, even if they themselves don't maticians. I think that's extremely important, and want to do it in the long term. I hope to try to think of ways in which we can in­ About ten years ago, a former University of crease membership. It is not a question of the Toronto student endowed a chair in mathematics. dues-dues are not a large part of our budget. He was a person who had gone to New York and What is much more important is to have as many had done very well in investment banking. He took people participating in the AMS as possible. It in­ first-year calculus in 1980 or 1981 at the Univer­ creases our strength as a community. We need to sity of Toronto, and then did not continue in math­ emphasize that we are part of a common enterprise ematics. He so liked his experience in calculus that that is a source of strength and energy for all of ten years later he came back to the university and us. endowed a chair in mathematics. This is really Notices: Are there other things that you want to quite scary, when you think of all those students try to address as president? in your calculus classes! Calculus is very pretty, and Arthur: I haven't completely formulated in my we forget how remarkable it is. Perhaps we can own mind what things I will try to do. So many sometimes try to indicate to students that calcu­ things are working well in the AMS that one has to lus is the starting point for much of mathematics. be a little bit conservative. But I will try to change I think there is a lot of good will that we don't tend or add to things where I think I can accomplish to see on the part of the students. What we see per­ something. Perhaps the area where one could try haps is their resentment athaving to take calcu­ to make the greatest difference is in, as I already lus when they don't want to take it, and also the said, the public understanding of mathematics. I stress they have in learning and passing the exams think we should consider sponsoring more public in what is quite frankly a difficult subject, compared lectures and more publications that are directed at to the other first-year courses that they might take. the general public. But we forget that, underneath, a lot of them have For public lectures, there are two aspects. We a good will towards mathematics. Not all of them, would first of all have to do the mechanics of ad­ but a significant number have an interest and an vertising it, getting people to come to it-then we've appreciation, even if they are not going to go on in got to deliver the goods! We would not regard giv­ the subject. We need to stimulate it. ing a public lecture as a reward for being a good I am very honored to be president of the Soci­ mathematician. We would instead have to make ety. It has had a wonderful history, and I have sure we find the people who do it best. I am told some very distinguished predecessors that I can try that other organizations actually script their pub­ to live up to! I am very much looking forward to lic lectures. Before the talk they have a practice starting my term.

352 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Mathematics People

for the Advancement of Teaching and the Council for TWAS Prizes Announced Advancement and Support of Education (CASE), Which The Academy of Sciences for the Developing World (for­ cosponsor the awards. The Professor of the Year Awards merly the Third World Academy of Sciences, TWAS) has are intended to reward outstanding professors for their announced its 2004 prizewinners. LONGYIMIN Gof the Nankai dedication to teaching, their commitment to students, and Institute of Mathematics at Nankai University was awarded their innovative instructional methods. the prize in mathematics for fundamental contributions The State Professors of the Year Award Program selects to Hamiltonian dynamics, in particular for establishing the outstanding educators in all fifty states, the District of index iteration theory for symplectic paths and deep stud­ Columbia, Guam, Puerto Rico, and the U.S. Virgin Islands. ies on periodic solution orbits of Hamiltonian systems. SPENTA R. WADIA, Tata Institute of Fundamental Research, Winners receive personalized award certificates and receive was awarded the physics prize for his contributions to non­ national and local media recognition. State and national perturbative quantum field theory and string theory. winners are chosen on the basis of their dedication to The Academy of Sciences for the Developing World an­ undergraduate teaching, determined by excellence in the nually awards prizes of $10,000 each to scientists from following four areas: impact on and involvement with developing countries who have made outstanding contri­ undergraduate students; scholarly approach to teaching butions in the fields of agricultural sciences, biology, and learning; contributions to undergraduate education chemistry, earth sciences, engineering sciences, mathe­ in the institution, community, and profession; and support matics, medical sciences, and physics. from colleagues and current and former undergraduate students. -From a TWAS announcement -From a Carnegie Foundation announcement Lemieux and Dick Win Information-Based Complexity Young Mathematidan Honored Young Researcher Award in Siemens Westinghouse CHRISTIANE LEMIEUX of the University of Calgary and JosEF DICK Competition of the University of New South Wales have been awarded the second Information-Based Complexity Young Researcher Po-LING LoH, a senior at James Madison Memorial High Award. The award recognizes significant contributions to School in Madison, Wisconsin, has been awarded a$ 50,000 information-based complexity by researchers who have scholarship in the national Siemens Westinghouse Com­ not reached their thirty-fifth birthday by September 30 of petition in Math, Science, and Technology for her project the year of the award. The prize consists of $500 and a "Closure Properties of D 2p in Finite Groups". She studied plaque. The prize committee consisted of Stefan Heinrich, a generalized version of a problem posed by the topolo­ Universitiit Kaiserslautern; Frances Kuo, University of New gist E. Farjoun about closed embeddings of finite groups South Wales; Joseph F. Traub, Columbia University; Arthur Werschulz, Fordham University; and Henryk Wozniakowski, in finite groups. Her mentor on her project was Michael Columbia University and University of Warsaw. Aschbacher of the California Institute of Technology. Loh is a member of the varsity math team, the Knowl­ -Joseph Traub, Columbia University edge Masters team, and the National Honor Society. She has participated in the American Mathematics Competi­ tions, writing a perfect paper in the 2003 competition, and Professor of the Year Awards was one of the top twelve students in the 2003 USA Math­ ematical Olympiad. She is also a MathCounts student Announced coach and a peer tutor. She aspires to become a math RoBERT L. DEvANEYof Boston University and STEPHEN]. GREEN­ teacher. FIELD of Rutgers University have been chosen to receive State Professor of the Year awards by the Carnegie Foundation -From a Siemens Foundation announcement

MARCH 2005 NOTICES OF THE AMS 353 Mathematics Opportunities

professional activities. It also provides the participants Collaboration in Mathematical with a network of peers and mentors as they assume these Geosciences responsibilities. Each year about sixty faculty members from colleges and universities throughout the country are The Division of Mathematical Sciences of the National Sci­ selected to participate in a workshop preceding the ence Foundation sponsors an activity called Collaboration Mathematical Association of America (MAA) summer meet­ in Mathematical Geosciences (CMG). ing, in activities during the summer MAA meetings and the The purposes of the CMG activity are: (A) Interdiscipli­ Joint Mathematics Meetings in January, and in an elec­ nary Group Research Projects- to enable collaborative tronic discussion network. Faculty for whom the 2005-2006 research at the intersection of mathematical sciences and academic year will be the first or second year of full-time geosciences, and to encourage cross-disciplinary education teaching (post-Ph.D.) at the college or university level are through (B) Interdisciplinary Postgraduate Summer invited to apply to become Project NExT Fellows. Training-summer graduate training activities and (C) The application deadline is April 15, 2005. For more Interdisciplinary Postdoctoral Research- opportunities for information, see the Project NExT website, http: I I interdisciplinary postdoctoral research. Research topics archives. math. utk. edu/proj next/. Project NExT is a under (A) should fall within one of three broad themes: program of the MAA. It receives major funding from the (1) mathematical and statistical modeling of large, complex ExxonMobil Foundation, with additional funding from the geosystems; (2) representing uncertainty in geosystems; or Dolciani-Halloran Foundation, the American Mathematical (3) analyzing large geoscience data sets. Research projects Society, the Educational Advancement Foundation, the supported under this activity must be essentially collaborative American Statistical Association, the National Council of in nature. Research groups must include at least one math­ Teachers of Mathematics, Texas Instruments, the Associ­ ematical scientist and at least one geoscientist. ation of Mathematics Teacher Educators, the Association Please see the webpage http://www.nsf.gov/ for Symbolic Logic, and the Greater MAA Fund. pubsys/ods/getpub. cfm?nsf05 53 5 for details. The dead­ line for proposals is March 14, 2005. -Christine Stevens, Project NExT -From an NSF announcement International Mathematics Project NExT: New Experiences Competition for University in Teaching Students The Twelfth International Mathematics Competition for Project NExT (New Experiences in Teaching) is a professional University Students (IMC) will be held July 22-28, 2005, at development program for new and recent Ph.D.'s in the the American University in Bulgaria in Blagoevgrad, mathematical sciences (including pure and applied math­ Bulgaria. The competition is coorganized by the American ematics, statistics, operations research, and mathematics University and University College, London. education). It addresses all aspects of an academic career: Participating universities are invited to send several improving the teaching and learning of mathematics, students and one teacher each. Individual students are engaging in research and scholarship, and participating in welcome. The competition is intended for students

354 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Mathematics Opportunities completing their first, second, third, or fourth years of uni­ mathematics for an educated public. Articles should be versity education and will consist of two sessions of five submitted both in the original language (the published ver­ hours each. Problems will be taken from the fields of sion) and preferably also in an English translation. Article algebra, analysis (real and complex), and combinatorics. translations may, however, also be submitted in French, The working language will be English. German, Italian, or Spanish. The English or alternative The deadline for registration is May 31, 2005. For more language version should be submitted both in paper and complete information, see the website http: I /www. electronic forms. The deadline for submissions to Com­ i me-math. org or contact John E. Jayne, Department of petition 2005 is August 1, 2005. Mathematics, University College London, Gower Street, Competition 2006 focuses on articles for the general London WClE 6BT, United Kingdom; telephone +44-20- public. Entries must be already published or about to be 7679-7322; fax +44-20-7419-2812; email: j .jayne@ published in a daily newspaper or some other widely read i me-math. o rg. If you would like a copy of the competi­ general magazine. Articles should be submitted both in the tion poster, please send your request with postal address original language (the published version) and preferably to j . j ayne@i me-math. o rg. also in an English translation. Article translations may, - From an IMC announcement however, also be submitted in French, German, Italian, or Spanish. The English or alternative language version should be submitted both in paper and electronic forms. The deadline for submissions to Competition 2006 is Summer Program for Women january 1, 2006. lJndergraduates For each competition, prizes of 200 euros (about US$270), 150 euros (about US$200), and 100 euros (about The 2005 Summer Program for Women in Mathematics US$135) respectively will be awarded to the three best (SPWM) will take place at George Washington University in articles. Many of the winning articles will be published in Washington, DC, from July 2 to August 6, 2005. This is a the Newsletter of the EMS. five-week intensive program for mathematically talented Articles should be identified with the name of the com­ undergraduate women who are completing their junior petition to which they are being submitted: Competition years and may be contemplating graduate study in math­ ematical sciences. Goals of this program are to commu­ 2005 or Competition 2006. The address for submission is: nicate an enthusiasm for mathematics, to develop research Vagn Lundsgaard Hansen, Department of Mathematics, skills, to cultivate mathematical self-confidence and in­ Technical University of Denmark, Matematiktorvet, Build­ dependence, and to promote success in graduate school. ing 303, DK-2800 Kongens Lyngby, Denmark; email: Sixteen women will be selected. Each will receive a travel V. L. Hansen@mat. dtu. dk. For more information see the allowance, campus room and board, and a stipend of website http: I jwww. mat. dtu. dk/peop l e/V. L. Hansen/ $1,500. The deadline for applications to be received is rpa/secondartcomp.html. March 1, 2005. Early applications are encouraged. For further information please contact the director, -From an EMS announcement Murli M. Gupta, email: mmg@gwu. edu, telephone 202-994- 4857; or visit the program's website at http: I jwww. gwu. edu/-math/spwm.html. Application material may be Call for Nominations for Luden printed from the website. Godeaux Prize - Murli M. Gupta, George Washington University The Liege Royal Society of Sciences is seeking nominations for the Lucien Godeaux Prize in Mathematics, one of four prizes to be awarded in honor of the 150th anniversary European Mathematical Sodety of the founding of the society. Announces New Article The prize will be awarded to a researcher or researchers under thirty-five years of age for a body of work pub­ Competitions lished by a single author or by collaborating authors. The The Committee for Raising Public Awareness of Mathe­ prize carries a cash award of 2,500 euros (about US$3,350). matics (RPA) of the European Mathematical Society (EMS) The deadline for applications is October 1, 2005. is seeking submissions of articles for two new competi­ For information on the nominating process, please con­ tions: one for articles aimed at the educated layperson and tact]. Aghion, c/o Secretariat of the Royal Society of Sci­ the other for articles that address a general audience. ences of Liege, Institute of Mathematics of the University Competition 2005 focuses on articles for the educated of Liege, 12 Grande Traverse, Sart Tilman Bat. B 3 7, B-4000 layperson. These must be articles that have been or are Liege 1, Belgium; email: j aghi on@ul g. ac. be. about to be published in an international magazine or in a specialized national magazine that publishes articles on -From a Royal Society of Sciences announcement

MARCH 2005 NOTICES OF THE AMS 355 Mathematics Opportunities

educational researchers, mathematicians, and policy Call for Nominations for makers to examine what is known about the knowledge needed for teaching mathematics, and how it can be de­ Information-Based Complexity veloped and learned. Funding may be available to support Prize workshop attendees. Students, recent Ph.D.'s, women, and minorities are particularly encouraged to apply. For more The Prize for Achievement in Information-Based Com­ information, visit http: I lwww. ms ri . o rgl (see "Work­ plexity is awarded every year for work done in a single year, shops" listing). in a number of years, or over a lifetime. The work can be published in any journal, in a number of journals, or in -From an MSRI announcement monograph form. The prize consists of $3,000 and a plaque. The dead­ line for nominations is March 31, 2005. However, a person does not need to be nominated to win the award. For more information, contact Joseph Traub at [email protected].

-joseph Traub, Columbia University NSF Computing Equipment and Instrumentation Programs The Division of Mathematical Sciences of the National Sci­ ence Foundation (NSF) plans a limited number of awards for the support of computing environments for research in the mathematical sciences. SCREMS (Scientific Com­ puting Research Equipment for the Mathematical Sciences) supports computing environments dedicated to research in the mathematical sciences. Proposals may request support for the purchase of computing equipment and limited support for professional systems administrators New Journals from or programmer personnel for research computing needs. WORLD SCIENTlFlC - These grants are intended to support research projects of high quality that require access to advanced computing INTERNATIONAL JOURNAL OF CALL FOR PAPERS! resources. Requests for routine upgrades of standard NUMBER THEORY (IJNT) Managing Editors desk-environment workstations or laptop computers are JOURNAL OF HYPERBOLIC not appropriate for this program. Awards are made to Bruce C Berndt, DIFFERENTIAL EQUATIONS Urbana-Champaign, IL OHDE) provide support for specific research projects rather than Dipendra Prasad, Tata, India to provide general computing capacity. Proposers are Michel Waldschmidt, Paris, France Managing Editors This journal publishes original Philippe Lefloch, Paris, France encouraged to include projects involving symbolic and research papers and review Jian-Guo Liu, College Park, MD algebraic computations, numerical computations and articles on all areas of Number This journal publishes original simulations, and graphical representations (visualization) ' Theory. research papers on non I i near hyperbolic problems and related in aid of the research. Effective in FY 2005, cost sharing topics, of mathematica l and/or is no longer required and will not be considered by NSF in physical interest. the evaluation of SCREMS proposals. Please see http: I lwww. nsf. gov lpubsyslodsl _...... ~ getpub. cfm?nsf05538 for details. The deadline for __ HYPERBOLIC proposals is April 4, 2005. DIFFERENTIAL EQUATIONS - From an NSF announcement News from MSRI Please visit worldscinet.com for more information. The Mathematical Sciences Research Institute (MSRI) will For subscription, please contact sales®wspc.com. host a workshop focused on Mathematical Knowledge for ~ ~--. Teaching from May 25-28, 2005 in Berkeley, California. The lii) World ScientifiC ' ,_ et conference is designed to bring together K-12 educators, '~ ~ "'"'""'""ld> cll~

356 NOTICES OF THE AMS VOLUME 52, NUMBER 3 For Your Information

Everett Pitcher Lectures DMS Employment The next series of Everett Pitcher Lectures will be held Opportunities March 14, 15, and 1 7, 2 00 5, on the campus of Lehigh University in Bethlehem, Pennsylvania. The speaker will be Several of the technical staff of the Division of Mathematical Sciences (DMS) of the National Science Foundation (NSF) Peter Sarnak of Princeton University and the Courant serve one- to two-year visiting scientist or Intergovernmental Institute of Mathematical Sciences. The title of his lecture Personnel Act appointments as program directors while on series is "Quantum Chaos and Spectra of Arithmetic leave from universities, colleges, industry, or national lab­ Manifolds". oratories. Since the timing of these positions is staggered, The lectures, which are open to the public, are held in the division continually seeks talented applicants. In 2005 honor of Everett Pitcher, who was secretary of the AMS from the division will be seeking to make appointments in all 196 7 until 1988. Pitcher served in the mathematics de­ areas. Permanent program director appointments will also partment at Lehigh from 1938 until1978, when he retired be considered. as Distinguished Professor of Mathematics. The positions involve responsibility for the planning, For further information, contact the Everett Pitcher coordination, and management of support programs for Lecture Series, Department of Mathematics, Lehigh Uni­ research (including multidisciplinary projects), infra­ versity, Bethlehem, PA, 18015; telephone 610-758-3788; or structure, and human resource development for the see the website http://www.lehigh.edu/~math/ mathematical sciences. Normally, this support is provided pitcher. html. through merit-reviewed grants and contracts that are awarded to academic institutions and nonprofit, nonaca­ -From a Lehigh University announcement demic research institutions. Applicants should have a Ph.D. or equivalent training in a field of the mathematical sciences, a broad knowledge of one of the relevant disciplinary areas of the DMS, some IClAM Establishes Su Buch:in administrative experience, a knowledge of the general scientific community, skill in written communication and Prize preparation of technical reports, an ability to communi­ The International Council for Industrial and Applied Math­ cate orally, and several years of successful independent ematics (ICIAM) has announced the creation of the Su research normally expected of the academic rank of Buchin Prize to provide international recognition of an out­ associate professor or higher. Skills in multidisciplinary standing contribution by an individual in applying math­ research are highly desirable. Qualified women, ethnic/ ematics to emerging economies and human development, racial minorities, and/or persons with disabilities are strongly urged to apply. No person shall be discriminated in particular at the economic and cultural levels in devel­ against on the basis of race, color, religion, sex, national oping countries. origin, age, or disability in hiring by the NSF. The prize, in honor of the distinguished Chinese math­ Applicants should send a letter of interest and a vita to ematician Su Buchin, was created on the initiative of the Deborah Lockhart, Executive Officer, Division of Mathe­ China Society of Industrial and Applied Mathematics matical Sciences, National Science Foundation, 4201 Wilson (CSIAM) and will be awarded for the first time at the I ClAM Boulevard, Suite 102 5, Arlington, VA 2 2 2 3 0; telephone 703- Congress in 2007. The prize is being funded by the CSIAM. 292-4858; fax 703-292-9032; email: dl ockhar@nsf. gov.

-From an ICIAM announcement -NSF announcement

MARcH 2005 NOTICES OF THE AMS 357 Inside the AMS

For the 2004 awards the AMS chose six geographically AMS Sponsors NExT Fellows distributed schools to receive one-time awards of $4,000 Project NExT (New Experiences in Teaching) is a profes­ each. The mathematics departments at those schools then sional development program for new or recent Ph.D.'s in chose students to receive the funds to assist them in pur­ the mathematical sciences (including pure and applied suit of careers in mathematics. The schools are selected mathematics, statistics, operations research, and mathe­ in a random drawing from the pool of AMS institutional matics education). It addresses all aspects of an academic members. . career: improving the teaching and learning of mathe­ Waldemar ]. Trjitzinsky was born in Russia in 1901 matics, engaging in research and scholarship, and and received his doctorate from the University of Califor­ participating in professional activities. It also provides nia, Berkeley, in 1926. He taught at a number of institu­ the participants with a network of peers and mentors as tions before taking a position at the University of Illinois, they assume these responsibilities. Each year sixty to Urbana-Champaign, where he remained for the rest of his seventy new Ph.D.'s receive Project NExT Fellowships, professional life. He showed particular concern for students which allow them to attend special events at the MAA of mathematics and in some cases made personal efforts MathFest and at the Joint Mathematics Meetings. to ensure that financial considerations would not hinder Each year the AMS sponsors six AMS NExT Fellows who their studies. Trjitzinsky was the author of about sixty are affiliated with Ph.D.-granting institutions and who mathematics papers, primarily on quasi-analytic functions show promise in mathematics research. The AMS also and partial differential equations. A member of the AMS holds activities for the AMS NExT Fellows at the Joint for forty-six years, he died in 1973. Mathematics Meetings. Following are the names of the selected schools for The names, affiliations, and areas of research of the 2004, the names of the students receiving Trjitzinsky 2004-2005 AMS NExT Fellows are: KATHARINE GURSKI, George Awards, and brief biographical sketches of the students. Washington University, applied mathematics, mathemat­ Beloit College: LAURA WOLFGRAM. Wolfgram grew up in ical modeling; ELIZABETH KlODGINSKI, University of Califor­ Madison, Wisconsin. She became interested in mathemat­ nia, Davis, low-dimensional topology; MARK W. LuciANovrc, ics during her sophomore year in college. She was an ath­ Stanford University, number theory, automorphic forms; lete in high school and continues to compete in college. KATHRYN L. NYMAN, Loyola University, Chicago, combina­ Lafayette College: PRINCE CHIDYAGWAI, EKATERINA JAGER, and torics; JoHN STARRETI, New Mexico Institute of Mining and BLERTA SHTYLLA. Chidyagwai was born in Zimbabwe and Technology, dynamical systems and knot theory; and came to the United States to study at Lafayette. He majors PETER ]. THOMAS, Oberlin College, mathematical biology in both mathematics and computer science. He has par­ and computational neuroscience. ticipated in the National Science Foundation's Research For further information about Project NExT, visit the Experiences for Undergraduates (REU) Program. He is vice website http: I /archives. math. utk. edu/proj next/. president of Pi Mu Epsilon; a teaching assistant in the mathematics department; and a member of the Interna­ -Elaine Kehoe tional Students Association, the student chapter of the As­ sociation of Computing Machinery, and the Mathematics Club. He plans to pursue a Ph.D. in applied mathematics. Trjitzinsky Memorial Awards Jager was born in Tashkent, Uzbekistan, and came to Lafayette as an engineering student. She is now a double Presented major in mathematics and electrical and computer engi­ The AMS has made awards to nine undergraduate stu­ neering. She presented a poster at the Joint Mathematics dents through the Waldemar]. Trjitzinsky Memorial Fund. Meetings in Phoenix in 2004 and was a member of The fund is made possible by a bequest from the estate Lafayette's team in the 2004 Putnam Competition. She is of Waldemar ]., Barbara G., and Juliette Trjitzinsky. The the treasurer of the Society of Women Engineers and is a will of Barbara Trjitzinsky stipulates that the income from member of the Math Problem Solving Group, the Interna­ the bequest should be used to establish a fund in honor tional Students Association, the Investment Club, and the of the memory of her husband to assist needy students Hispanic Society. She enjoys Latin and swing dancing. She in mathematics. plans to continue to study mathematics in graduate school.

358 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Inside the AMS

Shtylla was born in in Tirana, Albania, and spent her se­ nior year of high school studying in Switzerland. She ma­ Deaths of AMS Members jors in mathematics and minors in bioengineering. She has SHIING-SHEN CHERN, professor emeritus of the University of participated in the Program for Women in Mathematics at California, Berkeley, and Nankai University, People's the Institute for Advanced Study and held a summer re­ Republic of China, died on December 3, 2004. Born in search fellowship at the Mayo Clinic. She also participated October 1911, he was a member of the Society for 61 years. in the National Science Foundation's REU Program. She KARIN GATERMANN , of the University of Western Ontario, plans to study applied mathematics in graduate school. died in 2004. Born on December 17, 1961, she was a mem­ Michigan State University: ANTONIO VELOZ. Veloz grew ber of the Society for 3 years. up in Detroit, Michigan. At Michigan State, according to MICHAEL GRUTER, of the Universitat des Saarlandes, Peter Bates, chair of the Department of Mathematics, "he Germany, died on November 13, 2004. Born on September caught sight of the beauty and power of mathematics." He 8, 1953, he was a member of the Society for 24 years. is a nearly straight-A student in advanced mathematics and ]. HARRisoN HANcocK, retired, from Charlottesville, VA, plans to continue to study mathematics in graduate school. died on October 14, 2004. Born on July 31, 1916, he was University of Pennsylvania: DANIEL M. POMERLEANO. a member of the Society for 7 years. Pomerleano was born in Falls Church, Virginia. He at­ FRANK HARARY, professor emeritus, New Mexico State tended the Thomas Jefferson High School for Science and University, Las Cruces, died on january 5, 2005. Born on Technology, where he took upper-level math classes. As a March 11, 1921, he was a member of the Society for 60years. result he has already begun to take graduate courses as GREGERS KRABBE, professor emeritus, of Concord, CA, a sophomore at Pennsylvania. He plans to combine his died on October 31, 2004. Born on January 5, 1919, he was degree in mathematics with a degree in finance from the a member of the Society for 49 years. ]EAN E. LEFEBVRE, retired, from Antony, France, died on Wharton School of Business. He has research experience September 25, 2004. Born on june 29, 1914, he was a mem­ in the simulation of stochastic processes as well as dif­ ber of the Society for 43 years. ferential games. He is a master in chess and is interested FREDERICK W. LIPPS, retired, from Bellaire, TX, died on in a mathematical representation of chess; he applies his July 30, 2004. Born on February 18, 1929, he was a mem­ current theories against the highest level of competition. ber of the Society for 28 years. He is interested in researching the relationship between MICHAJLO YADRENKO, of Kiev University, Ukraine, died on mathematics and finance. September 28, 2004. Born on April 16, 1932, he was a Portland State University: KATHRYN CARR and CASS BATH. member of the Society for 9 years. Carr is a senior mathematics major at Portland. After grad­ uation she plans to enter the Peace Corps for two years and hopes to work in health education in Africa. When she completes this experience, she plans to enroll in graduate school in advanced mathematics with the goal of becom­ ing a high school or community college teacher. Bath carries a double major in mathematics and women's studies and a minor in psychology. She plans to study sta­ tistics in graduate school with the goal of earning a Ph.D. Her career goal is to combine research in the social sciences with work in statistics, and she is particularly interested in studying gender equity, as well as other inequities, in mathematics and science. The mathematics department at Portland State Univer­ sity will contribute $1,000 in matching funds to these students' awards. Santa Clara University: OLMA NICHOL GISTAND. Gistand began her college career as a computer science major but became fascinated with mathematics and changed her major. She plans to become an actuary and is also inter­ ested in teaching high school mathematics. The mathe­ matics department at Santa Clara will contribute $500 in matching funds to her award. For further information about the Trjitzinsky Memor­ ial Fund, contact the AMS Development Office, 201 Charles Street, Providence, RI 02904-2294; email: development@ ams.org; telephone 401-455-4111.

-Elaine Kehoe

MARCH 2005 NOTICES OF THE AMS 359 Reference and Book List

The Reference section of the Notices Upcoming Deadlines March 1, 2005: Applications for is intended to provide the reader with February 15, 2005: Nominations for Summer Program for Women in Math­ frequently sought information in Clay Mathematics Institute Liftoff ematics. See "Mathematics Opportu­ an easily accessible manner. New Program. See the website http: I I nities" in this issue. information is printed as it becomes claymath.org/fas/liftoff_ March 1, 2005: Applications for summer program of the Christine available and is referenced after the fellows/. Mirzayan Science and Technology Pol­ first printing. As soon as information February 28, 2005: Applications is updated or otherwise changed, it icy Graduate Fellowship Program of for Clay Mathematics Institute 2005 will be noted in this section. the National Academies. See http: I I Summer School. See http: I /www. www7.nationalacademies.org/ Contacting the Notices claymath.org/sum~erschool; pol i cyfe ll ows or contact The Na­ The preferred method for contacting telephone: 617-995-2600; email: tional Academies Christine Mirzayan the Notices is electronic mail. The [email protected]. Science and Technology Policy editor is the person to whom to send Where to Find It articles and letters for consideration. Articles include feature articles, A brief index to information that appears in this and previous issues of the Notices. memorial articles, communications, opinion pieces, and book reviews. The AMS Bylaws-November 2003, p. 1283 editor is also the person to whom to AMS E-mail Addresses-December 2004, p. 1365 send news of unusual interest about AMS Ethical Guidelines-]unej]uly 2004, p. 675 other people's mathematics research. AMS Officers 2002 and 2003 (Council, Executive Committee The managing editor is the person Publications Committees, Board of Trustees)-May 2004, p. 566 ' to whom to send items for "Mathe­ AMS Officers and Committee Members-October 2004, p. 1082 matics People", "Mathematics Op­ Conference Board of the Mathematical Sciences-September 2004, portunities", "For Your Information", p. 921 "Reference and Book list", and "Math­ Information for Notices Authors-]une/]uly 2004, p. 670 ematics Calendar". Requests for Mathematics Research Institutes Contact Information-August 2004, p. 810 permissions, as well as all other inquiries, go to the managing editor. National Science Board-january 2005, p. 76 The electronic-mail addresses are New Journals for 2003-]unej]uly 2004, p. 672 noti ces@math. ou. edu in the case of NRC Board on Mathematical Sciences and Their Applications-March 2005, the editor and noti ces@ams. org in p. 361 the case of the managing editor. The NRC Mathematical Sciences Education Board-April 2004, p. 446 fax numbers are 405-325-7484 for NSF Mathematical and Physical Sciences Advisory Committee-February the editor and 401-3 31-3842 for the 2005, p. 261 managing editor. Postal addresses Program Officers for Federal Funding Agencies-October 2004, p. 1078 (DoD, DoE); December 2004, p. 1368 (NSF) may be found in the masthead.

360 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Reference and Book List

Graduate Fellowship Program, 500 Graduate Fellowship Program, 500 Lawrence Craig Evans, University Fifth Street, NW, Room 508, Wash­ Fifth Street, NW, Room 508, Wash­ of California, Berkeley ington, DC 20001; telephone: 202- ington, DC 20001; telephone: 202- John E. Hopcroft, Cornell Univer­ 334-2455; fax: 202-334-1667. 334-245 5; fax: 202-334-1667. sity March 1, 2005: Applications for June 2, 2005: Applications for NSF Robert E. Kass, Carnegie Mellon EDGE Summer Program. See http: I I University-Industry Cooperative Re­ University www.edgeforwomen.orglindex. search Programs in the Mathematical Kathryn B. Laskey, George Mason html. Sciences (UICRP). See http: I lwww. University March 1, 2005: Applications for nsf.govl pubsyslodsl getpub. C. David Levermore, University of IMA New Directions visiting profes­ cfm?nsf05504. Maryland sorships at IMA. See http: I lwww. june 30, 2005: Nominations for Robert Lipshutz, Affymetrix, Inc. ima.umn.edul docslmembership. the 2005 Fermat Prize. See http: I I Charles M. Lucas, American Inter­ html. www.ups-tlse.friACTUALITESI national Companies March 14, 2005: Collaboration in Sciences1 Prix_Fermat_2004l Charles Manski, Northwestern Uni­ Mathematical Geosciences. See "Math­ Aregl ement. html. versity ematics Opportunities" in this issue. July 31, 2005: Nominations and David McLaughlin, New York Uni­ March 31, 2005: Nominations for applications for the Monroe H. Martin versity (chair) Information-Based Complexity Prize. Prize. Contact R. Roy, Director, Insti­ Joyce R. McLaughlin, Rensselaer See "Mathematics Opportunities" in tute for Physical Science and Tech­ Polytechnic Institute this issue. nology, University of Maryland, Prabhakar Raghavan, Stanford April1, 2005: Applications for College Park, MD 20742-2431. University IMA New Directions Short Course. August 1, 2005: Submissions for Stephen M. Robinson, University of See http: llwww.ima.umn.edul Competition 2005 of the European Wisconsin, Madison new-directionsi2005NDshort­ Mathematical Society. See "Mathe­ Edward J. Wegman, George Mason coursei NDcourse-app.php. matics Opportunities" in this issue. University April4, 2005: Scientific Computing October 1, 2005: Nominations for Detlof von Winterfeldt, University Research Equipment in the Mathemat­ Lucien Godeaux Prize. See "Mathe­ of Southern California ical Sciences (SCREMS). See "Mathe­ matics Opportunities" in tbis issue. The postal address for BMSA is: matics Opportunities" in this issue. January 1, 2006: Submissions for Board on Mathematical Sciences and April 8, 2005: Proposals for 2005 Competition 2006 of the European Their Applications, National Academy NSF-CBMS Regional Conferences. See Mathematical Society. See "Mathe­ of Sciences, Room K974, 500 Fifth http: I l www. cbms. org or contact: matics Opportunities" in this issue. Street, NW, Washington, DC 20001; Conference Board of the Mathematical January 1, 2006: Applications for telephone: 202-334-2421; fax: 202- Sciences, 1529 Eighteenth Street, NW, ICM 2006 Travel Grants. See http: I I 334-242212101; email: bms@nas. edu; Washington, DC 20036; telephone: 202- www. i cm2006. org or email: grants@ World Wide Web: http: I l www7. 293-1170; fax: 202-293-3412; email: i cm2006. org. nationalacademies.orgl bmsl kol be@math. georgetown. edu or BMSA_Members.html. [email protected]. April15, 2005: Project NExT Fel­ lowships. See "Mathematics Opportu­ Board on Mathematical Sciences nities" in this issue. and Their Applications Book List May 1, 2005: Applications for AWM The Board on Mathematical Sciences The Book List highlights books that Travel Grants. See the AWM website, and Their Applications (BMSA) was have mathematical themes and are http:ll www.awm-math.org l established in November 1984 to lead aimed at a broad audience potentially travel grants. html; telephone: 301- activities in the mathematical sciences including mathematicians, students, 405-7892; email: awm@math. umd. edu. at the National Research Council and the general public. Wh en a book May 31, 2005: Registration for In­ (NRC). The mission of BMSA is to sup­ has been reviewed in the Notices, a ternational Mathematics Competition port and promote the quality and reference is given to the review. Gen­ for University Students. See "Mathe­ health of the mathematical sciences erally the list will contain only books matics Opportunities" in this issue. and their benefits to the nation. Below published within the last two years, June 1, 2005: Applications for fall are the current BMSA members. though exceptions may be made in program of the Christine Mirzayan Tanya Styblo Beder, Tribeca In­ cases where current events (e.g., the Science and Technology Policy Grad­ vestments LLC death ofa prominent mathematician, uate Fellowship Program of the Na­ Patrick L. Brockett, University of coverage of a certain piece of mathe­ tional Academies. See http : I I Texas at Austin matics in the news) warrant drawing www7.nationa l academies .orgl Avarinda Chakravarti, Institute of readers' attention to older books. Sug­ pol i cyfe ll ows or contact The Na­ Genetic Medicine gestions for books to include on the list tional Academies Christine Mirzayan Philip Colella, Lawrence Berkeley may be sent to noti ces-bookl i st@ Science and Technology Policy National Laboratory ams. org.

MARCH 2005 NOTICES OF THE AMS 361 Reference and Book List

1'Added to "Book List" since the Cape, September 2002. Pantheon 0-555953-407-9. (Reviewed February list's last appearance. Books, January 2003. ISBN 0-375- 2005.) 42221-8. (Reviewed November 2004.) Karl Pearson: The Scientific Life in 13: The Story of the World's Most Count Down: Six Kids Vie for Glory a Statistical Age, by Theodore M. Popular Superstition, by Nathaniel at the World's Toughest Math Compe­ Porter. Princeton University Press, Lachenmeyer. Thunder's Mouth Press, tition, by Steve Olson. Houghton Mif­ February 2004. ISBN 0-691-11445-5. October 2004. ISBN 1-568-58306-0. flin, April2004. ISBN 0-618-25141-3. Kepler's Conjecture: How Some of 1089 and All That. A journey into (Reviewed August 2004.) the Greatest Minds in History Helped Mathematics, by David Acheson. Ox­ The Curious Life of Robert Hooke, Solve One of the Oldest Math Prob­ ford University Press, July 2002. ISBN the Man Who Measured London, by lems in the World, by George G. Szpiro. 0-19-851623-1. (Reviewed February Lisa Jardine. HarperCollins, February Wiley, January 2003. ISBN 0-471- 2005.) 2004. ISBN 0-060-53897-X. 08601-0. (Reviewed January 2005.) Across the Board: The Mathematics Everything and More: A Compact The Knot Book: An Elementary In­ of Chessboard Problems, by John ]. History of Infinity, by David Foster troduction to the Mathematical Theory Watkins. Princeton University Press, Wallace. W. W. Norton, October 2003. of Knots, Colin C. Adams. AMS, Sep­ April2004. ISBN 0-691-11503-6. ISBN 0-393-00338-8. (Reviewed tember 2004. ISBN 0-8218-3678-1. Adam Spencer's Book of Numbers, June/July 2004.) The Liar Paradox and the Towers by Adam Spencer. Four Walls Eight The Fabric of the Cosmos, by Brian of Hanoi: The Ten Greatest Math Puz­ Windows, January 2004. ISBN 1-568- Greene. Knopf, February 2004. ISBN zles of All Time, by Marcel Danesi. 58289-7. 0-375-41288-3. Wiley, August 2004. ISBN 0-471- Alan Turing: Life and Legacy of a Fields Medalists' Lectures, edited 64816-7. Great Thinker, edited by Christof by Sir Michael Atiyah and Daniel Masters of Theory: Cambridge and Teuscher. Springer, 2004. ISBN 3-540- Iagolnitzer. World Scientific, 2nd the Rise of Mathematical Physics, by 20020-7. edition, December 2003. ISBN 9-812- Andrew Warwick. University of Alfred Tarski: Life and Logic, by 38259-3. Chicago Press, July 2003. ISBN 0-226- Anita Burdman Feferman and From Eudoxus to Einstein: A His­ 87375-7. Solomon Feferman. Cambridge Uni­ tory of Mathematical Astronomy, by Math Magic: How to Master Every­ versity Press, October 2004. ISBN C. M. Linton. Cambridge University day Math Problems, by Scott Flansburg. 0-521-80240-7. Press, August 2004. ISBN 0-521- Perennial Currents, revised edition, Alpha and Omega: The Search for 82750-7. August 2004. ISBN 0-060-72635-0. the Beginning and End of the Universe, From Newton to Hawking: A History by Charles Seife. Viking, July 2003. of Cambridge University's Lucasian Math through the Ages: A Gentle ISBN 0-6 70-03179-8. Professors of Mathematics, edited by History for Teachers and Others, by Automated Reasoning and the Dis­ Kevin C. Knox and Richard Noakes. WilliamP. Berlinghoff and Fernando Q. covery of Missing and Elegant Proofs, Cambridge University Press, Novem­ Gouvea. Oxton House, 2002. ISBN by Larry Wos and Gail Pieper. Rinton ber 2003. ISBN 0-521-66310-5. 1-881929-21-3. (Reviewed October Press, December 2003. ISBN 1-58949- Gamma: Exploring Euler's Constant, 2004.) 023-1. by Julian Havil. Princeton University The Mathematical Century: The 30 Beyond Coincidence, by Martin Press, May 2003. ISBN 0-691-09983-9. Greatest Problems of the Last 100 Plimmer and Brian King. Icon Books, (Reviewed August 2004.) Years, by Piergiorgio Odifreddi, trans­ March 2004. ISBN 1-840-46534-4. Geometry: Our Cultural Heritage, lated by Arturo Sangalli. Princeton The Changing Shape of Geometry: by Audun Holme. Springer, April University Press, May 2004. ISBN Celebrating a Century of Geometry 2002. ISBN 3-540-41949-7. (Reviewed 0-691-09294-X. and Geometry Teaching, edited by May 2004.) Mathematical journeys, by Peter D. Chris Pritchard. Cambridge Univer­ The Golden Ratio: The Story of Phi, Schumer. Wiley InterScience, February sity Press, January 2003. ISBN 0-521- the World's Most Astonishing Number, 2004. ISBN 0-471-22066-3. 53162-4. by Mario Livio. Broadway Books, ''A Mathematician at the Ballpark: Cogwheels of the Mind: The Story of October 2002. ISBN 0-767-90815-5. Odds and Probabilities for Baseball Venn Diagrams, by A. W. F. Edwards. (Reviewed in this issue.) Fans, by Ken Ross. Pi Press, July 2004. Johns Hopkins University Press, April A Handbook of Mathematical Dis­ ISBN 0-131-47990-3. 2004. ISBN 0-801-87434-3. course, by Charles Wells. Infinity Mathematicians as Enquirers: Learn­ Constantin Caratheodory: Mathe­ Publishing Company, 2003. ISBN ing about Learning Mathematics, edited matics and Politics in Turbulent Times, 0-7414-1685-9. (Reviewed September by Leone Burton. Kluwer, April 2004. by M. Georgiadou. Springer, Septem­ 2004.) Hardbound, ISBN 1-4020-7853-6; pa­ ber 2004. ISBN 3-540-44258-8. The Heart of Mathematics: An perback, ISBN 1-4020-7859-5; eBook, The Constants of Nature: From Invitation to Effective Thinking, by ISBN 1-4020-7908-7. Alpha to Omega-The Numbers That Edward B. Burger and Michael A Mathematician's Survival Guide: Encode the Deepest Secrets of the Uni­ Starbird. Key College Publishing Graduate School and Early Career De­ verse, by John D. Barrow. Jonathan (Springer-Verlag), April 2000. ISBN velopment, by Steven G. Krantz. AMS,

362 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Reference and Book List

August 2003. ISBN 0-8218-3455-X. Henry Press, March 2003. ISBN (Reviewed April 2004.) 0-309-08549-7. Mathematicians under the Nazis, by Probability Theory: The Logic of Sanford L. Segal. Princeton University Science, by E. T. Jaynes, edited by Press, July 2003. ISBN 0-691-00451-X. G. Larry Bretthorst. Cambridge Uni­ Mathematics: A Very Short Intro­ versity Press, April2003. ISBN 0-521- duction, by Timothy Gowers. Oxford 59271-2. University Press, October 2002. ISBN The Reader of Gentlemen's Mail: 0-192-85361-9. (Reviewed February Herbert 0. Yardley and the Birth of 2005.) American Codebreaking, by David Mathematics and War, edited by Kahn. Yale University Press, March Bernheim Booss-Bavnbek and ]ens 2004. ISBN 0-300-09846-4. H0yrup. Birkhauser, December 2003. The Riemann Hypothesis: The ISBN 3-764-31634-9. Greatest Unsolved Problem in Mathematics in Nature: Modeling Mathematics, by Karl Sabbagh. Farrar Patterns in the Natural World, by John Straus & Giroux, April 2003. ISBN Adam. Princeton University Press, No­ 0-3 74-2 5007-3. vember 2003. ISBN 0-691-11429-3. The Saga of Mathematics: A Brief Meta Math! The Quest for Omega, History, by Marty Lewinter and William by Gregory J. Chaitin. April 2004. Widulski. Prentice Hall, January 2002. Available at http: I jwww. cs. ISBN 0-130-34079-0. umaine.edu/~chaitin/omega.html. Shooting the Sun, by Max Byrd. '' The (Mis)Behavior of Markets: A Bantam, December 2003. ISBN 0-553- Fractal View ofRisk, Ruin and Reward, 80208-9. by Benoit Mandelbrot and Richard Signs of the Inka Khipu: Binary Cod­ Hudson. Basic Books, August 2004. ing in the Andean Knotted-String ISBN 0-465-04355-0. Records, by Gary Urton. University of '' More Damned Lies and Statistics: Texas Press, August 2003. ISBN 0-292- How Numbers Confuse Public Issues, 78540-2. Strange Curves, Counting Rabbits, by Joel Best. University of California and Other Mathematical Explorations, Press, August 2004. ISBN 0-520- by Keith Ball. Princeton University 23830-3. Press, November 2003. ISBN 0-691- More Mathematical Astronomy 113 21-1. (Reviewed in December Morsels, by JeanMeeus. Willmann-Bell 2004.) Inc., 2002. ISBN 0-943396-743. Towards a Philosophy ofReal Math­ The Music of the Primes: Searching ematics, by David Corfield. Oxford to Solve the Greatest Mystery in University Press, April 2003. ISBN Mathematics, by Marcus du Sautoy. 0-521-81722-6. HarperCollins, April 2003. ISBN The Universal Book ofMathematics: 0-066-21070-4. From Abracadabra to Zeno's Para­ The Number rr, by Pierre Eymard doxes, by David Darling. Wiley, July and Jean-Pierre Lafon. AMS, 2004. 2004. ISBN 0-471-27047-4. ISBN 0-8218-3246-8. '' A World without Time: The For­ Number Theory from an Analytic gotten Legacy of Gddel and Einstein, Point of View, by Badih Ghusayni. by Palle Yourgrau. Basic Books, Janu­ Komati, December 2003. ISBN 9953- ary 2005. ISBN 0-465-09293-4. 0-0282-7. You Can Do the Math: Overcome Phase Change: The Computer Rev­ Your Math Phobia and Make Better olution in Science and Mathematics, Financial Decisions, by Ron Lipsman. by Douglas S. Robertson. Oxford Praeger Publishers, November 2004. University Press, March 2003. ISBN ISBN 0-275-98341-2. 0-195-15748-6. Portraits of the Earth: A Mathe­ matician Looks at Maps, by Timothy G. Feeman. AMS, September 2002. ISBN 0-8218-3255-7. Prime Obsession: Bernhard Rie­ mann and the Greatest Unsolved Problem, by John Derbyshire. Joseph

MARCH 2005 NOTICES OF THE AMS 363 AMERICAN MATHEMATICAL SOCIETY

The selection committee for these prizes requests nominations for con­ sideration for the 2006 awards. Further information about the prizes can be found in the November 2003 Notices, pp. 1288-1302 (also avail­ able at http://www.ams.org/prizes-awards).

Three Leroy P. Steele Prizes are awarded each year in the following cat­ ·- egories: (I) the Steele Prize for Lifetime Achievement: for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) the ·- Steele Prize for Mathematical Exposition: for a book or substantial sur­ vey or expository-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 2006 the prize for Seminal Contribution to Research will be awarded for a paper in Applied Mathematics.

Nominations with supporting information should be submitted to the Secretary, Robert J. Daverman, American Mathematical Society, 3 120 Ayres Hall, University ofTennessee, Knoxville, TN 37996-1330.1nclude a short description on the work that is the basis of the nomination, includ­ ing complete bibliographic citations. A curriculum vitae should be included. The nominations will be forwarded by the Secretary to the prize selection committee, which will, as in the past, make final decisions on the awarding of prizes.

Deadline for nominations is March 3 I, 2005. Mathematics Calendar

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

March 2005 ysis, Dartmouth College, Hanover, New Hampshire. (Feb. 2005, p. 288) 1-2 DIMACS Short Course: A Field Guide to GenBank and NCBI Molecular Biology Resources, DIMACS Center, Rutgers University, 14- 1 8 Deterministic and stochastic Navier-Stokes equations, Piscataway, New Jersey. (Jun/Jul. 2004, p. 692) AIM Research Conference Center, Palo Alto, California. (Oct. 2004, p . 1095) 2- 5 Representing Unresolved Degrees of Freedom for the Atmosphere and Ocean, Centre de Recherches Mathematiques, 1 8- 1 9 AMS Southeaste rn Section Meeting, Western Kentucky Universite de Montreal, Montreal, Q uebec, Canada. (Aug. 2004, University, Bowling Green, Kentucky. (May 2004, p. 576) p. 833) 19-20 Twenty-Eighth Annual Texas Partial Differential Equa­ 3-5 International Conference on Environmental Fluid Mechanics tions Conference, University of Texas, Pan American, Edinburg, (ICEFM'OS), Indian Institute of Technology Guwahati, Guwahati, Texas. (Oct. 2004, p . 1095) India. (Sept. 2004, p. 971) 1 9- 22 2005 ASLAnnual Meeting, Stanford, California. (Jun/ Jul. 2004, 6-1 2 International Conference on Algebras, in Memory of Kostia p . 692) Beidar,NationalChengKungUniversity, Tainan, Taiwan. (Nov. 2004, 2 1- 2 5 Extens ions of Hil bert's Tenth Problem, AIM Research p . 1264) Conference Center, Palo Alto, California. (Sept. 2004, p . 970) 7-9 DIMACS Working Group on Order Theoretic Aspects of 21 -2 5 Workshop o n N= 1 Compactifications, The Fields Institute, Epidemiology, DIMACS Center, Rutgers University, Piscataway, Toronto, Ontario, Canada. (Mar. 2004, p . 361) New Jersey. (Jun/ Jul. 2004, p. 692) 22- 26 Conference on Algebra and its Applications, Center of 7-1 1 Third Internat ional Conference o n Pattern Avoiding Permu­ Ring Theory and Its Applications, Department of Mathematics, tations, University of Florida, Gainesville, Florida. (Jun/ Jul. 2004, Ohio University, Athens, Ohio. (Oct. 2004, p . 1095) p . 692) 24- 2 7 Geometric Representation Theory, University of Arizona, ?-June 1 0 IPAM Long Progra m: Grand Challenge Problems Tucson, Arizona. (Aug. 2004, p. 833) in Computational Astrophys ics, UCLA, Los Angeles, California. 2 7-31 The 3rd International Conference on Sciences of Elec­ (Dec. 2004, p. 1375) tronic, Technologies of Information and Telecommunications 8-11 IPAM Program in Grand Challenge Problems in Compu­ (SETIT 2005), Susa, Tunisia. (Nov. 2004, p. 1264) Tutorials, UCLA, Los Angeles, California. tational Astrophysics- 28- 30 IMA Tutorial/Workshop: New Paradigms in Computation, 1375) (Dec.2004,p. University of Minnesota, Minneapolis, Minnesota. (Jun/ Jul. 2004, 9- 1 3 Minima l Surfaces, Sub-Elliptic POE's and Geometric Anal- p . 692)

respect to participation in the meeting, this fact should be noted. and conferences This section contains announcements of meetings All communications on meetings and conferences in the mathematical public, including ad of interest to some segment of the mathematical 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 notices@ams . org to specialized topics, as well as announcements of regularly scheduled or mathcal@ams. org . meetings of national or international mathematical organizations. A In order to allow participants to arrange their travel plans, organizers of on the last page of complete list of meetings of the Society can be found 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 Notices it contains a call An announcement will be published in the if 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 li sting 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/ July, and ry third issue until has appeared, the event will be briefly noted in eve 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 c ontaining 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/.

MARCH 200 5 NOTICES OF THE AMS 365 Mathematics Calendar

28-April 1 Generalized Kostka Polynomials, AIM Research Con­ Tennessee. (Sept. 2004, p. 972) ference Center, Palo Alto, California. (Jun/Jul. 2004, p. 692) May 2005 28-April 1 Topology and geometry of the moduli space of curves, AIM Research Conference Center, Palo Alto, California. 2-6 IMA Workshop: Experiments in Physical Biology, Part I, (Dec. 2004, p. 1375) University of Minnesota, Minneapolis, Minnesota. (Jun/Jul. 2004, p. 693) 28-April 1 Workshop on String Phenomenology, The Perimeter Institute, Waterloo, Ontario, Canada. (Apr. 2004, p. 461) 2-61PAM Workshop: Relativistic Astrophysics, UCLA, Los Angeles, California. (Dec. 2004, p. 1376) 29-April 1 14th International Workshop on Matrices and Statis­ tics, Massey University, Albany Campus, Auckland, New Zealand. 2-6 Workshop on Gravitational Aspects of String Theory, The (Nov. 2004,p. 1264) Fields Institute, Toronto, Ontario, Canada. (Apr. 2004, p. 461) 29-April 2 The 2005 UAB International Conference on Differ­ '' 4-5 Latent Variable Models and Survey Data for Social and ential Equations and Mathematical Physics, The University of Health Sciences, Centre de recherches mathematiques, Universite Alabama at Birmingham, Alabama. (Jan. 2005, p. 79) de Montreal, Montreal, Quebec, Canada. Organizer: Mary E. Thompson (University of Waterloo). April 2005 1-July 8 Special Semester on "Modern Methods of Time­ 5-6 DIMACS Workshop on Security of Web Services and E­ Frequency Analysis", Erwin Schroedinger Institute (ESI) for Math­ Commerce, DIMACS Center, CoRE Bldg, Rutgers University, Piscat­ ematical Physics, Vienna, Austria. (Apr. 2004, p. 461) away, New Jersey. (Oct. 2004, p. 1096) 2-3 AMS Eastern Section Meeting, University of Delaware, Newark, 6-9 Statistical Inferences on Shape Manifolds, AIM Research Delaware. (May 2004, p. 576) Conference Center, Palo Alto, California. (Jun/Jul. 2004, p. 693) 4-8 Stiff Sources and Numerical Methods for Conservation 9-20 The Third Annual Spring Institute on Noncommutative Laws, AIM Research Conference Center, Palo Alto, California. (Nov. Geometry and Operator Algebras in conjunction with the 20th 2004, p. 1264) Shanks Lecture, Vanderbilt University, Nashville, Tennessee. (Jan. 2005, p. 80) 4-9 IPAM Workshop: Astrophysical Fluid Dynamics, UCLA, Los Angeles, California. (Dec. 2004, p. 1376) 11-1 5 Integrative Multiscale Modeling and Simulation in Mate­ rials Science, Fluids and Environmental Science, Centre de Re­ 6-1 0 Extracting Macroscopic Information from Molecular Dy­ cherches Mathematiques, Universite de Montreal, Montreal, Quebec, namics, Centre de Recherches Mathematiques, Universite de Canada. (Aug. 2004, p. 834) Montreal, Montreal, Quebec, Canada. (Aug. 2004, p. 833) 1 3-14 Sixth Mississippi State-UAB Conference on Differen­ 7-9 The 2005 Missouri Mathematical Association of Two Year tial Equations & Computational Simulations, Mississippi State Colleges Conference, Lake Ozark, Missouri. (Jan. 2005, p. 79) University, Mississippi State, Mississippi (Oct. 2004, p. 1096) 8-1 0 AMS Central Section Meeting, Texas Tech University, Lub­ 14-1 5 Conference in Honor of Heydar Radjavi's 70th Birthday, bock, Texas. (May 2004, p. 5 76) Hotel Golf, Bled, Slovenia. (Aug. 2004, p. 834) 9-1 0 Third Annual Graduate Student Topology Conference, '' 14-20 Workshop on Symplectic Field Theory, University of Northwestern University, Evanston, Illinois. (Dec. 2004, p. 1376) Leipzig, Germany. 1 0-1 5 2005 USENIX Annual Technical Conference, Marriott Topic: Lecture series by Helmut Hofer on "Fredholm theory in Anaheim, Anaheim, California. (Dec. 2004, p. 1376) Polyfolds with Applications". 11-1 5 IMA Workshop: Atomic Motion to Macroscopic Models: Organizers: Kai Cieliebak, Helmut Hofer, Dusa McDuff, Klaus The Problem of Disparate Temporal and Spatial Scales in Matter, Mohnke, Matthias Schwarz. University of Minnesota, Minneapolis, Minnesota. (Jun/Jul. 2004, Deadline: January 17, 2005. p. 692) Information: http: I /www .math. uni -leipzig. de/ws/. 14-1 5 DIMACS Workshop on Intellectual Property Protec­ 1 5-l 8 HPCS 2005: The New HPC Culture in Canada, The 19th tion, DIMACS Center, Rutgers University, Piscataway, New Jersey. Annual Symposium on High Performance Computing Systems (Jun/Jul. 2004, p. 692) & Applications, University of Guelph, Guelph, Ontario, Canada. 14-1 6 30th Annual Arkansas Spring Lecture Series in the (Dec. 2004, p. 1376) · Mathematical Sciences: Recent Developments in Tight Closure, 1 5-l 8 OSCAR'OS: The 3rd Annual Symposium on Open Source University of Arkansas, Fayetteville, Arkansas. (Jan. 2005, p. 80) Cluster Application Resources (OSCAR), University of Guelph, 16-1 7 AMS Western Section Meeting, University of California, Guelph, Ontario, Canada. (Nov. 2004, p. 1264) Santa Barbara, California. (May 2004, p. 576) 1 5-21 43rd International Symposium on Functional Equations, 1 8-22 IPAM Workshop: N-body Problems in Astrophysics, UCLA, Batz-sur Mer, France. (Nov. 2004, p. 1265) Los Angeles, California. (Dec. 2004, p. 13 76) 1 5-21 ICMI Study 1 5: The Professional Education and Develop­ 1 8-July 1 3 Time at Work, Institut Hemi Poincare, Paris, France. ment of Teachers of Mathematics, Aguas de Lind6ia, Sao Paulo, (Aug. 2004,p. 833) Brazil. (Jun/Jul. 2004, p. 693) 25-29 Analytical Methods in Number Theory, Probability Theory 1 6-20 IMA Workshop: Experiments in Physical Biology, Part II, and Mathematical Statistics, Euler IMI, St. Petersburg, Russia. University of Minnesota, Minneapolis, Minnesota. (Jun/Jul. 2004, (Feb. 2005, p. 288) p. 693) 27-May 1 Multiscale Modeling in Solids, Centre de Recherches 1 6-20 IPAM Workshop: Transport Phenomena in Computational Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. Astrophysics, UCLA, Los Angeles, California. (Dec. 2004, p. 1376) (Aug. 2004, p. 834) 1 6-20 Stability Criteria for Multi-dimensional Waves and Pat­ 28-30 Barrett Lectures: New Developments in Nonlinear Par­ terns, AlM Research Conference Center, Palo Alto, California. (Nov. tial Differential Equations, University of Tennessee, Knoxville, 2004, p. 1265)

366 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Mathematics Calendar

17-20 Graph Theory with Altitude, University of Colorado at 6-8 SEM Annual Conference & Exposition on Experimental Denver, Denver, Colorado. (Jun/Jul. 2004, p. 693) and Applied Mechanics, Marriott Portland Downtown, Portland, Oregon. (May 2004, p. 576) 1 7-21 Lie algebras, Vertex operator algebras and their appli­ cations, North Carolina State University, Raleigh, North Carolina. 6-1 0 (REVISED) International Mediterranean Congress of Math­ (Feb. 2005, p. 288) ematics Almeria 2005, Almeria, Spain. (Dec. 2004, p. 1377) 1 8-21 Combinatorial and additive number theory (CANT 2005): 6-10 Moduli Spaces of Properly Embedded Minimal Surfaces,AlM A conference in celebration of Mel Nathanson's 60th birthday, Research Conference Center, Palo Alto, California. (Jun/Jul. 2004, CUNY Graduate Center, 365 Fifth Avenue, New York, New York. p. 693) (Feb. 2005, p. 288) 6-11 14th Summer St. Petersburg Meeting in Mathematical 1 9-21 CTS Conference on Combinatorics and Its Applications Analysis, Euler IMI, St. Petersburg, Russia. (Feb. 2005, p. 289) in Honor of Frank K. Hwang's 65th Birthday, National Chiao Tung ,., 6-24 Summer School and Conference on Geometry and Topology University (NCTU), Hsin Chu, Taiwan. (Sept. 2004, p. 973) of 3-Manifolds, ICTP, Trieste, Italy. 22-2 5 ICCS 2005: International Conference on Computational Organizers: Michel Boileau, Carlo Petronio, Bruno Zimmermann. Science, Atlanta, Georgia. (Oct. 2004, p. 1096) Deadline: January 20, 2005. Information: http: I /agenda. ictp. trieste. it/ agenda/ current/ 23-25 DIMACS Workshop on The Epidemiology and Evolution fullAgenda.php?email=O&ida=a04195. of Influenza, DIMACS Center, CoRE Bldg, Rutgers University, Piscataway, New Jersey. (Oct. 2004, p. 1096) 7-10 SIAM Conference on Mathematical and Computational 23-29 International conference "Function Spaces, Approxima­ Issues in the Geosciences, Palais des Papes, The International tion Theory and Nonlinear Analysis" dedicated to the 1 OOth Conference Center, Avignon, France. (Nov. 2004, p. 1265) birthdayofS. M. Nikolskii, SteklovMathematicalinstitute, Moscow, 7-12 25th Great Plains Operator Theory Symposium (GPOTS-05), Russia. (Jan. 2005, p. 80) University of Central Florida, Orlando, Florida. (Feb . 2005, p. 289) 27-30 Scattering theory and singular spaces, Northwestern 7-1 7 Fields Institute Summer School on Operator Algebras, University, Evanston, Illinois. (Feb. 2005, p. 289) University of Ottawa, Ottawa, Ontario, Canada. (Nov. 2004, p. 1265) 29-June 35th Conference on Stochastic Analysis, Random Fields 8-1 0 Eleventh Conference on Integer Programming and Com­ and Applications and Minisymposium on Stochastic Methods in binatorial Optimization (IPCO XI), Berlin, Germany. (Dec. 2004, Financial Models V, Centro Stefano Franscini, Ascona, Switzerland. p. 1377) (Dec. 2004, p. 1377) 8-11 IMA Workshop: Effective Theories for Materials and 31-June 3 Discrete Groups and Geometric Structures, with Ap­ Macromolecules, University of Minnesota, Minneapolis, Minne­ plications II. (Crystallographic Groups and their Generalizations sota. (Jun/Jul. 2004, p. 693) IV), Oostende, Hotel Royal Astrid, Belgium. (Jan. 2005, p. 80) 8-12 Computability in Europe 2005 (CiE 2005): New Compu­ 31-June 4 2nd Conference on Analysis and Probability on tational Paradigms, Amsterdam, The Netherlands. (Feb. 2005, Fractals, Cornell University, Ithaca, New York. (Feb. 2005, p. 289) p. 289) june 2005 1 0-12 Groups, Rings and Algebras-A Conference in Honor of Donald S. Passman, University of Wisconsin, van Vleck Hall, 1-4 ACMS 15th Biennial Conference, Huntington College, Hunt­ Madison, Wisconsin. (Jan. 2005, p. 80) ington, Indiana. (Feb. 2005, p. 289) 1 2-24 Foliations 2005, Lodz, Poland. (Sept. 2004, p. 973) 1- 5 Stochastic Modeling in Financial Mathematics Uoint with 12-July 23 DIMACS Reconnect Conferences 2005: Reconnect­ SAMSI), Centre de Recherches Mathematiques, Universite de Montreal, ing Teaching Faculty to the Mathematical Sciences Research Montreal, Quebec, Canada. (Aug. 2004, p. 834) Enterprise, DIMACS, CoRE Building, 4th Floor, Rutgers, the State 2-1 0 Seventh International Conference on Geometry, Integrabil­ University of New Jersey, 96 Frelinghuysen Road, Piscataway, New ity and Quantization with a special session on Multi symplectic Jersey. (Feb. 2005, p . 290) Geometry and Classical Field Theory, Sts. Constantine and Elena 1 3-1 7 Boundaries in Geometric Group Theory, AIM Research resort (near Varna), Bulgaria. (Feb. 2005, p. 289) Conference Center, Palo Alto, California. (Jan. 2005, p. 80) 4-8 International Conference on Scientific Computing (ICSC05), 1 3-1 8 Computational Methods and Function Theory (CMFT Nanjing, P.R. China. (Oct. 2004, p. 1096) 2005), Joensuu, Finland. (Feb. 2004, p. 279) '' 5-1 0 31st International Conference "Applications of Mathemat­ 1 3-2 5 (REVISED)CIMPASummerSchooiAGAHF 2005-Arithmetic ics in Engineering and Economics", Sozopol, Bulgaria. and Geometry around Hypergeometric Functions, Galatasaray Topics: Potential Theory and Partial Differential Equations, Mathe­ University, Ortakoy, Istanbul, Turkey. (Aug. 2004, p. 834) matical Analysis and Appl., Operation Research, Numerical Methods and Mathematical Modeling, Computer Science. '' 1 4-1 8 Random Media and Stochastic Partial Differential Equa­ Organizer: Faculty of Applied Mathematics and Informatics by the tions, University of Southern California, Los Angeles, California. Technical University of Sofia, Bulgaria. Organizer: Sergey Lototsky (lototsky©math. usc . edu). Speakers: M. I. Freidlin (University of Maryland, College Park); Information: http: I /www. tu-sofia. bg/ FPMI/amee05. R. Z. Khasminskii (Wayne State University); N. V. Krylov (Uni­ 6-8 DIMACS Workshop on Polyhedral Combinatorics of Random versity of Minnesota); G. Papanicolaou (Stanford University); A. N. Utility, DIMACS Center, CoRE Bldg, Rutgers University, Piscataway, Shiryaev (Steklov Mathematics Institute, Moscow, Russia); R. Sowers New Jersey. (Oct. 2004, p. 1096) (Univesity of Illinois, Urbana-Champaign). Information: Program to be posted at http: I /math. usc. edu. 6-8 Nonlinear Partial Differential Equations and Applications: Conference in honor ofJim Serrin on the occasion oft he awarding '' 1 5-1 8 Algebraic and Topological Methods in Non-classical of the title of Doctor Honoris Causa of Universite Fran~;ois Logics II, University of Barcelona, Barcelona, Spain. Rabelais, Faculty of Sciences-Universite, Fran<;:ois Rabelais-Tours, Description: The semantic study of non-classical logics is a field France. (Feb. 2005, p. 289) where no single overarching paradigm has been established, and

MARCH 2005 NOTICES OF THE AMS 367 Mathematics Calendar where a variety of techniques are currently being explored. An Illinois. (Feb. 2005, p. 290) important goal of this meeting is to promote the cross-fertilization 26-29 12th International Linear Algebra Society Conference, between the fundamental ideas connected with these approaches. Regina Inn Hotel and Conference Centre, Regina, Canada. (Dec. 2004, Thus, we aim to bring together researchers from various fields p. 1378) of non-classical logics and applications, as well as from lattice theory, universal algebra, category theory and general topology, in 26-July 1 ERLOGOL-2005: Intermediate Problems of Model The­ order to foster collaboration and further research. The scientific ory and Universal Algebra, State Technical University/Mathe matics programme of the congress will include a few invited lectures and Institute, Novosibirsk, Russia. (Jun/Jul. 2004, p. 694) will provide ample time for contributed papers and interaction 26-July 1 30th Conference on Stochastic Processes and Their between participants. Researchers whose interests fit the general Applications, University of California at Santa Barbara (UCSB), aims of the conference are encouraged to participate. Santa Barbara, California. (Jun/Jul. 2004, p. 694) Information: The names of invited speakers, and further details about travel grants, hotels, etc., will be posted at the congress' web 27-July 1 Nonlinear Modelling and Control, An International page http : I lwww .mat. ub . esrlogicalmeeting2005l. Please visit Seminar, Nayanova University, Samara, Russia. (Feb. 2005, p. 290) it for any other information on the meeting. '' 27-July 5 Probability and Mathematical Physics, Centre de re­ 1 6-1 9 Second Joint International Meeting with the Deutsche cherches mathematiques, Montreal, Quebec, Canada. Mathematiker-Vereinigung (DMV) and the Oesterreichische Description: Conference celebrating the 65th birthday of Stanislav Mathematische Gesellschaft (OMG), Mainz, Germany. (May 2004, Molchanov. p. 576) Organizers: Don Dawson (Carleton&McGill), Vojkan]aksic (McGill), Boris Vainberg (UNCC). 17-23 (NEW DATE) Algebraic Geometry and Number Theory, Invited Speakers: G. Ben Arous (Courant), L. Bogachev (England), R. Euler IMI, St. Petersburg, Russia. (Feb. 2005, p. 290) Carmona (Princeton), K. Chen (N. Carolina), M. Cranston (Rochester), 19-24 33rd Canadian Operator Symposium (COSy), dedicated G. Derfel (Ben Gurian), E. Dynkin (Cornel), A. Figotin (Irvine), M. to George Elliott's 60th birthday, University of Ottawa, Ottawa, Freidlin (Maryland), ]. Gaertner (Berlin), I. Goldsheid (England), A. Ontario, Canada. (Nov. 2004, p. 1265) Gordon (Rochester), R. Grigorchuk (Texas A&M), D. Hundertmark 19-24 First Announcement: Conference on Applied Mathematics (Illinois), S. Jitornirskaya (Irvine), R. Kashrninski (Detroit),* K. Khanin and Scientific Computing 4: On the occasion of Professor (England), W. Kirsch (Ruhr), A. Kiselev (Wisconsin), A. Klein (Irvine), Aganovic's 70th birthday, Brijuni, Croatia. (Nov. 2004, p. 1265) L. Koralov (Princeton), S. Kotani (Japan)," P. Kuchment (Texas A&M), A. Laptev (Stockholm), Y. Last (Hebrew), V. Malyshev (Paris), 19-24 Conference on Applied Mathematics and Scientific Com­ M. Menshykov (England), N. Minami (Japan), L. Pastur (Ukraine), puting 4: On the occasion of Professor Aganovic's 70th Birthday, A. Ramirez (Chile), B. Simon (Caltech), A. Soshnikov (Davies), T. Brijuni, Croatia. (Nov. 2004, p. 1265) Spencer (lAS). (*) To be confirmed. 20-22 DIMACS Workshop on Detecting and Processing Reg­ ularities in High Throughput Biological Data, DIMACS Center, 28-July 1 DAYS on DIFFRACTION-2005, St. Petersburg Branch of CoRE Bldg, Rutgers University, Piscataway, New Jersey. (Dec. 2004, Steklov's Math. Inst. and St. Petersburg Univ., St. Petersburg, Russia. p. 1377) (Dec. 2004, p. 1378) 20-24 Second Conference on Self-Similarity and Applications, * 30-July 2 Primeras Jornadas de Teoria de Numeros, EPSVG-UPC, INSA Toulouse, Toulouse, France. (Aug. 2004, p. 834) Vilanova i la Geltru (Barcelona), Spain. Description: The "Jornadas de Teo ria de Numeros" (Number Theory 2 0-2 4 The Fourth International Conference on High Dimensional Conference) are born to become a periodic meeting point for the Probability (HDP), St. John's College, Santa Fe, New Mexico. arithmetical community working in Spanish language, so that it (Feb. 2005, p. 290) can show the state of the art of research in Number Theory in Spain 20-25 Asymptotic and Probabilistic Methods in Geometric Group and promote relationships between different research groups. Theory, Geneva, Switzerland. (Jun/Jul. 2004, p. 693) Invited Speakers: P. Bayer,]. Cilleruelo, E. Friedman,]. Guardia,]. 20-25 FPSAC'OS : 17th Annual Conference on Formal Power C. Peral, F. Rodriguez-Villegas. Series and Algebraic Combinatorics, Taormina, Sicily (Italy). Scientific Committee:]. Gonzalez-Rovira, ]. Muiioz-Porras, E. Nart, (Dec. 2004, p. 1377) A. Quiros, N. Vila. Organizing Committee: G. Cardona, ]. Fernandez, ]. Gonzalez­ 20-26 Sixth International Conference "Symmetry in Nonlin­ Rovira, ]. Guardia, V. Rotger. ear Mathematical Physics", Institute of Mathematics, National Information: http: I landuril. epsevg. upc. esrjtn05. Academy of Sciences of Ukraine, Kyiv (Kiev), Ukraine. (Dec. 2004, p. 1377) July 2005 20-July 8 (REVISED) Random matrices, random processes, and integrable systems (C.R.M. Short Program), Centre de recherches 2-7 SRTL-4: The Fourth International Research Forum on Sta­ mathematiques, Universite de Montreal, Montreal. (Nov. 2004, tistical Reasoning, Thinking, and Literacy, The University of p. 1265) Auckland, Auckland, New Zealand. (Jun/Jul. 2004, p. 694) 20-July 1 5 Clay Mathematics Institute Summer School: Ricci Flow, 2-9 Mile High Conference on Quasigroups, Loops and Nonas­ 3-Manifolds and Geometry, MSRI, Berkeley, California. (Feb. 2005, sociative Systems, University of Denver, Denver, Colorado. (Nov. p. 290) 2004, p. 1266) 20-August 1 5 Computational Prospects of Infinity, Institute for 3-9 XXIViemes Journees Arithmetiques, Marseilles, France. (Sept. Mathematical Sciences, National University of Singapore, Singapore. 2004, p. 975) (Jun/Jul. 2004, p. 693) 4-8 8th International Symposium on "Generalized Convexity 21-24 MAM 5-Fifth International Conference on Matrix Analytic and Monotonicity", Insubria University, Varese, Italy. (Dec. 2004, Methods in Stochastic Models, Pisa, Italy. (Sept. 2004, p. 974) p. 1378) 25-July 1 The Twentieth IEEE Symposium on Logic In Computer 4-8 Conference on Universal Algebra and Lattice Theory, Uni­ Science (LICS 2005): Call for Workshop Proposals, Chicago, versity of Szeged, Szeged, Hungary. (Dec. 2004, p. 1378)

368 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Mathematics Calendar

9-11 Joint Meeting of the Chinese Society of Probability and theory. The Andre-Oort conjectures: Equidistribution of CM-points Statistics (CSPS) and the Institute of Mathematical Statistics and Heeke points, points of small height. Quantum ergodicity: (IMS), Beijing, China. (Aug. 2004, p. 834) Quantum maps and modular surfaces. Note: This summer school is primarily targeted at senior grad 1 0-1 3 The 20th Summer Conference on Topology and Its students, postdocs and junior faculty. For full consideration, Applications, Denison University, Granville, Ohio. (Jan. 2005, p. 80) requests for participation or financial assistance must be received before February 28, 2005. Financial support available. 1 0-14 12th International Conference on Mathematical Modelling Information: http: I lwww. dms. umontreal . calsmsl; email: and Applications (ICTMA 12), City University, London, England. belanger©dms.umontreal . ca. (Jan. 2004, p. 64) '' 16-August 1 The Eighth International Diffiety school, Santo 1 0-1 5 20th British Combinatorial Conference, University of Stefano del Sole (Avellino), Italy. Durham, United Kingdom. (Sept. 2004, p. 975) Aim: The aim of the School is to introduce undergraduate and 1 0-1 5 SampTAOS (Sampling Theory and applications), Ondokuz Ph.D. students in Mathematics and Physics as well as post-doctoral May)is University Samsun, Turkey. (Nov. 2004, p. 975) researchers in a recently emerged area of Mathematics and Theo­ 1 0-16 Stochastic Modelling of Complex Systems (SMOCS-05), retical Physics: Secondary Calculus. A diffiety is a new geometrical Daydream Island resort (Whitsundays, Queensland, Australia). (Nov. object that properly formalizes the concept of the solution space 2004, p. 1266) of a given system of (nonlinear) PDEs, much as an algebraic variety does with respect to solutions of a given system of algebraic 10-23 Cornell Summer School in Probability, Cornell University, equations. Secondary Calculus is a natural diffiety analogue of the Ithaca, New York. (Aug. 2004, p. 834) standard Calculus on smooth manifolds, and as such leads to a 11-14 Sixth SIAM Conference on Control and its Applica­ very rich general theory of nonlinear PDEs. Moreover, it appears to tions, Hilton New Orleans Riverside Hotel, New Orleans, Louisiana. be the unique natural language for quantum physics, just as the (Dec. 2004, p. 1378) standard Calculus is the natural language for classical physics. Organizer: Diffiety Institute (Russia). 11-1 5 Gravitational Lensing in the Kerr Spacetime Geometry, Deadline: June 15, 2005. AIM Research Conference Center, Palo Alto, California. (Nov. 2004, Contact: Prof. A. M. Vinogradov, Dipartimento di Matematica e p. 1266) Informatica, Universita' di Salerno, Via Ponte del Melillo, 84084 '' 1 1 -1 5 Novi Sad Algebraic Conference 2005, University of Novi Fisciano (SA), Italy; email: school05 © diffiety. org; http: I I Sad, Novi Sad, Serbia/Serbia and Montenegro. diffiety. ac. ru. Program: In order to attract younger researchers and others who are not experts in the area, there will be longer plenary talks, 17-30 Advances in Sensing with Security Applications: A NATO 60 minutes in duration. Many of them should be of expository, Advanced Study Institute, Ciocco Resort, Tuscany. (Jan. 2005, survey-like nature, which should make them understandable to p. 81) a wider audience. The afternoon sessions will be reserved for 18-22 Algorithms for Approximation V, University College, contributed talks in several special sessions, the number of which Chester, UK. (Nov. 2004, p. 1266) will depend on the number of speakers. The duration of these will 18-22 VI Brazilian Workshop on Continuous Optimization, West be 20 minutes, and it is expected that the speakers will present Side Hotel Residence, Av. Republica do Libano, 2526, Setor Oeste, their most interesting recent research. in Goiania, Brazil. (Jan. 2005, p. 81) Topics: Universal algebra, Lattices and ordered structures, Model theory and set theory, Clone theory, Algebraic methods in computer 20-27 The Sth International Algebraic Conference in Ukraine, science. Odessa I. I. Mechnikov National University, Odessa, Ukraine. (Nov. Plenary speakers: Z. Esik (Univ. of Szeged), R. Freese (Univ. of 2004, p. 1266) Hawaii), P. Idziak (Jagiellonian Univ., Krakow), ]. Jezek (Charles 22-27 AMSI Workshop entitled "Noncom mutative Geometry and Univ., Prague), K. Kearnes (Univ. of Colorado at Boulder), R. Index Theory", Australian National University, Canberra, Australia. Mckenzie (Vanderbilt Univ.), R. P6schel (Technical Univ., Dresden), (Feb. 2005, p. 291) I. Rosenberg (Univ. of Montreal), A. Szendrei (Univ. of Colorado at Boulder), S. Todoreevice(Univ. of Toronto & Universite Paris 7), F. 24-27 International Symposium in Symbolic and Algebraic Wehrung (Univ. of Caen), R. Willard (Univ. of Waterloo). Computation ISSAC'2005, Beijing, China. (Feb. 2005, p. 291) Deadlines: The deadline for the registration is April 30, 2005; The 24-28 25th European Meeting of Statisticians, University of Oslo, abstract submission deadline is May 31, 2005. Oslo, Norway. (Dec. 2004, p. 1379) lnformation:http: I lwww. im . ns . ac . yuleventsiNSAC05Idefault. 25-29 IMA Workshop: Mixed Integer Programming, University htm1; email: nsac05©im. ns. ac . yu. of Minnesota, Minneapolis, Minnesota. (Jun/Jul. 2004, p. 694) 1 1-1 6 Strings 2005, Toronto, University of Toronto, Toronto, 25-30 First Announcement: International Conference on Dif­ Ontario, Canada. (Nov. 2004, p. 1266) ference Equations, Special Functions and Applications, Munich, '' 11-22 SMS 2005-NATO Advanced Summer Institute: Equidis­ Germany. (Feb. 2005, p. 291) tribution in Number Theory, Universite de Montreal, Montreal, 28-August 3 Logic Colloquium '05: ASL European Summer Canada. Meeting, Athens, Greece. (Oct. 2004, p. 1097) Speakers: Yuri Bilu (Bordeaux I), William Duke (UCLA), John 30-August 6 Groups St Andrews 2005, University of St Andrews, Friedlander (Toronto), Andrew Granville (Montreal), Roger Heath­ St Andrews, Scotland. (Jan. 2005, p. 81) Brown (Oxford), Elon Lindenstrauss (NYU), ]ens Marklof (Bristol), Zeev Rudnick (Tel Aviv), Wolfgang Schmidt (Colorado at Boulder and 31-August 3 Bridges: Mathematical Connections in Art, Music, Vienna), K. Soundararajan (Michigan), Yuri Tschinkel (G6ttingen), and Science, The Banff Centre, Banff, Canada. (Jan. 2005, p. 81) Emmanuel Ullmo (Paris-Sud 11), Akshay Venkatesh (MIT). August 2005 Features: Uniform Distribution and beyond: Applications to Cryp­ tographic protocols, distribution of integers, level statistics. Integer * 1-9 XVI Coloquio Latinoamericano de Algebra, Colonia, Uruguay. and rational points on varieties: Geometry of numbers, the circle Description: This biannual event is the premier conference linking method, homogeneous varieties via spectral theory and ergodic algebraists and algebraic geometers from all of Latin America.

MARCH 2005 NOTICES OF THE AMS 369 Mathematics Calendar

Topics: Besides the plenary talks and general courses this meeting '' 8-1 2 NSF-CBMS Regional Conference on Algebraic and Topologi­ will have seven thematic parallel sessions on the following topics: cal Combinatorics of Ordered Sets, San Francisco State University, Commutative Algebra and Algebraic Geometry, Non-associative San Francisco, California. Algebras and Ring theory, Group Theory, Hopf Algebras and Speaker: Anders Bjoorner will give ten lectures to introduce Algebraic Combinatorics, Homological Methods and Representation background material, fundamental results, and recent advances in Theory, Number Theory, Operator Algebras. A special session on the field of algebraic and topological combinatorics of partially order Applications of Algebra will also be held. sets, (oriented) matroids, subspace arrangements, and algebraic Speakers: A list of a few of the confirmed speakers is the following: shifting etc. Most of the 40-50 participants should expect to obtain Nicolas Andruskiewitsch, Raymundo Bautista, Michel Brion, Ken funding. Graduate students and postdocs are highly encouraged Brown, Antonio Campillo, Max Karoubi, Jean-Louis Loday, Susan to apply. Montgomery, Adrian Ocneanu, Jose Antonio de la Pefi.a, Vladimir Other Speakers: Alexander Barvinok, Winfried Bruns, Gunnar Popov, Hans-Jurgen Schneider, Aron Simis, Frank Sottile, Richard Carlsson, Persi Diaconis, Isabella Novik, Bernd Sturmfels, Michelle Stanley, Boris Tsygan, Mariusz Wodzicki. Wachs, Neil White. Organizing and Scientific Committee: Walter Ferrer Santos (Co­ Organizers: ]. Gubeladze, email: soso©math. sf su. edu; S. Hosten; ord.), Gerardo Gonzalez-Sprinberg, Alfredo Jones, Alvaro Rittatore, email: serkan©math. sfsu. edu. Andrea Solotar. Information: http: I /math. sfsu. edu/ gubeladze/ cbms. html. Deadline: May 1st, 2005. Information: http: I /www. cmat. edu. uy/ cmat/eventos/16cla/en; 8-1 3 XX Nevanlinna Colloquium, ETH Lausanne, Lausanne, Switz­ Walter Ferrer: email: wrferrer©cmat. edu. uy erland. (Nov. 2004, p. 1266)

!-December 23 Pattern Formation in Large Domains, Isaac 8-December 23 Global Problems in Mathematical Relativity, Isaac Newton Institute for Mathematical Sciences, Cambridge, England. Newton Institute for Mathematical Sciences, Cambridge, England. (Aug. 2004, p. 834) (Aug. 2004, p. 834) 2-6 Eighth IMS North American New Researchers Conference, * 1 7-21 Third Pacific Rim Conference on Mathematics, Fudan Minneapolis, Minnesota. (Jan. 2005, p. 81) University, Shanghai, China. Topics: All areas of mathematics with focus topics on: Algebra 3-5 DIMACS Workshop on Yield Management and Dynamic and Combinatorics; Algebraic Aspects of Lie Theory and Geometry; Pricing, DIMACS Center, CoRE Bldg, Rutgers University, Piscataway, Applied Differential Geometry; Asymptotics and Riemann-Hilbert New Jersey. (Dec. 2004, p. 1379) Problems; Computational Approach to Complex Dynamical Sys­ '' 5-11 Logic In Hungary, 2005, Budapest, Hungary. tems; Kinetic Theory; Low Dimensional Topology and Geometry; Topics: Set Theory, Foundations of Space-Time, Algebraic Logic, Nonlinear Analysis Nonlinear Phenomena, Symmetry and Integrable but contributions from all other branches of symbolic logic are Structures; Partial Differential Equations and Applications. welcome. Plenary Speakers: Gerard Jennhwa Chang (Taiwan), Shuxing Chen Organizing Committee: A. Hajnal, J Suranyi (honorary chair), (China), Philippe G. Ciarlet (Hong Kong), Konstantin Mischaikow H. Andreka, I. Juhasz, P. Komjath, I. Nemeti (co-chair), G. Sagi (USA), Colin Rogers (Australia), Minoru Wakimoto, (Japan), Shicheng (secretary), L. Csirmaz, M. Ferenczi, M. Redei, I. Sain and L. Soukup Wang (China), Roderick Wong (Hong Kong), Shih-Hsien Yu (Hong (members). Kong). Information: Contact: email: lh0511lrenyi . hu; http : I /www. renyi. Local Organizers: X. Chen, ]. Cheng, ]. Hong, T. Li (Chairman), L. hu/lh05. Lu, Y. Tan, Z. Wu. Supporter: Fudan University, Mathematical Center of Ministry of '' 7-1 2 High-dimensional Partial Differential Equations in Science Education of China, National Natural Science Foundation of China, and Engineering, Centre de recherches mathematiques, Universite Chinese Mathematical Society, China Society for Industrial and de Montreal Montreal, Quebec, Canada. Applied Mathematics, Liu Bie Ju Centre for Mathematical Sciences Description: High dimensional spatio-temporal partial differential (City University of Hong Kong), Sino-French Institute of Applied equations are a major challenge to scientific computing of the Mathematics future. Up to now deemed prohibitive, they have recently become Information: Contact: Zhou Chunlian, Sino-French Institute of manageable by combining recent developments in numerical tech­ Applied Mathematics, Fudan University, Shanghai 200433, China; niques, appropriate computer implementations, and the use of tel: 81-21-6564 2469; fax: 86-21-6564 8274; email: clzhou©fudan. computers with parallel and even massively parallel architectures. edu . cn;http : //PRCM3.fudan.edu.cn. This opens new perspectives in many fields of applications. Ki­ netic plasma physics equations, many body Schrodinger equation, September 2005 Dirac and Maxwell equations for molecular electronic structure and nuclear dynamic computations, options pricing equations in * 5-9 Workshop on: Modular Forms, Automorphic Forms and mathematical finance, and Fokker-Planck and fluid dynamics equa­ Related Moduli Spaces, INdAM-institute on the campus of the tions for complex fluids, are examples of equations that can now Universita di Roma "La Sapienza", Rome, Italy. be handled. The objective of the workshop is to bring together Organizers: Riccardo Salvati Manni (Roma) and Bert van Geemen experts of international stature in that broad spectrum of areas to (Milano). confront their approaches and possibly bring out common problem Invited Speakers: F. Andreatta (Padova); S. Bocherer (Mannheim); formulations and research directions in the numerical solutions of ]. H. Bruinier (Heidelberg) to be confirmed; D. Doud (Provo) to high dimensional partial differential equations in various fields of be confirmed; C. Faber (Stockholm) to be confirmed; V. Gritsenko science and engineering with special emphasis on chemistry and (Lille); S. Grushevsky (Princeton); E. Freitag (Heidelberg); K. Hulek physics. (Hannover); T. Ibukiyama (Osaka); W. Kohnen (Heidelberg); S. Scientific Program Committee and Organizers: Andre Bandrauk Kondo (Nagoya); S. Kudla (College Park) to be confirmed; E. (CRC, Chimie, Universite de Sherbrooke); Michel Delfour (CRM/ DMS, Looijenga (Utrecht); D. Pollack (Middletown); G. K. Sankaran (Bath); Universite de Montreal, Canada); Claude Le Bris (CERMICS, Ecole R. Schmidt (Oklahoma); N. I. Shepherd-Barron (Cambridge, UK); Nationale des Ponts et Chaussees, France). W. Stein (Cambridge, USA); A. M. Uludag (Bursa); R. Weissauer (Heidelberg) to be confirmed. 7-12 International Conference: Mathematics in Finance, Kruger­ Information: http : I /www. mat. uniroma1. it/modular/; email: National Park, South Africa. (Dec. 2004, p. 1379) modular©mat.uniroma1.it.

370 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Mathematics Calendar

5-1 1 The Seventh International Workshop on Differential Ge­ Information: SIAM's conference on Mathematics for Industry fo­ ometry and its Applications, Deva, Romania. (Feb. 2005, p. 291) cuses attention on the many and varied opportunities to promote applications of mathematics to industrial problems. Since the '' 1 2-16 p-Adic Representations, Centre de recherches mathemati­ SIAM community encompasses enormous talent for integrating ques, Univ. de Montreal, Montreal, Quebec, Canada. work and academic research, this Topics: The main topics are related to a p-adic Langlands corre­ and enriching both industrial conference will stress interactions within the context of mathe­ spondence and its relationship to p-adic families of motives. More systems, and will encourage other precisely the p-adic Langlands correspondence is a correspondence matical models and complex mathematical themes of interest to industry, government, business between p-adic Galois representations of dimension n (of the abso­ and finance. lute Galois group of Qp) and certain representations of GLn(Qp) on The multidisciplinary nature of challenging manufacturing p-adic topological vector spaces. This correspondence is supposed and development problems inspires the first thematic focus on to be compatible with p-adic families on both sides. es encountered in manufacturing. Organizers: Adrian Iovita (Concordia); Henri Darmon (McGill). mathematical models of process In addition to validating models for consistency and computational Information: http://www. crm. umontreal. ca/Number2005/. correctness, and verifying them against real world data, these 12-19 Small Deviation Probabilities and Related Topics, Euler models must be joined into larger, more complex and interacting IMI, St. Petersburg, Russia. (Feb. 2005, p. 291) mathematical models. The second theme focuses on complex sys­ tems, which can vary from interactions among simplistic individual * 1 3-1 7 5th International Conference on Words, Centre de re­ agents to complex mathematical models of behaviors. Each theme cherches mathematiques, Universite de Montreal, Montreal, Quebec, will include attention to the challenges that arise in coping with Canada. enormous amounts of data. Organizers: Srecko Brlek (Univ. du Quebec a Montreal); Cedric Information: http: //widen©siam. org . Chauve (Univ. Bordeaux I, UQAM); Annie Lacasse (Univ. du Quebec a Montreal); Genevieve Paquin (Univ. du Quebec a Montreal). '' 24- 281nternational Conference on Computing and Mathematical Modeling for Environmental, Social-Economical, and Technical 16-20 International Conference of Numerical Analysis and Systems-2005, East China Normal University, Shanghai, China. Applied Mathematics 2005 (ICNAAM 2005), Hotel Esperides, Topics: Include, but not limited to: Computing and Modeling of Rhodes, Greece. (Feb. 2005, p. 291) Earth Systems, Coastal and Marine Systems, GIS and Spatial Models, 19-23 IMA Tutorial: Radar and Optical Imaging, University of Neural Networks, Statistics and Statistical Modeling, Fuzzy Sets and Minnesota, 207 Church St. SE, 400 Lind Hall, Minneapolis, Minnesota. Systems, Numerical Methods and Applications, Optimization and (Jan. 2005, p. 82) Decision Making, Environmental Modeling, Allocation of Resources, Mathematics and Computing ofRobotics, Sensors and Measurement, 1' ? International Conference Harmonic Analysis and Approx­ 20-2 Distance Learning and Educational technologies, Mathematics and Ill, Tsahkadzor, Armenia. imations, Science Education. Deadline for Application: March 31, 2005. Information: http: //sci. t amucc. eduri ccmm/index .html . Description: The program of the conference will consist of invited plenary lectures and contributed 20-minutes talks. The 40-minutes November 2005 following mathematicians have agreed to give a plenary lecture at the conference: Borislav Bojanov (Bulgaria), Carl de Boor (USA) , 7- 11 IMA Workshop: Frontiers in Imaging, University of Minne­ Ronald DeVore (USA), Nira Dyn (Israel), Hakop Hakopian (Armenia), sota, 207 Church St. SE, 400 Lind Hall, Minneapolis, Minnesota. Kazaros Kazarian (Spain), Gerard Kerkyacharian (France), Sergey (Jan. 2005, p. 82) Konyagin (Russia), Michael Lacey (USA), Konstantin Oskolkov (USA), 1 2-1 3 AMS Western Section Meeting, University of Nebraska, Allan Pinkus (Israel), Gerald Shmieder (Germany), Przemyslaw University of Oregon, Eugene, Oregon. (Dec. 2004, p. 1379) Wojtaszczyk (Poland). Contact Information: Artur Sahakian, Institute of Mathematics, 25- December 1 Reform, Revolution and Paradigm Shifts in Marshal Bagramian ave, 24-B, 375019, Yerevan, Armenia; email: Mathematics Education, Johor Bharu, Southern Malaysia (very mathconf©ysu . am; http://math.sci.am; fax: (3741) 524801. On­ close to Singapore). (Feb. 2005, p. 291) //math. sci. am/ conference/ line registration is available: http: December 2005 sept2005/registration.html. 5- 9 IMA Workshop: Integration of Sensing and Processing, Uni­ 26- 30 49th Annual Meeting of the Australian Mathematical versity of Minnesota, 207 Church St. SE, 400 Lind Hall, Minneapolis, Society, The University of Western Australia, Perth, Australia. Minnesota. (Jan. 2005, p. 82) (Jan. 2005, p. 82) '' 1 2- 16 Intersection of Arithmetic Cycles and Automorphic October 2005 Forms, Centre de recherche mathematiques, Universite de Montreal, Montreal, Quebec, Canada. 8- 9 AMS Eastern Section Meeting, Bard College, Annandale-on­ Purpose: To explore the relationship between intersection numbers Hudson, New York. (Dec. 2004, p. 1379) for arithmetic cycles on Shimura varieties, Fourier coefficients of 1 5- 16 AMS Southeastern Section Meeting, East Tennessee State automorphic forms, and special values of L-functions. University, Johnson City, Tennessee. (Dec. 2004, p. 1379) Organizers: Eyal Goren (McGill) and Henri Darmon (McGill). Information: http: //www. crm. umontreal. ca/Number2005/. 1 7- 21 IMA Workshop: Imaging from Wave Propagation, Univer­ sity of Minnesota, 207 Church St. SE, 400 Lind Hall, Minneapolis, 14- 1 8 First joint International Meeting with the Taiwanese Minnesota. (Jan. 2005, p. 82) Mathematial Society, Taiwan, Taiwan. (Dec. 2004, p. 1379) 1 7-21 Nonlinear Parabolic Problems, Helsinki, Finland. (Jan. 2005, 1 5- 1 7 International Conference on Computer & Information p. 82) Science (ICCIS'2005), Fort Panhala, Kolhapur, India. (Dec. 2004, 21 - 2 2 AMS Central Section Meeting, University of Nebraska, p. 1379) Lincoln, Nebraska. (Dec. 2004, p. 1379) '' 1 5- 1 9 Conference on Low-dimensional Topology, University of * 24- 26 SIAM Conference on Mathematics for Industry: Challenges Virginia, Charlottesville, Virginia. and Frontiers, Detroit Marriott Renaissance Center, Detroit, Michi­ Objective: To bring together researchers working on different gan. aspects of low-dimensional topology, including Floer homology,

MARCH 2005 NOTICES OF THE AMS 371 Mathematics Calendar

the Ricci flow program, and more "classical" geometric topology, May 2006 with the goal of looking for connections and unifying perspectives. '' 1 3-1 8 Analytic methods for Diophantine equations, Banff Inter­ A theme in several of these areas is that subtle relationships between national Research Station, Banff, Alberta, Canada. a 4-manifold and its boundary constrain and illuminate both 3- Description: This meeting brings together the participants of the and 4-dirnensional topology. We propose this as a framework for MSRI and CRM workshops The meeting will be held at the Banff commparing existing approaches and formulating key problems. International Research Station. Organizers: Slava Krushkal (univ. Va), Frank Quinn (Virginia Tech). Organizers: Andrew Granville (Montreal), Yuri Tschinkel (Gottingen), Information: http: I /www. math . vi rginia. edu/topol ogy I . Michael Bennett (UBC), Chantal David (Concordia) and Bill Duke (UCLA). january 2006 Information: email: paradi s@cr m. umontreal . ca. 9- 1 2 IMA Workshop: New Mathematics and Algorithms for 3-D Image Analysis, University of Minnesota, 207 Church St. SE, 400 June 2006 Lind Hall, Minneapolis, Minnesota. (Jan. 2005, p. 82) '' 2 7-July 3 International Commission on Mathematical Instruc­ tion: Challenging Mathematics In and Beyond the Classroom 9-June 30 Principles of t he Dynamics of Non-Equilibrium Sys­ , Trondheim, Norway. tems, Isaac Newton Institute for Mathematical Sciences, Cambridge, Scope: The scope of this England. (Aug. 2004, p. 835) study will be wide. It will look at, for instance, the impact of mathematical challenges both inside and 16-July 7 Logic and Algorithms, Isaac Newton Institute for Math­ outside of the classroom, the role of mathematical challenges in ematical Sciences, Cambridge, England. (Aug. 2004, p. 835) supporting the curriculum for students of all levels of ability, vehicles for propagating mathematical challenges and assessment February 2006 of their effectiveness. We would like to emphasize that we are 6- 1 0 IMA Workshop: The Mathematics and Art of Film Editing interested in students and activities of all type, and want to go far and Restoration, University of Minnesota, 207 Church St. SE, 400 beyond contests for talented students. Lind Hall, Minneapolis, Minnesota. (Jan. 2005, p. 82) Discussion document: Has been prepared by an international committee chaired by Ed Barbeau of the University of Toronto " 1 3-18 L-functions and Related Themes, Centre de recherche (barbeau@math. utoronto . ca) and Peter Taylor of the University of mathematiques, Univ. de Montreal, Montreal, Quebec, Canada. Canberra in Australia who is the executive-director of the Australian Focus: The workshop will focus on the analytic theory of L­ Mathematical Trust (pjt@ol ympiad. or g). This document defines functions and how they are used in a variety of questions ranging terms, describes issues, provides sample situations, and poses from arithmetic geometry to classical analytic number theory. questions for discussion. Finally, it indicates how to become Lecturers: Philippe Michel (Montpellier II); Kumar Murty (Toronto); involved in the Study Conference. Would-be participants will be K. Soundararajan (Michigan). asked to submit a brief curriculum vita and a 6-10 page document Organizers: Chantal David (Concordia) and Ram Murty (Queen's). addressing matters relevant to the study no later than August 31, Information: http : I /www. crm . umontreal . ca/Number 2005/. 2005. The committee plans to send out invitations by January 31, 2006. The Conference will be followed by a publication. A copy of the discussion document can be obtained by going to the website http : I /www. amt. canberra. edu, clicking on "LINKS" and The following new announcements will not be repeated until then on "ICMI Study 16". the criteria in the next to the last paragraph at the bottom of Deadline: August 31, 2005. the first page of this section are met. Information: http: I /www. amt . canberr a. edu/ i cmi s 16 .html /. March 2006 july 2006 " 1 3-1 7 Anatomy of Integers, Centre de recherche mathematiques '' 1 0- 1 4 1nternational Conference on Analytic Topology, Lake Plaza Universite de Montreal, Montreal, Quebec, Canada. Hotel, Rotorua, New Zealand. Organizers: Jean-Marie de Koninck (Laval) and Andrew Granville Description: The main goal of this conference is to bring together (Montreal). a group of researchers from around the world, who are working Workshop Focus: On multiplicative number theory, divisors, prime at the interface between Topology and Analysis, to discuss recent factors, distribution of prime divisors, multiplicative functions, developments and future directions of Analytic Topology. smooth/ friable numbers etc. Organizers: Warren B. Moors (Auckland University, email: moor s@ math . au ckl and . ac. nz); and Jiling Cao (Auckland University, email: Lecturers: Kevin Ford (Urbana-Champaign); K. Soundararajan cao@math. au ckl and. ac. nz). (Michigan); Gerald Tenenbaum (Institut Elie Cartan Nancy). Information: http : I /www .math . auckl and. ac . nzrcao/ Information: http : I /www. crm . umontreal . ca/Number2005/. conference06.html. April 2006 ' 6- 1 2 Additive Combinatorics, Centre derecherchemathematiques, Universite de Montreal, Montreal, Quebec. Topics: The topics covered will include: the Freiman-Ruzsa the­ orem, the structure of set theory addition, Gowers' approach to Szemeredi's theorem and Green and Tao's approach to combinato­ rial sets with structure. A mini-school will be organized before this workshop to introduce more people to this vibrant subject. More information will be made available on this site. Lecturers: Tim Gowers (Cambridge), Ben Green (Cambridge), Irnre Ruzsa (Alfred Renyi Institute) and Terence Tao (UCLA). Organizers: JozsefSolymosi (UBC) and Andrew Granville (Montreal). Information: http: I /www. crm. umontr eal . ca/Number 2005/.

372 N OTICES OF THE AMS V OLUME 52, N UMBER 3 New Publications Offered by the AMS

points on elliptic curves; V. Shpilrain and J.-T. Yu, Test poly­ Algebra and Algebraic nomials, retracts, and the Jacobian conjecture; D. Wright, The Jacobian Conjecture: ideal membership questions and recent Geometry advances. Contemporary Mathematics, Volume 369 Affine Algebraic March 2005, 276 pages, Softcover, ISBN 0-8218-3476-2, LC 2004062382, 2000 Mathematics Subject Classification: 14Rxx, Geometry 14H50, 14]50, 14E05, 13P10, 12Y05, All AMS members $63, List $79, Order code CONM/ 369 Affine Algebraic Jaime Gutierrez, University of Geometry Cantabria, Santander, Spain, Jaime Gutierrez Vladimir Shpiroin Vladimir Shpilrain, City Jle·ToiYu Editors College of New York, and Analysis Jie-Tai Yu, University of Hong () Kong, Editors Geometric Analysis A Special Session on affine and of PDE and Several algebraic geometry took place at the Geometric Analysis first joint meeting between the American Mathematical Society of POE and Several Complex Variables (AMS) and the Real Sociedad Matematica Espanola (RSME) held Complex Variables in Seville (Spain). This volume contains articles by partici­ Dedicated to Fronyois Treves Dedicated to Fran<;ois Sagun Chonlllo Paulo D. Cordaro pating speakers at the Session. Nicholas Honges Treves JOfgeHounle Abdelhomld Mezloni The book contains research and survey papers discussing Editors Sagun Chanillo, Rutgers '',- recent progress on the Jacobian Conjecture and affine alge­ University, Piscataway, N], braic geometry and includes a large collection of open problems. It is suitable for graduate students and research Paulo D. Cordaro, Instituto de mathematicians interested in algebraic geometry. Matematica e Estatistica, Contents: Open problems in affine algebraic geometry Universidade de Sao Paulo, IME-USP, Brazil, (collected by G. Freudenburg and P. Russell); T. Asanuma, Nicholas Hanges, Herbert H. Lehman College, Purely inseparable k-forms of affine algebraic curves; CUNY, Bronx, NY, Jorge Hounie, Universidade T. Asanuma, S.M. Bhatwadekar, and N. Onoda, Generic fibra­ Federal de Sao Carlos, Brazil, and Abdelhamid tions by A 1 and A* over discrete valuation rings; M. de Bondt Meziani, Florida International University, Miami, and A. van den Essen, Hesse and the Jacobian Conjecture; P. Cassou-Nogues, Bad field generators; V. Drensky, Coordi­ Editors nates in ideals of polynomial algebras; H. Flenner and This volume is dedicated to Franc;:ois Treves, who made M. Zaidenberg, On the uniqueness of ([* -actions on affine substantial contributions to the geometric side of the theory surfaces; T. Kambayashi and M. Miyanishi, On two recent of partial differential equations (PDEs) and several complex views of the Jacobian Conjecture; T. Kishirnoto, Singularities variables. One of his best-known contributions, reflected in on normal affine 3-folds containing AJ-cylinderlike open many of the articles here, is the study of hypo-analytic subsets; H. Kraft, Free ([ +-actions on affine threefolds; structures. L. Makar-Limanov, Again x + x 2y + z 2 + t 3 = 0; K. Masuda and M. Miyanishi, Equivariant cancellation for algebraic vari­ An international group of well-known mathematicians eties; R. Peretz, Constructing polynomial mappings using contributed to the volume. Articles generally reflect the non-commutative algebras; T. Shaska and J. L. Thompson, On interaction of geometry and analysis that is typical of the generic curve of genus 3; I. E. Shparlinski, Orders of Treves's work, such as the study of the special types of

MARCH 2005 NOTICES OF THE AMS 373 New Publications Offered by the AMS partial differential equations that arise in conjunction with CR-manifolds, symplectic geometry, or special families of The p-Harmonic vector fields. There are many topics in analysis and PDEs Equation and Recent covered here, unified by their connections to geometry. The p-Harmonic Advances in The material is suitable for graduate students and research Equation and Recent mathematicians interested in geometric analysis of PDEs and Advances in Analysis Analysis several complex variables. PletroPoggi-corrodini Edit Of Pietro Poggi-Corradini, Kansas This item will also be of interest to those working in geometry State University, Manhattan, and topology. Editor Contents: P. Ahern and X. Gong, Cusp-type singularities of real analytic curves in the complex plane; S. Berhanu and Comprised of papers from the Illrd J. Hounie, The F. and M. Riesz property for vector fields; Prairie Analysis Seminar held at A. Bove, Gevrey hypo-ellipticity for sums of squares of vector Kansas State University, this book fields: Some examples; H. Brezis, P. Mironescu, and reflects the many directions of current research in harmonic A. C. Ponce, W1•1 -maps with values into S1; S. Chanillo, analysis and partial differential equations. Included is the Analytic hypoellipticity and spectral problems for work of the distinguished main speaker, Tadeusz Iwaniec, his Schrodinger's equation; H. Chen and z. Luo, Formal solutions invited guests John Lewis and Juan Manfredi, and many other for higher order nonlinear totally characteristic PDEs with leading researchers. irregular singularities; F. Colombini and N. Lerner, Unique­ The main topic is the so-called p-harmonic equation, which is ness of L "" solutions for a class of conormal BV vector fields; a family of nonlinear partial differential equations general­ P. D. Cordaro and N. Hanges, Impact of lower order terms on izing the usual Laplace equation. This study of p-harmonic a model pde in two variables; M. Derridj and D. S. Tartakoff, equations touches upon many areas of analysis with deep Global analytic hypoellipticity for a class of quasilinear sums relations to functional analysis, potential theory, and calculus of squares of vector fields; M. Eastwood, Representations via of variations. overdetermined systems; G. Francsics and P. D. Lax, A semi­ explicit fundamental domain for a Picard modular group in The material is suitable for graduate students and research complex hyperbolic space; S. Gindikin, Complex horospherical mathematicians interested in harmonic analysis and partial transform on real sphere; J. Gorsky and A. A. Hirnonas, On differential equations. analyticity in space variable of solutions to the KdV equation; This item will also be of interest to those working in differential L. Hormander, The multinomial distribution and some equations. Bergman kernels; X. Huang, S. Ji, and D. Xu, Several results for Contents: F. H. Beatrous, T. J. Bieske, and J. J. Manfredi, The holomorphic mappings from Bn into BN; H. Jacobowitz, maximum principle for vector fields; I. Blank, A partial classi­ Whitney and Mizohata structures; A. E. Kogoj and fication of the blowups of the singularities in a composite E. Lanconelli, One-side Liouville theorems for a class of membrane problem; A. Domokos and J, J. Manfredi, C1·"'­ hypoelliptic ultraparabolic equations; L. Lempert, Acyclic regularity for p-harmonic functions in the Heisenberg group sheaves in Banach spaces; A. Li and Y. Y. Li, A Liouville type for p near 2; L. D'Onofrio and T. Iwaniec, Notes on p­ theorem for some conformally invariant fully nonlinear equa­ harmonic analysis; M. Foss, A condition sufficient for the tions; S. T. Melo, Norm closure of classical pseudodifferential partial regularity of minimizers in two-dimensional nonlirlear operators does not contain Hormander's class; G. Metivier, elasticity; C. Frosini, Dynamics on bounded domains; Remarks on the well-posedness of the nonlinear Cauchy K. E. Hare and A. M. Stokolos, On the rate of tangential problem; A. Meziani, Representation of solutions of planar convergence of functions from Hardy spaces, 0 < p < 1; elliptic vector fields with degeneracies; L. Nirenberg, Some P. A. Hasto, Counter-examples of regularity in variable expo­ recollections of working with Fran<;ois Treves; M.-C. Shaw, nent Sobolev spaces; L. V. Kovalev and D. Opela, Boundary value problems on Lipschitz domains in IR\n or e n; Quasiregular gradient mappings and strong solutions of S. Spagnolo, Hyperbolic systems well posed in all Gevrey elliptic equations; R. S. Krausshar, Y. Qiao, and J. Ryan, classes. Harmonic, monogenic and hypermonogenic functions on some Contemporary Mathematics, Volume 368 conformally flat manifolds in Rn arising from special arith­ March 2005, 414 pages, Softcover, ISBN 0-8218-3386-3, LC metic groups of the Vahlen group; J. L. Lewis, On symmetry 2004055409, 2000 Mathematics Subject Classification: 35-XX, and uniform rectifiability arising from some overdetermined 32-XX, All AMS members $87, List $109, Order code elliptic and parabolic boundary conditions; L. Forzani and CONM/ 368 D. Maldonado, Recent progress on the Monge-Ampere equa­ tion; J. Onninen, Mappings of finite distortion: Future directions and problems; M. Stawiska, Riemarm-Hurwitz formula and Morse theory. Contemporary Mathematics, Volume 3 70 March 2005, 211 pages, Softcover, ISBN 0-8218-3610-2, LC 2004062294, 2000 Mathematics Subject Classification: 30-06, 31-06, 32-06, 35-06,46-06,47-06, All AMS members $47, List $59, Order code CONM/370

374 NOTICES OF THE AMS VOLUME 52, NUMBER 3 New Publications Offered by the AMS

Proceedings of Symposia in Pure Mathematics, Volume 73 Mathematical Physics April 2005, 418 pages, Hardcover, ISBN 0-8218-3666-8, LC 2004062363, 2000 Mathematics Subject Classification: 81 Txx, 57-XX, 18-XX, 53Dxx, 55-XX, 37-XX, 17Bxx, All AMS members Proceedings of Symposia In Graphs and Patterns $71, List $89, Order code PSPUM/73 PURE MATHEMATICS in Mathematics and Graphs and Patterns Theoretical Physics in Mathematics and Theoretical Physics Mikhail Lyubich, Stony Brook New AMS-Distributed University, NY, and University Mlkha.ll. Lyublch LeonTakhtajan Editors of Toronto, ON, Canada, and Publications Leon Takhtajan, Stony Brook University, NY, Editors The Stony Brook Conference, "Graphs Geometry and Topology and Patterns in Mathematics and Theoretical Physics", was dedicated to Dennis Sullivan in honor of his sixtieth birthday. The event's scientific content, Non-linear Elliptic which was suggested by Sullivan, was largely based on mini-courses and survey lectures. The main idea was to help Equations in researchers and graduate students in mathematics and Conformal Geometry theoretical physics who encounter graphs in their research to overcome conceptual barriers. Sun-Yung Alice Chang, The collection begins with Sullivan's paper, "Sigma models and Princeton University, NJ string topology," which describes a background algebraic struc­ Non-linear elliptic partial differential ture for the sigma model based on algebraic topology and equations are an important tool in the transversality. Other contributions to the volume were organized study of Riemannian metrics in into five sections: Feynman Diagrams, Algebraic Structures, differential geometry, in particular for Manifolds: Invariants and Mirror Symmetry, Combinatorial problems concerning the conformal Aspects of Dynamics, and Physics. These sections, along with change of metrics in Riemannian geometry. In recent years the more research-oriented articles, contain the following surveys: role played by the second order semi-linear elliptic equations in "Feynman diagrams for pedestrians and mathematicians" by M. the study of Gaussian curvature and scalar curvature has been Polyak, "Notes on universal algebra" by A. Voronov, "Unimodal extended to a family of fully non-linear elliptic equations maps and hierarchical models" by M. Yampolsky, and "Quantum associated with other symmetric functions of the Ricci tensor. A geometry in action: big bang and black holes" by A. Ashtekar. case of particular interest is the second symmetric function of This comprehensive volume is suitable for graduate students the Ricci tensor in dimension four closely related to the Ffaffian. and research mathematicians interested in graph theory and In these lectures, starting from the background material, the its applications in mathematics and physics. author reviews the problem of prescribing Gaussian curvature This item will also be of interest to those working in discrete on compact surfaces. She then develops the analytic tools mathematics and combinatorics. (e.g., higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully Contents: D. Sullivan, Sigma models and string topology; nonlinear equation in prescribing the Chern-Gauss-Bonnet Feynman diagrams: M_ Polyak, Feynman diagrams for pedes­ integrand on compact manifolds of dimension four. trians and mathematicians; D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries; Algebraic structures: The material is suitable for graduate students and research A. A. Voronov, Notes on universal algebra; D. Tamarkin and mathematicians interested in geometry, topology, and differ­ B. Tsygan, The ring of differential operators on forms in ential equations. noncommutative calculus; v_ Gorbounov, F. Malikov, and This item will also be of interest to those working in differential V. Schechtman, Twisted chiral de Rham algebras on IP' 1 ; Mani­ equations. folds: Invariants and mirror symmetry: R. Kashaev and Distributed within the Americas by the American Mathematical Society. N. Reshetikhin, Invariants of tangles with flat connections in their complements; S. Garoufalidis and J. Levine, Tree-level Contents: Gaussian curvature equation; Moser-Trudinger invariants of three-manifolds, Massey products and the inequality (on the sphere); Polyakov formula on compact Johnson homomorphism; K. Fukaya, Multivalued Morse surfaces; Conformal covariant operators-Paneitz operator; theory, asymptotic analysis and mirror symmetry; Combinato­ Functional determinant on 4-manifolds; Extremal metrics for rial aspects of dynamics: R. Forman, Some applications of the log-determinant functional; Elementary symmetric func­ combinatorial differential topology; A. de Carvalho, Exten­ tions; A priori estimates for the regularized equation (*)s; sions, quotients and generalized pseudo-Anosov maps; Smoothing via the Yamabe flow; Deforming u2 to a constant M. Yampolsky, Unimodal maps and hierarchical models; function. Physics: A. Ashtekar, Quantum geometry in action: big bang Zurich Lectures in Advanced Mathematics and black holes; P. van Nieuwenhuizen, Supersymmetry, supergravity, superspace and ERST symmetry in a simple October 2004, 100 pages, Softcover, ISBN 3-03719-006-x, 2000 model. Mathematics Subject Classification: 53Axx, 58Jxx, All AMS members $22, List $28, Order code EMSZLEC/2

MARCH 2005 NOTICES OF THE AMS 375 New AMS-Distributed Publications

group; I. Satake, The Laplacian for a Frobenius manifold; Frobenius Manifolds B. Siebert, Virtual fundamental classes, global normal cones Quantum Cohomology and Fulton's canonical classes; A. Takahashi, A note on BPS and Singularities invariants on Calabi-Yau 3-folds; List of Participants. Vieweg Aspects of Mathematics, Volume 36 Claus Hertling, Universitat September 2004, 378 pages, Hardcover, ISBN 3-528-03206-5, Mannheim, Germany, and 2000 Mathematics Subject Classification: 53D45, 14N35; Matilde Marcoli, Max Planck 37K10, 37K20, 34M35, 32S40, 14B05, All AMS members $75, Institute for Mathematics, List $83, Order code VWAM/36 Bonn, Germany, Editors Quantum cohomology, the theory of Metric Spaces, Frobenius manifolds and the relations to integrable systems have been flourishing areas since the Convexity and early 1990s. A conference was organized at the Max-Planck­ Nonpositive Institute for Mathematics to bring together leading experts in these areas. This volume originated from that meeting and Curvature presents the state of the art in the subject. Athanase Papadopoulos, Frobenius manifolds are complex manifolds with a multiplica­ Institut de Recherche tion and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in Mathematique Avancee, 1991 by Dubrovin, motivated by physics results. Another Strasbourg, France source of Frobenius manifolds is singularity theory. Duality This book covers metric spaces of between string theories lies behind the phenomenon of mirror nonpositive curvature in the sense of symmetry. One mathematical formulation can be given in Busemann, that is, metric spaces whose distance function terms of the isomorphism of certain Frobenius manifolds. A satisfies a convexity condition. Also contained is a systematic third source of Frobenius manifolds is given by integrable introduction to the theory of geodesics as well as a detailed systems, more precisely, bihamiltonian hierarchies of presentation of some facets of convexity theory, which are evolutionary PDE's. As in the case of quantum cohomology, useful in the study of nonpositive curvature. here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be deter­ Concepts and techniques discussed in the volume are illus­ mined implicitly by the underlying Frobenius manifolds. trated by many examples from classical hyperbolic geometry and from the theory of Teichmiiller spaces. It is useful for This volume is suitable for graduate students and research graduate students and researchers in geometry, topology and mathematicians interested in geometry and topology. analysis. A publication of Vieweg Verlag. The AMS is exclusive distributor in North America. Vieweg Verlag Publications are available worldwide Distributed within the Americas by the American Mathematical Society. from the AMS outside of Germany, Switzerland, Austria, and Japan. Contents: Introduction: Some historical markers; Lengths of Contents: A. Douai and C. Sabbah, Gauss-Manin systems, paths in metric spaces; Length spaces and geodesic spaces; Brieskorn lattices and Frobenius structures (II); J. Fernandez Maps between metric spaces; Distances; Convexity in vector and G. Pearlstein, Opposite filtrations, variations of Hodge spaces; Convex functions; Strictly convex normed vector structure, and Frobenius modules; E. Getzler, The jet-space of spaces; Busemann spaces; Locally convex spaces; Asymptotic a Frobenius manifold and higher-genus Gromov-Witten invari­ rays and the visual boundary; Isometries; Busemann functions, ants; A. B. Givental, Symplectic geometry of Frobenius co-rays and horospheres; References; Index. structures; C. Hertling and Yu. I. Manin, Unfoldings of mero­ IRMA Lectures in Mathematics and Theoretical Physics, morphic connections and a construction of Frobenius Volume 6 manifolds; R. Kaufmann, Discrete torsion, symmetric products and the Hilbert scheme; X. Liu, Relations among universal December 2004, 300 pages, Softcover, ISBN 3-03719·010-8, equations for Gromov-Witten invariants; A. Losev and 2000 Mathematics Subject Classification: 26-01, 30F25, 30F45, Yu. I. Manin, Extended modular operad; S. Merkulov, Operads, 30F60, 32Q45, 32G15, 51-01, 51K05, 51K10, 51M09, 51M10, deformation theory and F-manifolds; A. Polishchuk, Witten's 51F99, 52-01, 52A07, 52A41, 53-01, 53C70, 54-01, 54E35,All AMS members $43, List $54, Order code EMSILMTP/6 top Chern class on the moduli space of higher spin curves; K. Saito, Uniformization of the orbifold of a finite reflection

376 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Classified Advertisements Positions available, items for sale, services available, and more

1,300 minority and 500 international stu­ needs business-savvy mathematicians. See KENTUCKY dents, and has a strong commitment to http: //www.sciencemasters.com/. achieving diversity among faculty, staff, 000019 WESTERN KENTUCKY UNIVERSITY and administration. The university is in Ogden College of Science & Engineering Bowling Green, between Nashville, TN, and Department of Mathematics Louisville. NEW JERSEY Applications are invited for the position of Review of applications will begin Feb­ ruary 1, 2005, and will continue until the Head of the Department of Mathematics KEAN UNIVERSITY'S NEW JERSEY starting July 1, 2005. position is filled. Please send a letter of ap­ FOR SCIENCE & TECHNOLOGY Applicant must have a doctorate in plication, vita, a statement of administra­ CENTER mathematics or a mathematical science tive leadership philosophy, and at least EDUCATION with appropriate credentials for a tenured three letters of recommendation to: Faculty Position appointment at the rank of professor. We Dr. Frank Conley, Interim Chair, are seeking a dedicated and effective leader Search Committee Kean University's New Jersey Center f or who can help promote and strengthen the Department of Mathematics Science & Technology Education 5 year department's academic, research, and ser­ Western Kentucky University combined bachelor/ master program; As­ vice programs. Qualified candidates must 1 Big Red Way sistant Professor, tenure-track, Ph.D. re­ have a commitment to recognizing and Bowling Green, KY 42101 quired; specializing in mathematics. Ef­ encouraging excellence in teaching, have email: frank [email protected] a familiarity with current issues involving fective September 1, 2005. Primary the mathematics curriculum and technol­ For more information about the Depart­ teaching responsibilities include under­ ogy, and have an established record of re­ ment of Mathematics at Western Kentucky graduate mathematics and computational search/scholarly activity and a commit­ University, visit our Web page at http: I I methods of science and probabilistic meth­ / math. ment to encouraging such activity, and www.wku.edu ods in science. Other undergraduate and an have a history of significant professional Western Kentucky University is graduate courses appropriate to area of ex­ service. Evidence of additional adminis­ EO/ AA employer. pertise. Commitment to teaching excel­ trative expertise is also desired. 000039 Mathematics is one of the nine depart­ lence, an integrated science curriculum ments in the College of Science and Engi­ and scholarly research required. Send cur­ neering. With 34 full-time positions (23 MICHIGAN riculum vitae, statement of research in­ tenured/tenure-track), the department of­ terests, statement of teaching philosophy fers baccalaureate and masters programs MICHIGAN STATE UNIVERSITY and three recommendation letters to: Dr. in mathematics. In addition, mathematics Laura Lorentzen, Acting Chairperson, Dept. is included in the university general edu­ East Lansing, Ml48824 cation requirements for all undergraduate proMSc Program in of Science & Technology, Kean University, degrees. Industrial Mathematics 1000 Morris Ave., Union, NJ 07083. Kean Western Kentucky University enrolls ap­ University is an Affirmative Action/Equal proximately 18,000 undergraduate and Direct your students toward one of the Opportunity Employer. graduate students, including more than professional M.Sc. programs. Industry 000037

Suggested uses for classified advertising are positions available, books or 2005 issue-April 27, 2005; August 2005 issue-May 26, 2005; September lecture notes for sale, books being sought, exchange or rental of houses, 2005 issue-June 2 7, 2005; October issue-July 25, 2005. and typing services. U.S. laws prohibit discrimination in employment on the basis of color, age, The 2005 rate is $100 per inch or fraction thereof on a single column(one­ sex, race, religion, or national origin. "Positions Available" advertisements inch minimum), calculated from top of headline. Any fractional text of 1/2 from institutions o utside the U.S. cannot b e published unless they are inch or more will be charged at the next inch rate. No discounts for multi­ accompanied by a statement that the institution does not discriminate on ple ads or the same ad in consecutive issues. For an additional $10 charge, these grounds whether or not it is subject to U.S. laws. Details and spe· announcements can b e placed anonymously. Correspondence will be cific wording may be found on page 13 73 (val. 44). forwarded. Situations wanted advertisements from involuntarily unemployed math­ Advertisements in the "Positions Available" classified section will be set ematicians are accepted under certain conditions for free publication. Call with a minimum one-line headline, consisting of the institution name above toll-free 800-3 21-4AMS (321-4267) in the U.S. and Canada or 401-455-4084 body copy, unless additional h eadline copy is specified b y the advertiser. worldwide for further information. Headlines will be centered in boldface at n o extra charge. Ads will appear Submission: Promotion s Department, AMS, P.O. Box 6248, Providence, in the language in which they are submitted. Rh ode Island 02940; or via fax: 401·331-3842; or send email to There are no member discounts for classified ads. Dictation over the c l assads@ams . or g. AMS location for express delivery packages is telephone will not be accepted for classified ads. 201 Charles Street, Providence, Rhode Island 20904. Advertisers will be Upcoming deadlines for classified advertising are as follows: April 2005 billed upon publication. issue-January 2 8, 2005; May 2005 issue- February 25, 2005; June/ July

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NEW YORK GREECE

BRONX COMMUNITY COllEGE NATIONAl TECHNICAl UNIVERSITY OF OF THE CITY COllEGE OF NEW YORK ATHENS (CUNY) Department of Mathematics Full-Time Tenure-Track Faculty Positions 2005 The Department of Mathematics invites Visit our website: http: I /www. cuny. edu/ applications for two tenure-track assis­ abtcuny/cunyjobs/ or http://www.bcc. tant professor positions. One in "Stochastic cuny. edu. Processes and Finance" and one in "Nu­ Mathematics & Computer Science: The merical Analysis", starting December 2 00 5. Mathematics & Computer Science Depart­ The applicant must be of Greek national­ ment of Bronx Community College, City ity, and must have completed all require­ University of New York, seeks to fill out six tenure-track positions in the ranks of Lec­ ments, according to the Greek law, by the turer/ Assistant/ Associate Professor. time of appointment. Responsibilities in­ Lecturer/Assistant Professor: Descrip­ clude teaching undergraduate and gradu­ tion and duties: Full-time tenure-track fac­ ate courses, research, and service. For ulty position to teach developmental and deadlines and more information, visit the upper level mathematics and computer website http: / /www.math.ntua.gr/ science courses. Evening, Saturday, and Sunday teaching required. Qualification Thesei sDEP or communicate via email: requirements: Assistant Professor: Doc­ [email protected]. torate in Mathematics/ Computer Science 0000 36 or Mathematics Education required. Lec­ turer: Master's degree preferred. Record of commitment to excellence in teaching de­ ISTANBUL sired. Knowledge of databases and/ or in­ structional technologies and grant writ­ ISIK UNIVERSITY ing an asset. Good interpersonal skills required. Must be available for evening Department of Mathematics and weekend work. Salary Range: Lecturer: $32,997-$53,113, Assistant Professor: The Department of Mathematics at Isik $35,031-$56,014. University invites applications for faculty Assistant/ Associate Professor: Position positions at the assistant professor level description and duties: Full-time tenure­ from all areas of mathematics. The ap­ track faculty position teaching mathe­ pointment begins September 1, 2005. In matics courses, participating in interdis­ addition, multi-year visiting assistant pro­ ciplinary course development; com­ mitment to improving students' writing fessor positions will be available. Appli­ and thinking skills; student recruitment; cants should have a Ph.D. in mathematics student advisement; research in mathe­ and should show outstanding promise matics education; and service to the col­ and/ or accomplishments in both research lege and community. Evening, Saturday, and teaching. Duties include undergradu­ and Sunday teaching required. Qualifica­ ate and graduate teaching and independent tion requirements: Ph.D. or Ed.D. in math­ ematics or mathematics education, with a research. research agenda in mathematics educa­ The University is located in Sile, forty tion. Candidates with experience in teacher miles east of Istanbul, at the coastline of preparation for secondary school mathe­ the Black Sea. Applicants should send a let­ matics instruction will be given special ter of application, a curriculum vitae and consideration. the names of three references to: Salary Range: Assistant Professor: $41,974-$61,111, Associate Professor: Saadet Erbay, Chair $45,651-$73,028. Department of Mathematics Send cover letter and resume, listing Isik University current salary and three letters of reference by 04/ 08/ 05 to: Ms. Shelley B. Levy, Di­ 34398 Maslak rector of Human Resources, Office of Istanbul, Turkey Human Resources, Bronx Community Col­ lege, University Avenue & West 181st Electronic submissions may be forwarded Street, Bronx, NY 10453. EEO/ / AA/ IRCA/ to email: serbay@i si kun. edu. tr. ADA Employer. 0 0 0038 0000 35

378 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences oftheAMS

IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information withlinkstotheabstractforeachtalkcanbefoundon theAMSwebsite.See http: I /WWN. 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.

Robert J. McCann, University of Toronto, Optimal Bowling Green, convergence rates for the fastest conservative nonlinear diffusions. Kentucky M. Susan Montgomery, University of Southern California, Some applications of group theory to classifying Hop{ Western Kentucky University algebras. March 18-19, 2005 James J. Zhang, University of Washington, Searching for Friday - Saturday quantum projective spaces.

Meeting #1 004 Special Sessions Southeastern Section Advances in the Study of Wavelets and Multiwavelets, Associate secretary: Matthew Miller Douglas P. Hardin, Vanderbilt University, and Bruce Announcement issue of Notices: January 2005 Kessler, Western Kentucky University. Program first available on AMS website: February 3, 2005 Commutative Ring Theory, Michael C. Axtell, Wabash Program issue of electronic Notices: March 2005 College, and Joe Alyn Stickles Jr., University of Evansville. Issue of Abstracts: Volume 26, Issue 2 Dynamic Equations on Time Scales and Applications, Deadlines Ferhan M. Atici and Daniel C. Biles, Western Kentucky For organizers: Expired University, and Billiir Kaymakc;:alan, Georgia Southern For consideration of contributed papers in Special Sessions: University. Expired Geometric Topology and Group Theory, ]ens E. Harlander, For abstracts: Expired Western Kentucky University. Graph Theory, Mustafa Atici, Western Kentucky University. The scientific information listed below may be dated. For the latest information, see www. ams. org/amsmtgs/ Hop{ Algebras and Related Topics, David E. Radford, sectional . html. University of Illinois at Chicago, and Bettina Richmond, Western Kentucky University. Invited Addresses Knot Theory and Its Applications, Yuanan Diao, University Bennett Chow, University of California San Diego, Title to of North Carolina, Charlotte, and Claus Ernst, Western be announced. KentUcky University.

MARCH 2005 NOTICES OF THE AMS 379 Meetings & Conferences

L-Functions, Heather Russell, Nilabh Sanat, and Dominic Special Sessions Lanphier, Western Kentucky University. Arithmetic Groups and Related Topics, Alexander Nonlinear Analysis and Applied Mathematics, Robert ]. Lubotzky, Hebrew University of Jerusalem, and Andrei McCann, University of Toronto, and Daniel P. Spirn, Uni­ Rapinchuk, University of Virginia. versity of Minnesota. Asymptotic Behavior of Evolution Equations, Gaston M. Numerical Analysis, Approximation, and Computational N'Guerekata, Morgan State University, and Nguyen Van Complexity: Interdisciplinary Aspects, David Benko, West­ Minh, James Madison University. ern Kentucky University, and Steven B. Damelin, Georgia Southern University. Designs, Codes, and Geometries, James A. Davis, Univer­ sity of Richmond, Keith E. Mellinger, University of Mary Partial Differential Equations and Their Applications, Zhang­ wei Shen and Changyou Wang, University of Kentucky. Washington, and Qing Xiang, University of Delaware. Recent Advances in Noncommutative Algebra, Ellen E. Frontiers on Complex Fluid Flows: Analytic and Computa­ Kirkman, Wake Forest University. tional Methods, L. Pamela Cook and Louis F. Rossi, Uni­ versity of Delaware. Representation Theory, Markus Hunziker, University of Georgia. Geometric Analysis, Xiuxiong Chen, University of Wis­ consin, Madison, Pengfei Guan, McMaster University, Semigroups of Operators and Applications, Khristo Boy­ adzhiev, Ohio Northern University, Lan Nguyen, Western Zhiqin Lu, University of California Irvine, and Jeff A. Kentucky University, and Quoc-Phong Vu, Ohio University. Viaclovsky, Massachusetts Institute of Technology. Topology, Convergence, and Order, in Honor of Darrell High Dimensional Probability, Wenbo Li, University of Kent, Gary Richardson, University of Central Florida, and Delaware, and Joel Zinn, Texas A&M University. Thomas A. Richmond, Western Kentucky University. Homotopy Theory (in Honor of Donald M. Davis's and Martin Bendersky's 60th Birthdays), Kenneth G. Monks, University of Scranton, and W. Stephen Wilson, Johns Newark, Delaware Hopkins University. Integral and Operator Equations, Charles W. Groetsch, University of Delaware University of Cincinnati, and M. Zuhair Nashed, Univer­ April2-3, 2005 sity of Central Florida. Saturday - Sunday Mathematical Biology, David A. Edwards, University of Delaware. Meeting #1 005 Mathematical Methods for Efficient Simulation of Stochas­ Eastern Section tic Nonlinear Optical Systems, Richard 0. Moore, New Associate secretary: Lesley M. Sibner Jersey Institute of Technology, and Tobin A. Driscoll, Announcement issue of Notices: February 2005 University of Delaware. Program first available on AMS website: February 17, 2005 Program issue of electronic Notices: April 2005 Mathematical Methods in Electromagnetic Wave Propaga­ Issue of Abstracts: Volume 26, Issue 2 tion, Fioralba Cakoni and Peter B. Monk, University of Delaware. Deadlines Probabilistic Paradigms in Combinatorics, Joshua N. For organizers: Expired Cooper, Courant Institute of Mathematics, NYU, and Jozef For consideration of contributed papers in Special Sessions: Skokan, Universidade de Sao Paulo. Expired Recent Progress in Thin Fluid Flows, Richard J. Braun, For abstracts: Expired University of Delaware. The scientific information listed below may be dated. For Singular Analysis and Spectral Theory of Partial Differen­ the latest information, see www. ams. org/amsmtgs/ tial Equations, Juan B. Gil, Pennsylvania State University, sectional . html. Altoona, and Gerardo A. Mendoza, Temple University. Spectral and High-Order Discretization Methods for Partial Invited Addresses Differential Equations, Tobin A. Driscoll, University of Xiuxiong Chen, University of Wisconsin, Madison, Title to Delaware. be announced. Symmetry Methods for Partial Differential Equations, Philip Anna C. Gilbert, AT&T Labs-Research, Analysis, approxi­ Broadbridge, University of Delaware, and Danny Arrigo, mations, and algorithms. University of Central Arkansas. Alexander Lubotzky, Hebrew University of Jerusalem, Counting primes, groups, and manifolds. Lorenz Schwachhoefer , University of Dortmund, Special symplectic connections.

380 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences

Graph Theory(Code: SS 12A), John C. George, Eastern New Lubbock, Texas Mexico University, and Walter D. Wallis, Southern Illinois University at Carbondale. Texas Tech University Homological Algebra and Its Applications (Code: SS 4A), Alex Martsinkovsky, Northeastern University, and Mara D. April8-1 0, 2005 Neusel, Texas Tech University. Friday - Sunday Invariants of Links and 3-Manifolds (Code: SS 8A), Meeting #1 006 Mieczyslaw Krzysztof Dabkowski, University of Texas at Dallas, Razvan Gelca, Texas Tech University, and Jozef Central Section Henryk Przytycki, George Washington University. Associate secretary: Susan ]. Friedlander Announcement issue of Notices: February 2005 Partial Differential Equation and Its Application in Biomedical Study (Code: SS 16A), Jay R. Walton, Texas A&M University, Program first available on AMS website: February 24, 2005 and Padmanabhan Seshaiyer and Akif Ibragimov, Texas Program issue of electronic Notices: April 2005 Tech University. Issue of Abstracts: Volume 26, Issue 3 Real Algebraic Geometry (Code: SS 6A), Anatoly Korcha­ Deadlines gin and David Weinberg, Texas Tech University. For organizers: Expired Recent Advances in Complex Function Theory (Code: SS SA), For consideration of contributed papers in Special Sessions: Brock Williams, Roger W. Barnard, and Kent Pearce, Texas Expired Tech University. For abstracts: February 15, 2005 Statistical Image Processing and Analysis and Applications (Code: SS 13A), Victor Patrangenaru, Texas Tech University. The scientific information listed below may be dated. For Theory and Application of Stochastic Differential Equa­ the latest information, see www. ams. o rg/ amsmtgs/ tions (Code: SS 9A), Edward J. Allen, Texas Tech Univer­ sectional . html. sity, and Armando Arciniega, University of Texas at San Antonio. Invited Addresses Topology of Continua (Code: SS lA), Wayne Lewis, Texas Nikolai Ivanov, Michigan State University, Title to be Tech University. announced. Topology ofDynamical Systems (Code: SS 7A), Brian Raines, Mattias Jonsson, University of Michigan, Title to be Baylor University. announced. Undergraduate and Graduate Student Research (and related Nicolas Monod, University of Chicago, Title to be poster session organized by Ali Khoujmane and Mara D. Neusal, Texas Tech) (Code: SS 14A), Ali Khoujmane, announced. Edward W. Swim, Edward ]. Allen, and Padmanabhan Hee Oh, California Institute of Technology, Title to be Seshaiyer, Texas Tech University. announced. Special Sessions Santa Barbara, Classical and Differential Galois Theory (Code: SS 3A), Lourdes Juan and Arne Ledet, Texas Tech University, and California Andy R. Magid, University of Oklahoma. Differential Geometry and Its Applications (Code: SS 2A), University of California Santa Barbara Josef F. Dorfmeister, Munich University of Technology, April16-1 7, 2005 Magdalena D. Toda, Texas Tech University, and Hongyou Saturday - Sunday Wu, Northern Illinois University. Discrete Groups, Homogeneous Spaces, Rigidity (Code: SS Meeting #1 007 15A), Alex Gorodnik, University of Michigan, Ann Arbor, Western Section Hee Oh, California Institute of Technology, and Nicolas Associate secretary: Michel L. Lapidus Monod, University of Chicago. Announcement issue of Notices: February 2005 Extinction, Periodicity, and Chaos in Population and Epidemic Program first available on AMS website: March 3, 2005 Program issue of electronic Notices: April 2005 Models (Code: SS lOA), Linda J. S. Allen, Texas Tech Uni­ Issue of Abstracts: Volume 26, Issue 3 versity, Sophia Ruey-Jen Jang, University of Louisiana at Lafayette, and Lih-lng W. Roeger, Texas Tech University. Deadlines Future Directions in Mathematical Systems and Control For organizers: Expired Theory (Code: SS llA), David Gilliam and W. P. Dayawansa, For consideration of contributed papers in Special Sessions: Texas Tech University. Expired

MARCH 2005 NOTICES OF THE AMS 381 Meetings & Conferences

For abstracts: February 22, 2005 Representation Theory of Algebras (in Honor of Claus Michael Ringel) (Code: SS 7A), Alex Martsinkovsky, North­ The scientific information listed below may be dated. For eastern University, Dan Zacharia, Syracuse University, the latest information, see www. ams. org/amsmtgs/ Birge K. Huisgen-Zimmermann, University of California sectional.html. Santa Barbara, and Edward L. Green, Virginia Polytech Institute & State University. Invited Addresses Ricci Flow/Riemannian Geometry (Code: SS 9A), Guofang Mei-Chu Chang, University of California Riverside, Set Wei and Rugang Ye, University of California Santa Barbara. addition and set multiplication. Mischa Kapovich, University of California Davis, Title to be announced. Mainz, Germany Mihai Putinar, University of California at Santa Barbara, Positive polynomials, a Hilbertian perspective. June 16-19,2005 James Sethian, University of California Berkeley, Advances Thursday - Sunday in advancing interfaces: New techniques for propagating fronts in wave propagation and materials sciences. Meeting #1 008 ]oint International Meeting with the Deutsche Mathematiker­ Special Sessions Vereinigung (DMV) and the Oesterreichische Mathematis­ Algebraic Geometry and Combinatorics (Code: SS 14A), che Gesellschaft (OMG) Alexander Yong and Allen Knutson, University of Cali­ Associate secretary: Susan]. Friedlander fornia Berkeley. Announcement issue of Notices: February 2005 Arithmetic Geometry (Code: SS 13A), Adebisi Agboola, Program first available on AMS website: Not applicable University of California Santa Barbara, and Cristian Program issue of electronic Notices: Not applicable Dumitru Popescu, University of California San Diego. Issue of Abstracts: Not applicable Automorphisms of Surfaces (Code: SS 4A), Anthony Deadlines Weaver, Bronx Community College of the City University For organizers: Expired of New York, and Peter Turbek, Purdue University Calumet. For consideration of contributed papers in Special Sessions: Complexity of Computation and Algorithms (Code: SS lOA), March 15, 2005 Mark Burgin, University of California Los Angeles. For abstracts: March 31, 2005 Curvature in Group Theory and Combinatorics (Code: SS llA), Laura M. Anderson, State University of New York at The scientific information listed below may be dated. For Binghamton, Noel Patrick Brady, University of Oklahoma, the latest information, see www. ams. org/amsmtgs/ Robin Forman, Rice University, and Jonathan P. McCam­ i nternmtgs. html. mond, University of California Santa Barbara. Dynamical Systems in Neuroscience (Code: SS lA), Eugene M. Watch the website maintained by the local organizers Izhikevich, The Neurosciences Institute. athttp://www.mathematik.uni-mainz.de/mainz2005/ Function Theory (Code: SS 12A), Mihai Putinar and for additional program details and links to sites for Stephan R. Garcia, University of California Santa Barbara. hotels, tours, and other local information. Geometric Methods in Three Dimensions (Code: SS 6A), Daryl Cooper, David Darren Long, and Martin G. Scharle­ Invited Addresses mann, University of California Santa Barbara. Helene Esnault, University of Essen, Deligne's integrality Geometry and Physics (Code: SS 8A), Xianzhe Dai, University theorem in unequal characteristic and rational points over of California Santa Barbara, and Zhiqin Lu, University of finite fields. California Irvine. Richard Hamilton, Columbia University, The Ricci flow. History of Mathematics (Code: SS 2A), Shawnee L. Michael J. Hopkins, Massachusetts Institute of Technology, McMurran, California State University, San Bernardino, Title to be announced. and James J. Tattersall, Providence College. Christian Krattenthaler, University of Lyon-I, Exact and as­ Noncommutative Geometry and Algebra (Code: SS SA), ymptotic enumeration of vicious walkers with a wall inter­ Kenneth R. Goodearl, University of California Santa Bar­ bara, J. T. Stafford, University of Michigan, and J. J. Zhang, action. University of Washington. Frank Natterer, University of Muenster, Imaging and in­ Recent Advances in Combinatorial Number Theory(Code: verse problems for partial differential equations SS 3A), Mei-Chu Chang, University of California Riverside, Horng-Tzer Yau, New York University and Stanford Uni­ and Van Ha Vu, University of California San Diego. versity, Dynamics of Bose-Einstein condensate.

382 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences

Special Sessions Mathematics Education, Gunter Tomer, Universitat Duis­ burg-Essen, and Alan Schoenfeld, School of Education, Affine Algebraic Geometry, Shreeram Abhyankar, Pur­ Berkeley. due University, Hubert Flenner, Fakultat fiir Mathematik, and Makar Limanov, Wayne State University. Modules and Comodules, Sergio L6pez-Permouth, Ohio University, and Robert Wisbauer, University of Dusseldorf. Algebraic Combinatorics, Patricia Hersh, University of Michigan, Christian Krattenthaler, University ofLyon-I, and Multiplicative Arithmetic of Integral Domains and Monoids, Volkmar Welker, Philipps University Marburg. Scott Chapman, Trinity University, San Antonio, Franz Halter-Koch, University of Graz, and Ulrich Krause, Uni­ Algebraic Cryptography, Dorian Goldfeld, Columbia versitat Bremen. University, Martin Kreuzer and Gerhard Rosenberger, Universitat Dortmund, and Vladimir Shpilrain, City College Nonlinear Elliptic Boundary Value Problems, Thomas of New York. Bartsch, Universitat Giessen, and Zhi-Qiang Wang, Utah State University. Algebraic Cycles, Eric Friedlander and Marc Levine, North­ western University, and Fabien Morel, Universite Paris. Nonlinear Waves, Herbert Koch, University of Dortmund, and Daniel I. Tataru, University of California Berkeley. Algebraic Geometry, Yuri Tschinkel, Georg-August­ Ordinary Differential, Difference, and Dynamic Equations, UniversiUit Gottingen, and Brendan E. Hassett, Rice Werner Balser, Universitat Ulm, Martin Bohner, University University. of Missouri-Rolla, and Donald Lutz, San Diego State Dirac Operators, Clifford Analysis and Applications, Klaus University. Gtirlebeck, University of Weimar, Mircea Martin, Baker Uni­ Quantum Knot Invariants, Anna Beliakova, Universitat versity, John Ryan, University of Arkansas, and Michael Zurich, and Uwe Kaiser, Boise State University. Shapiro, IPN Mexico. Representations and Cohomology of Groups and Algebras, Discrete Geometry, jacob Eli Goodman, The City College Dave Benson, University of Georgia, and Henning Krause, of New York (CUNY), Emo Welzl, Eidgen Technische Universitat Paderborn. Hochschule, and Gunter M. Ziegler, Technical University of Berlin. Set Theory, joel Hamkins, City University of New York, Peter Koepke, Universitat Bonn, and Benedikt Lowe, Function Spaces and Their Operators, Ernst Albrecht, Uni­ Universiteit van Amsterdam. versitat des Saarlandes, Raymond Mortini, Universite de Metz, and William Ross, University of Richmond. Spectral Analysis of Differential and Difference Operato~s, Evgeni Korotyaev, Humboldt-University Berlin, Bons Functional Analytic and Complex Analytic Methods in Mityagin, The Ohio State University, and Gerald Teschl, Linear Partial Differential Equations, R. Meise, University University of Vienna. of Dusseldorf, B. A. Taylor, University of Michigan, and Dietmar Vogt, University of Wuppertal. Stochastic Analysis on Metric Spaces, Laurent Saloff-Coste, Cornell University, Karl-Theodor Sturm, University of Geometric Analysis, Victor Nistor, Pennsylvania State Bonn, and Wolfgang Woess, Graz Technical University. University, and Elarnr Schrohe, Universitat Hannover. Topics in Applied Mathematics and Mechanics: Mathemat­ Geometric Topology and Group Theory, Cameron MeA ical Control Theory and Numerical Methods, Peter Benner, Gordon, The University of Texas at Austin, Cynthia Fakultat fur Mathematik. Hog-Angeloni, Johann Wolfgang Goethe-Universitat, and Topics in Applied Mathematics and Mechanics: Mechanics, Wolfgang Metzler, University of Frankfurt. Friedrich Pfeiffer, Technical University of Munich. Group Theory, Luise-Charlotte Kappe, SUNY at Bingham­ Topics in Applied Mathematics and Mechanics: Multiscale ton, Robert Fitzgerald Morse, University of Evansville, Problems, Oscillations in PDEs, and Homogenization, and Gerhard Rosenberger, University of Dortmund. Alexander Mielke, University of Hannover. Hilbert Functions and Syzygies, Uwe Nagel, University of Topics in Applied Mathematics and Mechanics: Numerical Kentucky, Irena Peeva, Cornell University, and Tim Romer, PDEs, Equations with Inherent Conditions, Rolf Jeltsc~, Universitat Osnabruck. Eidgen Technische Hochschule, Maria Lukacova, Techni­ History ofMathematics: Mathematics and War, Thomas W. cal University of Brno, and Mac Hyman, Los Alamos Archibald, Acadia University, John H. McCleary, Vassar National Laboratory. College, Moritz Epple, University of Stuttgart, and Norbert Topics in Applied Mechanics: Algebraic Approaches to Schappacher, Technische Universitat Darmstadt. Preconditioning, Heike Fassbender, Technical University Homotopy Theory, PaUl G. Goerss, Northwestern University, of Braunschweig, and Andreas Frommer, University of Hans-Werner Henn, Institut de Recherche Mathematique Wuppertal. Avancee, Strasbourg, and Stefan Schwede, Universitat Topology of Manifolds, Matthias Kreck, University of Bonn. Heidelberg, and Andrew Ranicki, University of Edmburgh. Hop{ Algebras and Quantum Groups, Susan Montgomery, University of Southern California, and Hans-Jurgen Contributed Papers Schneider, University of Munich. There will be no sessions of contributed papers.

MARCH 2005 NOTICES OF THE AMS 383 Meetings & Conferences

Abstracts Hilton Mainz City: First-class hotel in the city center; Abstracts must be submitted electronically; mathematical breakfast is an additional €18 per person/day; €159/ displays can be accommodated in LaTeX only. See the con­ €179 or €179/€199; 3500 m. ference website and follow the links to the abstract sub­ Hotel Stiftswingert'""': Three-star hotel, not far from the mission web form at (http: I /atlas-conferences. old town; €74/€100.20; 3000 m. com/ cgi -bin/abstract/submit/ caoz-01). The deadline Ibis Hotel'"': Centrally located in the old town; €81/ for abstract submission is March 31, 2005. €90; 3500 m. Favorite ParkhoteF'"''': Four-star hotel, completely ren­ Accommodations ovated, situated in the public park of Mainz; €120/€152; Participants should make their own accommodation 5500 m. arrangements. To obtain special meeting rates, you must Giinnewig Bristol Hotel'"""': Four-star hotel, in the utilize the online reservation form at http: I /www. pleasant suburb of Mainz-Weisenau; €113/€130; 6000 m. info-mainz.de/tourist/ams/reservierungeng.htm. Hotel Kaiserstiibchen: Small personal hotel in the sub­ The deadline for reserving hotel rooms at special rates urb of Mainz-Finthen; €55/€75; 4800 m. through the conference website is May 4, 2005. After that, Atrium Hotel Mainz'"""': Four- star superior hotel, calm participants must make reservations on their own. The and nicely situated in Mainz-Finthen; €113/€118 or recommended reservation service after May 4 is at €143/€148; 4800 m. http: I jwww. rna i nz. de (for the English version, click on Hotel Am Lerchenberg: Hotel in a park-like setting in the small round button showing the pattern of the British the suburb of Lerchenberg; €73/€99; 6400 m. flag). The AMS is not responsible for rate changes or the Restaurants/Food Service quality of the accommodations. All rates are per room per Several restaurants and bistros are scattered across cam­ night and include private facilities, shower, VAT, and pus. For evening meals there are many restaurants and breakfast (except Hilton hotels). Rates are quoted in euros pubs in the downtown area around the main cathedral. for single/double occupancy; the distance to the univer­ There is a small cafeteria next to lecture hall RW-1 open sity is shown in meters (m). on Thursday and Friday, 8:00 a.m. to 7:00 p.m. The main Links to the homepages of many of these hotels and cafeteria on Staudinger Weg is close to the Institute of others are found at http://www.mainz.de/ Mathematics and is open on Thursday and Friday, 8:00 a.m. WGAPublisher/online/html/default/ to 6:00p.m. hthn-5v7kpk.en.O. Hotel Romerstein"'"': Comfortable hotel in the suburb Registration and Meeting Information Mainz-Bretzenheim in an idyllic and calm atmosphere; The meeting will take place at Johannes Gutenberg Uni­ €75/€120; 400 m. versity in Mainz (near Frankfurt). Plenary sessions will be Contel Mainz: Large hotel with various kinds of rooms held in the main lecture hall RW-1 (street address is Jakob­ and apartments; €101.50/€121.50; 1350 m. Welder-Weg 9). Special Sessions will take place at (or just Hotel Konigshof: Newly renovated middle-class hotel nearby) the Institute of Mathematics (street address is at the station; €57 /€82; 1800 m. Staudinger Weg 9). On Friday, June 17, there will be an Hotel Schottenhof: Standard hotel centrally located Opening Ceremony at 8:30a.m. in the plenary lecture hall. next to the station; €73/€98; 1800 m. There will be signs on campus to all sessions. The con­ Central Hotel Eden'""'': Traditional middle-class hotel ference begins on Thursday, June 16, at 1:00 p.m. and ends with the charm of former times, at the station; €80/€95; on Sunday, June 19, at 3:00p.m. 1800 m. There are two conference offices where participants Hotel Hammer'""'•: Modern, newly renovated hotel with may pick up conference materials, the booklet of abstracts, upper class comfort, at the station; €84.50/€103; 1800 m. and city maps: in front of lecture hall RW-1 on Wednes­ Hotel Stadt Mainz'""': Newly renovated hotel on a calm day, 4:00 p.m. to 8:00 p.m., on Friday, 9:30 a.m. to 1:00 street not far from the station; €85/€100; 2000 m. p.m., on Saturday, 8:30a.m. to 1:00 p.m., and on Sunday, Hotel Austria: Nice family hotel built in 1997, next to noon to 3:00p.m.; at the Institute of Mathematics in Room the station; €80/€100; 2000 m. 05-432 (Hilbertraum) on the fifth floor on Thursday, 1:00 Advena Europa Hotel: Newly renovated hotel, not far p.m. to 8:00p.m., on Friday, 1:00 p.m. to 8:00p.m., on Sat­ from the station; €88/€98; 2000 m. urday, 1:00 p.m. to 8:00p.m., and on Sunday, 8:30a.m. to City Hotel Neubrunnenhof: Hotel in the city center noon. next to the pedestrian area; €72/€98; 2500 m. Registration fees paid before April1 are €50 for mem­ Dormotel Mainz: Comfortable and quiet accommoda­ bers of any of the five participating societies (AMS, DMV, tions in Mainz-Bretzenheim; €66/€78; 3000 m. OeMG, GAMM, SIAM), €15 for students, and €80 for all oth­ Hotel Mainzer Hoi"''''': Private hotel near the river; ers. Registration fees after April1 are €80, €15, and €120, €85/€115; 2800 m. respectively. As of January 12, 2005, US$1=€.76272. Hilton Mainz: Comfortable first -class hotel on the river­ side; breakfast is an additional €18 per person/day; Social Events €159/€179/€194 or €179/€199/€214; 3800 m. Space is limited for some of these events, so reserve early.

384 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences

Evening reception in the castle 'Kurfurstliches Schloss' By car: Take highway A60 toward Mainz and exit at in Mainz: Friday, June 17, at 8:00p.m.; cost is €10/person. Saarstasse toward Innenstadt. Follow the signs for Uni­ Take city buses from Mainz-Hauptbahnhof #9 toward versitat. The visitors' parking lot is at the main university Schierstein or #6 toward Wiesbaden; leave at bus stop entrance at Ackermannweg (street address). Other park­ Land tag. ing is available south of the campus on Albert -Schweitzer­ Mathematics Museum in Giessen: Friday, June 17, 1:00 Strasse and at Dahlheimer Weg. Both of these lots are p.m.-7:00p.m.; cost is €20/person for museum entry only within walking distance of the lecture halls (parking on cam­ (food and beverages available at extra cost). Departure pus is restricted to employees only by permit). and return is at the Institute of Mathematics, Staudinger Weg 9, parking lot. Boat trip on the Rhine River: Saturday, June 18, 6:30 p.m.-11:30 p.m.; cost is €30/person, food and bev­ Annandale-on­ erages are available on board at extra cost. Departure and return is at landing stage 'Am Fischtorplatz'. Take city buses Hudson, New York from Mainz-Hauptbahnhof #60 toward Ginsheim, #61 to­ Bard College ward Laubenheim; leave at bus stop Fischtor. October 8-9,2005 Travel and Maps Saturday - Sunday Maps of the area and city may be found on the conference website cited above. Also, city maps will be distributed with Meeting #1 009 your conference registration materials. Your air travel des­ Eastern Section tination should be the Frankfurt/ Main Airport or Frank­ Associate secretary: Lesley M. Sibner furt/Hahn Airport. Announcement issue of Notices: August 2005 From Frankfurt/Main Airport to Mainz main train sta­ tion (Hauptbahnhof): Take the train from platform Program first available on AMS website: August 25, 2005 Regionalbahnhof by S-Bahn S8, Regonalbahn (RB), Program issue of electronic Notices: October 2005 Regionalexpress (RE); travel time is about 30 minutes. Get Issue of Abstracts: Volume 26, Issue 4 off at Mainz Hauptbahnhof (not Mainz Sud or Mainz­ Deadlines Kastel). Or, take the train from platform Fernbahnhof by Eurocity (EC), Intercity (IC), Intercity Express (ICE); travel For organizers: March 8, 2005 time is about 20 minutes. For consideration of contributed papers in Special Sessions: From Frankfurt/Hahn Airport to Mainz main train June 21, 2005 station (Hauptbahnhof): Take the ORN-Bus which runs For abstracts: August 16, 2005 every three hours, 24 hours a day. By car, take state route B50 to highway A61 at the entrance Rheinbollen toward The scientific information listed below may be dated. For Ludwigshafen, change at Dreieck Nahetal to highway A60 the latest information, see www.ams.org/amsmtgs/ toward Mainz. Leave highway A60 at the Saastrasse exit sectional . html. toward Innenstadt. Follow the signs for Hauptbahnhof or Universitat; travel time is about 90 minutes. Invited Addresses From the Mainz train station (Hauptbahnhof) to the Persi Diaconis, Stanford University, Title to be announced university: There are frequent buses and trams around the (Erdos Memorial Lecture). city and surrounding area. You should consider buying a Harold Rosenberg, University of Paris VII, Title to be ticket for €17.80 entitling you to travel on the public trans­ announced. portation network 24 hours/ day for a week, at the infor­ mation center of MVG-Mainzer Verkehrsgesellschaft, Boni­ Alice Silverberg, University of California Irvine, Title to be fazius Strasse 2, in front of the Mainz train station. announced. To get to campus by bus take the #54 toward Lerchen­ Christopher Sogge, Johns Hopkins University, Title to be berg or #55 toward Finthen or # 58 toward Wackernheim announced. or #68 toward Klein-Winterheim/Koblenzer Strasse. Get off Benny Sudakov, Princeton University, Title to be at Freidrich-von-Pfeiffer-Weg. The Institute of Mathemat­ announced. ics and the main lecture hall RW-1 are about a 10-minute walk. Special Sessions To get closer to the main lecture hall RW-1 (but about a 20-minute walk from the Institute of Mathematics) by bus, Geometric Group Theory (Code: SS 1A), Sean Cleary, The take #6 toward Marienborn, or #6A toward Bretzenheim City College of New York, and Melanie I. Stein, Trinity or #56, #57, #65 toward Hartenberg or #64 toward Bu­ College. denheim or # 54 toward Lerchenberg or #55 toward Fin then Geometric Transversal Theory (Code: SS 3A), Richard or #58 toward Wackernheim or #68 toward Kelin-Win­ Pollack, Courant Institute, New York University, and Jacob ternheim/Koblenzer Strasse and get off at Universitat. Eli Goodman, The City College of New York.

MARcH 2005 NOTICES OF THE AMS 385 Meetings & Conferences

Global Theory of Minimal Surfaces (Code: SS 6A), David A. Hoffman, Mathematical Sciences Research Institute, and Lincoln, Nebraska Harold Rosenberg, University of Paris VII. University of Nebraska in Lincoln The History ofMathematics (Code: SS 2A), Patricia R. Allaire, Queensborough Community College, CUNY, Robert E. October 21-23,2005 Bradley, Adelphi University, and Jeff Suzuki, Bard College. Friday - Sunday Homological Aspects of Commutative Algebra (Code: SS 4A), Alexandre Tchernev, University of Albany, SUNY, and Meeting #1 011 Janet Vassilev, University of Arkansas. Central Section Associate secretary: Susan]. Friedlander Special Functions and Orthogonal Polynomials: Theory and Announcement issue of Notices: August 2005 Applications (Code: SS 7 A), Diego Dominici, State Univer­ Program first available on AMS website: September 8, 2005 sity of New York at New Paltz. Program issue of electronic Notices: October 2005 Theory of Infinite-Dimensional Lie Algebras, Vertex Oper­ Issue of Abstracts: Volume 26, Issue 4 ator Algebras, and Related Topics (Code: SS SA), Antun Deadlines Milas, SUNY at Albany, Alex J. Feingold, Binghamton University, and Yi-Zhi Huang, Rutgers University. For organizers: March 22, 2005 For consideration of contributed papers in Special Sessions: July 5, 2005 Johnson City, For abstracts: August 30, 2005 The scientific information listed below may be dated. For Tennessee the latest information, see www. ams. o rg/amsmtgs/ sectional . html. East Tennessee State University Invited Addresses October 15-16,2005 Howard Masur, University of Illinois at Chicago, Title to Saturday - Sunday be announced. Meeting #1 010 Alejandro Uribe, University of Michigan, Title to be Southeastern Section announced. Associate secretary: Matthew Miller Judy Walker, University of Nebraska, Title to be announced. Announcement issue of Notices: August 2005 Jack Xin, University of Texas, Title to be announced. Program first available on AMS website: September 1, 2005 Program issue of electronic Notices: October 2005 Special Sessions Issue of Abstracts: Volume 26, Issue 4 Algebraic Geometry (Code: SS 1A), Brian Harbourne, Uni­ versity of Nebraska-Lincoln, and Bangere P. Purnaprajna, Deadlines University of Kansas. For organizers: March 15, 2005 Recent Progress in Operator Algebras (Code: SS 2A), Allan P. For consideration of contributed papers in Special Sessions: Donsig and David R. Pitts, University of Nebraska. June 28, 2005 For abstracts: August 23, 2005 Eugene, Oregon The scientific information listed below may be dated. For the latest information, see www. ams. o rg/amsmtgs/ University of Oregon sectional.html. November 12-13, 2005 Invited Addresses Saturday - Sunday Alberto Bressan, Pennsylvania State University, Title to be Meeting #1 012 announced. Western Section Assaf Naor, Microsoft Research, Title to be announced. Associate secretary: Michel L. Lapidus Announcement issue of Notices: September 2005 Prasad V. Tetali, Georgia Institute of Technology, Title to Program first available on AMS website: September 29, be announced. 2005 Rekha R. Thomas, University of Washington, Title to be Program issue of electronic Notices: November 2005 announced. Issue of Abstracts: Volume 26, Issue 4

386 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences

Deadlines For organizers: April12, 2005 San Antonio, Texas For consideration of contributed papers in Special Sessions: Henry B. Gonzalez Convention Center July 26, 2005 For abstracts: September 20, 2005 January 12-1 5, 2006 Thursday - Sunday The scientific information listed below may be dated. For the latest information, see www. ams. org/amsmtgs/ Meeting #1 014 sectional . htm l. ]oint Mathematics Meetings, including the 112th Annual Invited Addresses Meeting of the AMS, 89th Annual Meeting of the Mathe­ matical Association ofAmerica, annual meetings of the As­ Matthew Foreman, University of California Irvine, Title to sociation for Women in Mathematics (A VVM) and the National be announced. Association of Mathematicians (NAM), the winter meeting Mark Haiman, University of California Berkeley, Title to be of the Association for Symbolic Logic (ASL), with sessions announced. contributed by the Society for Industrial and Applied Math­ ematics (SIAM). Wilhelm Schlag, California Institute of Technology, Title Associate secretary: John L. Bryant to be announced. Announcement issue of Notices: October 2005 Hart H. Smith, University of Washington, Title to be Program first available on AMS website: November 1, 2005 announced. Program issue of electronic Notices: January 2006 Issue of Abstracts: Volume 27, Issue 1 Special Sessions Noncommutative Algebra and Noncommutative Birational Deadlines Geometry(Code: SS 3A), ArkadyDmitrievichBerenstein, Uni­ For organizers: April12, 2005 versity of Oregon, and Vladimir Retakh, Rutgers University. For consideration of contributed papers in Special Sessions: Partial Differential Equations with Applications (Code: SS To be announced 4A), Alexander Panchenko, Washington State University, For abstracts: To be announced R. E. Showalter, Oregon State University, and Hong-Ming For summaries of papers to MAA organizers: To be Yin, Washington State University. announced Regular Algebras and Noncommutative Projective Geometry (Code: SS 2A), Brad Shelton, University of Oregon, Michaela Vancliff, University of Texas at Arlington, and James J. Miami, Florida Zhang, University of Washington. Florida International University Resolutions (Code: SS 1A), Christopher Alan Francisco, Uni­ versity of Missouri, and Irena Peeva, Cornell University. April1-2, 2006 Saturday - Sunday Southeastern Section Associate secretary: Matthew Miller Taiwan Announcement issue of Notices: To be announced December 14-1 8, 2005 Program first available on AMS website: To be announced Wednesday- Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1 013 Deadlines First ]oint International Meeting between the AMS and the Taiwanese Mathematical Society. For organizers: September 1, 2005 Associate secretary: John L. Bryant For consideration of contributed papers in Special Sessions: Announcement issue of Notices: May 2005 To be announced Program first available on AMS website: To be announced For abstracts: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced For abstracts: To be announced

MARCH 2005 NOTICES OF THE AMS 387 Meetings & Conferences

Deadlines Notre Dame, Indiana For organizers: To be announced For consideration of contributed papers in Special Sessions: University of Notre Dame To be announced For abstracts: To be announced April8-9, 2006 Saturday - Sunday The scientific information listed below may be dated. For Central Section the latest information, see www. ams. org/amsmtgs/ Associate secretary: Susan]. Friedlander sectional . html. Announcement issue of Notices: February 2006 Program first available on AMS website: To be announced Special Sessions Program issue of electronic Notices: To be announced History ofMathematics (Code: SS lA), Shawnee L. McMurran, Issue of Abstracts: To be announced California State University, San Bernardino, and James J. Tattersall, Providence College. Deadlines For organizers: September 9, 2005 For consideration of contributed papers in Special Sessions: Fayetteville, To be announced For abstracts: To be announced Arkansas Durham, University of Arkansas November 3-4, 2006 Friday - Saturday New Hampshire Southeastern Section University of New Hampshire Associate secretary: Matthew Miller Announcement issue of Notices: To be announced April22-23, 2006 Program first available on AMS website: To be announced Saturday - Sunday Program issue of electronic Notices: To be announced Eastern Section Issue of Abstracts: To be announced Associate secretary: Lesley M. Sibner Deadlines Announcement issue of Notices: To be announced Program first available on AMS website: To be announced For organizers: April 3, 2006 Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Sessions: To be announced Issue of Abstracts: To be announced For abstracts: To be announced Deadlines For organizers: September 22, 2005 For consideration of contributed papers in Special Sessions: New Orleans, To be announced For abstracts: To be announced Louisiana New Orleans Marriott Hotel and Sheraton San Francisco, New Orleans Hotel January 4-7, 2007 California Thursday - Sunday ]oint Mathematics Meetings, including the 113th Annual San Francisco State University Meeting of the AMS, 90th Annual Meeting of the Mathe­ April29-30, 2006 matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A HIM) and the Saturday - Sunday National Association of Mathematicians (NAM), and the Western Section winter meeting of the Association for Symbolic Logic (ASL), Associate secretary: Michel L. Lapidus with sessions contributed by the Society for Industrial and Announcement issue of Notices: To be announced Applied Mathematics (SIAM). Program first available on AMS website: To be announced Associate secretary: Susan]. Friedlander Program issue of electronic Notices: To be announced Announcement issue of Notices: October 2006 Issue of Abstracts: To be announced Program first available on AMS website: To be announced

388 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Meetings & Conferences

Program issue of electronic Notices: January 2007 Issue of Abstracts: To be announced Washington, District Deadlines of Columbia For organizers: April 4, 2006 For consideration of contributed papers in Special Sessions: Marriott Wardman Park Hotel and Omni To be announced Shoreham Hotel For abstracts: To be announced For summaries of papers to MAA organizers: To be january 7-10,2009 announced Wednesday - Saturday joint Mathematics Meetings, including the 115th Annual Meeting of the AMS, 92nd Annual Meeting of the Mathe­ Oxford, Ohio matical Association of America (MAA), annual meetings of the Association for Women in Mathematics (A VVM) and the Miami University National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL). March 16-1 7, 2007 Associate secretary: Lesley M. Sibner Friday - Saturday Announcement issue of Notices: October 2008 Central Section Program first available on AMS website: November 1, 2008 Associate secretary: Susan]. Friedlander Program issue of electronic Notices: January 2009 Announcement issue of Notices: To be announced Issue of Abstracts: Volume 30, Issue 1 Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: To be announced For organizers: April 7, 2008 For consideration of contributed papers in Special Sessions: Deadlines To be announced For organizers: To be announced For abstracts: To be announced For consideration of contributed papers in Special Sessions: For summaries of papers to MAA organizers: To be To be announced announced For abstracts: To be announced San Francisco, San Diego, California California San Diego Convention Center Moscone West Convention Center january 6-9, 2008 january 6-9, 201 0 Sunday - Wednesday joint Mathematics Meetings, including the 114th Annual Wednesday - Saturday Meeting of the AMS, 91st Annual Meeting of the Mathe­ joint Mathematics Meetings, including the 116th Annual Meeting of the AMS, 93rd Annual Meeting of the Mathe­ matical Association ofAmerica (MAA), annual meetings of matical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AVVM) and the the Association for Women in Mathematics (A VVM) and the National Association of Mathematicians (NAM), and the National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL). winter meeting of the Association for Symbolic Logic (ASL). L. Associate secretary: Michel Lapidus Associate secretary: John L. Bryant Announcement issue of Notices: October 2007 Announcement issue of Notices: October 2009 Program first available onAMS website: November 1, 2007 Program first available on AMS website: November 1, 2009 Program issue of electronic Notices: January 2008 Program issue of electronic Notices: January 2010 Issue of Abstracts: Volume 29, Issue 1 Issue of Abstracts: Volume 31, Issue 1

Deadlines Deadlines For organizers: April 6, 2007 For organizers: April 5, 2009 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced For summaries of papers to MAA organizers: To be For summaries of papers to MAA organizers: To be announced announced

MARCH 2005 NOTICES OF THE AMS 389 Meetings & Conferences New Orleans, Louisiana New Orleans Marriott Hotel and Sheraton New Orleans Hotel January 5-8,2011 Wednesday - Saturday ]oint Mathematics Meetings, including the I 17th Annual Meeting of the AMS, 94th Annual Meeting of the Mathe­ matical Association ofAmerica, annual meetings of the As­ sociation for Women in Mathematics (A HIM) and the National Association ofMathematicians (NAM), and the winter meet­ ing of the Association for Symbolic Logic (ASL). Associate secretary: Susan]. Friedlander 1.. ,.. Announcement issue of Notices: October 2010 Program first available on AMS website: November 1, 2 010 Program issue of electronic Notices: January 2 011 ~· Issue of Abstracts: Volume 32, Issue 1 Deadlines Q:t"'** a 4444 {!9iif$flW . '· For organizers: April 2, 2010 ~ A k ...d For consideration of contributed papers in Special Sessions: To be announced 1.. For abstracts: To be announced "' For summaries of papers to MAA organizers: To be announced

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390 NOTICES OF THE AMS VOLUME 52, NUMBER 3 ~AY RESEARCH FE AMS members may Ma~;,~~~ Venkartesh of aryam Mirzakhan; of H PPOinted four Research F subscribe to receive Fellows• resear~~~ Matwssachusetts Institute~~';!d University, and Ande llowvs: Ciprian or o years. echnology. The lnsr ras . asy and twice-a-month email MAI-':!.f.MATICA ltute Will fully support The Nationa~Cl.fiDISTil..m.,f.C.~ notifications of news, associates in . emy of Engmeerin (N ..IQNAE about professor ~f cludmg these individual~ . AE) has elected 76 new announcements ~~~i~;o::n~~:~~~~n~~~~~~~~~~!:e~i~~~~:~~hfn~~:~~= l ~fc~e~~~:~~e~~ rt;~~~~~~:n programs, publications, ngmeenng and computatio:;;ng programs, and profes~9j' · lmda R Petzold, SCience, University of C 1.:· mechanical and and events, as well as a IIOrnla, Santa alerts about deadlines for fellowship and grant applications, calls for proposals, and meeting registrations. AMS members can sign up for the service at ~~~~1~~~Z~~:~~~~~i~~C~~~~h~~ext rouenrd2~6~h~:t~r~~~~~:;~~sff~ ~~~v~l - ~o ;dank: e 5 d mic year or summ · h f m Ch1na an www.ams.org/enews. ~=~:~~:;Y t:~P~~~~cf~n~~av~~ t~~;~e;~ ~us~~u~i~~si~~oat~~~~~;t~~~~~:br a :~~~~~i~u . s . math departments .'n Ina m. f d ed by a gift from Ky and Yu-Fen a . mathematician. ThiS progra~ '\ u~ __ due MARCH 15, 2004. includes details and the app !Ca 10 THE YEAR PROGRAM . 1 ENTRIES FOR U.S. PROFESSORS OF - of Education (CASE) and The Carnegie The Council for Advancement and S~~~~~the U.S. Professors of the Year P rog r~~i a for the Advancement of Te~: nrr:~ed certificate of recognition, and a ~e3 h ~na~~e professors winners recelldve a~;~~~~g~on DC. State winners are also nam~~~m2~nity College District) luncheon he. 1n N D 1 th) and Judy Kasab1an (EI Cam1no · Joseph Galhan (UM u u

The deadlines for consideration of contributed papers in Special Sessions at the first AMS fall Section Meetings are as follows: Southeastern Section meeting at Vanderbilt University, Nashville, TN, October 16-17: JUNE 29,2004. See http:// www.ams.orglamsmtgs/210 8_ deadlines.html. Western Section meeting at the University of New Mexico, Albuquerque, October 16- 17: JUNE 29,2004. See http://www.ams.orglamsmtgs/211 2_deadlines.html.

Headlines & Deadlines is a twice-a-month email news service available to AMS members. www.am s.org To unsubscribe from this mailing list visit http://www.ams.orglenews. Meetings and Conferences of the AMS

Associate Secretaries of the AMS Western Section: MichelL. Lapidus, Department of Math­ Eastern Section: Lesley M. Sibner, Department of Mathe­ ematics, University of California, Sproul Hall, Riverside, CA matics, Polytechnic University, Brooklyn, NY 11201-2990; 92521-0135; e-mail: l a pi dus@math. uc r. edu; telephone: 951- e-mail: lsi bne r@duke. poly. edu; telephone: 718-260-3 505. 827-5910. Southeastern Section: Matthew Miller, Department of Math­ Central Section: Susan J. Friedlander, Department of Math­ ematics, University of South Carolina, Columbia, SC 29208- ematics, University of Illinois at Chicago, 851 S. Morgan (M/C 0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690. 249), Chicago, IL 60607-7045; e-mail: susan@math. nwu. edu; tele­ phone: 312-996-3041.

The Meetings and Conferences section of the Notices March 16-17 Oxford, Ohio p. 389 gives information on all AMS meetings and conferences 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.389 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. 389 for electronic abstract submission. For some meetings the list Annual Meeting may be incomplete. Information in this issue may be dated. 2010 Up-to-date meeting and conference information can be January 6-9 San Franciso, California p. 389 found at www. ams. orglmeeti ngsl. Annual Meeting 2011 Meetings: January 5-8 New Orleans, Louisiana p. 390 Annual Meeting 2005 March 18-19 Bowling Green, Kentucky p. 379 Important Information regarding AMS Meetings April 2-3 Newark, Delaware p. 380 Potential organizers, speakers, and hosts should refer to page 100 in the January 2005 issue of the Notices for general April8-10 Lubbock, Texas p. 381 information regarding participation in AMS meetings and April16-17 Santa Barbara, California p. 381 conferences. June 16-19 Mainz, Germany p. 382 October 8-9 Annandale-on-Hudson, Abstracts New York p. 385 Speakers should submit abstracts on the easy-to-use interactive October 15-16 Johnson City, Tennessee p. 386 Web form. No knowledge of mpc is necessary to submit an October 21-23 Lincoln, Nebraska p. 386 electronic form, although those who use .0Tpc may submit November 12-13 Eugene, Oregon p. 386 abstracts with such coding, and all math displays and simi­ December 14-18 Taiwan p.387 larily coded material (such as accent marks in text) must be typeset in LAfpc. Visit http:llwww.ams.orglcgi-binl abstractslabstracts.pl. 2006 Questions about abstracts and requests for paper forms January 12-15 San Antonio, Texas p.387 may be sent to abs-i nfo@ams. or g. Annual Meeting Paper abstract forms must be sent to Meetings & Confer­ April1-2 Miami, Florida p.387 ences Department, AMS, P.O. Box 6887, Providence, RI 02940. April8-9 Notre Dame, Indiana p. 388 There is a $20 processing fee for each paper abstract. There April 22-23 Durham, New Hampshire p.388 is no charge for electronic abstracts. Note that all abstract dead­ April29-30 San Francisco, California p. 388 lines are strictly enforced. November 3-4 Fayetteville, Arkansas p. 388 Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated. 2007 January 4-7 New Orleans, Louisiana p. 388 Annual Meeting Conferences: (see http: I lwww. ams. o rglmeeti ngsl for the most up-to-date information on these conferences.) June 5-July 21, 2005: Joint Summer Research Conferences in the Mathematical Sciences, Snowbird, Utah (see November 2004 Notices, page 1294). July 25-August 12, 2005: Summer Research Institute on Algebraic Geometry, Seattle, Washington (see November 2004 Notices, page 1293). Co-sponsored conference: June 2006: Fifth Conference on Poisson Geometry, Tokyo, Japan (watch http: 1ltmugs. math. metro-u. ac. j pi general. html for future information).

392 NOTICES OF THE AMS VOLUME 52, NUMBER 3 ADVERTISERS' FORUM AMERICAN MATHEMATICAL SOCIETY

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