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of the American Mathematical Society AugustAugust20042004 Volume 51, Number 7

The Status of the Classification of the Finite Simple Groups pagepag~ 736 Intimations of Infinity page 741 Interview with Joseph Keller pagpagee 775151 Nashville Meeting~lceting page 849 Albuquerque Meeting page 851 Evanston Meeting papagege 8538 53

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KNAPP, State University ofNew York at Stony PROGRESS IN MATHEMATICS, VOL. 231 Variational Methods Brook, NY in Shape Optimization This work systematically develops the concepts and Geometric Mechanics on Problems tools of real analysis, whether pure, applied, aspiring, or established. It presents a comprehensive, global Riemannian Manifolds DORIN BUCUR, Universite de Metz, Metz, France; and treatment of the subject, emphasizing connections Applications to Partial Differential Equations GIUS~PPE BUTTAZZO, Universita di Pisa, Pisa, Italy between analysis and other branches of mathematics. OVIDIU CALIN, Eastern Michigan University, East Lansing, MI,' The study of shape optimization problems involves a Rich in examples as well as problems with hints and and DER-CHEN CHANG, Georgetown University, Washington, DC wide area of academic research and applications to the solutions, the book is ideal as a course text or refer­ Differential geometry techniques have very useful and real world. 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Feature Articles 736 The Status of the Classification of the Finite Simple Groups ~ichaelAschbacher The classification of the finite simple groups is one of. the great theorems of recent mathematics. One of its principal participants reviews the result and current progress on understanding it.

741 Intimations of Infinity Kirk Weller, Anne Brown, Ed Dubinsky, ~ichael ~cDonald, and Cynthia Stenger The authors report on their research into student understanding of the concept of infinity. 751 Interview with Joseph Keller The distinguished applied mathematician talks to the Notices about his work and career.

Communication5 Commentary

762 WHAT IS...anExpander?' 733 Opinion PeterSarnak 734 Letters to the Editor 771 Happy 100th, Henri Cartan! 764 CountDown-ABook Review 772 Doctorate Degrees inMathematics Reviewed byDaniel Ullman EarnedbyBlacks, Hispanics/ 768 Gamma-ABook Review Latinos, and Native Americans: Reviewed byDan Segal A Look at the Numbers HerbertA. Medina 776 Has the Women-in-Mathematics ProblemBeen Solved? AllynJackson 784 WomeninAcademia: Are We Asking the Right Questions? Carolyn Gordon andBarbara Lee Keyfitz 787 Gromov Receives Nemm.ers Prize 789 2003 Annual Survey ofthe Mathematical Sciences (SecondReport) Ellen E. Kirkman, James HZ Maxwell, and ColleenA. Rose Notices Departments oftheAmericanMathematical ~ociety Mathematics People 802 Joyce Wins Adams Prize, PECASE Awards Announced, Ferran EDITOR: Andy Magid Sunyer i Balaguer Prize Awarded, Mathe Receives 2004 Prize ASSOCIATE EDITORS: for Achievement in Information-Based Complexity, Allgower Susanne C. 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with much enthusiasm within the scientific community, Investing in the Future since NSF supports science research across all disciplines (e.g. over 65 percent of all mathematical research carried Each February the U.S. government budget season begins out in academic institutions is supported through the when the Administration presents its budget request for NSF). the fiscal year beginning October 1. This presentation ini­ So far PL107-338 has had little effect, as the FY 2004 tiates activity in the thirteen corresponding House and NSF budget is $5.58 billion, while the authorized amount Senate appropriations subcommittees. Several of these is $6.39 billion, and the FY 2005 budget request sets it at subcommittees oversee the budgets of agencies that sup­ $5.75 million, much less than the authorized amount of port science research and education. $7.38 billion. It is unlikely that the NSF budget will reach Over the course of the summer these subcommittees try $9.84 billion in FY 2007. to come up with agreements on the budgets of all programs Of course, the NSF budget should grow to $9.84 million and agencies falling under the discretionary part of the U.S. sooner rather than later. But what happens after the goal federal budget. Once the Congress finishes its work on is reached? What's the plan for future funding? Nothing the bills that contain these program and agency budgets, in the law indicates how funding levels are established or the bills are sent to the president for his signature. Once how they should be maintained over time other than this signed, these bills become law and these budgets are five-year span. As we see with the NIH after "the dou­ operational. Rarely are these budgets ready by October 1. bling", Congress, the Administration, and the biomedical Observing this budget process year after year, I have community are haggling over how to proceed with future come to the conclusion that the U. S. lacks a consistent, funding-never mind all the young scientists entering the stable, transparent, year-to-year funding mechanism for biomedical pipeline who will need to gain research sup­ supporting basic research across all disciplines of science port. and engineering. Not having such a mechanism inhibits A consistent method of funding basic research across scientific progress, quashes the morale of scientists, and all fields of science on a year-to-year basis is needed. Dou­ deters young people from becoming scientists. bling one agency at a time is not such a plan. Establishing For example, basic research is increased by only 0.6% a stable growth model that will enable all fields of science over fiscal year (FY) 2004 in the Administration's recent to prosper is critical. Such a model will support the needed budget request. The year-to-year rate ofincrease of the total scientific infrastructure that facilitates advances in many federal basic research budget has been decreasing since fields. Furthermore, this infrastructure will contribute to 2001, going up by 11.7% fromFY 2001 to FY 2002, by 6.3% our national security. from FY 2002 to FY 2003, by 5.5% from FY 2003 to 2004, The federal government needs to take note here. In­ and now by 0.6%. vesting in basic research is much like individuals putting Looking more closely at the Administration's FY 2005 money into their retirement accounts. Even though we federal basic research budget is eye-opening. Basic re­ may have debts or other pressures on our incomes, pru­ search funded by agencies other than the Department of dent individuals continue to invest, knowing in time their Health and Human Services (including the National Insti­ foresight will payoff. Society will also benefit from our fore­ tutes of Health (NIH)) decreases by 2.46% over FY 2004. If sight ifwe make steady, systematic, adequate investments the Department of Homeland Security funds are also sub­ now and over time. tracted, basic research drops by 3.36%. History has shown that basic research is the basis of The country's most recent model for funding science is technological invention and economic growth as well as the doubling model-more precisely, doubling infive years. being critical to security. Congress and the Administration This model was used successfully to double the budget of need to address the issue of science funding with the idea NIH. More recently this model was put forth in the guise of of developing a model that works fiscally as well as mak­ the National Science Foundation (NSF) AuthorizationAct of ing sure that our basic research enterprise runs robustly. 2002, now Public Law 107-338. This established a schedule The scientific community should advocate for such a for doubling the NSF budget over the next five fiscal years. process and help to develop a feasible method for taking Beginning with the FY 2003 budget, the NSF budget was to it forward. increase by 15% a year over the preceding year, until it dou­ bled the FY 2002 NSF level of approximately $4.8 billion to -Samuel M. Rankin III $9.84 billion in FY 2007. Passage of PL107-338 was greeted AMS Associate Executive Director, Director, AMS Washington Office This is a modified version ofan editorial that first appeared in the [email protected] April 2004 issue (volume 2, number 3) ofFrontiers in Ecology and the Environment. Reprinted with permission of the Ecological Society ofAmerica.

AUGUST 2004 NOTICES OF THE AMS 733 Letters to the Editor blindly in this intriguing unknown obtained. And mastery of a standard world of numbers. algorithm is portable when a child Paper-and-Pencil Math Recently a student ofmine came to moves from one school district to an­ While ostensibly a critique of mathe­ my office for help in baby calculus. other. Each of these is an important maticians who express interest in One of the problems had the expres­ consideration. mathematics education (particularly sion 3(2/3). Naturally, I cancelled the .I am surprised to learn that Tony those disagreeing with his views), two factors of 3 and said that the an­ made no reference to one of his ear­ Tony Ralston's article in the April No­ swer was 2. The student did not see lier and more provocative papers tices has a second but hardly subor­ ["The really new college mathe­ dinate theme: an updating of his ideas matics and its impact on the high expressed in other writings on school school curriculum," The Sec- mathematics. I think Tony is ondary School Mathematics wrong in many of his pro­ Curriculum (C. Hirsch and nouncements on school mathe­ M. Zweng, eds.), Reston matics, starting with arithmetic. VA: National Council of On page 407 of his Notices ar­ Teachers of Mathemat- ticle, Ralston tells us: "It may be ics (1985), pp. 200-210], that the teaching of pencil-and­ where he writes: "No paper arithmetic, which has sound argument can be been the gateway to the study adduced to support a of school mathematics for thesis that claims that more than a century, is as im­ high school students portant as it has ever been." must be very skillful at This cautionis not observed in polynomial algebra, his paper "Let's abolish pencil­ trigonometric identities, and-paper arithmetic" Uournal the solution of linear of Computers in Mathematics and quadratic equa­ and Science Teaching, volume tions or systems of 18, number 2 (1999),173-194]. equations, or any of Although there we learn that the myriad manipu­ Ralston wants youngsters to lative tasks that are have some knowledge of mental part of the current arithmetic, when the going gets high school cur­ tough-when, for example, stu­ riculum." Where dents might need a technique to are the statistical add two 4-digit numbers or three studies support- 2-digit numbers that they cannot ing Ralston's more do mentally-then Tony would why. So I wrote the equivalent im­ radical conclusions? demur. Calculators to the rescue! At proper fraction 6/3, and then it was Do most mathematicians really the very moment when addition is clear to the student that the answer share this view? I believe that they do about to blossom into an algorithm­ was indeed 2. This otherwise intelli­ not. Rather, I believe that the prepon­ and perhaps the first algorithm that gent student had used calculators ex­ derance of mathematicians want stu­ a child will see-Ralston declares it ed­ tensively since fifth grade at a very dents to internalize the procedures ucationally unnecessary, writing in good school. This is an example of and processes of the traditional basics this 1999 article that "children should what I mean by "calculator-assisted of algebra, geometry, trigonometry, not be expected to learn these algo­ mathematical incompetence". and, yes, arithmetic before coming to rithms." However unhappy Ralston is I support the intelligent use of com­ college. with the Klein-Milgram paper on the puter or calculator technology. Yet, Although Tony Ralston will and long-division algorithm, his real op­ while not every use of a calculator should be castigated by some readers position appears to be mastery by constitutes an abuse, students need of the Notices, at least he has taken the children of any pencil-and-paper al­ to be held accountable for mastering time to express his views on school gorithm of arithmetic. the mathematics that they study. And mathematics. His views have made their part of that accountability should in­ way into mathematics education cir­ Mastery of addition and the other clude homework and in-class exami­ cles because many mathematicians algorithms of basic arithmetic act as nations-the latter with at least have left it to others to address the a flashlight, allowing the young stu­ restricted calculator use. content crisis in school mathematics. I dent to move freely about in the world The standard arithmetic algorithms appreciate very much the work of re­ of numbers and basic numeric oper­ allow a teacher to communicate with searchmathematicians, butI pleadwith ations. Without such mastery a young students, and students with each other, them to allocate some of their time to student is condemned to move about to show hoyY a given answer was school mathematics. In addition to

734 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Letters to the Editor some of the mathematicians cited in "Rational selfish prisoners always let U be a unipotent subgroup of G, Ralston's article, we need more math­ choose the one strategy pair [Le., the and let N be the normalizer of U in ematicians with good judgment to Nash equilibrium (D,D)] that both can G. If U coincides with the unipotent speak out on this matter of national agree is undesirable-in the sense radical of N, then N is a parabolic urgency. that they would both prefer (C,C)." subgroup of G. This theorem was Where are they? (Strategy D is to defect and C is to co­ provedbyBoris Weisfeiler inthe paper operate.) "On a class ofunipotent subgroups of -Richard H. Escobales Jr. Rational selfish prisoners should semisimple algebraic groups", Uspekhi Canisius College not choose the Nash equilibrium. Be­ Mat. Nauk 21:2 (1966), 222-3 (in Russ­ [email protected] cause the game is symmetrical for the ian). For an English translation ofWe­ two players andbecause bothplayers isfeiler's paper and related comments, (Received April 9, 2004) are rational, then whichever strategy see the arXiv: http://www.arxi v . Player 1 decides is best, Player 2 will org/math.AG/0005149. also decide is best. Thus, the only Geometry Texts for Teachers possibilities are (D,D) and (C,C). Since -Victor Kac Hung-Hsi Wu, in his review of Audun (C,C) is better for each player than MIT Holme's Geometry: Our Cultural Her­ (D,D), rational selfish prisoners should itage, laments the lack of the foster­ choose (C,C). The reason the Nash (Received April 29, 2004) ing of geometric intuition in geome­ equilibrium is not relevant is that its try texts and offers his own work with definition considers pairs of strategies B. Braxton (available on the Web) as which are impossible if both players one way to achieve that goal. There are are rational, Le., (C,D) and (D,C). at least two other published geome­ This is discussed in detail in Chap­ try texts I know of that pay careful at­ ter 30 of Metamagical Themas: Quest­ tention to that goal. One, meant for ing for the Essence of Mind and Pat­ high school students but easily adapt­ tern, by Douglas R. Hofstadter (Basic able for future teachers, is EDC's (Ed­ Books, March 1996, ISBN 0-465-04566­ ucation Development Center's) Con­ 9). Hofstadter notes that most people nected Geometry, one of the NSF when presented with the above argu­ curriculum projects of the 1990s. The ment still say they would choose D. other is David Henderson's Experi­ encing Geometry, which has gone -David Marcus, Ph.D. through various iterations (distin­ Northrop Grumman Information guishable by their subtitles), gradually Technology becoming more and more compre­ Reading, MA hensive. Both books have high stan­ [email protected] dards of mathematical correctness, mathematical depth, and careful at­ (Received April 19, 2004) tention to how students actually learn, and both are writtenwith remarkable clarity. In fact, years ago while I was Work of Morozov, Weisfeiler, reviewing a draft ofpart of Connected and Borel Geometry, my seatmate on the air­ Regarding the article onArmand Borel plane, who identified herself as some­ inthe May 2004 issue ofNotices, Iwould one ordinarily not interested inmath­ like to comment on the related impor­ ematics, got so intriguedwhile reading tant earlier contributions of Vladimir over my shoulder that she asked if it v. Morozov and Boris Weisfeiler, two was available as a Christmas gift. eminent Russian mathematicians who The Notices invites readers to are not with us anymore. submit letters and opinion pieces - Judy Roitman Comment 1 (cf. p. 510 ofMay 2004 on topics related to mathematics. University ofKansas issue): The conjugacy ofmaximal solv­ Electronic submissions are pre­ [email protected] able subalgebras of a complex finite­ ferred (noti ces-l etters@ dimensional Lie algebra was provedby ams . 0 rg); see the masthead for (Received April 15, 2004) Vladimir V. Morozov in the paper "On postal mail addresses. Opinion a nilpotent element in a semisimple pieces are usually one printed page Lie algebra", Doklady USSR 36:3 inlength (about 800 words). Letters Prisoner's Dilemma (1942), 83-86 (in English). are normally less than one page Regarding the Prisoner's Dilemma, Comment 2 (cf. pp. 517-8 of May long, and shorter letters are pre­ Steven E. Landsburg ("Quantum game 2004 issue): Let G be a semisimple al­ ferred. theory", April 2004,. page 395) says, gebraic group over an arbitrary field,

AUGUST 2004 NOTICES OF THE AMS 735 The Status ofthe Classificationofthe FllllteSllnpleGroups A1ichaelAschbacher

ommon wisdom has it that the theorem Assume G is finite, and write H :s;j G to indi­ classifying the finite simple groups was cate that H is a normal subgroup of G. A normal proved around 1980. However, the proof series for G is a sequence

of the Classification is not an ordinary 1 = Go Michael Aschbacher is professor ofmathematics at the Cal­ satisfy an important generalization of Sylow's The­ ifornia Institute of Technology. His email address is orem, and indeed this property characterizes solv­ [email protected]. able groups. This generalization says: If G is solv­ This work was partially supported by NSF-0203417. able of order nand m is a divisor of n with

736 NOTICES OF THE AMS VOLUME 51, NUMBER 7 (m, n/m) = 1, then G has a subgroup of order m, make the following observation: With the exception all subgroups of G of order m are conjugate in G, of some of the sporadic groups, each group G ap­ and each subgroup of order dividing m is con­ pearing in the theorem can be regarded as essen­ tained in some subgroup of order m. tially the group of automorphisms of some fairly accessible mathematical object X. The existence of One can conceive of an analysis of the finite X gives an existence proof for G, but, more im­ groups based on a solution to the following two portant, the representation of G on X gives a means problems: for studying G and obtaining information about The Classification Problem. Determine all finite representations of the simple groups invarious cat­ simple groups. egories, most particularly representations as per­ mutation groups (Le. subgroup structure of G) and The Extension Problem. Given groups X and Y, de­ linear groups. The verification of some special termine all extensions ofX by Y; i.e. determine all property of such representations is usually the groups G with a normal subgroup H such that sort of extra input necessary to apply the Classi­ H~XandG/H~Y. fication to solve a given problem. In practice the Extension Problem is too hard, Indeed some suchinformation was necessary to except in special cases. It seems better not to look prove the Classification in the first place. That is too closely at the general finite group, but instead to say, in proving the Classification Theorem, one when faced with a problem about finite groups, to begins with the list X of simple groups appearing attempt to reduce the problem or a related prob­ in the statement of the theorem and considers a lem to a question about simple groups or groups minimal counterexample to the Classification closely related to simple groups. Then using the Theorem: A finite simple group G minimal subject Classification of the finite simple groups and knowl­ to G $. X. Thus each proper subgroup] of G is a edge of the simple groups, solve the reduced prob­ X -group: if K ~ H ~ ] with H/K simple, then the lem. Note this procedure works only if one knows section H/K is in X. The proof of the Classifica­ enough about simple groups to solve the problem tion depends very heavily on facts about the for simple groups; this is where the Classification subgroup structure and linear representations of comes in: it supplies an explicit list of groups which X-groups. can be studied in detail using the effective de­ During this article I will be discussing two dif­ scription of the groups supplied by the Classifica­ ferent efforts: The "original proof" of the Classifi­ tion. cation and work done since about 1980 aimed at This approach to solving group theoretic prob­ improving, simplifying, and writing down carefully lems has been in use since about 1980, when the in one place a proof of the Classification. The orig­ finite simple groups were deemed to have been clas­ inal proof sought only to make sure the literature sified. It has been extremely successful: virtually actually contains all the pieces of some program none of the major problems in finite group theory purporting to prove the Classification Theorem. The that were open before 1980 remain open today. second effort aims to produce a more readable Moreover, finite group theory has been used to treatment that inspires a higher level of confi­ solve problems in manybranches of mathematics. dence. In short, the Classification is the most important There is a two-volume exposition of part of the result in finite group theory, and it has become in­ original proofby Gorensteinin [Gl] and [G2]. Goren­ creasingly important in other areas of mathemat­ stein's books do not attempt to give details of the ics. proof, but only to give an outline ofwhat is entailed. Now it is time to state the: Further, Gorenstein died before completing the Classification Theorem. Each finite simple group third volume of the series. is isomorphic to one of the following groups: The largest part of the second effort is the pro­ gram begun by Gorenstein, Lyons, and Solomon 1. A group ofprime order. (GLS). While this program does not attempt to ad­ dress all parts of the proof, if completed as envi­ 2. An alternating group. sioned, it would deal with most parts. The work is 3. A group ofLie type. being publishedby the American Mathematical So­ ciety (AMS), and at the time I write this article, five 4. One of26 sporadic groups. volumes in the series have appeared in [GiS]. In Observe that the statement of the Classifica­ [GLS] you will also find references to other parts tion Theorem given above is deceptively simple. In of the second effort. order for it to have real content, one must define I have described the Classification as a theorem, what one means by "group of Lie type" and "spo­ and at this time I believe that to be true. Twenty radic group". Constraints on space preclude in­ years ago I would also have described the Classi­ cluding such definitions here, so instead I will fication as a theorem. On the other hand, ten years

AUGUST 2004 NOTICES OF THE AMS 737 ago, while I often referred to the Classification as r = p from when r =f. p. To implement Step 1 we a theorem, I knew formally that that was not the must translate these notions for linear groups into case, since experts had by thenbecome aware that related notions defined in abstract groups. a significant part of the proof had not been com­ Let p be a prime, G a finite group, and H a pletely worked out and written down. More pre­ p-Iocal subgroup of G. Define H to be of charac­ cisely, the so-called "quasithin groups" were not teristic p if dealt with adequately in the original proof. Steve Smith and I worked for seven years, eventually classifying the quasithin groups and closing this where Op(H) is the largest normal p-subgroup of gap in the proof of the Classification Theorem. We H and for U f; G, CH(U) is the subgroup of all el­ completed the write-up of our theorem last year; ements of G commuting with each element of U. it will be published (probably in 2004) by the AMS. Define G to be of characteristic p-type ifeach p-Iocal Later I will state the result; it should be viewed as subgroup of G is of characteristic p, and define G part of both the original proof and the second ef­ to be of even characteristic if for each 2-local sub­ fort. group H of G containing a Sylow 2-subgroup of G, It is time for some specifics: The proof of the H is of characteristic 2. It turns out that each group Classification proceeds by studying the so-called of Lie type and characteristic p is of characteristic local subgroups of G. Let p be a prime. A p-local p-type. subgroup of G is the normalizer of a nontrivial For reasons I will not go into, the prime 2 plays p-subgroup of G. a special role in the local theory of finite simple Let G be our minimal counterexample. The proof groups. (For example, by the Feit-Thompson The­ of the Classification canbe thought of as made up orem [FT], nonabelian finite simple groups are of of two steps: even order.) Thus in the original proof the follow­ ing partition appears: Step 1. Prove the local structure of G resembles that of some G E X. Case I. The minimal counterexample G is of char­ acteristic 2-type. Step 2. Use the resemblance in Step 1 to prove G ~ G. Case II. G is not of characteristic 2-type. Step 2 for the sporadic groups is one of the The generic group appearing in Case I is a group parts of the second effort that [GLS] does not ad­ of Lie type and characteristic 2, while almost all dress. However, there has been quite a bit of other simple groups appear in Case II. progress inimproving the treatment of Step 2 since There is also a partition according to size that the original proof. Many of the original proofs of corresponds roughly to the notion of size for the existence and uniqueness of the sporadic groups of Lie type: Given a prime p and a finite groups were machine aided, and the mathematics group G, define the p-rank mp(G) of G to be the involved was often unnecessarily complicated. In maximal dimension of an abelian subgroup of G the last twenty years, methods with the flavor of of exponent p, regarded as a vector space over the combinatorial group theory and/or algebraic topol­ field of order p. In Case II the "size" of G can be ogy have emerged that provide simpler, more con­ taken to be the 2-rank m2(G) of G. In Case I the size ceptual, treatments of uniqueness in Step 2 and is the parameter e(G) defined by Thompson in the have eliminated almost all computer calculations. n-group paper [T] (the model for all later work on I believe, however, that some of the old computer­ Case I): aided existence and uniqueness proofs have not e(G) = max{mp(H) : H is a 2-local of G and been superseded; e.g. the proof of the uniqueness of Thompson's group and the existence proof for p is an odd prime}. O'Nan's group are probably still machine aided. I Define G to be quasithin if e(G) :::; 2. The "small" will not saymore than that about Step 2 but instead groups in Case I are the quasithin groups. Thus in will focus on Step 1. the original proof we have four blocks in our par­ The generic finite simple group is a finite group tition corresponding to the large and small groups of Lie type. Each such group G is described via a of characteristic 2-type and to the large and small representation as a linear group, say G :::; GL(V) for groups not of characteristic 2-type. The quasithin some finite-dimensional vector space V over some groups of characteristic 2-type constitute one of the finite field F. Thus G has a characteristic which is four blocks. a prime: The characteristic p ofF. Similarly the Lie In the GLS program, one of the partitions is rank of G is a measure of the "size" of G, and for changed by altering the definition of "characteris­ our purposes we can think of the Lie rank as roughly tic". I will not give the GLS definition, since it is tech­ the dimension of V. Finally, given a prime r, the r­ nical. However, to accommodate the GLS change, local structure of G is qualitatively different when Steve Smith and I also work with a different notion

738 NOTICES OF THE AMS VOLUME 51, NUMBER 7 of characteristic in our study of quasithin groups: class of groups in the extended Case I have at least Recall that G is of even characteristic if been overcome for the small groups in Case I. CH(02(H)) :::; 02(H) for each 2-local containing a In the original proof of the Classification, chrono­ Sylow 2-subgroup of G. Define a finite group G to logically Case IT was treatedbefore Case I, more time be a QTKE-group if G is quasithin of even charac­ was spent dealing with Case II, and more people teristic and each proper simple section of G is in worked on this case. Probably as a result, the treat­ X. ment of Case II in the original proof is in better shape than the treatment of Case I. Volume 6 in the Classification ofQTKE Groups. (Aschbacher-Smith) [GLS] series treats the small groups in (their rede­ Let G be a nonabelian simple QTKE-group. Then G fined) Case II. After a certain point, GLS treat the is isomorphic to one of the following: large groups in Cases I and II together; this work 1. A group ofLie type ofcharacteristic 2 andLie rank is begun involume 5 of [GLS] and will be completed at most 2, but not Us(q). in the next few volumes. Thus the part of the sec­ ond effort that is furthest from completion is the 2. L4(2), Ls(2), Sp6(2), or Us(4). initial phase of Case I. 3. The alternating group Ag. The approach used to treat large groups in Case I is to focus on p-Iocals for odd primes p but 4. L2(p) for p a Mersenne or Fermat prime, L3(3), to keep 2-locals in the picture. This leads us to the U3(3), L4(3), U4(3), or G2(3). following definitions: Given a 2-local H, define 5. One of 11 sporadic groups: a Mathieu group, a cr(H) = {p : p is an odd prime and mp(H) > 2}, Janko group other than Jl, HS, He, or Ru. and let cr be the union of the sets cr(H), as H varies The proof of this theorem will be published by over the 2-locals of G. We work with p-Iocals for the AMS in two volumes in [AS]. It is roughly 1,200 p E cr. pages in length. In part this reflects the complex­ In the original proof, two problems in Case I re­ ity of the proof, but it also reflects a style of ex­ quired special treatment: position that includes more detail than one usually A. The uniqueness case: There exists a 2-local H finds in the original proof of the Classification, with cr(H) =f 0 and H strongly p-embedded in G for and it reflects our decision to keep our treatment each p E cr(H); that is, IH n Hg I is prime to p for as self-contained as possible. Indeed, one of the two eachg E G -H. volumes is devoted to group-theoretic infrastruc­ B. The case e(G) = 3. ture, such as proofs of folk theorems and facts Initially GLS hoped that new methods devel­ about X-groups. oped after 1980 (e.g. the so-called amalgam To my knowledge the main theorem of [AS] method) could be used to treat both cases (A) and closes the last gap in the original proof, so (for the (B) for the larger class of groups of even type, and moment) the Classification Theorem can be re­ their program was based on the assumption that garded as a theorem. On the other hand, I hope I specialists in those methods would handle the two have convinced you that it is important to complete cases. However, to date this has not happened, so the program by carefully writing out a more reli­ the treatment of cases (A) and (B) in groups of able proofin order to minimize the chance of other even type remains as perhaps the greatest obsta­ gaps being discovered in the future. Thus our dis­ cle to the completion of the second effort. The cussion of the status of the Classification would not work on these cases in groups of characteristic be complete without some indication of what re­ 2-type in the original proof is available as a model, mains to be done in that program. but it is not clear how much more difficult the Recall that the condition that G be of even char­ problems are in groups of even type. Gernot Stroth acteristic is weaker than the condition that G be and Inna Korchagina have done preliminary work of characteristic 2-type; thus if one changes the par­ on (A) and (B), respectively, under the hypothesis tition in the original proof to a division based on that G is of even type. Case (B) requires special treat­ the even characteristic condition, more groups ap­ ment, because different signalizer functors (cf. pear in Case I. GLS work with so-called groups of Volumes 1 and 2 of [GLS]) are used in p-rank 3 and even type; again, this condition is weaker than the p-rank greater than 3. characteristic 2-type condition, so more groups Finally, in addition to the original proof of the also appear in Case I in their partition, and hence Classification and the second-generation approach this part of the problem becomes more difficult. of GLS, there is also a third-generation program in­ However, as a corollary to our main theorem, Steve volving a number of people, particularly Ulrich Smith and I also determine in [AS] the quasithin Meierfrankefeld, Bernd Stellmacher, and Gernot groups of even type, the result on quasithin groups Stroth, that would treat all groups of characteris­ required in the GLS approach to the Classification. tic 2-type (and perhaps eventually all groups of even Thus any difficulties involved in treating the larger characteristic) using the amalgam method. Steve

AUGUST 2004 NOTICES OF THE AMS 739 Smith and I made use of variations of this method in our work in [AS]. It is possible that this approach will give a better treatment of Case I or at least of Case (B).

References [AS] M. ASCHBACHER and S. SMITH, The Classification ofQua­ sithin Groups, Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, to appear. [FT] ]. THOMPSON and W. FElT, Solvability of groups of odd order, Pacific]. Math. 13 (1963), 775-1029. [Gl] D. GORENSTEIN, Finite Simple Groups; An Introduction to Their Classification, Plenum, New York, 1982. [G2] __ , The Classification ofthe Finite Simple Groups, Volume I, Plenum, New York, 1983. [GLS] D. GORENSTElN, R. LYONS, and R. SOLOMON, The Classi­ fication ofthe Finite Simple Groups, vol. 40, Math. Sur­ veys Monogr., Amer. Math. Soc., Providence, RI; Num­ ber 1: 1995, Number 2: 1996, Number 3: 1997, Number 4: 1999, Number 5: 2002. [T]]. THOMPSON, Nonsolvable finite groups all ofwhose local subgroups are solvable, Bull. Amer. Math. Soc. (N.S.) 74 (1968), 383-437; II, Pacific ]. Math. 33 (1970), 451-536; III, Pacific]. Math. 39 (1971), 483-534; IV, Pa­ cific]. Math. 48 (1973),511-592; V, Pacific]. Math. 50 (1974),215-297; VI, Pacific]. Math. 51 (1974), 573-630.

\V\V\V. an1S. 0 r g

740 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Intimations ofInfinity Kirk Weller, Anne Brown~ Ed Dubinsky, MichaelMcDonald, and Cynthia Stenger

The natural numbers? The natural num­ understanding the thinking of both novices and bers are 1,2,3 ... ,00, but there is no practitioners as they grapple with the notion ofin­ such number as 00 ; there is nothing you finity. In APOS theory, which will be more fully ex­ can think ofas a concrete value. plained later, an individual develops an under­ standing of a concept by employing certain There is no actual infinity; and when we mechanisms called interiorization, encapsulation, speak of an infinite collection, we un­ and thematization. These mechanisms are used to derstand a collection to which we can build and connect mental structures called actions, add new elements unceasingly. processes, objects, and schemas. To get a feeling for the complexity of how peo­ ple grapple with infinity, see how you and perhaps Overture some of your colleagues would answer the fol­ The comments above (see the next page for who lowing questions. How do you think your answers made them and when) represent one type of think­ compare with what has been said by mathemati­ ing about infinity. There are other types, as we will cians and philosophers over the last 3,000 years see, and they all create difficulties for students, or by students today? philosophers, and even mathematicians. The pur­ • If the slow tortoise starts a little ahead of the pose of this article is to show how a particular the­ swift Achilles, how can this demi-god ever catch ory about how people come to understand math­ up? For Achilles must first advance to where the ematics, APOS Theory, can be helpful in tortoise started, by which time the plodder has moved on a little, so Achilles must then advance Kirk Weller is associate professor of mathematics at to that spot, and so on, forever. the University of North Texas. His email address is • How can the quantity dx be treated both as a well erk@unt . edu. positive quantity with which calculations can Anne Brown is associate professor of mathematics at be made, and something that can be ignored as Indiana University South Bend. Her email address is abrown@; usb. edu. if it were O? • Is 0.999 ... = I? Ed Dubinsky is a partially retired professor of mathe­ • Suppose you put two tennis balls numbered 1 matics at Kent State University. His email address is [email protected]. and 2 in Bin A and then move ball 1 to Bin B, Michael McDonald is associate professorofmathematics at Acknowledgement. The authors would like to thank Judy Occidental College. His email address is mi ckey@oxy . edu. Roitman for many helpful comments about the work Cynthia Stenger is associate professor ofmathematics at described in this article and about the article itself the University of North Alabama. Her email address is Work on this article was partially supported by the [email protected]. ExxonMobil Foundation.

AUGUST 2004 NOTICES OF THE AMS 741 then put balls 3 and 4 in Bin A and move 2 to Tennis Balls. Bin B, then put balls 5 and 6 into Bin A and move During interviews with college students for a cur­ 3 to Bin B, and so on without end. How many rent research project, one said: "... they're both balls are in Bin A when you are done? Infinitely gonna contain half the balls." Another claimed that many, because the number increases by one Bin A contained "two infinity minus infinity which each time, or none, since every ball is eventually would be infinity." Yet another said: "A doesn't re­ removed? ally have a limit on how big it is...so A goes to in­ • If you build a setby putting in the integer 1, then finity." One student felt that "... you cannot decide 2, then 3, and so on, how do you get from this what's gonna be in A." Only one student thought unending process to a conception of the full set that Bin A would be empty [11]. of natural numbers? Getting N from a Process. • Is the infinite union Uk=lP({1,2, ... ,k}) equal "It [the natural numbers] means this is the collec­ to the power set of the natural numbers P(N)? tion of things 1, 2, 3, and...then I keep adding Here, P stands for the power set operator. 1...and now I'm going to take the union over all • Is there any sense in which an uncountable set those sets 1 through n. ... What annoys me about can be the outcome of a countable algorithm? this is that when I take that union ... somehow you For each of these questions, rigorous formal have to know in advance what the integers are be­ thought provides answers on which mathemati­ fore you can take the union over all the finite trun­ cians can agree. But agreement has not come eas­ cations of the natural numbers." (A research math­ ily; students often have a hard time accepting the ematician's response during an interview for a formal solutions, and describing the infinite canbe current research project.) difficult. For example, the first statement at the very The Result ofTakinganInfinite Union. beginning of this article is a paraphrase of what a During an interview a college student tried to de­ student said recently in describing her conception termine whether the infinite union of the natural numbers. The second statement is Uk=lP({1,2, ... ,k}) is equal to P(N). She noted a quote from H. Poincare [10] almost a century that the power sets are nested and thought of the ago. union iteratively in terms of an infinite sequence The following examples illustrate the thinking P ({ 1}), P({ 1, 2, }), ..., P({ 1, 2, ... , k}), ..., but then of students, mathematicians, and philosophers on remarked, "and then you keep adding one and the above bulleted questions. They indicate some you'll still have finite sets, but eventually you have of the many aspects of the infinity concept, the va­ to, I just still want to include that infinity!" [3] riety of approaches taken to deal with these aspects, Getting anUncountable Set from a Countable and some of the difficulties encounteredby experts Algorithm. and students in their efforts to understand the in­ "I don't believe the power set of the natural num­ finite. bers is countable, and you have presented what ap­ Achilles andthe Tortoise. pears to be a countable process for creating this All objections to the infinite, Aristotle insisted, are set." (A college teacher's response during an in­ objections to the actual infinite. The potential in­ terview for a current research project.) finite, on the other hand, is a fundamental feature "A procedure that purports to construct P(N) of reality. Aristotle used this distinction between only gives an illusion of construction." (From a pri­ the two types of infinity to resolve paradoxes'like vate conversation with a set theorist.) Achilles and the tortoise [7]. The issues raised here are controversial for both mathematicians and philosophers and have been Infinitesimals. for centuries. In this article we describe several of "And what are these fluxions? The velocities of our investigations (some completed, some ongoing, evanescent increments. And what are these same and some only contemplated) into the issues raised evanescent increments? They are neither finite by the variety of thinking about infinity that his­ quantities, nor quantities infinitely small, nor yet tory and current discourse provide. We illustrate nothing. May we not call them the ghosts of de­ how research in mathematics education can con­ parted quantities?" [1, p. 83] tribute to resolving classical issues related to the Is 0.999...=1? concept of infinity and point out how this same re­ Responses to this question from university stu­ search can also explain certain difficulties we see dents included: in students who are trying to understand infinite "Just less than one, but it is the nearest you can processes and objects in mathematics. get to one without actually saying it is one." We will explain how the APOS theory of learn­ "It is just less than one, but the difference be­ ing helps us analyze the type of thinking exempli­ tween it and one is infinitely small." fied in the quotes given above. Specifically, we use "The same, because the difference between them the theory to argue that human beings can and do is infinitely small" [12]. conceive of an infinite process as a totality and

742 NOTICES OF THE AMS VOLUME 51, NUMBER 7 think about actual infinity, that one can apply cer­ infinite iterative process that yielded an infinite se­ tain mental mechanisms to think about the set of quence of power sets of the form P( {1, 2, ... , k}). natural numbers as a totality, and that there is a Subsequently, everyone of them seriously enter­ sense in which those students who claim that tained the notion that the sequence would "even­ .999 ... is not the same as 1 are right. And, though tually" yield the power set of an infinite,set. we will leave the full details for a future article, we In [11] a different set of students was asked to will also suggest that the same mental mechanism work on the tennis ball problem mentioned in the that allows an individual's thoughts to shift from "Overture". Even though everyone of the thirteen enumerating the natural numbers one-by-one to students could articulate the time at which the nth considering the entire set of natural numbers can ball would be dropped into Bin A and then trans­ also be used to make the leap from thinking about ferred to Bin B, only one student concluded that a countable process to seeing an uncountable set Bin A wouldbe empty after all of the balls hadbeen such as P(N) as a mental object. dropped. Their conflicting thoughts about how to deal effectively with anunending process appeared Motivation to be at the root of their difficulties. The concept of mathematical infinity appears In addition to playing a role in understanding throughout the collegiate mathematics curricu­ infinite processes (such as those arising in con­ lum, especially in precalculus and calculus courses, sidering an infinite union or the tennis ball prob­ where students consider topics such as limits, the lem), students' conceptions ofinfinity have a bear­ asymptotic behavior of rational functions, infinite ing upon their ability to solve other types of sequences and series, and improper integrals. How­ problems. In order to act on an infinite set, say to ever, this represents but a small portion of all the compare the cardinality of two infinite sets or to situations where the infinite appears. For instance, show that the set of all linear combinations of a set many of the mathematical structures studied inlin­ of vectors in a vector space over R is a subspace, ear algebra, abstract algebra, real analysis, and one must be able to think of these infinite sets as topology are infinite sets; existence proofs fre­ mathematical objects, or entities, which can be quently require the construction of infinite men­ transformed. Helping students formulate object tal procedures; and problem situations involving conceptions of mathematical concepts requires collections of mathematical objects indexed by an the development and implementation of carefully infinite set occur throughout the undergraduate designed instruction. In the sections that follow we curriculum. discuss how APOS Theory can provide an expla­ Although the concept of infinity permeates the nation of how human beings conceive of the infi­ undergraduate curriculum, students experience nite. This is a first step toward the development little if any formal instruction on the concept prior ofpedagogical strategies intended to help students to a study of cardinality in a "bridge" course or a to understand and apply the kinds of transforma­ formal study of Cantorian theory in an upper­ tions required for the successful solution of vari­ division set theory course. Our experiences as ous problems involving the infinite. teachers and researchers convince us that this is There are two reasons why we believe APOS insufficient. As one can see from the quotes given Theory might be a useful tool in these endeavors. in the "Overture", students continue to struggle First, APOS Theory has been used over the past with the concept of infinity, despite their experi­ twenty years to analyze student thinking about var­ ences in lower-division courses. Two projects ([3] ious concepts in the undergraduate mathematics and [11]) show the degree to which students with curriculum. The results of these analyses have fairly strong mathematics backgrounds struggle gUided the development of effective pedagogical with infinity even after having completed at least strategies [13]. Second, our initial attempts to one course where aspects of mathematical infin­ apply APOS Theory to understand how individu­ ity were considered in some depth. als think about infinity have been encouraging. In the study reported in [3], students were asked Based on an analysis of interview data of students to prove or disprove the statement attempting to solve the above infinite union prob­ 00 lem, the research study [3] describes how stu­ P({I, 2, ... , k}) = P(N), U dents appear to construct their conceptions ofin­ k=l finite iterative processes. We have also recently where P indicates the power set operator. Of the thir­ prepared a comprehensive report to discuss how teen students interviewed, only one student solved APOS Theory can be used to explain, and in some the problem correctly, and even this studentneeded cases to propose resolutions of, many of the issues significant prompting before providing the correct and paradoxes of the infinite that have plagued solution. In trying to interpret the meaning of the philosophers and historians of mathematics for infinite union, all of the students constructed an centuries [2].

AUGUST 2004 NOTICES OF THE AMS 743 Inspiration experience in mathematical thinking. For example, Before we describe the investigations that have en­ one might wish to add two functions f and 9 to ob­ sued from thinking about problems like those men­ tain a new function f + g. Thinking about doing this tioned above, we discuss the mental mechanisms requires that the two original functions and the re­ and structures to which we will be referring. This sulting function are conceived as objects. The actual brief explanation of APOS Theory is meant to fa­ transformation is imagined by de-encapsulating miliarize the reader with the terminology used in back to the two underlying processes and coordi­ subsequent sections. nating them by thinking about all of the elements The interiorization of actions is an "everyday" x of the domain and all of the individual transfor­ activity in the mathematics classroom. For exam­ mations f(x) and g(x) at one time so as to obtain, ple, an algebra student may wish to describe the by adding, the new process, which consists of trans­ behavior of a quadratic function over a given in­ forming each x to f(x) + g(x). This new process is terval to see whether it increases for a while and then encapsulated to obtain the new function f + g. then decreases. The transformation of calculating The mental mechanisms of interiorization and functional values over the interval is first con­ encapsulation allow one to think about what hap­ ceived as an action, in that it requires specific in­ pens after a process is completed. In many cases, structions, e.g., a formula. Repeating this action and the domains and the ranges of functions are infi­ reflecting on the relationship between functional nite sets, so these mechanisms allow an individual values as x varies over the interval, the student may to think about infinity in these contexts. begin to interiorize the action into a mental struc­ While these mental structures describe how an ture called a process. This is a structure that im­ individual constructs a single transformation, a plements the action, not externally, but internally, mathematical topic often involves many actions, in the individual's mind. A process enables the in­ processes, and objects that need to be organized dividual to imagine the calculation of several val­ and linked into a coherent framework that is ues of the function and to think about these cal­ called a schema. The mental structures of action, culations all at the same time. Thus, the individual process, object, and schema constitute the can observe the behavior of the functional values acronym. APOS. In this article we use these men­ as x varies over the interval without having to eval­ tal structures and the mental mechanisms of in­ uate f(x) for explicit values of x. At this point, if teriorization and encapsulation to analyze from the student becomes aware of the process as a to­ a cognitive perspective various issues raised as a tality, realizes that transformations can act on that result of careful thinking about the infinite. For totality, and can actually construct such transfor­ more information about APOS Theory and a sum­ mations explicitly or in her or his imagination (e.g., mary of how it has been used in mathematics ed­ think about horizontal and vertical shifts or com­ ucation research, see [5]. pressions and expansions), then we say that the in­ dividual has encapsulated the process into a cog­ Application nitive object. Our discussion of applications of APOS Theory to There are two aspects of encapsulation that are the issues listed at the beginning of this essay will important to keep in mind. First, according to APOS be divided into four parts. First, we consider some Theory, an encapsulation occurs because the indi­ problems that have been around for a long time. vidual desires to perform an action (or process) on These include the classical paradoxes, disputes a process. This is not possible, because a process about the infinitely small, and the value of an in­ is dynamic, something that is in progress, and as finite, repeating decimal. Next, we consider the such is not susceptible to being acted upon. For ex­ tennis ball problem as an example of a situation ample, students who have not encapsulated the involving infinite iterative processes inwhich one's process of set formation into an object will think conception of the set of natural numbers plays a that a set such as {2, {5, 6, 8}} has cardinality 4 role. This is followed by a description of a more gen­ (and not 2). This may be because their thinking eral consideration of infinite iterative processes about 2,5,6,8 does not go further than the process based on a recently completed research project. Fi­ of inserting these four numbers into a set. Encap­ nally, we describe a new research project that be­ sulating the process of forming {5, 6, 8} into an ob­ gins to look at mental constructions of uncount­ ject eliminates this difficulty and allows one to able sets. perform the desired action (in this example, de­ ClassicalParadoxes, the InfinitelySmall and termining cardinality). Second, it is oftenimportant RepeatingDecimals in a mathematical activity to de-encapsulate an ob­ Aristotle's resolution of paradoxes such as Achilles ject, that is, to go back to the process from which and the tortoise consisted of making a distinction it came. between actual and potential infinity and then re­ The encapsulation and de-encapsulation of jecting the former. While he believed humanbeings processes in order to perform actions is a common could conceive of potential infinity, Aristotle

744 NOTICES OF THE AMS VOLUME 51, NUMBER 7 considered the idea of an actual infinity to be be- the tortoise; the question of the existence of infi- yond the understanding ofmortals ([8, pp. 34-44]). nitely small, nonzero quantities; and the relation Although there were dissenters, Aristotle's idea between a sequence and its limit. We hope the persisted for millenia, and it has been expressed unity of our explanations makes them clearer and quite explicitly in modern times byvarious writers, more satisfactory than other explanations that including mathematicians such as Poincare ([10, have been offered. But more importantly, we feel pp. 46-7]). that these analyses could lead to pedagogical strate- In our view, however, the rejection of actual in- gies that will be effective in helping students un- finity is unnecessary and ignores important math- derstand the mathematics of the infinite. ematical notions such as actually infinite sets, com- Achilles andthe Tortoise parison of infinite cardinalities, mathematical An individual can think about an infinite iterative induction, etc. Infact, there are explanations ofhow process using the mental structure of process as an individual might think ------described in APOS Theory. In about infinity that incorpo- terms of that theory, per- rate the mind's ability to con- It is important to forming a small number of template actual infinite ob- h iterations constitutes an ac- jects. note t at, given an tion.Byinteriorizingtheseac- Today's formal definition - -Pi- -t th tions, an individual can use of limit (in terms of epsilons zn,znz e process, e the resulting process struc- and deltas) provides a satis- mental ture to imagine repeating the factory mathematical expla- actions indefinitely or "for- nation of how a symbol such mechanisms of ever", so to speak. This cor- as dx canbe used in one part responds to potential infin- of a calculation as if it were a interiorization and ity. Using the process mental positive quantity and in an- structure, an individual can other part, ignored (or "ne- encapsulation see the process as a totality, glected") as if it were zero. allow one to think even if it is inconvenient or Extensive research shows, impossible to think explicitly however, that the dispute be- about what about each step in the tween Newton's "evanescent process, and decide to per- quantities" and Berkeley's happens after the form actions on the total "ghosts of departed quanti- process. Here, the mental ties" [1] is alive andwell in the process IS structure of encapsulation minds of many (most?) of comes into play. Encapsula- today's calculus students [4], completed. tionconsists in transforming [12], [15], [16]. Again, we feel the process to an object and that there is an alternative applying the desired action. cognitive explanation that could be more helpful In the case of Achilles and the tortoise, we have to students struggling to resolve the paradoxes of two coordinated processes of Achilles repeatedly the "infinitely small". covering the previous distance traveled by the tor- The importance of the pervasive difficulty in toise while the tortoise continually moves farther dealing with the relation 0.999 ... = 1 and the per- along. As processes, we can imagine this going on sistent idea that there is an "intermediate state" be- forever, and we encapsulate the completed tween a sequence ofvalues and its limit [4] lies not processes in order to perform on them the action only in the specific mathematical errors that stu- of comparing the total distances covered by the two. dents make but also in what it tells us about how With a cognitive grasp of the question, we can then far so many students are from having a useful in- do the calculations (which amount to summing in- tuitive understanding of the concept of limit of a finite series) and see that in a finite time, the total sequence. As nearly as we can tell from our review distance cqvered by Achilles exceeds that covered of the literature [2], no explanations other than by the tortoise. those based on APOS Theory have been offered re- It is important to note that, given an infinite garding the cognition of 0.999 ... = 1, that is, re- process, the mental mechanisms ofinteriorization garding how an individual might think about this and encapsulation allow one to think about what relation in ways that fit with the mathematics. happens after the process is completed. The ob- Moreover, the only pedagogical strategy available jection that this cannot be done, since one can seems to be a reiteration of the mathematical facts. never actually perform an infinite number of steps, We will use a single kind of analysis, based on is precisely what the structure ofprocess takes care APOS Theory, to propose resolutions of cognitive of, since one does not have to actually perform all issues as disparate as the paradox of Achilles and of the steps, whether there be finitely or infinitely

AUGUST 2004 NOTICES OF THE AMS 745 many of them. And, in our view, the ability to en­ why many people had such difficulty in under­ capsulate an infinite iterative process requires standing what he was saying. In a finite process, thinking about actual infinity. Thus, we claim that there is always an object produced in the last step human beings can conceive of actual infinity. of the process. Even though there is no last step Infinitesimals in an infinite process, such a process can still re­ The dispute about the meaning of dx (Newton ac­ sult in an object. This requires, however, a much tually used 0) in the expression more powerful mental mechanism than imagining a last step. According to APOS Theory that mech­ f(x + dx) - f(x) anism is encapsulation, and the resulting object is dx what is called in [3] the transcendent object of the can be represented by the ideas of Newton and process. Our interpretation of Newton's thinking Berkeley. In [2] we consider the ideas of these two is that he understood the distinction between thinkers in some detail and try to show how our process and object in this context and realized interpretations are based on their actual writings. that a more powerful mental step was required in Here, we briefly summarize our interpretations order to go from an infinite process to an ultimate and explain in APOS terms how an individual can value. We conjecture that he had encapsulated the think about infinitesimals in calculus. limiting process but had no mathematical tools In terms of an APOS analysis, it appears that the (such as the formal concept oflimit) to express pre­ crux of the issue was that when writing the cisely his object conception. difference quotient, Newton intended dx to rep­ Drawing on an analogy with determining the resent a process of approaching O. That is, the velocity of a body "at the very instant it arrives", ' symbol dx in the difference quotient stands for the Newton emphasized what happens "ultimately", process of replacing the symbol dx by smaller and which we interpret as an attempt to apply an ac­ smaller positive numbers. For each one of these pos­ tion of evaluation to a completed process, leading itive numbers, one can compute the difference to an encapsulation. He clearly distinguished the quotient. Hence, that expression represents the objects produced by the process from the tran­ process of obtaining values by replacing dx with scendent object produced by encapsulating the smaller and smaller positive numbers and making process. the calculations. On the other hand, when Newton Newton's critics, however, insisted that dx and wrote the difference quotient itselfmust always be viewed as static objects. Thus, when Berkeley insisted that Quantities, and the ratios of quantities, Newton's evanescent increments were "neither fi­ which in any finite time converge con­ nite quantities, nor quantities infinitely small, nor tinually to equality, and before the end yet nothing" [1], our view is that he was not dis­ of that time approach nearer the one to tinguishing between an object produced by a the other than by any given difference, become ultimately equal [9, p. 29], process and an object that is brought into being by encapsulating the process as a result of applying the action "What is the ultimate value of the he was seeing those processes as totalities and en­ process?" capsulating them in order to obtain the ultimate value of dx (which is 0) and of the difference quo­ 0.999... and 1 tient (its limit, in contemporary parlance). In the ex­ Maybe the students are right, maybe 0.999 ... is not amples that he considered, Newton was generally the same as 1, at least not cognitively. APOS The­ able to obtain the latter value by simplifying the ory can offer an explanation of such thinking. Math­ difference quotient expression and then replacing ematicians consider 0.999 ... to stand for the limit dx by 0, that is, "neglecting" it. of an infinite sequence. However, the ability to Thus, in our interpretation, when dx represents think of this expression in that way requires cer­ a process, it is a positive number, so the difference tain mental constructions that some students may quotient makes sense mathematically. But then, not yet have made. For such individuals, the sym­ with the encapsulation, dx represents an object to bol 0.999 ... appears to represent a process (that which Newton referred as its ultimate value, which is the only possible explanation of the "..." or is zero. The ultimate value of the difference quo­ phrases such as "and so on"). It is an infinite tient, again no longer as a process but represent­ process, and so there is no object produced by a ing an object, is its limit. This distinction between last step. The symbol 1, however, refers to an ob­ process and object is our resolution of the "con­ ject. Since a process is something different from tradiction" of dx being sometimes positive and an object, it makes sense to say that the process sometimes O. 0.999 ... cannot be the same as the object 1. What Our interpretation of Newton's thinking has an makes this particularly difficult is that the num­ additional attribute that we think helps explain ber 1 is not an object produced by any step in the

746 NOTICES OF THE AMS VOLUME 51, NUMBER 7 process but is the result of encapsulating it and so manyballs at noon. However, the kthball is moved transcends the process. from Bin A to Bin B at the kth step, from which it In fact, we conjecture that certain mental con­ follows that Bin A will be empty at noon. structions are necessary before it is possible to This paradox is resolved by comparing the con­ even think about the mathematical solution, much tents of the bins. The mental mechanisms of inte­ less understand it. Specifically, to understand that riorization and encapsulation allow one to think 0.999 = 1, the individual must first realize that about what happens after the process is completed. 0.999 is an infinite process and, as such, does Inthe tennis ball problem there are two coordinated not produce an object (a numerical value in this processes: one inwhich Bin A receives the next two case) directly. Rather, the process must be encap­ balls, and in the other, the lowest numbered ball sulated to an object in order to find this value. Once in Bin A drops into Bin B. One imagines these co­ this encapsulation has been made, one can then do ordinated processes continuing forever and then some mathematics to determine the value. For ex­ encapsulates the completed processes in order to ample, one might argue that there are two numbers perform the action of comparing the contents of Land 1, L being the object produced by the en­ Bin A and Bin B. At this point, with a cognitive un­ capsulation. Since L is understood to be a value that derstanding of the question of the contents of Bin is determined after the process is finished, there A after the process is completed, one can see that might be less of a tendency to try to force it to come at noon Bin A will be empty. Even though there is from the process andbe very close to 1 or "the num­ no last step in the infinite process offilling the bins, ber just before 1". One might then calculate that one can use encapsulation to imagine a resulting IL - 11 is smaller than any positive real number, transcendent object and then determine mathe­ so it must be 0, and so L = 1. matically that it is the empty set, e.g. by checking We have no data to support the idea that stu­ that Bin A and the empty set contain exactly the dents are aware of the subtle distinction between same elements. infinite processes and their transcendent obJects. PreliminaryResults ofStudentInterviews We plan to design experiments to investigate this To unravel the basic notions undergraduates hold question as well as the conjectures above. This concerning infinite processes, students (with ma­ might lead to the design of new pedagogy focused jors in mathematics, mathematics education, or on the process/object distinction that is intended computer science) were asked to solve the tennis to help students overcome difficulties, such as ball problem. A preliminary analysis of the data sug­ those reported in [4] and [12], having to do with gests that students who had difficulty solving the understanding infinite decimals. problem did not see the underlying infinite itera­ Constructionofthe NaturalNumbers and tive processes as completed totalities, which is a StudentThinking onthe Tennis Ball Problem necessary precursor to encapsulation. Thus, they College students' thinking on the tennis ball prob­ did not see the "ultimate" contents of each bin as lem is the focus of a current study [11]. During in­ an object, and so the question of how many balls terviews, students were asked to imagine three each contains was meaningless to them. bins of unlimited capacity: a holding bin, where Nearly every student could articulate that the kth balls numbered k = 1,2,3 ... would originate; Bin ball would be moved from Bin A to Bin B at the k th A, where each ball would be held temporarily; and step. However, only one student argued that Bin A Bin B, the final destination for each ball. Their task is empty. This student seemed to grasp that the was to determine the contents of Bins A and B at movement of the kth ball not only describes what noon, if at 2\ seconds before noon, balls numbered happens to a single, randomly selected ball but to 2k - 1 and 2k are placed in Bin A, while the ball every ball in the holding bin. Several of the students numbered k is moved from Bin A into Bin B. who had difficulty with the problem believed Bin The data in [11] is part of an analysis of students' A would never be empty because the iterative thinking on this problem. The aim is to investigate process would continue beyond the kth step. Al­ how they think about the set of natural numbers though they understood that every ball would and to test the generalizability of the description "eventually move to B," they believed that "there of the mental construction of infinite iterative will always be more to come." processes developed in [3]. This latter study, which Other students argued on the basis of cardi­ we describe later in this article, looks at a more com­ nality. At each succeeding step the number ofballs plex mathematical situation. in each bin increases by one. At the kth step balls A TheoreticalAnalysis ofthe Tennis Ball numbered 1 through k are in B, and balls numbered Problem k + 1 through 2k are in A. Some of the students gen­ There are two competing conceptions of the prob­ eralized the finite case to assert that each bin lem which make it paradoxical. On one hand, the would contain "half of the balls," the "upper half" number of balls in Bins A and B increases by one in A and the "lower half" in B. Others substituted at each step, suggesting thatbothbins have infinitely 00 for k to conclude that Bwould contain 1 through

AUGUST 2004 NOTICES OF THE AMS 747 00 and A would contain 00 + 1 to 200. When sense that it differs from and is not produced by prompted, these students generally acknowledged, any step of the process. as did those who did not focus on cardinality, that These preliminary findings suggest that a cor­ each ball would "eventually" be moved to Bin B. rect solution is dependent upon the student's abil­ However, they did not find this information use­ ity to see the underlying infinite iterative process ful. In their view, the problem could not be solved. as a completed totality. Without making this men­ Because the procedure would continue indefinitely, tal construction, the student finds the problem one could never identify a particular number (other difficult or impossible to solve, because the process than 00 ) that could be substituted for k to identify "does not stop," from which it follows that "there the precise "halfway" point that would denote are always more balls to come." which numbered balls would be in A and which Conceptions ofInfinite Iterative Processes would be in B at noon. The ongoing study concerning the natural numbers A preliminary analysis of the data suggests that and the tennis ball problem is closely related to the the students who had difficulty may not have made research study reported in [3]. In this investigation certain mental constructions. Those who simply the authors interviewed students solving a partic­ substituted 00 for k and used this to conclude that ular elementary set theory problem and developed Bin B contains 1 through 00 and A contains 00 + 1 a description of the mental constructions an indi­ through 200 may not have constructed a useful in­ vidual might make and use to understand infinite finite iterative process from their conceptions of iterative processes. As stated earlier, students were finite iterative processes. They generalized from the asked to prove or disprove the following equality:

finite case to the infinite in a manner that did not 00 account for the way inwhich the infinite case tran­ U P({I, 2, ... , k}) = P(N). l scends the finite. Those students who did not k=l substitute 00 for k, but asserted that the "ones Although one can resolve the question by noting that are higher" remain in Bin A because "there's that the set on the left side is countable and the an infinite number of natural numbers," likely did set on the right is not, in the context of the course not see the process as being complete. Because the authors expected students to compare the two the process "does not stop," or "noon can never be sets on the basis of set inclusion. In particular, reached," they could not imagine that all the steps they expected students to note that the union on of the process could be carried out and finished. the left side contains only finite sets as elements, The student who correctly proved the result ap­ whereas P(N) contains infinite sets as elements. peared to make this latter construction. In his While mathematicians may see both sides of proof he argued that if the nthball were contained this proposed equality as static objects, the stu­ inA at noon, a contradictionwould result, because dents interviewed saw these as processes they had the ball would appear in Bin B at any time within constructed or needed to construct. Thus, the au­ 1/2n seconds before noon. In making this argu­ thors decided to look carefully at the actions and ment, the student realized that all of the balls processes that individuals might construct in order would be in Bin Bat noon. This required him to see to understand the formal notation represented in the process as a totality or a single operation. The this problem. The central role of one's conceptions student in question gave evidence of having made of infinite iterative processes was brought to light this construction when he said: "So any time I by this analysis. We give a short description of the choose an n, like say I choose an n out of the hold­ mental construction of infinite iterative processes ing thing. Well, after the nth time, it's going to be proposedby the authors and then make some brief in B, and so that for me was like saying, okay, if I observations. have an n, n has to be in B, so that means all the The construction of a mathematically useful holding bin will end up in B." Because he could see conception of infinite iterative processes appears the process as a completed totality, he was able to to be based on one's process conception of finite encapsulate it and argue that Bin A is empty. In iteration. One must be able to apply the relevant making this construction, he understood that the finite process to an initial object and understand situation at noon transcends the process, in the generally how an object is produced via the process from the preceding object or objects. A process con­ lOne possible explanation is that some students inappro­ ception of infinite iteration develops as the indi­ priately felt that this process was Ucontinuous" in the sense vidual becomes able to coordinate multiple in­ that the number ofballs at the end would be the limit, as k goes to infinity, of the number of balls at the k th step. stantiations of this finite process. A successful Such overgeneralization is common. For example, some cal­ coordination leads to the individual becoming able culus students will overgeneralize the "zero product prop­ to conceive of this infinite process as being com­ erty" and reason that limx-o x cot(x) = 0, even though plete, even though there is no final step of the limx-o cot(x) is not finite. process and no last object. Once the process can

748 NOTICES OF THE AMS VOLUME 51, NUMBER 7 be imagined as being complete, the individual may Conceptions of P(N) reflect upon it and begin to see it as a totality, in While one may come to understand P(N) through the sense of seeing it as a single operation that can its formal definition as the set of all subsets of N, be carried out and finished. Depending on the sit­ we would argue that the definition alone is not suf­ uation, the individual might attempt to construct ficient, at least not for some undergraduate math­ an action of evaluation on the process, typically with ematics students. To apply actions or processes to the goal of determining the state at infinity for the this set in certain problem situations, APOS The­ process. A successful application of an action of ory posits that an individual needs to be able to ac­ evaluation happens in tandem with the encapsu­ cess a rich process conception through de­ lation of the process into an object, called its tran­ encapsulation of an object conception. The set scendent object. This object is understood to be re­ P(N) cannot, of course, be constructed by encap­ lated to, but beyond the objects produced by the sulating the infinite iterative process of taking the process, in the sense that it cannotbe producedby union of the power sets of initial segments of N. applying the iterative process to any of the previ­ Nevertheless, we may ask if it is possible to con­ ously produced objects. The objects produced by ceive of this set as arising from any infinite itera­ the process, followed by the transcendent object, tive process, given that the set itselfis uncountable. are then conceived of as forming an extended se­ Our investigation of mental constructions of un­ quence (Le., an ordered set indexed by N u {oo} ). countable sets begins with the question: How do To consider the infinite union above, many stu­ experts get a rich conception of P(N)? Is it simply dents began by constructing a process of the through mathematical formalism, or are there following form, noting that successive power sets underlying mental constructions that can be re­ are nested: vealed through careful research? These questions, as well as our interest in studying conceptions of P({I}) = P({ 1}) uncountable infinity, led us to investigate how mathematicians come to understand P(N) and P({l}) u P({l, 2}) = P({l, 2}) whether their construction of this understanding admits an APOS analysis. This is a project that we are currently in the midst of conducting. P({I}) U P({I, 2}) UU Is the mental construction of P(N) made by a so­ P({l, 2, , k}) = P({l, 2, ... , k}). phisticated mathematical thinker, for example, a research mathematician, derived mainly from the Students who were successful realized that they formal definition? Certainly this definition states could co'ntinue this process and see it as being completely what the set is, and one could imagine complete, even though there is no final step to the various infinite processes for building a given set process and no final object produced. Many of the in P(N). But the question still remains as to how students' difficulties were associated with this one would think of the set "all at once", as a cog­ issue, in that it was hard for them to consider the nitive object. According to APOS Theory, when one infinite process without imagining that the set uses this object in a problem situation, a de­ P(N) or an entity that they referred to as encapsulation to a process is needed. But what is P({1,2,3, ... ,00}) was produced by the process. the process that was encapsulated to give the set Only a few students could see the process as com­ in the first place? Could this process be an infinite plete, and in order to be successful with the prob­ iterative process even though the set itself is un­ lem, they also had to be able to see the completed countable? In our current research, we have in fact process as a totality. This means that they had to developed a countable, iterative process along with understand that every set constructed within the multiple encapsulations which we believe yields the process contains only finite sets. With that under­ uncountable set P(N). This process seems to be re­ standing, they could see that the transcendent ob­ lated to the binary tree construction ofP(N), where ject for this process is the set of all finite subsets P(N) is the set of all branches of an infinite binary of N, an object that is not constructed anywhere tree.2 Our concern, however, is with the cognitive in the process but rather is constructed only as the meaning of "the set of all branches" and how an process is encapsulated. individual might construct this set in her or his The above analysis focused on the infinite union mind. that appears in the problem. Students' struggles in As of this writing, we are conducting interviews determining what that union is equal to tell us a with mathematicians who do have rich conceptions great deal about their construction of this set. Be­ cause this set is not equal to the uncountable set 2 This construction is familiar to set theorists. See, for ex­ P(N), however, this investigation tells us little about ample, the proofthat the Cantor set is homeomorphic to the mental construction of P(N). This is a com­ 2No in [14J, the identification of the Baire space NN with pletely different matter to which we now turn. the irrationals in [6J, etc.

AUGUST 2004 NOTICES OF THE AMS 749 of uncountable sets to compare the mental mech­ [6] K. HRBACEK and T. ]ECH, Introduction to Set Theory, anisms they appear to use in developing these con­ M. Dekker, New York, 1978. ceptions with the process we have constructed and [7] A. W. MOORE, A brief history of infinity, Sci. Amer. 272 its encapsulation. Preliminary results are encour- (1995),112-116. [8] __ , The Infinite (2nd edition), Routledge, London , aging, and we hope to produce reports on this and New York, 1999. work in the near future. [9] 1. NEWTON, The Principia: Vol. 1. The Motion ofBodies A particularly interesting aspect of this current (translated into English by A. Motte in 1729. Transla­ work has to do with our suggestion that a count­ tions revised and supplied with an historical and ex­ able process can yield an uncountable set. Mathe­ planatory index by F. Cajori), University of California matically, this is not possible, but it may be that Press, Berkeley, 1934. in our minds we can make such a leap. We prOpose [10] H. POINCARE, Mathematics and Science: Last Essays that what makes the difference cognitively is en­ (J. W. Bolduc, trans.), Dover, New York, 1963. capsulation, which is not a mathematical tool but [11] C. STENGER, D. VIDAKOVIC, and K. WELLER, Students' in­ terview responses to a problem involving an infinite a mental mechanism. We believe that encapsulation iterative process (data for a study in progress), 2003. may just be a sufficiently powerful mechanism to [12] D. TALL and R. L. E. SCHWARZENBERGER, Conflicts in the allow the mind to make such a leap from the count­ learning ofreal numbers and limits, Math.Teaching 82 able to the uncountable. In future reports we hope (1978),44-49. to provide data and analyses that tend to support [13] K. WELLER,]. CLARK, E. DUBINSKY, S. LOCH, M. A. McDONALD, (or, as the case may be, not support) this conjec­ and R. MERKOVSKY, Performance and attitudes in courses ture. based on APOS Theory, Research in Collegiate Math­ ematics V, CBMS issues in Mathematics Education, Finale Volume 12 (A. Selden et aI., eds.), Amer. Math. Soc., Providence, RI, 2003, pp. 97-131. Some of the explanations we have given in this ar­ [14] S. WILLARD, General Topology, Dover, Mineola, NY, ticle are at variance with those of other commen­ 1970. tators oninfinity in mathematics, bothpast and pre­ [15] S. R. WILLIAMS, Models of limit held by college calcu­ sent. There is, of course, no issue here of lus students, ]. Res. Math. Ed. 22 (1991), 219-236. determining who is correct (whatever that may [16] __ , Predications of the limit concept: An appli­ mean). Rather, we hope that our explanations ex­ cation of repertory grids, ]. Res. Math. Ed. 32 (2001). hibit a coherence, unity, and simplicity that may render them worth thinking about when trying to understand how human beings can and do think about infinity. Perhaps more important, certainly from a prac­ tical point of view, is the hope that our explana­ tions of student difficulties with infinity will point to pedagogical strategies that can lead to im­ provement inlearning. The reason for our optimism is that explanations of other mathematical con­ cepts using these mechanisms and the totality of APOS Theory of which they are a part have led to effective pedagogy that has been reported in the literature. Whether a similar outcome will occur for infinity only time and future research and devel­ opment can tell.

References [1] C. B. BOYER, The history of the calculus, The Two-Year College Mathematics Journal! (1970),60-86. [2] E. DUBINSKY, K. WELLER, M. A. McDONALD, and A. BROWN, An APOS perspective on historical paradoxes and di­ chotomies of the infinite (under review). [3] A. BROWN, M. A. McDONALD, and K. WELLER, Students' con­ ceptions of infinite iterative processes (under review). [4] B. CORNU, Limits, Advanced Mathematical Thinking (D. O. Tall, ed.), Kluwer, Dordrecht, 1991, pp.153-166. [5] E. DUBINSKY and M. A. McDONALD, APOS: A constructivist theory of learning in undergraduate mathematics ed­ ucation research, The Teaching andLearning ofMath­ ematics at University Level: An ICMI Study (D. Holton et aI., eds.), Kluwer, Dordrecht, 2001, pp. 273-280.

750 NOTICES OF THE AMS VOLUME 51, 'NUMBER 7 Interviewwith JosephKeller

he moved to Stanford University, where he is now an emeritus professor. Keller has received many awards and honors throughout his career, including the von Karman Prize of the Society for Industrial and' Applied Mathematics (1979), the Timoshenko Medal of the American Society of Mechanical Engineers (1984), the National Medal of Science (1988), the National Academy of Sciences Award in Applied Mathe­ (1995), , Joe Keller at home in Stanford, CA. matics and Numerical Analysis the Nemmers Prize from Northwestern University (1996), and joseph B. Keller is one of the premier applied math­ the Wolf Prize (1997). ematicians of recent times. The problems he has What follows is the edited text of an interview worked on span a wide range of areas, including with Keller conducted in March 2004 by Notices se­ wave propagation, semiclassical mechanics, geo­ nior writer and deputy editor Allyn jackson. physical fluid dynamics, epidemiology, biome­ chanics, operations research, finance, and the math­ Early Years ematics of sports. His best-known work includes Notices: How did you get interested in mathemat­ the Geometrical Theory of Diffraction, which is an ics? extension of the classical theory of optics and in­ Keller: I was always good at mathematics as a spired the introduction of geometric methods in child. My father, who was not educated but was a the study of partial differential equations, and the smart guy, used to give my Einstein-Brillouin-Keller Method, which provided brother and me mathematical new ways to compute eigenvalues in quantum me­ puzzles when we were kids. chanics. Keller has had about fifty doctoral students Here's an example. A goose met and many collaborators. With a publication list of a flock of geese, and the goose over four hundred articles, he remains active in re­ said, "Hello, 100 geese." The search; his most recent paper appeared this year leader of the flock said, "We are in the Proceedings ofthe National Academy ofSci­ not 100 geese. But if we were ences. twice as many as we are, plus Keller was borninPaterson, New jersey, in 1923. half of that, and you, then' we He received his bachelor's degree from New York would be 100. How many were University in 1943. He was an instructor in physics there?" That was a typical ex­ at Princeton during 1943-44 and thenbecame a re­ ample, which I happen to re­ search assistant in the Columbia University Divi­ member. sion of War Research during 1944-45. After re­ In my high school, which was ceiving his Ph.D. from NYU in 1948, he joined the East Side High School inPaterson, faculty there and participated in the building of the New jersey, I had several good Courant Institute ofMathematical Sciences. In 1979 mathematics teachers. One of Keller at age 4.

AUGUST 2004 NOTICES OF THE AMS 751 them was Mr. Dougherty. After I left high school, Chevalley. One course I had was with Pauli. Of I didn't hear from him until maybe twenty years course many of the people were away during the later. I was a faculty member at the Courant Insti­ war, some at Los Alamos. tute. I got a letter from him saying that he hadjust I was able to continue studying at Princeton by purchased a copy of one of Morris Kline's books. teaching in programs in which the armed services On the dust jacket there was a curve, and he had put people to be trained as engineers before the end tried to figure out the equation of that curve, and of the war. After I was at Princeton for one year, he didn't succeed. He wrote 1943-44, it was recognized that the war would end that he knows that Kline is a before the students ever came out as engineers. The , very busy man, but perhaps I programwas terminated, so I had to find a new po­ could ask him about the equa­ sition. I went to the American Institute of Physics, tion of that curve. So I took which sent me to the Columbia University Divi­ the letter to Kline, and he re- sion ofWar Research. The positionwas working for membered Dougherty because the Office of Scientific Research and Development, Doughtery used to bring a in offices on the fiftieth floor of the Empire State team from our high school for Building. The work was to analyze the use of sonar the Pi Mu Epsilon examination, in submarine detection. That's what I worked on which was held at NYU every during the rest of the war. year. I had been a member of I had very good colleagues there, primarily physi­ that team. When Kline read cists. My boss was a physicist named Henry Pri­ Dougherty's letter, he said, makoff, and I shared an office with another grad­ "Joe, there is something funny uate student, Martin Klein, who is now a professor about that figure. I had given in the history of science at Yale. I got to meet a lot the draftsman a portion ofit in of people there who were working in various as­ the first quadrant, and he was pects of submarine detection. For example, I met supposed to flip it over to fill Conyers Herring, now an emeritus professor of Joe Keller, top right, with his out the whole picture. But he physics here at Stanford. brother Herbert B. Keller and did it incorrectly, and conse­ Incidentally, during the time Iwas working there, quently there is no simple their parents, early 1950s. equation that would give that an airplane hit the Empire State Building. It hap­ pened on a Saturday, and I was late to work. By the figure." Kline was amazed that time I arrived, the building was cordoned off and Dougherty had found this, and he sent him a copy I couldn't get in. That was a precursor of the World of the book as a reward. Dougherty was then teach­ Trade Center disaster, although this one was an ac­ ing at a state college in Pennsylvania. He was no cident. It was due to the fact that the cloud level longer at the high school-he had "graduated". was very low and the pilot couldn't see. I was always interested in mathematics. When I We worked there on the following kind of ques­ went to college, I thought I would major in math­ tion: How much sound is reflected back from a sub­ ematics, but the first year I took a course in physics. marine? We had devices, called projectors, which I found that so attractive that I switched my major are like underwater loudspeakers and which sent to physics. By the end of my stay I had majored in out soundwaves. The waves would hit whatever was both mathematics and physics. Then I got an in­ out there, and some of the waves would be re­ structorship in physics at Princeton-that was dur­ flected back and picked up by a device called a re­ ing World War II. ceiver. The sonar officer would listen, oftentimes Notices: Somebody told me that you were in­ with earphones, to the reflected sound, if there volved in a course thatEinstein was teaching. Is that was any. When he heard something, then he con­ true? cluded there was an object out there, and he would Keller: No. When I was at Princeton, I had two try to locate it precisely. The problem was to cal­ contacts with Einstein. Once I walked past him on culate how the waves traveled through the water, the street. Another time I went to a lecture by spread out, became weaker, hit the object, and Bertrand Russell, and Einstein was in the audience. then how much was scattered back, and what the I fell asleep, but Einstein didn't. So I figured it was strength of the signal would be when it got back easy for me, and hard for Ein~tein. to the receiving sonar. We wanted to know, for ex­ Notices: Did you hang around much with the ample, at what range would it be possible to de­ mathematicians in Princeton? tect a submarine, and what was the best frequency Keller: Oh sure, because during the war there to use? The projector sent out a beam, and the beam were very few graduate students, so I hung around spread out, and ifyou didn't detect anything, then with all of them. I was taking half mathematics and you would change direction and do it again. How half physics courses. I had courses from Lefschetz much should you move each time, through what and Tucker and Church and Bohnenblust and angle? How long should you wait for the signal to

752 NOTICES OF THE AMS VOLUME 51, NUMBER 7 come back? How deep is the submarine, and should you look down and up? There were all kinds of questions like that. We had a laboratory in Mountain Lakes, New Jersey. Occasionally I was sent out there to test the projectors and receivers. One of the nicest jobs I had was testing an "underwater flashlight". It was an underwater, handheld sonar device the size of a large flashlight, and it was to be used by troops who were coming up on a beach to locate land mines and other obstructions underwater. I would have to go out swimming in the lake, which was great in nice weather, to test this device. It sent out a signal, and when the signal bouncedback, I would hear it on earphones. With practice I could tell how far away the object was, and it worked with objects up to about 50 feet away. That was the fun part of the laboratory work. The theoretical work that I did in the Empire State Building was on the analysis of sonar, scattering from objects, and things like that, which is what I did for many years during the rest of my career. I wrote my Ph.D. thesis with Primakoff, who moved to NYU, although he left NYU before I finished, so Copyright. 1948, Courant ended up being my nominal advisor. What by Gabriele Wasow I did in my thesis was to use some of the methods we had worked on during the war in connection Cartoon illustrating a shock wave, from Supersonic with sound wave propagation. I did the analogous Flow and Shock Waves by Richard Courant and K. o. things with electromagnetic wave propagation. Friedrichs, Interscience Publishers, 1948. some results in that paper by an elementary Explosions and Shock Waves method, which surprised them, and that cemented Notices: Did you also work on underwater explo- my position. About fifteen years ago Spencer re­ sions? ' ceived the National Medal of Science, so I called him Keller: That was later, when I came to NYU. to congratulate him. I mentioned to him that the There had been a lot of work done there on ex­ first problem I was given to work on when I went plosions. Supersonic Flow and Shock Waves, by to NYU was that paper of his and Shiffman's. He Courant and Friedrichs, was the book that resulted said, "You know, Joe, you're the first person who from their study of all kinds of explosions. In a car­ ever mentioned that paper to me!" toon that appeared in the book [see upper right], When an explosion occurs under water, the ex­ it's possible to recognize Courant and Friedrichs. The cartoon was drawn by Gabi Wasow, the wife plosive gets converted into a gas. The gas is under ofWolfgang Wasow. He was a mathematician, a stu­ very high pressure, so it expands and produces a dent ofFriedrichs, who later was a professor at Wis­ bubble, which gets bigger and bigger, until finally consin. You'll notice that the cartoon is not bound the pressure in the bubble has gone down so low in the book. Probably someone thought it was too that it stops expanding. Then the water pressure frivolous. outside forces it back in. So the bubble oscillates. When I got to NYU, I worked on underwater ex­ I worked on that problem with a colleague, Ignace plosions with Bernard Friedman and Max Shiff­ Kolodner. A theory of that oscillation had already man. Shiffman and Donald Spencer had written a been developed by Rayleigh during the First World paper onwater entryby aerial torpedoes. The ques­ War. What we did was to modify the theory to in­ tion is, What are the forces exerted by the water clude the loss of energy by the shock waves that on the torpedo? In order to analyze that problem, are sent out from the bubble, which makes its os­ they studied a related problem. When part of a cillations decay in size. Also, when the bubble os­ sphere is in the water, they reflected that part cillates, the water motion interacts with the top and across the surface of the water. The reflection, to­ the bottom, and that causes the bubble to move. gether with the part under water, they called a It turns out that often the first shock wave that the lens. They studied the flow ofwater around a lens. bubble sends out weakens the plates of the ship, When I got to NYU, Shiffman and Friedman gave and the second one breaks them. Before the sec­ me that paper to read. I found I was able to obtain ond one occurs, the bubble will move. If you are

AUGUST 2004 NOTICES OF THE AMS 753 Keller: That's right; it spreads out, and Bikini was not near any inhabited place. By the time the waves reached anywhere with a reasonable population, the amplitude of the waves had gone way down. Later, after the Bikini tests, in 1950 or 1951, a commit­ tee was formed to look at tests of underwater ex­ plosion of atomic bombs. Among the questions were, What damage would they do to a ship and at what distance? The committee was asked to advise whether or not such experiments should be done. von Neumann was a member of the committee. He appeared at the first meeting and at the last meet­ ing. His point ofview was interesting. He said, "The Keller, right, with Vic Twersky at the Sylvania admirals want to have the tests, so we should vote Research lab in Mountain View, CA, 1954. yes; we should recommend it." Notices: Is that the conclusion you came to? Keller: Yes. We had spent a lot of time looking at questions about how strong the shock waves would be at the location of the ship, what they would do to the equipment, and other things. During the summer of 1950 I spent one month at the Argonne National Laboratory, which was an Atomic Energy Commission Laboratory, and then another month at Los Alamos National Laboratory. When I got to Los Alamos I was asked to study ex­ plosions that were so strong that the shock waves would go far into the atmosphere, where the at­ mosphere becomes less dense. Obviously, the pur­ pose was to study the effect of hydrogen bombs. They didn't tell me that, because I didn't need to know. I was given the previous work that had been done. One of the people who had worked on that left to right: Hirsh Cohen, George Handelman, before was Klaus Fuchs, who was ultimately in­ and Keller after lecturing at RPI, late 1950s. dicted for having given information to the Rus­ trying to do the damage, you want it to be attracted sians about the atomic bomb project. So I worked to the ship and be closer when it emits its second on that problem. That same summer, Peter Lax, who shock. We analyzed questions like that. was a student with me at the Courant Institute, was My advisor, Primakoff, was working on under­ also at Los Alamos. He hadbeen there in 1945, dur­ water explosion of atomic bombs in connection ing the war when he was in the army. During that with the proposed Bikini tests of the atomic bomb. summer I worked with him on numerical methods A question we were asked, among others, was, for solving equations of gas dynamics, and that ul­ Would that underwater explosion produce a timately meant nonlinear hyperbolic partial dif­ tsunami that could cause damage all around the Pa­ ferential equations. He continued doing that kind cific Ocean? We calculated that it would not cause of work, but I didn't. any damage far away. Notices: During this time when you did a lot of Notices: It wouldn't be strong enough? research related to war, did any moral questions Keller: Yes, with the size of bombs that were come to your mind? used then. Keller: Well, yes and no. During World War II Notices: The atomic tests that were done at everyone who did war work felt that the atrocities Bikini-were they underwater explosions? committed by our enemies justified whatever we Keller: No, they were done above water. But the did to stop them. And later, we all believed that point is, the explosion was done relatively near what we were doing was for good. We didn't think the water, and the blast wave pushes the water it would be used inIraq, for example. We all believed down and away. It could blow all the water out, then that any work we did would be primarily for down to the ocean floor, and that water would be the defense of the United States. So although peo­ moving outward. ple thought about it occasionally, there didn't seem Notices: But the force of the wave would dissipate to be any serious doubt about whether or not it was over the distance? the right thing to do.

754 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Notices: After the Vietnam War and even into the Keller: They ulti­ 1990s, there were a lot of debates among mathe­ mately sent me a docu­ maticians about military funding in mathematics. ment describing our in­ Keller: The Vietnam War was a breaking point. terview, and that's no Up until thenwe all believed that the American mil­ doubt in my file some­ itary would only be used for what we thought of where. But it was not re­ as the "right" purposes, the defense of the coun­ ally a problem, because try. After the Vietnam War, attitudes changed com­ later I was a member of pletely. By that time the shock wave and explosion the JASON group, which work that we hadbeen doing was over. But I worked was a group mainly of on electromagnetic wave propagation, which has physicists advising the many ramifications. For example, when I was work­ Defense Department. ing on those problems in the 1950s or early 1960s, Building ofthe I devised something called the Geometrical Theory Courant Institute of Diffraction, which was a method for solving all Pitching horseshoes at an outing kinds ofwave problems. At one meeting when I pre­ Notices: You were at with students in the late 1950s. sented this work for the first time, Kip Siegel from NYU even be- the University of Michigan got up and said, "Okay, fore the Cour­ ifyour theory is so good, canyou calculate the radar ant Institute started. back-scattering from a cone?" He was thinking of Keller: a truncated cone of finite size, because that would That's right. be the shape of the nose cone of a missile. There When I was an ' had been great difficulty'in calculating that. I ap­ undergradu­ plied my theory to it. It worked out very nicely and ate student, I gave exactly the experimental results. In that paper attended a I also looked at this question: If we were doing the class given by shooting and didn't want the opponent's radar to Courant. I had see our missile, what should we do to diminish the taken all the radar signal that would be sent back? I described undergradu­ methods for doing that. After that work was pub­ ate mathemat­ lished, it was picked up by aviation magazines. I ics courses, got feedback from the Air Force contract monitor, and I went to Cross-country skiing at Lake Placid with saying that the big brass were unhappy that I had take a gradu- George Morikawa, James J. Stoker, and a published that work, that it should have been clas­ ate course. I friend. sified. Those ideas became the basis for the so­ went with a called "stealth" technology. friend of mine, Harold Lewis, later a professor of Notices: Were there any other times when you re­ physics at Santa Barbara, who was in the same po­ ceived such letters from the military? sition. We attended Courant's initial lecture on Keller: After that I was more careful! I once got Methods of Mathematical Physics. At the end of the a dressing-down from the contract monitor be­ lecture we went up and told him we were under­ cause I always wrote the footnote, "Supported by graduate students, and we asked, "Can we sit in on the Air Force" or whoever was supporting me. I this course?" He said, "No, it's absolutely not al­ included such a footnote on a paper on the math­ lowed. However," he said, "if you sit in the front ematics of sports. There was a complaint, not row there, maybe I won't see you. But just because because I was giving away secrets, but because I can't see you doesn't mean you can't ask ques­ the Air Force or Defense Department didn't want tions." So we did attend the course, and obviously to be seen as supporting that kind of work. that was what he wanted, but he told us in that Notices: It seemed too frivolous? funny way to make evident the foolishness of the Keller: That's right. During the Vietnam War, university rules. when I was a faculty member at the uptown cam­ Notices: What was your impression ofCourant? pus of New York University, I, together with other Keller: I liked him very much. He always invited faculty members, signed an ad in favor of student us students to his house in New Rochelle to din­ demonstrations against the war. At that time I was ner-and to rake the lawn and do other chores! I visited by some agents; I don't know if they were took a few trips to Europe withhim. In 1950 or 1951 FBI or Army intelligence. They questioned me, be­ we visited various laboratories inEurope-inFrance, cause I probably then still had clearance. Germany, and England-that had worked on ex­ Notices: They just came and asked questions plosion problems during the war. We went to and left? That was the end ofit? interview people to find out what they had done,

AUGUST 2004 NOTICES OF THE AMS • 755 and we wrote a summary of our findings. That was one another. That made it especially attractive. very nice. Courant knew everyone, so when I tailed Also, Courant was helpful in that he worked hard along with him, I automatically got to meet them. on raising the money, getting research contracts, So, for example, we visited G6ttingen, and we and thus protected us to a certain extent from that passed the house ofHeisenberg. There was Heisen­ chore. Of course later we had to do it ourselves, but berg on the porch, and Courant spoke to him. for quite a while he and his staff did all the work. Courant had been helpful to Heisenberg when Getting contracts at that time had some significance Heisenberg was a young man. Heisenberg invited for faculty members. Our teaching load was origi­ us to come to a party at his house that night, which nally 12 hours a week. When we complained, we did. Friedrichs said that he had done his best work During that when he was teaching 12 hours a week. So thenwe trip our clear­ had to shut up! However, after a while the rule ance to visit a was that anyone who was doing research had to Frenchlabora­ teachjust 9 hours a week. Then, ifyou had research tory had not contracts that could pay for it, you could buy off come through, 3 hours and then get down to 6 hours a week, so while await­ which meant two courses. When people did get of­ ing it we went fers from other places, Courant worked hard to get skiing in them matched, so our salaries gradually moved Switzerland. I up. It was beneficial to all of us if one got an offer didn't know and his salary was raised. Courant could not keep how to ski, but it inequitable for very long, so ultimately we would Courant said, all get raises from one person's offer. "Oh, there's Notices: You had a long-running applied math­ nothing to it." ematics seminar at Courant. When did you start I rented skis that? and took a les- Keller receiving an honorary degree from son, and then Keller: I don't remember, but it must have been Northwestern University, 1988, with Bernard he took me up in the 1950s sometime. We always held the semi­ Mathowsky looking on. on the moun- nar on Friday afternoons, and then it was followed by tea, and that was very convenient, because then tain. I broke we could discuss the seminar topic at tea. Then it one pair of skis, and when I brought themback, the was followed by basketball. people apologized for having given me defective skis. Then I broke the next pair! Later we used to Notices: You went out to play basketball? go skiing all the time in Lake Placid in New York. Keller: We went out to playbasketball. Then we A whole group ofus would go-Courant, Friedrichs, went to Chinatown to make up for any weight we Stoker, Peter Lax, Anneli Lax, and lots of others. We might have lost playing basketball. Occasionally it did that every winter for years. was listed as the "Basketball Seminar" in the weekly Notices: You saw the Courant Institute from its bulletin. In the beginning, lots of young faculty beginnings to 1979. That must have been an ex­ members playedbasketball. As years wentby, most traordinary thing to witness. To what do you at­ of them dropped out. I was one of the last survivors. tribute the enormous success of the Courant Insti­ Until I came here to Stanford, I was playing bas­ tute? ketball at our regular weekly basketball outing. Keller: Partly due to Courant's ability to recog­ For the seminar we would get speakers from the nize good people, like Louis Nirenberg, who came Courant Institute, or from anywhere around the city as a graduate student from Montreal; Harold Grad, or the area, or people who were coming by. There a graduate student from Cooper Union; Peter Lax, were many visitors at NYU, because anyone going and later Cathleen Morawetz. Courant was able to from anywhere inEurope to anywhere inthe United attract all these good people and to recognize them States stopped in New York. We oftentimes had our andbe helpful to them so they wanted to stay. Many choice of speakers who were coming through. The people from other universities complained that seminar was in applied mathematics, and we often Courant was keeping all the applied mathematicians had people who worked in other fields. For exam­ at NYU, which they felt deprived them of applied ple, I remember we had an early talk on chaos by mathematicians. It was partly true, but on the other Bob May, who is now president of the Royal Society. hand other places weren't as congenial to applied He was then a faculty member at Princeton. He had mathematics at that time. People were offeredjobs. done numerical computation on some simple model I considered going elsewhere, but the Courant In­ of population growth that exhibited chaotic behav­ stitute was so congenial, and there was a group of ior. We always had a big audience, because there were us that worked on related things; we could talk to lots of people who were interested in applied

756 NOTICES OF THE AMS VOLUME 51, NUMBER 7 problems andpartial differential equations, so it was very effective. When I came to Stanford I contin­ ued the seminar, but without the basketball com­ ponent.

Geometrical Theory of Diffraction Notices: The Geometrical Theory ofDiffraction you mentioned is one ofyour major results. How did you come upon that? Keller: In World War II I worked on sonar, scat­ tering of waves from surfaces, like the surface of a submarine. When I came to NYU, I read the work of a colleague there, Rudolf Luneburg. Luneburg had been a student at Gbttingenbefore World War IT, and Receiving an honorary degree with luis Bonilla and the he helped Jews escape from the Nazis into Hol­ president of the Universidad Carlos Tercera, Madrid, 1997. land. Ultimately he came to the United States, and The Einstein-Brillouin-Keller Method he worked for the American Optical Company in Rochester, New York, for a while. Then he came to Notices: One of your major results was the Ein­ NYU, and he wrote a set of lecture notes-he also stein-Brillouin-Keller method. wrote some notes at Brown University-on the elec­ Keller: Yes. I devised a certain method for solv­ ing eigenvalue problems in quantum mechanics. tromagnetic theory of optics. I had found during my After I devised it a colleague of mine pointed out work in sonar that it was possible to describe the that one of the formulas had a great similarity to reflection of waves from objects by means of rays something that Einstein had done. So I mentioned and geometrical calculations. I could figure out the in my paper that Einstein had done a certain thing, strength of the signal reflected from an object just and also in the course of my method I used some by calculating how the rays hit the object and were arguments of Leon Brillouin. So subsequently peo­ spread apart and so on. In his theory Luneburg also ple called that method the EBK method, Einstein­ emphasized this idea. But then, about 1950, I did Brillouin-Keller. I did that work in 1953 and pub­ a calculation together with a student, Albert Blank, lished it in 1958. In that work I developed a certain and that calculation showed that indeed there were index, which subsequently was rediscovered by waves reflected, as Luneburg's theory said, but in the Russian mathematician Maslov, and ever since addition there were waves that weren't predictedby it has been called the Maslov index. the theory, waves that came off the edge of an ob­ Notices: What was this index? ject. I realized that that is a general feature and that Keller: It was an index that, in my theory, had ordinary geometrical optics was inadequate to de­ to do with the number of times a path touched a scribe all the rays. I introduced additional rays, caustic curve. A caustic is a point in optics where which accounted for the waves coming from the rays come together. That number plays a role in cal­ edges, for waves coming from corners, and for culating eigenvalues or, in quantum mechanics, other kinds of things. calculating the energy levels. Maslov discovered I found it is possible to use these rays to con­ the same number, and the Russian mathematician struct asymptotic approximations of solutions to V. 1. Arnold named it the Maslov index. Jean Leray, Maxwell's equations and other wave equations. the famous French mathematician, complained That led to a general theory for linear partial that it was really the Keller index, so for a while peo­ differential equations, namely, a method for con­ ple called it the Keller-Maslov index, but the Keller got dropped. Leray wrote a letter to a journal about structing various aspects of the solutions by this. I subsequently met Maslov, and he told me that geometrical methods. That work was subsequently he was sorry he didn't know about my work when developed much further by Donald Ludwig, Robert he did his work, which is something that happens Lewis, and Cathleen Morawetz, who were at NYU. all the time. Of course there were precursors in the work by Notices: How did you get interested in problems Gerard Friedlander in Cambridge, Vladimir Fock in in quantum mechanics? Russia, and others. My geometrical theory incor­ Keller: I had studied physics as an undergrad­ porated a lot of the work of those other people. uate and graduate student, and the problems of Later, Lars Hbrmander in Sweden, Michael Taylor quantum mechanics were long-standing. I found in this country, and many others carried it much that the methods that I had devised to solve wave further and made a more comprehensive theory of propagation problems, like scattering and diffrac­ linear partial differential equations in which these tion and so on, could with a slight twist enable me geometrical ideas played a helpful role. to solve quantum mechanical problems. This is

AUGUST 2004 NOTICES OF THE AMS 757 Einstein's paper. It hadbeen completely ignored up to then. · Notices: Why was it ignored? Keller: That's not clear. It described a more el­ egant way of formulating quantum conditions, and it also had precursors of concepts of chaos in it. But somehow or other it was ignored. It was pub­ lished in the Proceedings of the German Physical Society. be- Pure versus Applied cause the wave function satisfies a differen­ Notices: How do you see the supposed dichotomy tial equation like the differ­ between pure and applied mathematics?AtCourant, ential equations that I had the emphasis was on applied mathematics. Now been solving for wave prob­ you are in a pure mathematics department. lems. So it was natural. My Keller: Before 1900 there wasn't any sharp dis­ Signing the membership book long-term colleague Sol Rubi- tinction; many mathematicians worked on appli­ on becoming a Foreign Member now and I illustrated the cations. Poincare, Hilbert, and others did both pure of the Royal Society, 1988. method by applying it to a and applied mathematics. But from 1900 to about number of examples. One of 1950, a schism developed. I believe the schismwas the examples was finding the eigenvalues and greater in the United States than inEurope. But since eigenfunctions of the interior of a closed curve. In 1950, pure mathematics departments in the United quantum mechanics this corresponds to the mo­ States have begun to incorporate some applied tion of a billiard ball bouncing off the sides of a mathematicians. Here at Stanford the department billiard table. We introduced two kinds of solu­ has incorporated more applied mathematics, and tions. One corresponds to motions inwhich the ball the interests of people have shifted a little bit. goes almost along the boundary. We called them Notices: Do you think the schism is closing? "whispering gallery modes", after a phenomenon Keller: A little bit. String theory and gauge field that hadbeenknown for manyyears at Saint Paul's theories in theoretical physics have provided an­ cathedral inLondon. A person could put his mouth other link between mathematics and science so near the wall of the cathedral and speak, and he that many mathematicians see scope for what they could be heard by someone at the opposite side of are doing in the realm of physics. Also, many de­ the cathedral, but not by anyone in between. The partments have been hiring applied mathemati­ explanation, given by Lord Rayleigh, was that the cians because they feel they provide a needed link sound waves stayed in a layer, hugged the wall, and with engineering departments. But there is still a came around to the other side. So when we dis­ wide gap. I would- say that the attitude of pure covered these solutions for more general shapes, mathematics departments toward applied mathe­ we called them "whispering gallery modes", and matics has become more friendly. But when it that's what they are called nowadays. Similarly, we comes down to the question of whether to hire found another kind of mode that pertains to a ball another algebraist, or a person who works in bouncing back and forth inside the curve. We called applications of differential equations in fluid these "bouncing ball" modes. That terminology mechanics, it's not so clear how they will come has caught on, and that's what they are called down. nowadays. Those modes have beenused in solving Notices: Why do you think the schism was less in wave guide problems, designing laser mirrors, and Europe? things like that. Keller: I don't know the answer. In France, for ex­ Notices: There was a paper ofEinstein's thatfig­ ample, where Bourbaki developed, mathematicians ured in your work on the EBK method but that had have beenvery pure. But, stimulatedbyJacques-Louis been neglected for a long time. Lions, a big group of young Frenchmen who were Keller: That's correct. I had a colleague at NYU, trained in pure mathematics and then began work­ a physicist, Fritz Reiche. He had occupied the chair ing on applied mathematics have developed a very of physics at the University ofBerlin after Einstein. strong school of applied mathematics. Before that, He was at NYU first as a faculty member and then there weren't so many applied mathematicians in as a research associate; he had retired already. France. In England there has always been a strong Upon reading my manuscript, he was reminded of applied mathematics component. The same is true Einstein's 1917 paper, which he brought me. I read inRussia. Much ofmyworkwas studiedbyRussians, Einstein's paper, and I found that one of his for­ so whenever Russians came to this country, theywere mulas was similar to one of mine. I mentioned this muchmore familiar with mywork thanmanyAmer­ in my paper, and that's how people came to know ican mathematicians. Russians were doing pure

758 NOTICES OF THE AMS VOLUME 51, NUMBER 7 and applied mathematics together. In the United States the applied mathematicians would know about mywork, but the pure mathematicians would not. I had lots of correspondence and interactions with Russian scientists, mathematicians, and physi­ cists, which was very nice.

Runners and Crawlers Keller: I was always interested in athletics-I men­ tioned the "Basketball Seminar". One time I de­ vised a theory ofrunning. How should a runner ex­ pend his oxygen supply in a race to run it in the shortest possible time? Initially the runner has a certain amount of oxygen in his body, and he breathes at a steady rate. What limits the runner's speed is running out of oxygen, it turns out. So I made up a theory for how he should dole out his stored oxygen in order to run at the fastest rate. It led to a nice calculus ofvariations problem. I fit the theory to data on the world records to determine various physiological constants: how much oxygen is in the body, the breathing rate, the friction con­ stant. I plotted the theoretical average speed in a race against the distance. This gave a very nice curve. The average speed went up to a maximum and then down. The first part of the curve corre­ sponds to sprints: there is no strategy; you just run as fast as you can. But I found that when the race is longer than 291 meters, the average speed goes down. The world's records lie right on the curve I found. Notices: Did any runners use this theory? Keller: No, but it was discussed invarious news­ papers. Here is what a runner should do. Up to 291 Keller and family skiing in Alta, Utah. Top, with meters he should run all out. But for longer dis­ wife Alice, and below, with Alice, daughter tances, his speed as a function of time starts at zero; Sarah, and Alice's daughters, Gayle and Margot. then he gets up to speed and runs at a constant speed throughout the race. A second or so before Keller: I had several very good students in that the end he should slow down. That's what I found. field. One was John Rinzel, who is now a profes­ Now, why should he slow down? The answer is, he sor at the Courant Institute, and we worked on the should run out of oxygen~ and then he should coast propagation of nerve impulses. There are some during the last bit. equations, the Hodgkin and Huxley equations, Notices: Wouldn 'tyou think you should speed up which describe the propagation ofnerve impulses, at the end? and those equations are complicated to solve. We Keller: That's what the runners often say. No, solved a simplified version of them and were able because you shouldn't have any oxygen left at the then to describe various interesting features of end of the race. You should have used it all up. In nerve conduction and nerve pulse propagation. fact, it should all be used up a hair before the end Later I had another student, Ken Miller, who of the race; then you should coast. If you think of startedworking with me, and he ultimately finished driving ten miles on a limited amount of gas, you with an advisor in the neuroscience department should speed up, run along, and then run out of here at Stanford. He was interested in optical dom­ gas right before the end, because ifyou run out right inance columns in the visual cortex. It turns out that at the end, then that last bit of gas isn't doing you some cells in the visual cortex respond only to any good. You should have used it up to go a hair right-eye stimulation, and others right nearby re­ faster. Some coaches said, uOh, we knew it all spond only to left-eye stimulation. Those cells are along." And others said it was nonsense. arrayed in columns, and so they are called optical Notices: What kinds of problems have you worked dominance columns. It turns out that in a newborn on in the life sciences? kitten these cells respond to both right-eye and

AUGUST 2004 NOTICES OF THE AMS 759 left-eye stimulation. But after angles to it. That was contrary to the known the­ six or twelve weeks, these op­ ory of friction. tical dominance columns de­ Notices: How do you choose things to work on? velop, so that the cells that were Keller: That's a good question. First of all, I have originally responsive to stimu­ to understand the phenomenon, so that limits me lation from both eyes become right away. Then I have to recognize there is a responsive only to one or the mathematical aspect to it, and I have to be able to other. It has to do with the way make some progress on it. Oftentimes it has hap­ in which synapse strength pened that I have had students or colleagues or changes with usage. This postdocs who are interested in a certain subject and change only happens when the come around with problems, and I would see if I eyes are exposed to light. It could help them. The hard part is to pick out prob­ does not develop if the kittens lems that are interesting and for which the results are left in the dark. Miller made would be significant-problems that are not math­ up a model having to do with ematically impossible, on the one hand, nor math­ how the synaptic strength ematically trivial on the other hand. It has to be the changed with time and with right order of difficulty so that it is possible to make continued stimulation, and I progress, but it's not just routine so that anyone helped himwith it. It was tricky" could do it. because the stimulation that is Notices: Isn't it difficult to learn about all these Joe Keller hiking in France, applied is somewhat random, different areas? 1996. and that randomness had to be Keller: Sure, but that's the fun too. I find that taken into account in making having studied physics as an undergraduate and up the model. He is now a professor at the Uni­ graduate student and continuing to do that has versity of California at San Francisco in neuro­ enabled me to understand a lot of things that science, although he started out as a physicist. would have been prohibitive to learn otherwise. Bi­ I have worked on other biological problems: ology is a whole new discipline, and I have only , crawling of worms. learned those small areas where I have been able Notices: What is the question there? to work. But in coming years we will see many Keller: The way a worm crawls, sayan earth­ mathematicians who studybiology along with their worm, is that it stretches and contracts, stretches mathematics so that they will be right at home and contracts. How does it do that? If we want to with it. move, we have bones that enable us to do it. But a Notices: How have you seen computers change worm is soft; it has no bones. How does it extend applied mathematics? itself? It's like toothpaste: the worm has some mus­ Keller: Computing has become an automatic cles that go around its body, and when it squeezes part of applied mathematics, another technique them, it sort of extrudes, just like when you squeeze like analysis and differential equations. On the one toothpaste. Then it relaxes those muscles and con­ hand it's being used as it was in the old days to cal­ tracts other, longitudinal, muscles to pull up the culate results obtained by analytical methods. But, rear. It does that again and again, so it is sending going beyond that, computing is used to provide waves of expansion and contraction along its body. numerical results for problems that we can't do by To analyze that motion, we formulated an opti­ hand or by ordinary analysis. One of the big areas mization problem: What is the fastest that the where that is the case is fluid flow that involves nir­ worm can crawl, and what motion should it make bulence. There are now methods to compute so­ in order to do that? I worked on this with a very lutions of some of those problems, but the hard­ good postdoc, Meira Falkovitz, who came here from est problems. are still too hard. We need further the Weizmann Institute in Israel. The pictures we mathematical insight. The insight won't tell us the got looked very much like the photographs show­ solution to the fluid flow problem but will enable ing how worms actually crawl. us to develop new computational methods that Notices: A snake moves differently. will make more complex problems accessible to Keller: A snake has a backbone, and therefore computers. In that realm, the job of the mathe­ it can't stretch. So what it does is push off from matician is to assist the computer or to enable the pebbles and grooves and twigs. It needs something computations to be performed on problems that to push against. If you put a snake on a glass plate, are otherwise too difficult for ordinary computa­ it can't get anywhere, because it has nothing to push tional methods. That's where there will ultimately on. I studied that motion at one time, and I found be a lot of developments. that if a snake were to move on a flat surface, there would have to be some friction not only in the di­ rection opposite to its motion, but also at right

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construction offinitely gener­ ated groups which cannot be embedded uniformly in a Ililbertspace(Gromov) andre­ lated counterexamples to the Baum-Connes conjectures for group actions. However, it is in applications in theoretical computer science where ex­ panders have had their major impact. Among their applica­ tions are the design of explicit superefficient communication networks, constructions of error-correcting codes with very efficient encoding and de- coding algorithms, deran- domization of random algo­ This Ramanujan graph has 80 vertices,which is close to the largest known planar rithms, and analysis of Ramanujan graph of 84 vertices. Its girth is 5, its expansion constant 1/4 (as algorithms in computational indicated by the shaded circle), and Al has been calculated by A. Gamburd to be group theory (see for example 2.81811 ... It may be constructed by shrinking pentagons on a dodecahedron. [R-V-W] and the references therein). By a graph X = (V, E) we mean a finite set V ofver­ The formal definition of the mostbasic expander tices and a set E of pairs of these vertices called is as follows: Let k ~ 2 be an integer, and let X edges. In words, an expander is a highly connected be a k-regular graph (that is, each vertex v E V sparse graph X. It is this apparently contradictory has exactly k neighbors). The Cheeger constant h feature of being both highly connected and at the of X is defined to be same time sparse that on the one hand makes the loFI existence of such graphs counterintuitive and on the h(X) = 0 ~ v min(IFI,IV\FI)' other hand makes them so useful. Their uses inpure =f =f mathematics include, for example, in combinatorics the explicit construction of graphs with large girth where aF is the set of edges running from F to its (this being the length of the shortest nonback­ complement V\F and Iof I is its cardinality. X is also tracking closed circuit) and large chromatic num­ called an [n, k, h] expander where IXI = n. Note that ber (this being the least number of colors needed h > 0 if and only ifX is connected. Also, if IFI ::;; ~, to paint the vertices so that adjacent vertices have then /aF/ ~ h/F/, so that if h is not small, then distinct colors), and in functional analysis the every such subset F has many neighbors outside F (hence the name expander). By expanders we really Peter Sarnak is professor of mathematics at Princeton have in mind a sequence of such [n, k, h] 's with k University and New Ydrk University. His email address is and EO > 0 fixed, h ~ EO, and n - 00. It is easy to [email protected];nceton.edu. see that for this to happen k must be at least 3,

762 NOTICES OF THE AMS VOLUME 51, NUMBER 7 which we assume henceforth. That h ~ EO > 0 en­ sures that the graphs are highly connected, while Frequency distribution of eigenvalue Al of adjacency k being fixed (and lEI = k2n) ensures that they are sparse. matrices in samples of It is perhaps surprising that expanders exist. The random 3-regular graphs of size N first proof of their existence by Pinsker is based on (data supplied counting arguments. Consider the probability space by T. Novikoff) of all Xn,k'S (that is, k-regular graphs on n marked vertices) where each such Xn,k is chosenwith equal probability. Then for k ~ 3 there is E(k) > 0 such that the probability that h(Xn,k) ~ E tends to 1 as n - 00. For applications one wants explicit construc­ tions. A useful means to achieve this is the following spectral method. Given an X = X n k, let A be the n x n symmetric matrix whose ro~s and columns are indexedby the vertices v E V and for which the I I v, w entry is 1 if v is joined to w, and 0 otherwise. 2.75 2.80 The vector (1,1, ... ,1) is an eigenvector of A with eigenvalue k. The eigenvalues of A are real and lie asymptotically Ramanujan. Some very interesting in the interval [-k, k]. Let Al (X) denote the next­ numerical experimentation (T. Novikoff, to-largest eigenvalue of A after k. The following in­ http://www.math.nyu.edu/Courses/V63.0393/, equalities, which are the discrete analogues of in­ honors mathlab, projects) indicates that the prob­ equalities of Cheeger and Buser in differential ability that the random graph is Ramanujan is geometry, relate h(X) to the gap between Al (X) slightlybigger than 0.5, corresponding to the skew­ and k and are due to Tanner, Alon, and Milman: ness in the Tracy-Widom distribution in random matrix theory (specifically the "Gaussian Orthog­ k - ~1(X) ::; heX) ::; ~2k(k - AI(X)). onal Ensemble"). In [R-V-W] it is shown how one may use their Thus Xn,k is an expanding family if and only if it novel "zig-zag graph product" (a notion related to has a uniformlower bound on the spectral gap, and the semidirect product of groups when the graphs the larger the gap the better the expansion. Using are Cayley graphs) to give new explicit construc­ this and quotients of explicit infinite discrete tions of expanders. Their construction of the fam­ groups which enjoy Kazhadan's property T (which ily is inductive, with the zig-zag product being is the property that the trivial representationis iso­ used at each step. It has the extra flexibility that lated in the space of all unitary representations), allows them to construct explicit expanders that ex­ Margulis gave the first examples of explicit ex­ pand small sets almost optimally (they call these panders. There is a limit as to how big the spectral "lossless expanders"). This extra twist turns out to gap can possibly be (Alon-Boppana): For k fixed be very useful in a number of applications, and this feature cannot be deduced from a spectral gap liminf Al (X k) ~ 2~. analysis. n-.oo n' References The liminfis taken over all X n k'S. An Xn k (or rather a sequence of such Xn,k'S with n - (x< k fixed) is [L-P-S] A. LUBOTZKY, R. PHruJps, and P. SARNAK, Combinatorica 8 (1988),261-277. called a Ramanujan graphif Al (Xn,k) :::; 2.J7(=T. In [M] G. MARGUliS, Problems Inform. Transmission 24 (1988), view of the above, such a graph is optimal, at least 39-46. as far as the spectral gap measure of expansion is [R-V-W] O. REINGOLD, S. V ADHAN, and A. WIGDERSON, Ann. of concerned. It is a pleasant fact that if k = q + 1 Math. 155 (2002), 157-187. with q a prime power, then Ramanujan graphs exist. The constructions are in terms ofvery explicit descriptions as Cayley graphs of PGL(2, IFq) [L-P-S], [M] (Cayley graphs are those whose vertex sets are the elements of a group G and whose edges correspond to g - ggI with gI in a set ofgen­ erators). For k not of this form it is an interesting question as to whether Ramanujan graphs exist. J. Friedman has recently shown that for k fixed and E > 0 the probability that Al (Xn,k) :::; 2.J7(=T + E tends to 1 as n - 00. So the random graph is

AUGUST 2004 NOTICES OF THE AMS 763 Book Review

CountDown Reviewed byDaniel Ullman

CountDown What makes these Steve Olson competitors dif­ Houghton Mifflin Co., 2004 ferent from other 244 pages, $24.00 teenagers? Does ISBN 06-1825-1413 intellect arise from nature or It is a modern American obsession to follow nurture? Is math­ competitors innational championships. The much­ ematical competi­ heralded documentary film Spellbound follows tion healthy? Why eight- children competing in the National Spelling are there so few Bee. The New York Times bestseller Word Freak ex­ girls? And why so amines the strange lives of competitive Scrabble many Asian players. American television coverage of the Americans repre­ Olympics focuses more on the athletes ("Up close senting the USA? and personal!") than on the athletics. Witness also The author is the current fascination with the popular television not a mathemati­ show American Idol, which pits aspiring young cian, and the singers against each other in competition before book is not writ­ caustic judges. ten for mathematicians. In particular, mathemati­ If you did not enjoy Spellbound, then you very cians may feel frustrated at the author's rough likely would not enjoy the similarly titled Count sketches of the contestant's solutions to the IMO Down, the story of the six American competitors problems, which omit technical details necessary in the 2001 International Mathematical Olympiad for a complete understanding. Nonetheless, the (IMO), held in the Washington, DC, area. The author, book successfully explains to a nonmathematical Steve Olson, has put together a thoughtful and en­ audience why mathematics is something that can gaging book, tackling with careful research and be enjoyed. It succeeds also in conveying positive appropriate balance the many critical questions attitudes and dispelling negative stereotypes about that naturally arise in any discussion of the IMO: mathematically talented people. Olson has organized his book into six main Daniel Ullman is professor of mathematics at George chapters, each ofwhich is devoted to one of the six Washington University. He was deputy leader of the U.S. team members, one of the six 2001 IMO problems, delegation to the 1991 and 1992 International Mathe­ and one of what he sees as six critical attributes matical Olympiads in Sweden andRussia and was the DC director ofthe 2001 IMO in Washington, D.C. His email ad­ that these competitors display. A short interlude dress is dull man@gwu. edu. separates the first three of these chapters from the

764 NOTICES OF THE AMS VOLUME 51, NUMBER 7 last three, just as the IMO competitors get a night's experience much evidence supporting this model, rest betweenwork on the first three IMO problems and it allows me to understand many phenomena on the first day of the competition and the final that I observe in the classroom. three problems on the second. There are also a cou­ Olson continues in three more chapters to in­ ple of chapters at the beginning of CountDown ex­ troduce the three remaining participants, the three plaining how the IMO works and a couple of chap­ remaining 2001 IMO problems, and the attributes ters at the end revealing how things turned out in of creativity, breadth, and "a sense of wonder", by 2001. which he means, if I understand him, the aesthetic appreciation of mathematical beauty. In that chap­ Attributes ter, Olson relates the already legendary story ofAn­ The six attributes that Olson discusses are insight, drew Wiles's proof of Fermat's conjecture. This competitiveness, talent, creativity, breadth, and a seems to digress from the theme of Count Down, sense of wonder. In discussing these attributes, although it is true that Wiles gave an inspiring Olson is at his most successful. Each attribute is keynote address at the closing ceremony of the investigated in the context of a single competitor, 2001 IMO, and the gold medalists had the distinct a single 2001 IMO problem, and recent social sci­ and moving honor of receiving their medals at that ence research. ceremony (in the concert hall of the Kennedy Cen­ In the chapter on insight-the word is to be un­ ter of Performing Arts) from Wiles himself. derstood in a literal sense: the ability to see images with the mind's eye-Olson notes the importance Problems of visualization in mathematics. Indeed, mental In CountDown, the IMO problems are reprinted ver­ imagery is a critical aspect of mathematical abil­ batim, and their statements are explained. Even the ity, and this may explain why manymathematicians students' solutions are discussed to a degree, with are interested in music rather than visual arts: the some details postponed to an appendix and some development of the mind's eye reduces depen­ details suppressed entirely. Olson's judgement, it dence on the visual eye. Olson reflects on the work seems, is that complete solutions to the problems of psychologists Roger Shepard, Beth Casey, and would have been over the heads of his audience. others, who argue that visualization is a special cog­ Reluctantly, I confess that I concur with his view. nitive ability. Olson shows how this ability helped In not offering the details, though, "he does not one participant solve the first 2001 IMO problem. give his readers a full chance to appreciate math­ In the chapter on competitiveness, Olson takes ematics. The best one can say is that he gives his on the writings of Alfie Kohn, the "great guru of readers a chance to appreciate the appreciation of anticompetitiveness", who argues that competi­ mathematics. To Olson's credit, though, he does tion is a disease of our culture and is harmful to convey the impression that mathematical reason­ competitors. The IMO is a competition, and the ing is just like any other kind of reasoning and that participants find it immensely stressful. Is this all people have the capacity to learn, understand, healthy? Olson tackles this question head-on and and enjoy mathematics. And even if details are shows how the competitiveness of one participant missing, mathematical argument is by no means engaged him in mathematics and helped him solve omitted. In fact, for most of his readers Olson will the second 2001 IMO problem. have engaged them in mathematical reasoning to In the chapter on talent, Olson takes on the age­ a degree very likely exceeding any other reading old and politically charged question of whether they have ever done. Clearly Olson himself ad­ intellectual talent is innate or acquired. I find his mires these young people and is moved by their treatment fair and balanced. In the end, he argues beautiful and clever solutions to the IMO prob­ that the question hardly makes sense. "[G]enes are lems. 100 percent responsible for our traits, and expe­ riences are also 100 percent responsible for our Mathletes traits." This is reminiscent of the question of The six members of the USA 2001 IMO team were whether light travels as particles or waves. My own Tiankai Liu, Ian Le, David Shin, Oaz Nir, Reid Bar­ view is that it is prudent to subscribe to the model ton, and Gabriel Carroll, listed here in the order in that intellectual talent is acquired. We judge mod­ which they appear in CountDown. Where do these els not on the basis of their correctness per se but talented youngsters come from? What kinds of on the basis of their usefulness, their explanatory families? Who are the mentors, the teachers? When and predictive value. I choose to believe that stu­ was their talent first noticed? What other inter­ dents can learn, that studying pays off, that a stu­ ests do they pursue? dent's potential cannot always be predicted, and In the six central chapters Olson introduces us that we have free will. I maybe wrong about all this, to each of these young men, to their families, to their but I am not sure that I can live comfortably in the teachers, to their communities. His main message world of the alternative. Moreover, I find in my is that no stereotype appropriately captures these

AUGUST 2004 NOTICES OF THE AMS 765 kids. While I agree with him, it may be important of a type quite different from what is required to to point out one feature that they do have in com­ succeed at, say, the National Spelling Bee, where mon: In each case, some mentor-perhaps a judging the correctness of a solution is a routine teacher, a parent, or a friend-recognized their ex­ matter, requiring knowledge of the 26 letters ofthe traordinary potential and provided them with a alphabet and nothing more. The IMO promotes program outside of school to motivate their inter­ creativity and development of ideas rather than est and develop their talent. "No child left behind" memorization and computation. Inthe end, spelling has a mirror image for gifted students: "No child is justnot that important. What is important is rea­ prevented from going ahead". While we can ap­ soning, inventiveness, and communication, pre­ plaud the weak student who keeps up, we should cisely what the IMO promotes. not applaud the gifted student being kept down. I have observed that Olympiad team members Motivation tend to come from small families; they have fewer The next generation of mathematicians will come siblings on average. This supports the contention from young people who possess both the talent and that the environment in which a student is living the inclination to do mathematics. We must pay at­ is as important as a student's native ability. It is tention to both elements. The mathematical pre­ hard to see how children with few siblings could cocity of the Olympiad competitors is something have greater innate mathematical ability, but it is to behold, but more rare yet is their mathematical not hard to see how parents of a gifted only child motivation. It is no miracle that seventh-graders may have the resources to provide the special en­ who read number theory texts over summer vaca­ vironment that their child needs. tion develop their mathematical abilities to a higher degree than their classmates. The miracle here is Interlude that such students find the motivation, the support, An anecdote from the 1991 IMO in Sigtuna, Swe­ the rewards to engage in this kind of summer ac­ den. The participants faced this challenge: tivity in the first place. Problem 6: Given any real number a > 1, con­ This is why we need the Olympiads. Mathemat­ struct a bounded infinite sequence Xo, Xl, X2, ... ics competitions allow us to reward and celebrate ~ such that IXi - X j II i - j Ia 1 for every pair of dis­ mathematical talent in the same way that athletic tinct i, j. competitions allow us to reward and celebrate ath­ Lenny Ng, an American competitor (now a post­ letic talent. Young people naturally want to excel doc in mathematics at Stanford), provided a com­ and be recognized for their excellence. In our cul­ plicated construction accompanied by certain de­ ture, this draws children toward athletics. To a tails suggesting that the constructionworked. The written work encompassed a mere two pages, but lesser degree, it draws children toward music and the graders were stumped. Should this be regarded theatre, where performance is inherent. But math­ as an essentially correct solution or not? Should this ematical writing does not make a very good spec­ get full credit, nearly full credit, or no credit at all? tator sport, and we have a more difficult taskwhen Did the construction even work? No one was quite trying to acknowledge accomplishment in our dis­ certain. Eventually, the grading of the entire com­ cipline. petitionwas complete and the scores were posted, Gender except for this one problemfrom this one student. The graders convened a special session to exam­ Four hundred seventy-three young people repre­ ine Lenny's solution. A dozen mathematicians pon­ senting eighty-three countries participated in the dered the construction for several hours. A deci­ 2001 IMO. Of these, 44S were boys and 28 were girls. sionwas finally rendered: Because complete details The USA has competed in twenty-nine IMO's, start­ were not provided, Lenny would get only 3 out of ing in 1974. During that time a total of 127 young 7 points for this solution. people have had the opportunity to represent our I asked Lenny recently whether his solution to country at this event. Of these, 126 were boys and that problem 'had in fact been correct. Admitting just 1, Melanie Wood, was a girl. (Melanie served as that he did not recall the details, he responded: "I a coach for the 2001 IMO team from the U.S. and is remember being convinced that the solution was a central character in Count Down. She is a recent correct. However, there was a crucial bit of hand­ winner of the Morgan Prize for undergraduate re­ waving involved that I hoped would be obviously search and will be a doctoral student in mathe­ true to the graders. It seemed true to me. Needless matics at Princeton in the fall.) There is some irony to say, it turned out not to be obvious at all, maybe in the fact that, while we inthe U.S. regard ourselves not even true." So we still do not know. as more progressive than many other countries in I tell this story occasionally to nonmathematicians our beliefs about the equality of women, we have in order to convey the nature of the IMO: The IMO done a particularly poor job of promoting girls to calls for sophisticated, subtle, high-level reasoning, the top ranks of mathematics competition.

766 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Why is this? It is not an easy question to answer. problems are not mere exercises to be solved in one Once upon a time, adults instructed children that routine step. Participants get plenty of time; it is mathematics was for boys and not for girls, and no not a race, but the problems are challenging. Not one can be surprised that girls did not advance in every schoolchild is prepared to handle an mathematics in such a climate. To the extent that Olympiad, but many more ought to learn. In Rus­ such attitudes persist, girls continue to face ob­ sia, there is a network of regional and junior Math­ stacles today. But most teachers nowadays are sen­ ematical Olympiads. I would like to see the devel­ sitive to the dangers of stereotyping; the message opment of a similar network in this country, where that math is not for girls is no longer so overt. Yet currently the national Olympiad (USAMO) is of­ the Olympiad programs continue to be dominated fered to only about 250 high school students each by boys. Olson debunks several of the easy expla­ year. Because the grading of Olympiads is time in­ nations, such as the one that links mathematical tensive, the preliminary exams are best handled re­ ability to the Y-chromosome. Unfortunately, this gionally. There is no reason not to begin with stu­ points the finger in a yet more unpleasant direc­ dents in middle schools, although the questions tion: our American culture. The power of culture need to be at an appropriate level. A network of over us is too strong to be overcome by public re­ Olympiads would help promote mathematics as a lations messages. We can and should tell students discipline of reasoning rather than computation. I that girls can do math too, but on television the high propose a United States of America Junior Mathe­ school student who is good in math is a boy geek; matical Olympiad (USAJMO), whose competitors the girls do not emulate him or even date him. We would be top middle school performers from re­ can and should develop programs to advance girls gional or state junior Olympiads. Some may say that in mathematics, but when Barbie says, "Math is Math Counts already does this, but Olympiads are hard!" the battle may be lost. We can and should fundamentally different and in myview place value show positive images of successful, female math­ on a more critical attribute: the ability to create, to ematicians, but when Mom tells her daughter that reason, and to express results in a persuasive writ­ she can't help with homework ("Go ask your fa­ ten essay. ther"), a powerful message is communicated. I remain optimistic that cultural signals en­ Results couraging girls to avoid mathematics will become Olson's book concludes with the story of the 2001 less prevalent. Culture changes only slowly, though, IMO results. Who won and who lost? How did it turn and the time scale for these changes may be out? I would like to conclude here with the story decades. Still, the past few decades give us reason of how Count Down will fare. Will it sell? How will for hope, and we may expect in a few more decades it turn out? I am not sure, but here is my predic­ that Americans will come to see mathematicians as tion. There is less diversity among these six boys passionate and happy and engaging and female and than there is among the eight competitors in Spell­ male. In particular, the achievement of girls on the bound, so the IMO story may have somewhat less most recent USA Mathematical Olympiads has been appeal on this ground. Far more Americans play stronger than ever before. Scrabble than solve Olympiad problems, so read­ ers may relate less well to Count Down than they Mathematical Olympiads did to Word Freak. And far, far more Americans will Most Americans, of course, have no idea what the enjoy American Idol, which features plenty of com­ mathematical enterprise is about. They have no idea petitors devoid of talent to amuse its audience. either what sort of problems would be on an in­ Nonetheless, Steve Olson has provided a deeper and ternational competition in mathematics. Those more meaningful investigation of precocity and who equate mathematics with arithmetic probably talent than is found in these other works, and he imagine that Olympiad competitions involve ad­ can be credited with a fine and thoughtful book dition of fantastically long columns of numbers, or even if it does not appeal to millions. And should perhaps races in mental multiplication. Or their Count Down become a bestseller after all, it may image may be of Math Counts, the middle school help significantly to correct the misimpressions that national mathematics competition devised as our many Americans have of mathematics. profession's answer to the telegenic National Spelling Bee. But Math Counts is to the IMO as spelling is to writing. Olympiads are contests of ideas. The term "Mathematical Olympiad", as I use it, refers to a contest that requires participants-let's not be shy about it-to create and write mathe­ matical proofs. Olympiads are not multiple-choice exams, they are not graded by machine, and the

AUGUST 2004 NOTICES OF THE AMS 767 Book Review

Gamma Reviewed byDan Segal

Gamma story in clear and Julian Havil straightforward Princeton University Press, 2003 language. Actually, 266 pages, $29.95 it would be more ISBN 069-1 099-839 accurate to say "stories", since this This book is evidently a labour of love. The au­ is something like a thor, Dr. Julian Havil, is a mathematics teacher at picaresque novel; one of the grand old English public (= private) the hero, Euler's schools. It is a pleasant surprise to see that such constant )I, serves teachers still exist: every now and then we see of­ as the unifying ficial reports lamenting the state of mathematics motif through a education in Britain, but too little is done to make wide range of schoolteaching once again a tempting career choice mathematical ad­ for bright and enthusiastic mathematics gradu­ ventures. The book ates (let alone Ph.D.'s). differs from many There has been a recent spate of popular books others in the large about mathematics. Most of them assiduously avoid proportion of genuine IOO-proof maths: there is a the use of mathematical notation, out of defer­ particular kind of satisfaction, familiar to us and ence to Hawking's exponential decay law "n equa­ a mystery to nonmathematicians, that comes with tions ~ 2-n x (readership)". HaviI's book, by con­ really grasping a proof, and the reader is given trast, positively revels in equations, formulae, tables plenty of such moments. Of course, not all the re­ of numbers and graphs; it is not a book about sults stated can be proved within the constraints mathematics but a book ofmathematics. It is gen­ of such a book as this, but the author takes care uinelya "popular" book nonetheless, written in an never to blur the distinction. When reproducing engagingly enthusiastic style and aimed at a spe­ some of Euler's more cavalier arguments, for ex­ cific class of reader. The author neither talks down ample, he points out that they are not rigorous but nor goes over this reader's head: he assumes that can easily be made so: the (correct) implication you are comfortable withbasic calculus, infinite se­ being that they could be made rigorous by the ed­ ries, etc., and on that basis gets on with telling his ucated reader without introducing further new ideas. In other words, the reader is given the com­ pliment ofbeing addressed as a professional math­ Dan Segal is professor of mathematics at the University ofOxford and senior research fellow at All Souls College. ematician. His email [email protected] . uk.

768 NOTICES OF THE AMS VOLUME 51, NUMBER 7 The first chapter sets the historical scene in the A similar pattern governs the arrangement of the sixteenth century. We are reminded of the real dif­ book as a whole. There is a leading thread of ar­ ficulties faced by astronomers such as Tycho Brahe, gument concerning relations between the harmonic who needed to perform large arithmetical calcula­ series, the logarithmic function, and the Gamma tions, and are given a lively anecdotal account of function, the emphasis later shifting to the zeta various labour-saving techniques developed for function. This is interspersed with numerous di­ the purpose. The invention of logarithms is de­ gressions as the author picks up on a point that scribed in some detail; it was something of a rev­ catches his interest and follows it wherever it may elation, to this reader anyway, to see the contor­ lead. For example, writing tions that John Napier had to go through to give a "kinetic" definition of his logarithm before the y = lim (Hn - In(n + lX)) , modern concept of a function was around to make n-oo everything seem straightforward. This is contrasted we may ask what choice of lX makes this limit con­ with Euler's 1770 definition, the one familiar to us verge as fast as possible: a sensible question from today. The relation between the two is derived at the point of view of evaluating y as a decimal. It is some length (it is not obvious!): easy enough to see that lX = 0 gives an error at the 7 7 nth step of about 2~; some hefty (though as always NapLog(x) = 10 10g1/ e(10- x); elementary) calculation shows that lX = ~ gives a the slightly odd scaling factors in effect allow the much better error of about 2lnz.This is used to writing of 7-figure log tables without decimal introduce a whole chapter on the Euler-Maclaurin points, a notation that was only coming into use summationformula, where we get a full asymptotic around Napier's time. expansion for y in terms of Bernoulli numbers, This chapter is the only one that is primarily his­ taking in on the way Stirling's formula for n!. An torical. It gives an educational glimpse into the earlier chapter introduces the Gamma function. mind-set of a much earlier generation of mathe­ Here we see not only the familiar identity maticians-very interesting but requiring some f(n) = (n - I)! but the more startling formulae effort to follow mathematically. After this, the f(~) =)IT and emphasis is on the mathematics. The originators are introduced with more or less colourful f'(n) biographical snapshots, but the ideas are clearly f(n) = Hn-l - y. explained in modern notation. Chapter 2 introduces the harmonic series I ~ Results like these accumulate to give the reader a and its partial sums sense of the deep magic of numbers: the hidden n 1 threads that tie together familiar yet apparentlyun­ H n = L-· related things. r=l r The middle part of the book treats us to a cou­ The author first points out the exceedingly slow ple of chapters of lighter relief, covering a wide growth of these numbers: the least n for which range of topics where the harmonic series and/or the logarithmic function make an appearance. H n ~ 100 is a 44-digit number (taking nearly a whole line of text); he then shows that the series These are too many and varied to list here; all are does nonetheless diverge, giving not only the usual interesting and many quite surprising, such as argument by grouping of terms but also an analytic "Benford's law" governing the statistical distribu­ proof which evaluates tion of digits in apparently-but-not-quite random o eX lists of numbers. The last two chapters focus in on the Prime -00 1 - eX· f Number Theorem and the Riemann Hypothesis. At the age of fifteen, by studying tables ofprime num­ Further interesting titbits: H n is never an integer; indeed except for n = 1, 2, 6 the decimal expansion bers, Gauss made the extraordinary observation of H n is nonterminating. While the first fact is that the number rr(x) of primes less than x seems quite elementary, the second is reduced to to approach l~x as x increases without limit; it "Bertrand's postulate" (which is explained but not took more than a century to turn Gauss's insight proved). I mention these details because they typ­ into the Prime Number Theorem: ify the author's approach, which is carefully de­ x signed (1) to arouse the reader's curiosity in a rr(x) ~ lnx. topic, (2) to instill an awareness ofhow remarkable the true facts often turn out to be, and (3) to demon­ Havil builds up to the PNT with a mixture of his­ strate how often the answer to one question can tory and heuristics, including plenty of tables and generate a host of new directions for research. graphs; by the end of Chapter 15 the reader has a

AUGUST 2004 NOTICES OF THE AMS 769 good intuition of the nature of the problem and of function, the Riemann Hypothesis-nowfeel more its subtlety. like old friends. I would heartily recommend The final chapter bravely attempts to explain Gamma as a Christmas present for any student of the role of the Riemann zeta function, von Man­ mathematics or for any scientist or engineer who goldt's "explicit formula" for primes, and its rel­ might be pleasantly surprised to find that there is evance for PNT. The zeros of zeta are discussed more to pure mathematics than abstract pie-in­ at length. This chapter is the most sophisticated the-sky. mathematically. Of course the main results are not The only similar book that I know is From Fer­ proved, but one gets a clear understanding of matto Minkowski, by Scharlau and Opolka (Springer, what they mean and some feeling for the signif­ 1984). While also interweaving history with num­ icance of RH. As the necessary complex analysis ber theory, this is much harder mathematically: a is beyond the basic knowledge assumed of the reader who has caught the bug from Gamma and reader, the author provides a crash course (with­ is willing to work seriously at mathematics would out :r;nany proofs) in Appendix D. This is too brief enjoy moving on to the earlier book at a slightly to serve as a source for really learning this ma­ later stage in his or her education. ' terial, but there is just enough to make sense of the preceding chapter and enough (for the de­ termined student) to understand the proof of an­ alytic continuation and the functional equation of the zeta function, given in Appendix E. As far as I could see, there are no genuine math­ ematical errors in the booknor, impressively, typos in the many formulae. But there are a few unfor­ tunate and curious spots where the author has clearly written something different from what he meant: Table 16.1 puts the early zeros of zeta on the line Im(z) = i instead of Re(z) = i, and this is reproduced in a couple ofplaces in the written text. The boxed statements of Cauchy's Integral Theo­ rem and Integral Formula, inAppendix D, are both wrong, though correctly explained in the sur­ rounding text. My final two years at (high) school were largely devoted to training in the manipulation of series, integrals, and trigonometric identities. While any skills I once possessed in such matters have long since atrophied, it was a pleasure to be shown in this book how this bag of tricks can be moulded into, and often arose out of, beautiful and deep mathematics. Of course we did not see much ofthat beauty at school; having been seduced by the sim­ ple pleasures of abstract algebra at university, I was more than happy to putintegrals and series behind me. But it might have been different if I had been given Gamma to read: any bright 18-20-year-old with good training in calculus will not only be able to understand everything in the book but is likely to be inspiredbyit andwant to learn more. Indeed, anyone who knows a bit of calculus (and is not a professional number theorist or analyst) will enjoy this carefully planned tour through five centuries of mathematics. The relative beginner may find some of the solid analysis a bit daunting but hopefully will be en­ couraged to persevere by the engaging anecdotal style. As an aging mathematician having a nod­ ding acquaintance with analytic number theory, I found muchin the book quite illuminating. Things I had been vaguely aware of before-the Gamma

770 NOTICES OF THE AMS VOLUME 51, NUMBER 7 nriCartan!

O· I, entfCartan turns one"!' un re folIo "onth-a rigorous discipline, and it was years old. One of the outstanding figures of twen­ observed. Rather than the 'rotating seminars' of tieth-century mathematics, Cartan made contri­ today, where each week one goes, often without butions to several areas ofmathematics, including much conviction, to hear people speak on contin­ complex analysis, algebraic topology, and homo­ ually changing subjects, the Cartan seminar de­ logical algebra. He had an especially profound in­ manded a serious, long-term investment by the fluence on mathematics in France, particularly participants. It was in the Cartan seminar, and, through the so-called "Cartan Seminar", which he later, in the seminars of Grothendieck, which fol­ ran from 1948 to 1964 in Paris. His best-known lowed the same principle, that I learned the metier. work maybe Homological Algebra (1956), which he Nothing was left in the shadows. There was no wrote with Samuel Ellenberg and which ushered in 'black box.' The necessary preliminaries and back­ a new era ofmathematics. Cartanis also known for ground were presented in detail. The proofs were his efforts at reestablishing relations between not simply 'sketched' but presented completely. French and German mathematicians after World Cartanwas concerned that one should understand, War II, a role that demanded the utmost integrity a legitimate concern that is no longer so wide­ and sensitivity. spread, it seems to me. Many times I saw him in­ The April 2004 issue of Gazette des Mathe­ terrupt a lecture to ask the speaker to 'light the way'. maticiens, the news publication of the Societe Math­ The seminar was a privileged occasion for meetings ematique de France, carried some tributes to Car­ and discussions, made all the more fertile by strong tan on the occasion of his one hundredthbirthday. One of the contributors was iuc Illusie of the Uni­ common interests." versite de Paris-Sud, who as a young man in 1963 Pierre Samuel, an emeritus professor at Univer­ began participating in the Cartan Seminar. Illusie site de Paris-Sud, also wrote a tribute, as did Jean wrote in part: Cerf of the Centre National de la Recherche Scien­ "It is hard for us to appreciate today the im­ tifique, who noted Cartan's efforts in French-Ger­ portance that a seminar such as the Cartan semi­ man reconciliation and onbehalf of human rights. nar held for the formation of young mathemati­ "Thank you, Monsieur Cartan," Cerf wrote, "for cians. First of all, it was a 'thematic' seminar, lasting showing us by your example that it is possible to a year, of a type that has nowadays disappeared. become more and more humane as we age." Cartan chose among the recent results a theorem -Allyn Jackson or a theory sufficiently rich to justify devoting a An interview with Henri Cartan appeared in the seminar to it. At the beginning of the year, he di­ August 1999 issue of the Notices. See also llAndre vided the lectures among the volunteers. A lec­ Weil: Memories ofa Long Friendship", byHenri Car­ ture, once presented, had to be written up in the tan, Notices, June/July 1999.

AUGUST 2004 NOTICES OF THE AMS 771 DoctorateDegreesin MathematicsEarnedby Blacks, Hispanics/Latinos, andNative Americans: A Lookat the Numbers

HerbertA. Medina

Introduction August of year n + 2. In this article we cite the lat­ Many within the U.S. mathematics community have est NSF published data [3], [5].1 a sense that Blacks, Hispanics/Latinos, and Native Table 1 gives the ethnic breakdown of the doc­ Americans are underrepresented in earning doc­ torate recipients in the mathematical sciences in torates in the mathematical sciences. That is, these the U.S. during the past ten years, the respective ethnic groups do not earn doctorates at rates com­ percentages of the degrees earned by U.S. Citi­ parable to the percentage of the population that zen/U.S. Permanent Residents (USC/PR) that these they comprise. But what is the extent of the un­ represent, the total degrees earnedby citizens and derrepresentation? What type ofincreases are nec­ permanent residents, the percentage of total de­ essary to bring these groups to equitable repre­ grees that these represent, and the number of total sentation in mathematics? As these groups' degrees. representation in the general populationincreases, One answers the underrepresentation question is there reason to be concerned about the rate at posed in the introduction by comparing the per­ which they participate in mathematics? centages in Table 1 with the current ethnic make­ Our purpose here is to present and analyze Na­ up of the U.S. population. For analysis, we also com­ tional Science Foundation (NSF) and U.S. Census Bu­ pare the data in Table 1 with the projected reau data to study the questions posed above. For demographics of the U.S. population in the year the remainder of this article we will use BHNs to 2025 [9], [10]. These comparisons are contained in refer to Blacks, Hispanics/Latinos, and Native Amer­ Table 2. The table shows that the underrepresenta­ icans, and the term underrepresentation (unless tion is significant: Although BHNs represent one otherwise described) will be used only to refer to quarter ofthe u.s. population, they have earned less the underrepresentation of BHNs in earning doc­ than 5% ofthe doctorate degrees in the mathemati­ torates in the mathematical sciences. cal sciences. It also shows that if current trends con­ tinue, the underrepresentation will become much The Data worse for Hispanics/Latinos as they continue to We beginwith a few words about the data. The NSF increase their representation in the total U.S. publishes its data on science and engineering doc­ population. torate degrees for year n on or after October ofyear n + 1 and its data on the ethnicity of science and 1 The AMS publishes its data on mathematics doctorate de­ engineering degree recipients for year n on or after gree recipients based on academic years instead ofcalendar years. Its 2002-03 preliminary data is available [6J, so if Herbert A. Medina is professor of mathematics at we had chosen to use AMS data, we could have included Loyola Marymount University. His email address is uone more semester's worth" ofdoctorate data. We decided [email protected] to limit ourselves to NSF data.

772 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Table 1. U.S. Mathematical Sciences Doctorates 1993-2002: Ethnicity of Recipients

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 '93-'02 Asian/Pac. lsI. 79 142 207 140 97 71 56 70 48 29 940 % ofUSC/PR 13.4% 21.6% 26.8% 21.6% 15.4% 10.6% 9.3% 12.2% 9.2% 6.6% 15.4% Black 8 11 5 8 7 16 12 14 19 14 114 % ofUSC/PR 1.4% 1.7% 0.6% 1.2% 1.1% 2.4% 2.0% 2.4% 3.6% 3.2% 1.9% Hispanic/Latino 16 13 15 11 20 27 15 15 15 12 159 % ofUSC/PR 2.7% 2.0% 1.9% 1.7% 3.2% 4.0% 2.5% 2.6% 2.9% 2.7% 2.6% Native Amer. 1 2 2 1 1 3 1 2 2 3 18 % ofUSC/PR 0.2% 0.3% 0.3% 0.2% 0.2% 0.4% 0.2% 0.3% 0.4% 0.7% 0.3% White 476 479 535 478 479 526 506 464 428 369 4740 % ofUSC/PR 80.7% 72.9% 69.4% 73.8% 76.2% 78.6% 83.6% 80.8% 81.7% 83.5% 77.6% BHN 25 26 22 20 28 46 28 31 36 29 291 %ofUSC/PR 4.2% 4.0% 2.9% 3.1% 4.5% 6.9% 4.6% 5.4% 6.9% 6.6% 4.8% Tot. USC/PR 590 657 771 648 629 669 605 574 524 442 6109 % of Tot. Deg. 51.5% 58.8% 64.8% 57.8% 56.0% 56.8% 55.9% 54.7% 52.0% 48.2% 55.9% Tot. Degrees U.S. 1146 1118 1190 1122 1123 1177 1083 1050 1007 917 10933

USC/PR = u.s. Citizen/U.S. Permanent Resident

We point out that the penultimate row in Table 1 illustrates the concern that many in the U.S. Table 2. Ethnic Make-up of U.S. Population and mathematics community have about the nation's Doctorate Degrees (Percentages) ability to produce "home-grown" mathematicians. U.S. Population Math. Sci. Doc. Some comments related to this concern are made 2000 2025 1993-2002 at the end of this article. Asian/Pac. lsI. 3.8 6.2 15.4 Black 12.2 12.9 1.9 Efforts to Address the Hispanic/Latino 11.9 18.2 2.6 Underrepresentation: Past and Present Native Amer. 0.7 0.8 0.3 White 71.3 62.0 77.6 For several decades the U.S. mathematics commu­ BHN 24.8 31.9 4.8 nity has engaged in efforts to strengthen home­ grown participationin graduate mathematics. Many show that only 3.8% (63 of 1,668) of the students of these efforts have centered and continue to cen­ who participated in these programs were BHNs [2]. ter onundergraduate research summer programs, During the past few years, there have been and e.g., NSF Research Experience for Undergraduates there continue to be efforts within the U.S. math­ (REUs), whose aim is to develop the skills and ematics community to address and correct the un­ knowledge and thus cultivate the talent of mathe­ derrepresentation. At the undergraduate level these matically promising undergraduates. While we efforts also have centered on summer programs.3 know of no study that "proves" that these pro­ Like REUs, the primary aim of these summer pro­ grams work, there seems to be a general consen­ grams is the cultivation of mathematical talent. sus that these programs are indeed effective in Most of these programs also emphasize the de­ contributing to the mathematical development of velopment of a "road map" to graduate school as talented students.2 well as the establishment of a network for the pro­ Unfortunately, such programs have not been grams' participants that will facilitate admissions successful in recruiting significant numbers of and funding, and increase the likelihood of success BHNs. Indeed, data collected from many mathe­ in g~aduate programs [7], [8], [11]. Although it is matics undergraduate summer programs in 1999 too early to tell if these programs will result in a 2 We comment that to verify the effectiveness ofsuch ef­ significant increase in the representation of BHNs forts would be extremely difficult. One would need to take in mathematics graduate programs, a large per­ two groups of students with comparable mathematical centage of students who have participated in these promise, ambition, work ethic, etc., and have one partic­ ipate in a "typical" REU and the other not. The success of 3 At the graduate level there are several national fellow­ the groups in graduate mathematics programs would ship programs whose funding is aimed at BHNs; these in­ then have to be compared years later. It is highly unlikely clude the National Physical Sciences Consortium (NPSC), that such an "experiment" with proper controls could take Graduate Assistance in Areas ofNational Needs (GAANN), place. Ford Foundation, and Sloan Foundation fellowships.

AUGUST 2004 NOTICES OF THE AMS 773 Total Blacks, Hispanics/Latinos, Native Americans 1700

14000 1600

1500

13000 1400

1300

12000 1200

1991 1992 1993 1994 1995 1996 1997 1998 2000 1991 1992 1993 1994 1995 1996 1997 1998 2000

Figure 1. Mathematical Sciences Bachelor's Degrees. (The NSF did not publish 1999 data.)

programs have gone on to graduate school. For increase of8 degrees peryear each and every year example, 20 of the 49 students who participated in from now until 2025. the Summer Institute in Mathematics for Under­ Will this increase happen naturally? Looking at graduates (SIMU) at the University of Puerto Rico­ data from Table 1, we see that during the past ten Humacao, a program aimed at Hispanics/Latinos years there has been a slight upward trend in the and Native Americans, in 1998 and 1999 went on number of degrees awarded to BHNs, but the re­ to doctorate programs in the mathematical sciences. gression line for this data gives a slope of only 1.17. If we look at the yearly totals of degrees earned by Thus, it is not likely that the needed increase will members of these ethnic groups in Table 1, we see happen without some sort of intervention. that 20 is a significant number. A couple of summer Are current efforts to address the underrepre­ programs aimed at BHNs continue, and agencies sentation enough? Again, the data seems to sug­ such as the NSF and the National Security Agency gest that they are not. We first note that there was (NSA) continue to have an interest in funding them; a sharp increase in the number of degrees awarded but as is explained below, a larger effort is needed to BHNs from 1996 to 1998; it is probably not a co­ if the underrepresentation is to be seriously ad­ incidence that the first major mathematics summer dressed. (There are, of course, similar programs in program for BHNs was established at the Univer­ other sciences; some information on their structure sity of California, Berkeley, by Leon Henkin and Uri and effectiveness can be found in [1].) Treismann in 1989. The trend since then opti­ mistically suggests that this increase was more Addressing the Underrepresentation: The than a spike (mostly because degrees awarded to Future Blacks have remained in the teens), but it is im­ Let us suppose that we aim to solve the under­ possible that the programs still in existence would representation by the year 2025. How large do the result in the long run in 8 more doctorates in year yearly increases in the number of BHN doctorates n + 1 than in year n year after year, because these have to be? programs plainly and simply do not serve that Table 1 shows that the total number of doctor­ many students. ates awarded in the U.S. during the past ten years In other words, the data suggests that if the U.S. has a downward trend, but for the purposes of our mathematics community hopes to successfully ad­ analysis here, we will assume that the number in dress the underrepresentation, it needs to devise the year 2025 will be 1,093 and that 611 of these and implement a plan that will result in significantly degrees will go to U.S. citizens and permanent res­ more doctorates being awarded to BHNs year after idents, the averages during the past ten years. (Ex­ year. trapolating on the number of doctorates during the What should the plan be? An easy, perhaps suf­ past ten years gives 494 doctorates in 2025; we be­ ficient-in-the-short-term answer is to have more suc­ lieve this number is unrealistically too low to con­ cessful summer programs for BHNs. This approach sider.) Using the demographic predictions of the begs the question of whether or not there are­ census, the underrepresentation problem would be enough BHNs earning bachelor's degrees in math­ solved in the year 2025 if 196 or 32% of these de­ ematics to produce an increase in doctorates. The grees went to BHNs. During the last ten years an answer to this question appears to be affirmative. average of 29 degrees have been awarded to BHNs While many in the U.S. mathematics commu­ each year. In order to produce the increase from nity are well aware of the serious decline in the 29 to 196 degrees per year over a twenty-one-year number ofbachelor's degrees in the mathematical period in a linear fashion, we would need a yearly sciences during the past decade, few are aware of

774 NOTICES OF THE AMS VOLUME 51, NUMBER 7 the significant upward trend in the number of such Undergraduates: A Longitudinal Study of Program degrees being awarded to BHNs [3]. The graphs in Outcomes, 1986-1996, Council on Undergraduate Re­ Figure 1 show both the decrease in total degrees search Quarterly 21 (3) (2000),114-119. and the increase in degrees being awarded to BHNs.4 [2] Summer Program Survey, in Proceedings of the Con­ In the long-term an overarching plan for deal­ ference on Summer Undergraduate Mathematics Re­ search Programs (J. A. Gallian, ed.), Amer. Math. Soc., ing with the underrepresentation is for the entire Providence, RI, 2000. u.s. mathematics community to identify early [3] National Science Foundation, Division of Science Re­ promising BHN undergraduates and invest time, sources and Statistics, Science and Engineering De­ money, and energy in developing and cultivating grees, byRace/Ethnicity ofRecipients: 1991-2000, NSF their mathematical talent. Certainly this practice, 02-329 (Susan T. Hill), Arlington, VA, 2002. although perhaps not a coordinated effort, has re­ [4] National Science Foundation, Division of Science Re­ sulted in significant progress in solving the un­ sources and Statistics, Science and Engineering Doc­ derrepresentation of women earning doctorates torateAwards: 2000, NSF 02-305 (Susan T. Hill, Pro­ in the recent past. Indeed, in 1991 21.6% of doc­ ject Officer), Arlington, VA, 2001. torates in the mathematical sciences awarded to [5] National Science Foundation, Division of Science Re­ u.s. sources and Statistics, Science and Engineering Doc­ citizens/permanent residents went to women; that torate Awards: 2002, NSF 04-303 (Susan T. Hill, Pro­ had increased to 30.3% in 2002 [4], [5]. ject Officer), Arlington, VA, 2003. We close with some remarks on the penultimate [6] D. KIRKMAN, ]. MAxwELL, and C. ROSE, 2003 Annual Sur­ row of Table 1. From 1993 to 2002, 55.9% of doc­ vey ofthe Mathematical Sciences (First Report), Notices torates in the mathematical sciences went to U.S. Amer. Math. Soc. 51 (2) (2004),218-233. citizens and permanent residents. Some within the [7] H. A. MEDINA and I. RUBIO, The Summer Institute in mathematics community see this as a concern, as Mathematics for Undergraduates (SIMU) at the Uni­ the U.s. clearly is not able to produce enough home­ versity ofPuerto Rico-Humacao, Proceedings ofthe Con­ grown mathematicians to satisfy its mathematical ference on Summer Undergraduate Mathematics Re­ needs. The declining number of total bachelor's de­ search Programs (J. A. Gallian, ed.), Amer. Math. Soc., Providence, RI, 2000, pp. 169-180. grees in the mathematical sciences suggests that [8] S. TENNENBAUM, Research and EducationActivities of the the country's reliance on foreign mathematical tal­ Mathematical and Theoretical Biology Institute at Cor­ ent will increase (perhaps dramatically) in the next nell, Proceedings of the Conference on Summer Un­ few years. If this is indeed a policy issue that war­ dergraduate Mathematics Research Programs (J. A. rants attention, then a good strategy to generate Gallian, ed.), Amer. Math. Soc., Providence, RI, 2000, an increase in the number of U.s. citizens and per­ pp.41-49. manent residents earning doctorates may be to [9] United States Census Bureau, http://www . focus more attention on BHNs studying mathe­ census.gov/population/estimates/nation/ matics. As indicated above, there certainly is lots i ntfi 1e3-1. txt. [10] United States Census Bureau, http://www . of room for improvement in identifying and culti­ census.gov/population/www/projections/ vating mathematical talent within these ethnic natsum-TS. html. groups, and there are significantly more BHNs at [11] W. O. WILIlAMS, Summer Undergraduate Applied Math­ the undergraduate level studying mathematics now ematics Institute at Carnegie Mellon University, Pro­ than ever before. Finally, we mention that unless ceedings of the Conference on Summer Undergradu­ we address the underrepresentation issue, the de­ ate Mathematics Research Programs (]. A. Gallian, ed.), mographic shifts that the nation will undergo dur­ Amer. Math. Soc., Providence, RI, 2000, pp. 13-18. ing this century almost certainly will contribute dra­ matically to our dependence on foreign mathematical talent. Indeed, the u.s. Census Bureau predicts that in the year 2060, 40.7% ofthe U.s. pop­ ulation will be Black, Hispanic/Latino, and Native American [10]; what percentage ofthe total u.s. doc­ torates in the mathematical sciences will go to U.S. citizens and permanent residents if these groups continue to earn so few of these degrees?

References [1] ]. FOERTSCH, B. AlExANDER, and D. PENBERTHY, Summer Re­ search Opportunity Programs (SROPs) for Minority

4 We note that even though the trend is a very positive de­ velopment, BHNs continue to be seriously underrepre­ sented at the bachelor's-degree level. For example, in the year 2000, BHNs earned only 14.4% ofthe bachelor's de­ grees awarded to u.s. citizens and permanent residents.

AUGUST 2004 NOTICES OF THE AMS 775 Has theWomen-in­ MathematicsProblemBeen Solved?

Editor's Note: Beginning with our 1991 "special issue" on women in mathematics, the Notices has featured coverage of diversity and underrepre­ sentationinmathematics. This tradition continues in the present issue with articles by Allyn Jackson, Carolyn Gordon and Barbara Keyfitz, and Herbert Medina. -Andy Magid

Thirteen years ago, in June 1991, Science mag­ The data in the tables below were repor,ted by azine published an article with statistics on the departments responding to Annual AMS-ASA­ number ofwomen in the "top ten" mathematics de­ IMS-MAA. Surveys of New Doctorates, from partments in the United States. Out of 303 tenured academic year 1995-1996 through 2002-2003. professors in these departments, the article re­ AMS~ASA-IMS-MAA For AIlnlial Survey Reports, ported, just 4 were women. The article caused a departments are divided into groups according good deal of discussion in the mathematical com­ to the highest degree offered in the mathemat­ munity. The details were picked apart: Who says ical sciences. Doctoral-granting·departments of these are the top ten departments? Why weren't mathematics are further subdivided according tenure-track women counted? But the larger ques­ to their ranking of "scholarly quality of pro­ tion came across loud and clear: Why were so few gr~m faculty" as reported in the 1995 publica­ women rising to the top levels ofmathematics? The tion Research-Doctorate Programs in the United headline on the article, which was about a highly States: Continuity and Change (National A-cad­ publicized tenure case at Berkeley, suggested an ex­ emy'Press, 1995). Groups:.are,.referred to in the planation: "Does the Harrison Case Reveal Sexism following tables: in Math?" Group I is composed of 48 departments with The Science article hails from a different era, scores inthe 3.00-5.00 range. Group IPublicand whenin-your-face pronouncements about the lack GroupIPrivate are Group Idepartmentsatpublic ofwomeninmathematics were more common than institutionsandprivateinstitutionsrespectively. today. In those ten departments-Berkeley, Cal­ Group II is composed of 56 departments tech, Chicago, Columbia, Harvard, Michigan, MIT, with scores in the 2.00-2.99 range. Princeton, Stanford, andYale-there are still around Group ill contains the remaining u.S. depart­ 300 tenured professors, and 16 are women. The ments reporting a doctoralprogram, inclucling·a number of women has quadrupled, and yet the number ofdepartments notincludedinthe 1995 picture is largely the same; what has changed is the r~gofprogramfaculty. lack of public expressions of discontent. Has the GroupVais appliedmathematics/applied sci- issue of women in mathematics, once so urgent in ence: Group Vb,which was no longer surveyed the mathematical community, disappeared from the as of 1998-1999, was operations research and radar screen? management science. Top Producers ofWomen Doctorates Listings of the actual departments which comprise these groups are available on the AMS Before examining that question, a look at a differ­ website at http://www.-ams '. org/outreach/. ent set of data may provide some additional per­ spective. The September 1991 Noticeswas a "special

776 NOTICES OF THE AMS VOLUME 51, NUMBER 7 TABLE 1. Leading U.S. Doctorate-Granting Departments of Mathematics by Number of Women Doctorates Reported, from Academic Year 1995-1996 through 2002-2003

Total Women Total Doctorates % Doctorates Reported Reported Women

Maryland, University of (College Park) 49 179 27.37 California, University of (Los Angeles) 40 160 25.00 Illinois, University of (Urbana-Champaign) 40 161 24.84 California, University of (Berkeley) 39 244 15.98 Massachusetts Institute of Technology 37 174 21.26 Wisconsin, University of (Madison) 34 165 20.61 New York, State University of (Stony Brook)";'," 32 101 31.68 Michigan, University of (Ann Arbor) 30 155 19.35 Rutgers University, New Brunswick 26 100 26.00 Nebraska, University of (Lincoln) 26 63 41.27 California, University of (San Diego) 24 88 27.27 Texas, University of (Austin) 24 94 25.53 Illinois, University of (Chicago) 22 95 23.16 Purdue University 22 109 20.18 Boston University 22 55 40.00 Michigan State University 20 82 24.39 Minnesota, University of (Twin Cities) 20 108 18.52 Rice University";''" 20 44 45.45 Virginia, University of 18 49 36.73 Stanford University 18 70 25.71 Brown University";''" 18 73 24.66 Graduate Center, City University of New York 17 69 24.64 Syracuse University 17 32 53.13 New York, State University of (Stony Brook) 16 71 22.54 Pennsylvania State University (University Park) 16 75 21.33

-k Applied Mathematics Departments issue" on women in mathematics, with nine arti­ mathematics education, which are not separated in cles devoted to this topic. One of them, "Top Pro­ the Annual Survey statistics. ducers of Women Mathematics Ph.D.'s", provided Comparing these tables to the ones that ap­ data from the Annual Survey showing which math­ peared in 1991 shows that women are receiving doc­ ematics departments had produced the most torates in higher proportions in all kinds of de­ women doctorates in the preceding ten years. This partments. For example, for every department that data has been updated for the present article; see appeared both in the 1991 version of Table 1 and the accompanying tables. It is useful to note that in the updated version, the percentage of women in the eight-year period covered by the tables (aca­ is up, in some cases dramatically. In the 1991 ver­ demic years 1995-1996 through 2002-2003),26% sion of Table 2, there were no Group I depart­ of all mathematics doctorates went to women. For ments; now, two Group I departments appear there, the tenyears preceding 1991, the analogous figure Boston University and Duke University (see the was 17%. box for explanation of the Groups). Table 3 shows The tables provide various ways to view the the percentage of women doctorates in depart­ question of which departments produce the most ments of similar sizes; the percentages are gener­ women doctorates and require some care in inter­ ally much higher than in 1991. In Table 4, which preting. In Table 1 some departments are high on shows the percentages ofwomen doctorates in de­ the list because they produce so many doctorates partments in Groups I, II, III, and Va, all the per­ overall, even though the percentages of women centages are up from the 1991 figures. doctorates in those departments are lower than the A small number of departments that are pro­ percentages for all departments. In Table 2 some ducing a high proportion of women doctorates departments that rank highly in terms of percent­ were contactedby the Notices and asked about the age of women doctorates award mostly degrees in reasons for their success. One of these is the

AUGUST 2004 NOTICES OF THE AMS 777 TABLE 2. Leading U.S. Doctorate-Granting Departments of Mathematics by Percentage of Women Doctorates Reported for Academic Year 1995-1996 to 2002-2003 (Departments Granting an Average of at Least Two Doctorates per Year)

% Total Total Women Doctorates Reported Women Reported

Northern Colorado, University of 56.52 23 13 Illinois State University 56.00 25 14 American University 55.56 27 15 Lehigh University 54.17 24 13 Syracuse University 53.13 32 17 Dartmouth College 52.38 21 11 Rhode Island, University of 52.17 23 12 Western Michigan University 48.28 29 14 Rice University"!( 45.45 44 20 Colorado, University of (Boulder) 43.75 32 14 Cincinnati, University of 43.75 16 7 Oklahoma, University of 43.33 30 13 Nebraska, University of (Lincoln) 41.27 63 26 New Mexico, University of 40.54 37 15 Boston University 40.00 55 22 Washington State University 40.00 30 12 Washington, University of"!( 40.00 30 12 Colorado State University 40.00 25 10 Wesleyan University 39.13 23 9 Rensselaer Polytechnic Institute 39.02 41 16 Emory University 38.24 34 13 Oregon, University of 37.50 40 15 Duke University 37.50 32 12 Bowling Green State University 37.50 32 12 Vanderbilt University 37.14 35 13

-k Applied Mathematics Departments

Department of Computational and Applied Math­ has been on the faculty since 1997. The department ematics at Rice University, where 46 of the Ph.D.'s has an intellectually inclusive atmosphere that fa­ granted from 1995 to 2003 went to women. De­ cilitates interactions with researchers from all areas partment chair William Symes said several factors of science and engineering. This inclusiveness, to­ account for this large proportion. One of the main gether with the department's emphasis on men­ ones is the influence ofRichard Tapia, who as chair toring, has a positive effect on the graduate stu­ of graduate admissions in the late 1980s made a dents. Symes said he did not have precise figures, point of admitting women into the program; Tapia but he estimated that graduation rate of students has been a strong mentor for women and minor­ admitted to the Ph.D. program is about 80%. ity students inmathematics. Convinced that his ap­ Over 50% of the doctorates given by the math­ proach was effective, the department has contin­ ematics department at Dartmouth College in the ued to admit a large proportion of women. "The past eight years went to women. The small doctoral champion role, which Richard played for us, seems program-just 21 degrees were given between 1995 to be critical," Symes commented. The continued and 2003-seems to be welcoming for all students, emphasis on admitting a significant number of male and female. The department has no special women students seems to produce a snowball ef- programs aimed at women students, but "we have fect, inthat prospective graduate students visiting faculty members who try to encourage and support the doctoral program see that there are many all our grad students, who are aware of the special women around, which in turn encourages women concerns of women, and who consciously nurture to enter the program. Women faculty members the supportive nature of the grad program," com­ have provided role models: Mary Wheeler was in mented Dartmouth faculty member Marcia Groszek. the department for many years before moving to She is one of four women faculty members (three the University of Texas in 1995, and Liliana Borcea are tenured, one is tenure-track) out of a total of

778 NOTICES OF THE AMS VOLUME 51, NUMBER 7 TABLE 3. Percentage of Women among Doctoral Programs of Comparable Size For this table, departments of mathematics were divided into categories of comparably sized doctoral pro­ grams, where size is defined by the number of reported doctorates awarded by the department, from aca­ demic year 1995-1996 through 2002-2003. The size categories are given in the leftmost column. For each size category, the table lists the three departments having the highest percentage of women doctorates in the given period. The rightmost column gives additional information about the departments in each size category.

Size of Department Top Three Departments % Women Average % Women Doctor- ates by%ofWomen Doctorates for Departments in This Group

Depts. Granting New York, State University of (Stony Brook)'" 31.68 20.70% for 14 departments 100 Doctorates Maryland, University of (College Park) 27.37 (414 women out of a total and Above Rutgers University, New Brunswick 26.00 2,000 doctorates)

Depts. Granting California, University of (San Diego) 27.27 20.83% for 6 departments 80-99 Doctorates Texas, University of (Austin) 25.53 (111 women out of a total Michigan State University 24.39 533 doctorates)

Depts. Granting Nebraska, University of (Lincoln) 41.27 22.39% for 16 departments 60-79 Doctorates Stanford University 25.71 (241 women out of a total Kentucky, University of 25.40 1,078 doctorates)

Depts. Granting Rice University'" 45.45 23.26% for 37 departments 40-59 Doctorates Boston University 40.00 (401 women out of a total Rensselaer Polytechnic Institute 39.02 1,724 doctorates)

Depts. Granting Northern Colorado, University of 56.52 38.85% for 64 departments 20-39 Doctorates Illinois State University 56.00 (541 women out of a total American University 55.56 1,875 doctorates)

Depts. Granting New Hampshire, University of 53.33 . 24.46% for 42 departments 9-19 Doctorates Howard University 45.45 (132 women out of a total Southern Methodist University 45.45 544 doctorates)

TABLE 4. Percentage of Doctorates Granted to Women by u.S. Departments of Mathematics (Groups I, II, III and Va), 1995-2003.

Group I (Public) 22.24% (553 women/2,486 total doctorates) Group I (Private) 19.14% (272 women/1,421 total doctorates) Group II 26.24% (515 women/1,963 total doctorates) Group III 29.55% (341 women/1,154 total doctorates) Group Va 23.22% (163 women/702 total doctorates)

AUGUST 2004 NOTICES OF THE AMS 779 18; another is the current president of the Associ­ to meet women who have established careers in the ation for Women in Mathematics, Carolyn Gordon. field. The department also has a week-long sum­ Groszek said that the "critical mass" of women mer "math camp" called ALL GIRLS/ALL MATH for faculty and students in the department creates an female students in grades 10 through 12. The at­ "atmosphere in which being a woman in math is tention to female students continues at the grad­ ordinary and normal." The department provides fi­ uate level; in 1998 the department received the nancial support to all students and gives them Presidential Award for Excellence in Science, Math­ equal teaching responsibilities, policies that ematics and Engineering Mentoring based on its Groszek said help students to feel that they are all success in mentoring female graduate students "on an equal footing." Her colleague Dorothy Wal­ during the 1990s. It is the only mathematical sci­ lace attributed much of the friendliness of the de­ ences department in the nation ever to have re­ partment to the students themselves. "The gradu­ ceived this award. When it comes to encouraging ate students have developed a culture of graduate students, the department puts its money cooperation, preparing each other and the younger where its mouth is: each year it spends around ones for qualifying exams, making sure seminars $6,000 for awards to recognize outstanding grad­ happen, etc.," she said. uate students and around $15,000 to support stu­ Boston University, with 40% women doctorates dents attending professional meetings. from 1995 to 2003, had the highest percentage among Group Iinstitutions. Chair Steven Rosenberg Evolution, Not Revolution emphasized that the department does not give As the data show, there are encouraging signs that women students preferential treatment and does the obstacles women encountered in the past in es­ not lower standards in order to increase the num­ tablishing careers in mathematics are lessening. ber ofwomen. But he did point to a number of fac­ "Things are undeniably better than they were tors that may account for the department's success twenty-five years ago," commented Groszek. "The in producing women doctorates. Out of 31 tenured most egregious and visible problems (and hence the faculty, there are five women, including three full egregious and visible evidence that something professors: Gail Carpenter, Nancy Kopell, and Emma should be done) have been eliminated....The solu­ Previato. Kopell, a MacArthur Fellow and codirec­ tions to the problems that remain are not simple tor of the Center for BioDynamics, has been espe­ or straightforward (and thus, not easy to agitate cially visible. Another factor is the department's for)." It is therefore unsurprising that the women­ size. "It is neither so small that students feel a in-mathematics issue has become more quiescent. fishbowl effect nor so large that students feel un­ Radical calls to overthrow the "old boys' network" noticed," Rosenberg observed. Once students are seem to have given way to a more systematic, pro­ accepted into the doctoral program, faculty work fessional approach. hard to persuade them to come. "I believe this per­ This approach could be seen in a "leadership sonal effort convinces students of both sexes that workshop" sponsored by the Association for we are interested in them for their specific Women inMa'thematics (AWM). Held in March 2004 strengths, not just to fill a [teaching fellow] slot," at the University ofMaryland at College Park, where he said. The department probably casts a wider net the AWM offices are located, the workshop brought for students than departments that are, as Rosen­ together about forty women mathematicians from berg put it, at the "pinnacle" of Group I. "In our case, academic institutions all over the country. It was the wider net means considering many candidates dedicated to the memory of Ruth Michler, a math­ from other countries, e.g. former Eastern Bloc coun­ ematician at the University of North Texas in Den­ tries, where my nonscientific impression is that ton who died in an accident in 2000 at the age of women make up a greater percentage of math ma­ thirty-three. Some of her colleagues and friends jors thanin the U.S.," he explained. "This wider net came to the workshop, and her father, Gerhard has presumably helped increase both the quality Michler, a mathematician at the Universitat Essen, of our graduate program and the number ofwomen flew over from Germany to attend. The aim of th,e students, without any of the conflict an imposed workshop was to help women mathematicians who social agenda would entail." are at the early stages of their careers to prepare While Boston University, Rice, and Dartmouth for leadership positions, such as serving as de­ have no programs specifically aimed at women, the partment chair. University of Nebraska, with 41% women Ph.D.s, has Most of the workshop activities focused onpro­ become known for its special programs to en­ viding sound, practical advice that would be use­ courage women in mathematics. One of the de­ ful to anyone, male or female, taking on a leader­ partment's most visible events is its annual Ne­ ship role. Speakers from different kinds of braska Conference for Undergraduate Women in institutions discussed their experiences in admin­ Mathematics. Each conference brings together over istrative positions, and in small group sessions one hundred students to discuss mathematics and participants grappled with real-life examples of

780 NOTICES OF THE AMS VOLUME 51, NUMBER 7 problems that arise in mathematics departments. opportunities for women in the following areas: There was also a panel discussion on leadership in ... the Society will include more women on Society professional societies and research. Some of the is­ programs and panels, including invited speakers." sues that arose were particular to women, such as Blum expressed frustration that she and others the "tenure clock" and how having children affects have been raising these same issues repeatedly for women's careers. Things happened at the workshop the past thirty years and the message seems not that one could not imagine happening in a group to have sunk in. "It's really outrageous, and I am of men. For example, at one point the participants thinking ofresigning from the AMS," she said. "It's were asked to stand if they had ever been mis­ way past time that the leadership of the AMS deem taken for a department secretary; about half stood. top priority the full inclusion of women in the en­ For many of the women it was gratifying, inspir­ terprise-and that includes significant presence ing, and just plain fun to be surrounded by so in the prestigious and highly visible roles as invited many female mathematicians. speakers at national meetings." The basic assumption of the workshop seemed In a presentation at the AWM workshop, Blum to be that the problems women might encounter described the efforts at Carnegie Mellon to increase in establishing careers inmathematics could be ad­ the number of women students in the School of dressed by imparting to them the right advice and Computer Science (SCS). One of the things that the right skills-in other words, evolution, not rev­ made a difference in the Ph.D. program was a re­ olution. And yet some of the senior mathematicians orientation of the admission criteria: rather than at the workshop expressed a good deal of impa­ zeroing in on applicants with a lot of prior pro­ tience and exasperation at what they see as the slow gramming experience, she said, SCS now looks for pace of change. Joan Feigenbaum was a mathe­ "people who might be visionaries in computer sci­ matics major at Harvard University twenty-five ence." As a result, more women were admitted, years ago and thenwent into computer science. She and the men who were admitted were better­ was at Bell Labs for many years before moving re­ rounded individuals. Blum has also initiated a com­ cently to the computer science faculty at Yale Uni­ munity-building program called Women@SCS. versity. As a student she heard people at Harvard When she was a deputy director of the Mathemat­ talking about the lack of women on the mathe-· ical Sciences Research Institute in Berkeley, Blum matics faculty. Today "there are still no tenured noticed that Mondays were a good day for an­ women on the Harvard mathematics faculty, and nouncements of new theorems. The reason: Math­ yet I don't hear anyone talking about it," she re­ ematicians would get together socially on the week­ marked. "I find this shocking," she continued. "I find end and end up doing mathematics. It can be it amazing that this was not dealt with, that no one difficult for women to take part in these kinds of has boiled over about this." Why have such issues activities that combine the social and the profes­ faded over the years? Feigenbaum ticked off a sional. Programs like Women@SCS provide an in­ number of reasons: the women's movement losing frastructure through which women can build the steam, the backlash against affirmative action, and profeSSional, educational, and networking rela­ the erosion of Vietnam War-era social conscious­ tionships they might otherwise miss out on. "We ness. Nowadays, she remarked, people are more are providing things that guys take for granted," placid and willing to go with the status quo. Blum remarked. Another senior participant in the workshop was "I think things have improved a lot" for women Lenore Blum of Carnegie Mellon University, who is in mathematics, said Ruth Charney of Brandeis a founder of AWM and has been a longtime advo­ University, who was one of the organizers of the cate for women in mathematics. For the past three AWM workshop. Thanks to the work of organiza­ years Blum has served on the program committee tions such as the AWM, she noted, the problems for national meetings of the AMS, most recently are "gradually solving themselves." But she be­ as chair. She said that in her experience very few lieves there is still a need for programs, like the suggestions for women speakers arise unless the workshop or Women@SCS, that help talented young committee is prodded. For example, in the planning women remain and succeed in the field. Charney stages for the upcoming national meeting to be held pointed to the long-running AWM program of travel in Atlanta in January 2005, the initial suggestions grants for women as another example of a program for potential candidates for AMS invited addresses that is still very much needed. Some women in­ were all men. When Blum drew this to the com­ terviewed for this article said that participation in mittee's attention, it immediately responded with the annual Institute for Advanced StudyjPrinceton suggestions of women speakers. She pointed out University programfor women undergraduate and that the instructions to the program committee graduate students made a big difference in their quote a resolution passed at an AMS Business Meet­ careers. This program, originally associated with the ing in 1972, which states in part: "The American Park City Mathematics Institute, was begun in 1994 Mathematical Society will work actively for equal by Karen Uhlenbeck of the University of Texas at

AUGUST 2004 NOTICES OF THE AMS 781 Austin. Another well-established program is EDGE mathematicians: "No, but..." She went on to recount (Enhancing Diversity in Graduate Education), run the following story. She and a man in her depart­ by Sylvia Bozeman of Spelman College and Rhonda ment had each been awarded a research grant. A Hughes of Bryn Mawr College. Now six years old, few days before a faculty meeting, she had been in the program aims to prepare women to excel in communication with the department chair about graduate school in mathematics. her grant. At the meeting the chair announced the man's grant as ifit were big news; nothing was said "No, but..." about the woman's grant. When she later com­ If things have improved so much, what kinds of plained to the chair, he apologized and was quite problems do women in mathematics face today? embarrassed. It seemed to have been a genuine The following scenario provides one example: oversight, but she was left with the nagging sus­ picion that it is unlikely the man's grant would have Your department has been building up been overlooked. faculty in a new area. Kevin and Jim, two , The same kind of "No, but..." reply came from assistant professors ill that area, are another young woman mathematician attending a serving on a search committee for a recent workshop at the Mathematical Sciences Re­ third person in the area. You scan the search Institute in Berkeley. She is an assistant applications as they flood in and notice professor in a department at a public university a few unusual letters. For example, one with a strong emphasis onresearch. She sees tenure writer says: "I really admire Martha. She as the juncture where difficulties can arise for is always smiling. She looks after her women m.athematicians. As an example she pointed son, bakes for our department collo­ to a recent tenure case in her department con­ quia, and still finds time to do research!" cerning a male professor. He had a strong research In another letter, the writer explains record in an area not much appreciated in the de­ that Sally has taken a relatively long partment, which recommended against tenure. She time to complete her Ph.D. because she observed the ease with which objections could be has gone through a messy divorce and raised about him by, for example, reading his let­ custodybattle. Kevin and Jim quickly de­ ters of recommendation from a certain viewpoint; cide that Martha and Sally are "not re­ from a different viewpoint, the letters could be ally in the right field." seen as very laudatory. "People gain power bybeing Gail Ratcliff, chair of the mathematics depart­ negatively critical of the work of others even when ment at East Carolina University, presented this sce­ it is not in their area," she remarked. "Or they nario for discussion during a small group session make a taste judgment and pretend it's fac­ that she ran at the AWM workshop. The scenario tual....People are not always honest about when is real, though the details have been slightly altered. they are being objective and when they are not." Imagine a letter about a man that contained a com­ Her larger point is that, far from being based on ment such as "John is always smiling"; when one objective and impersonal evaluations ofindividual of the participants said this aloud, it sounded so candidates, tenure cases involve subtle judgments absurd the room broke out in laughter. There ap­ about the candidate's research area, future poten­ peared to be general agreement in the group that tial in research, ability to relate to students, colle­ it is fairly common for such inappropriate per­ giality, professionalism, and a whole host of other sonal remarks to be made in letters about women factors that are difficult to measure objectively. In mathematicians. There was also acknowledgment the complex process of weighing suchfactors, bias that in all likelihood the letter writers were gen­ against women could creep in. Some women also uinely trying to be positive and helpful. The group say that when it comes to granting tenure or fill­ discussed whether it really was the personal re­ ing a position, women mathematicians are scruti­ marks in the letters rather than the applicants' re­ nized more or in different ways than are men. As search records that triggered the negative response Chuu-lian Terng, a woman mathematician at the on the part of Kevin and Jim. It was suggested that University of California at Irvine, put it, "Uncon­ the department chair should discuss the letters sciously, some people have more questions when with them to be sure they ignored the remarks. it's a woman." The difficulties that women encounter in math­ ematics are akin to the scenario above; inways that "Whiners" Not Welcome are small and subtle but can be quite damaging, Concerning the problems women face in mathe­ women are treated differently from their male col­ matics, "It's not a dead issue," said Rebecca Goldin leagues. When asked if she had encountered obsta­ of George Mason University. "But it's an unpopu­ cles in mathematics because she is a woman, one of lar issue." The perceptionis that if one complains­ the participants at the AWM workshop gave a re­ "whines" is the word often used-about problems sponse that may be typical of many young women women face, "thenyou are doing it to get something

782 NOTICES OF THE AMS VOLUME 51, NUMBER 7 you don't deserve rather than in order to address they chafed if a woman did so. Among some of the actual problems." Young women mathematicians men, she said, "the need not to be taught something feel pressure not to be seen as "whiners". Part of by me was very strong." These kinds of problems this pressure may result from the common as­ lessenwhen there are other women around. "In my sumption that women have advantages in mathe­ experience, if you get 20 percent women in a lec­ matics that men do not have. This assumption ture hall or a classroom, it becomes a nonissue," sometimes comes into play when women are ac­ Goldin remarked. cepted into elite universities as graduate students Do women do mathematics differently from or are hired by top departments. men? Of course they work on the same kinds of Emma Carberry is a Moore Instructor at the mathematical problems, but their working style Massachusetts Institute of Technology. Five years may be quite different. Carberry observed that ago, when she was in the Ph.D. program at Prince­ women usually do not engage in the aggressive, tonUniversity, she and her fellow student Julianna highly competitive sparring that sometimes prevails Tymoczko started a group for women students inmathematical conversations among men. This ob­ called Noetherian Ring. The name comes from a servationis reinforced byJane Hawkins of the Uni­ similar group that has been in place for years at versity of North Carolina at Chapel Hill, who said the University of California at Berkeley and has she has seen differences in how women and men been replicated in other institutions. The Noe­ approach doing mathematics, especially in how therian Ring provides a way for women to build a they deal with competition and criticism. Women, community of collegial relationships and avoid she said, usually "don't want to get down in the isolation. One activity of the Princeton Noetherian mud," meaning that they shy away from competi­ Ring has been to invite women mathematicians to tive or aggressive confrontations over mathemat­ give talks open to the entire department; these lec­ ics. Having a critical mass of women in mathe­ tures proved to be quite popular. Carberry said that matics may mean that a wider range of working one key to the success of the Noetherian Ring was styles are accepted, which could benefit the whole the support of Princeton faculty member Ingrid field. Daubechies. Women are entering mathematics in greater Such "women only" activities can contribute to numbers, and they are progressing to higher lev­ the perception that women have special advan­ els of achievement. Their representationin the top tages, thereby leading to dissatisfaction among departments is still low, but given that there are a men. For example, Goldin noted that, although the lot of talented young women in the pipeline, that lAS/Princeton women's program is open to male is sure to change. In the meantime, women's groups, and female participants, only women receive fi­ special programs, and organizations like the AWM nancial support to attend. Some men expressed re­ still fill a need. "It is crucial that women's organi­ sentment over "not getting a foot in the door" the zations stay strong for a long while yet, because way women do, she said. She added that these men it's not a dead issue," said Carberry. But in the did not agree that the low number of women in end, what really makes a difference "is that quiet mathematics points to a need for special efforts to reminder that there are a lot of women doing re­ encourage women in the field. Complaints of dis­ ally interesting mathematics." crimination have led to changes in some programs. -Allyn Jackson For example, the National Science Foundation (NSF) used to have programs that were aimed at en­ couraging women in science and that allowed only women principal investigators. The NSF reoriented those programs to allow male principal investiga­ tors after after being suedby someone who claimed discrimination because he was ineligible to apply to the foundation's Minority Graduate Fellowships program. One young woman mathematician, who asked not to be named, said that when she was in grad­ uate school, she sometimes found that she would be the only woman in a group of students. "There was not open sexism, but a difference in the ex­ pectation ofwhat Iwas there for," she noted. "Some­ times I would walk in and the mathematics talk would cease." In addition, she observed that some of the men could handle another man taking the upper hand in a mathematical conversation, but

AUGUST 2004 NOTICES OF THE AMS 783 WomeninAcademia:Are We AskingtheRight Questions? Carolyn Gordon andBarbaraLee Keyfitz

The recent study A National Analysis ofDiversity Some comments are in order on the selection of in Science andEngineering Faculties atResearch Uni­ departments to be surveyed. In each field the au­ versities by Donna ]. Nelson and Diana C. Rogers thors chose the fifty departments whose institu­ [2] has been widely cited by the National Organi­ tions received the largest research expenditures zation for Women (NOW) and in Congressional tes­ from the National Science Foundation. One can see timony [3], [4]. From fifty departments in each of good reasons for this criterion: it can be applied fourteen fields of science and engineering, in­ across fields, the information on NSF expenditures cluding mathematics, the authors obtained demo­ is readily available, and NSF expenditures would be graphic data (gender, race/ethnicity) of the tenured expected to correlate highly with research strength. and tenure-track faculty at each rank. As the first The resulting list of "top fifty" departments in each unified study of such demographic data for all field is a good sample of active research depart­ fourteen fields, the study draws attention to the sig­ ments, although not necessarily the top depart­ nificant underrepresentation of women and mi­ ments as perceived within the field. We compared norities (Blacks, Hispanics, and Native Americans). the list ofmathematics departments surveyed with the rankings of mathematics departments used in In mathematics the percentage of women in the the Annual Survey of the Mathematical Sciences rank offull professor, associate professor, and as­ (see [7]). The latter divides the Ph.D.-granting in­ sistant professor in the departments surveyed was stitutions into three groups: Group I, comprising the 4.6%, 13.2%, and 19.6% respectively. The compari­ forty-eight highest-ranked departments; Group II, son of assistant professors (19.6% female in 2002) consisting of the next ranked fifty-six departments; with recent Ph.D. recipients (27.2% female over the and Group III, which includes all remaining Ph.D.­ period 1993-2002) reveals a pattern of attrition granting departments. Of the "top fifty" mathe­ which persists across many of the disciplines sur­ matics departments in the Nelson-Rogers study, veyed. The survey points out that when minority forty-seven are Ph.D.-granting departments. Among status is also taken into account, the picture is these, twenty-nine appear in Group I, fourteen in even more dismal: minority women are practically Group II, and four in Group III. invisible in all ranks of the faculty at the institu­ The demographic data for mathematics obtained tions surveyed. in the Nelson-Rogers survey are similar to those in the Annual Survey of the Mathematical Sciences if Carolyn Gordon is the Benjamin CheneyProfessor ofMath­ one combines Groups I-III and Va in the Annual ematics at Dartmouth College. Her email address is Survey. (These groups comprise all the doctorate­ [email protected]. granting departments in pure and applied mathe­ Barbara Lee Keyfitz is professor ofmathematics at the Uni­ matics.) Unfortunately, the Annual Survey data for versity ofHouston. Her email address is b1k@math .uh. edu. the Group I departments paint an even more

784 NOTICES OF THE AMS VOLUME 51, NUMBER 7 discouraging picture. Indeed, the 2002 survey (Table stop the clock or extend the proba­ 3 in the Third Report [5]) shows that the percent­ tionary period, with or without taking age of females among untenured tenure-track fac­ a full or partial leave of absence, if the ulty in Group I is only 11% and the percentage of faculty member (whether male or fe­ women among the tenured faculty is roughly 7%. male) is a primary or coequal caregiver Consistent with the Nelson-Rogers study, the An­ of newborn or newly adopted children. nual Survey figures for the percentage of women Thus, faculty members would be enti­ among new Ph.D.'s in 2002 was 31% from all doc­ tled to stop the tenure clock while con­ torate-granting institutions and 24% from the Group tinuing to perform faculty duties at full I institutions. Thus, in the Group I institutions we salary... .Institutions should also take see an even higher rate of attrition from Ph.D. to care to see that faculty members are tenure-track than the already discouraging rate re­ not penalized in any way for requesting ported in the Nelson-Rogers study. and receiving extensions of the proba­ It is surprising that the Nelson-Rogers study tionary period. makes no mention ofpostdoctoral positions, an im­ portant first step in establishing a research career Even with an extended pretenure period, a fac­ ulty member caring for young children will succeed in almost all the fields surveyed. The Annual Sur­ only if (s)he is strongly dedicated to her/his career vey of the Mathematical Sciences shows that the percentage of women among new Ph.D.'s and the and is very effective at time management-quali­ percentage among postdoctoral fellows are pretty ties that give promise of a highly productive career. comparable, whether one restricts attention only The period of caring for small children occupies to Group I departments or considers all doctorate­ only a small part of one's career span. If one looks granting departments (Table 5E, [6]). The conspic­ around most departments, including the best, one uous drop in the percentage ofwomen occurs only sees that even in judging male faculty at the time when one considers the tenure track. Another fac­ of tenure, promotion committees have not invari­ ably made decisions that were in the department's tor not considered in the survey but surely opera­ best interest. A careful reexamination of the meth­ tive in most disciplines is that hiring in all fields tak.es place on the international stage these days, ods by which we make the most important deci­ sions in hiring and granting tenure and of the sup­ while the reported rates ofPh.D. attainment are for u.S. institutions only. port given to untenured faculty will benefit the quality ofresearch, scholarship, teaching, and men­ Why Is Attrition So High? toring in academic institutions. The study asserts that "new and totally different Conclusions ofthe Nelson-Rogers Study approaches will be needed" to right the imbalance. The authors assert that disparities inhiring and re­ The study cannot and does not attempt to answer tention between male and female scientists im­ the question ofwhether hiring and work practices at the top universities actively discriminate against pactwomen at all levels, from undergraduate to full professor: undergraduate science majors are dis­ women and minorities. The authors remark that there is general agreement that women are not ap­ couraged from pursuing academic careers bothby lack of role models and by observing female fac­ plying for tenure-track positions at research insti­ ulty in their own discipline "marginalized, treated tutions at a rate consistent with their attainment of degrees, although there is no consensus on an poorly, or not promoted." The article concludes: "In order to diversify suc­ explanation. cessfully and openwide the doors for women uni­ The problem is clearly complex and many­ versities have to examine culture, attitudes,' and faceted. A recent survey of members of the Asso­ policies they have long followed assuredly. This is ciation for Women inMathematics suggests one (of a long-overdue and realistic response to a chang­ possibly many) factors that may discourage women ing world. As Princeton chemist George McLendon from entering the tenure track: over 40% of the re­ observed, 'Academic institutions are intrinsically spondents identified flexibility in the tenure clock to allow for a better balance between career and monastic institutions that were created in the 13th century. They might need a little fine-tuning.'" childcare responsibilities as the issue that they A comparison, not mentioned in the analysis, would most like the AWM to address. (The ques­ gives a strong warning ofwhat maybefall our pro­ tion was not multiple choice!) This inflexibility in fession if universities do not change their culture. the tenure probationary period is a problem for The same thirtyyears following Title IX (Education women across academia. The American Association Act Amendments, 1972) that opened up universi­ of University Professors makes the following rec­ ties to women entering science and engineering ommendation [8]: fields also opened up the fields of law, medicine, The AAUP now recommends that, upon andbusiness to womeninunprecedented numbers. request, a faculty member be entitled to While faculties inthese fields may not have changed

AUGUST 2004 NOTICES OF THE AMS 785 either, the composition of those professions has changed dramatically in the thirty-year period. Ap­ parently the marketplace in the professions has been friendlier to women than has the professori­ SCHOOL OF ate. In the competition for talent [1], universities have generally beenlosing ground to better-paying careers. It would be doubly unfortunate if univer­ PHYSICAL. sities, faced with a shrinking pool, are making that SCIENCES; pool even smaller through poor recruitment or re­ tention practices. The Nanyang Technological University (NTU) has, as its proud antecedent, the Nanyang University which was founded as a private The conclusions of the Nelson-Rogers study are university in 1955. Currently, NTU has an undergraduate and consistent with the experience of many academic graduate student population of over 25,000 from some 50 countries and an international faculty of over 1,360. mathematicians. We encourage our colleagues to In addition to its five well-established Schools of Engineering, NTU examine the academic climate at their institutions. also has a reputable Nanyang Business School, the School of The Academic Climate website [9], developed by the Communication and Information, the National Institute of Education, and the School of Biological Sciences. NTU is seeking Association for Women in Science, is an excellent to establish three more new Schools: Humanities and Social resource. The site includes tools to initiate and Sciences; Physical Sciences; and Art, Design and Media. provide a framework for institutional assessment The new School of Physical Sciences marks the transformation of NTU from a mainly technological to a comprehensive university of gender equality in science departments, model with emphasis on interdisciplinary sciences in Chemistry, programs and policies, and many other resources. Mathematics, and Physics. We invite members of the mathematics community The Divisions of Chemistry, Mathematics and Statistics, and Physics to seek solutions within their own institutions. invite applications for full-time tenure-track faculty positions at the rank of Assistant, Associate, or Full Professor, to prepare for the first intake of students in July 2005. The successful applicants References are expected to teach undergraduate and graduate courses, undertake curriculum development, supervise master and doctoral [1] DEREK BOK, The Cost ofTalent: HowExecutives andPro­ students, and develop outstanding, internationally recognized fessionals Are Paid and How It Affects America, The research programmes that generate external funding and intellectual output and perform essential administrative work. Free Press, Macmillan, New York, 1993. [2] DONNA J. NELSON and DIANA C. ROGERS, A NationalAnaly­ The research interest areas include, but are not limited to: sis ofDiversity in Science and Engineering Faculties at Division of Chemistry Research Universities, Final Report, http://www . • Biological & Medicinal Chemistry • Computational & Theoretical Chemistry now.org/issues/diverse/diversity_report. • Nanoscience and Nanotechnology pdf. • Synthesis and Catalysis [3] NOW staff, Study Shows Top Colleges Have Few Women • Organic Chemistry • Physical Chemistry Profs in Hard Sciences, National Organization for Division of Mathematics and Statistics;·~'i~<,'F Women, http://www. now. org/i ssues/diverse/ • Coding Theory , Oll504study.html. • Dynamical System [4] KIM GANDY, Universities Must Change Faculty Atmos­ • Computational Mathematics phere for Women and People of Color, press confer­ • Quantum Computing ence, http://www.now . 0 r 9/ issues/ di ve r s e/ Division of Physics 01l504report.html. • Biophysics / Medical Physics, Biomaterials And Bioimaging REMICK • Laser, Subatomic Physics, Photonics [5] ELLEN E. KIRKMAN, JAMES W. MAxwELL, and KINDA • Nanoscience And Nanotechnology PRIESTLEY, 2002 Annual Survey of the Mathematical • Quantum Information Science and Quantum Technology Sciences (Third Report), Notices Amer. Math. Soc. 50 Applicants for these positions should have a PhD in the respective (September 2003), 925-935. discipline from an accredited university and an outstanding record [6] ELLEN E. KIRKMAN, JAMES W. MAxwELL, and COLLEEN ROSE, in research and teaching. Preference will be given to individuals with demonstrated research and teaching excellence such as 2003 Annual Survey of the Mathematical Sciences evidence of sustained scholarship, influential research, dynamic (First Report), Notices Amer. Math. Soc. 51 (February teaching, international reputation, record of external funding, and 2004),218-233. an ability to mentor postgraduate students. [7]http://www.ams.org/employment/groups_des. Successful applicants will be offered attractive remuneration packages commensurate with their qualifications and work html. experience. Details on the benefits provided and the terms and [8] American Association of University Professors, conditions of service may be obtained from http://www.ntu.edu.sg/ http://www.aaup.org/statements/REPORTS/ personnel/terms.htm reOlfam. htm. To apply, please submit an application form (which can [9] Association for Women in Science, http://www . be downloaded from http://www.ntu.edu.sg/personnel/ Applnforms.htm) and a detailed curriculum vitae which should academicclimate.org/. include your areas of research interest and list of publications. Applications should be sent to: The Vice-President (Human Resources) NANYANG TECHNOLOGICAL UNIVERSITY Office of Human Resources Administration Building, Level 4 50 Nanyang Avenue Singapore 639798 Telefax: (65) 6791 9340 Email: [email protected]

786 NOTICES OF THE AMS VOLUME 51, NUMBER 7 GromovReceives NemmersPrize

MIKHAEL L. GROMOV, professor of mathematics at The Work of Mikhael Gromov the Institut des Hautes Etudes Scientifiques, Gromov's work covers all areas of Bures-sur-Yvette, France, and Jay Gould Profes­ geometry and its relations with sor of Mathematics at the Courant Institute of neighboring fields such as topol­ Mathematical Sciences at New York University, ogy and analysis. He brings a pro­ has been named the recipient of the Frederic foundly original and expansive Esser Nemmers Prize in Mathematics. viewpoint to any subject he works The prize recognizes outstanding achieve­ on. This viewpoint illuminates the ment in mathematics as demonstrated by major subject and opens spectacular vis­ contributions to new knowledge or the develop­ tas which cannot be seenwith the ment of significant new modes of analysis. The nose closer to the ground, and prize carries a stipend of $150,000. In connec­ sometimes it creates a whole new tion with the prize, Gromov is expected to deliver subject which is then exploredby public lectures and participate in other schol­ many other researchers for years, arly activities at Northwestern University during long after Gromov has moved on. the 2004-05 academic year. A hallmark of much of Gromov's work is the softening of The Selection Committee for the prize recog­ geometry, whereby equations are Mikhael L. Gromov nized Gromov "for his work inRiemannian geom­ replaced by inequalities or ap- etry, which revolutionized the subject; his theory proximate or asymptotic equations. Examples ofpseudoholomorphic curves in symplectic man­ include the "coarse" viewpoint on Riemannian ifolds; his solution of the problem of groups of geometry, which considers all Riemannian struc­ polynomial growth; and his construction of the tures at once; the "homotopic" viewpoint on theory of hyperbolic groups." His work has been partial differential equations, which solves revolutionary in a number ofbasic areas ofmod­ generic underdetermined and some determined ern geometry. and overdetermined systems via topology; and Gromov's work is "both tremendously elegant the "asymptotic" viewpoint on geometric group and immediately relevant to problems in applied theory. Gromov has revolutionized symplectic mathematics and mathematical physics in a way geometryby the introduction of methods from that reflects his tremendous creativity and excel­ complex analysis and has given important im­ lent taste," said Ezra Getzler, professor ofmathe­ pulses to index theory and to sub-Riemannian matics at Northwestern University. Getzler com­ (or Carnot-Caratheodory) geometry. He has in­ mented that "Gromov's work on symplectic troduced many important new concepts into manifolds has already played a central role in the geometry, most of which are outgrowths of his development of one of the most promising uni­ "coarse" or "soft" viewpoint: almost flatness of metrics and connections, simplicial volume, fied field theories of theoretical physics, string K-area, hyperbolicity of groups, etc. theory. He is a true successor to great geometers A few of the many concrete results provedby of the past, such as Felix Klein, who lectured at Gromov are: the existence of foliations on open Northwestern in 1893."

AUGUST 2004 NOTICES OF THE AMS 787 manifolds, the volume comparison estimate gen­ waukee. Erwin Nemmers, who persuaded his eralizing a result of Bishop, the finiteness of brother to join him in making a substantial con­ positively curved manifolds, the nonexistence of tribution to Northwestern, served as a member of metrics of positive scalar curvature on flat or the faculty of the Kellogg School of Management hyperbolic and other enlargeable manifolds (joint from 1957 until his retirement in 1986. Along with with H. B. Lawson Jr.), the construction ofmanifolds his brother, Frederic, he was a principal in a ofvariable negative curvature (with W. P. Thurston), Milwaukee-based, family-owned church music the construction of nonarithmetic lattices in publishing house. hyperbolic spaces (with I. Piatetski-Shapiro), and Their gifts, totaling $14 million, were designated the uniqueness of the symplectic structure on the by Erwin and Frederic Nemmers for two purposes: complex projective plane (completed by C. H. the establishment of four endowed professorships Taubes). in the Kellogg School of Management and the Gromov has had and will continue to have a establishment of the Nemmers Prizes. widespread influence on contemporary mathe­ A third award, the Michael Ludwig Nemmers matics. Prize inMusical Composition, was announced infall 2003 and will be awarded in 2004-05. Like the Biographical Sketch economics and mathematics prizes, the music prize Mikhael L. Gromov was born onDecember 23, 1943, will be awarded every other year; it will have a in Boksitogorsk, USSR, and has been a French cit­ value of $100,000. izen since 1992. He studied at the University of -Allyn Jackson Leningrad, where he received his doctorate in 1969. In 1974 he left the USSR and became a professor at the State University of New York at Stony Brook. In 1981 he moved to the Universite de Paris and the following year assumed his present position as a permanent professor at the Institut des Hautes Etudes Scientifiques. He is currently also a professor at the Courant Institute of Mathematical Sciences, New York University. Gromov has received many prizes and honors, including the Prize of the Mathematical Society of Moscow (1971), the Oswald Veblen Prize of the AMS (1981), the Prix Elie Cartan de l'Academie des Sciences de Paris (1984), the Prix de l'Union des Assurances de Paris (1989), the Wolf Prize (1993), the AMSLeroy P. Steele Prize for Seminal Contri­ bution to Research (1997), the Lobachevski Medal (1997), the Balzan Prize (1999), and the Kyoto Prize (2002). He was an invited speaker at the Interna­ tional Congress of Mathematicians in Nice (1970), Helsinki (1978), Warsaw (1982), and Berkeley (1986). He is a foreign member of the U.S. National Acad­ emy of Sciences and of the American Academy of Arts and Sciences, and a membre de /'institut of l'Academie des Sciences de Paris. He received an honorary doctorate from the Universite de Geneve in 1992.

About the Nemmers Prize Every other year Northwestern University presents the Frederic Esser Nemmers Prize in Mathematics and the Erwin Plein Nemmers Prize in Economics. The 2004 prize in economics went to Ariel Rubin­ stein, professor of economics at Tel Aviv Univer­ sity and New York University. The Nemmers Prizes are made possible through bequests from the late Erwin E. Nemmers, a former member of the Northwestern faculty, and his brother, the late Frederic E. Nemmers, both of Mil-

788 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 AnnualSurvey oftheMathematicalSciences (Second Report)

UpdatedReport onthe 2002-2003 U.S. Doctoral Recipients Starting Salary Survey ofthe 2002-2003 U.S. Doctoral Recipients

Ellen E. Kirkman, James ~ Maxwell, and Colleen A. Rose

Update on the 2002-2003 This Second Report of the 2003 Annual Survey gives an update of the 2002-2003 new doctoral recipients from the First Report, which ap­ U.s. Doctoral Recipients peared in the Notices ofthe AMS in February 2004, pages 218-33. Prior Introduction to 2000 this report included information about faculty size, depart­ The Annual Survey of the Mathematical Sciences col­ mental enrollments, majors, and graduate students for departments of mathematical sciences in four-year colleges and universities in the lects information each year about departments, United States. This information is now published as a third report in faculties, and students in the mathematical sci­ the September Notices of the AMS. The First Report gave salary data ences at four-year colleges and universities in the for faculty members in these same departments. It also had a section United States. Definitions of the various groups on new doctoral recipients in statistics that is not updated here. surveyed in the Annual Survey can be found on The 2003 Annual Survey represents the forty-seventh in an annual page 800 of this report. series begun in 1957 by the American Mathematical Society. The 2003 This Second Report includes data from two parts Survey is under the direction of the Data Committee, a joint commit­ of the 2003 Annual Survey. First, we update infor­ tee of the American Mathematical Society, the American Statistical mation about new doctoral recipients reported Association, the Institute of Mathematical Statistics, and the Mathematical AssQciation of America. The current members of this committee are earlier in the February 2004 issue. Second, we Amy Cohen-Corwin, Donald M. Davis, Lorraine Denby, Alexanderj. Hahn, present the starting salaries of the new doctoral Naresh jain, G. Samuel jordan, Stephen F. Kennedy, Ellen E. Kirkman recipients who responded to a follow-up survey. (chair), David j. Lutzer, and James 'W. Maxwell (ex officio). The com­ The names of the 2002-2003 doctoral recipi­ mittee is assisted by AMS survey analyst Colleen A. Rose. Comments ents and their thesis titles were published in or suggestions regardit:lg this Survey Report may b~ directed to the com­ "Doctoral Degrees Conferred" (Notices ofthe AMS, mittee. February 2004, pages 246-63). This list has been supplemented by twenty additional new doctor­ using data gathered with a questionnaire, ates. The supplemental listing appears at the end Employment Experiences of New Doctoral of this report on page 801. Recipients (EENDR). The EENDR was sent in early Information about recipients of doctoral degrees October 2003 to all new doctoral recipients whose awarded between July 1,2002, and June 30, 2003, address was known. When a new doctoral recipient was collected from doctorate-granting departments did not respond or no address was known, beginning in late spring 2003 and from a follow-up information supplied by the department was used. census of individual degree recipients beginning in October. The "2003 Annual Survey First Report" (Notices ofthe AMS, February 2004, pages 218-33) Ellen E. Kirkman is professor ofmathematics, Wake Forest presented survey results obtained about new University. James ~ Maxwell is AMS associate executive doctoral recipients from the departments. Here we director for Membership and Programs. Colleen A. Rose update information for new doctoral recipients is AMS survey analyst.

AUGUST 2004 NOTICES OF THE AMS 789 2003 Annual Survey of the Mathematical Sciences

Updated Employment Status of 2002-2003 Highlights U.S. Doctoral Recipients Table lA shows the fall and final counts of doctoral • There were 1,037 doctoral recipients from u.s. institutions for recipients in the mathematical sciences awarded by 2002-2003, up 77 (8%) from the previous year. Last year's number U.S. institutions in each year from 1993 through was the lowest since 1989-1990. 2003. Final counts include those new doctoral re­ • The number of doctoral recipients who are u.s. citizens is 499, cipients reported from departments who missed the up 71 (12% increase) from last year's number, which was the deadline for inclusion in the First Report. Numbers lowestfigure reported since 1989-1990.The percentage ofu.s. in this table have been revised from reports prior citizens among all doctoral recipients this year is 48%, up from to 1998-1999 to exclude new doctorates data from 45% last year. The number of new doctoral recipients who are Group Vb departments, which are no longer sur­ notu.s. citizens is 538, up 6 from last year's number; this reverses veyed. Reversing the downward trend of the past afive-year trend ofdecline from 639 non-U.S. citizens in 1997-1998. four years, this year the total number of new doc- • Females totaled 308 (30%) of all new doctoral recipients, up in number (and down in percentage) from 296 (31 %) last year. Of Table 1A: Annual U.S. Doctoral Recipients, the 499 U.S. citizen new doctoral recipients, 158 are female (32%, Fall and Final Counts, 1993 to 2003 up from 30 %last year). The highest percentage offemales among the annual counts ofu.s. doctoral recipients was 34%, reported Year Fall Final for 1998-1 999. 1993-1994 1025 1034 • The number ofdoctoral recipients whose employment status is 1994-1995 1148 1157 unknown is much higherthis year than it has been in recent years. This year's report includes 193 newdoctoral recipients ofunknown 1995-1996 1098 1099 employment status; last year this number was 94. This fact 1996-1997 1123 1130 should be considered in interpreting comparisons in employment 1997-1998 1163 1176 data from this year and previous years. 1998-1999 1133 1135 • The final unemployment rate for 2002-2003 doctoral recipients 1999-2000 1119 1127 was 5.0%, the highest reported since 1996, when it was 8.1 %. 2000-2001 1008 1065 • Ofthe 792 new doctoral recipients known to have employment in fall 2003, 682 (86%) newdoctoral recipients found employment 2001-2002 948 960 in the U.S; last year this percentage was 88%. 2002-2003 1017 1037 • The number of new doctoral recipients taking positions in U.S. business and industrywas 99 in fall 2003, a 27% decrease from Table 1B: Citizenship of Annual U.S. Doctoral last year's number; this number has been decreasing the past Recipients, 1998 to 2003 four years from the 223 reported in fall 2000 (a 56 %decrease). The percentage ofdoctoral recipients employed in the u.s. who Year u.s. Non-U.S. TOTAL were hired in business and industry in the u.s. is 19%'in fall 1998-1999 560 575 1135 2002-2003, down from .24% in 2001-2002 and from 30% in 1999-2000 566 561 1127 2000-2001. 2000-2001 532 533 1065 • The numberofdoctoral recipients taking U.S. academic positions has been decreasing each ofthe past 5 years, from 6 ~ 0 in 1999 2001-2002 428 532 960 to 551 in 2003. 2002-2003 499 538 1037 • The number of new doctoral recipients hired by master's and toral recipients is 1,037, up from the previous year bachelor's institutions was 158 this year. This number is a 7% by 77; last year's number was the lowest since increase in last year's figure of 148. This is the first increase 1989-1990. reported since 1998. Table IB shows trends in the number ofnew doc­ • There were 551 new doctoral recipients responding tothe EENDR toral recipients for the past five years broken down survey; ofthe 469who found employment in the U.S., 54% reported obtaining a permanent position (last year this percentage was Table 1C: 2002-2003 U.S. Doctoral Recipients 52%). by Type of Degree-Granting Department • Aftershowing asignificant increase overthe previous fouryears, there is a decline this year in the percentage of temporarily I (Pu) I (Pr) II III IV Va employed respondents who reported taking a postdoctoral Number 258 154 171 122 241 91 position, from 83% in fall 2002 to 76% in fall 2003. The number of respondents who reported taking a postdoctoral position in Percent 25 15 16 12 23 9 fall 2003 was 164, down from 203 for fall 2002. by U.S. citizens and non-U.S. citizens. There was a drop of 98 new doctoral recipients from 1998-1999 to 2002-2003, mostly explained by a drop of 61 U.S. citizen new doctoral recipients. This year the

790 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences

Table 2A: 2002-2003 U.S. Doctoral Recipients: Field of Thesis by Fall 2003 Employment Status, Updated April 2004

FIELD OF THESIS

Real, Comp., Discr. Math./ Numerical Linear Differential, Algebra Funct., & Combin./ Analysis/ Nonlinear Integral, & Number Harmonic Geometry/ Logic/ Statistics/ Applied Approxi- Optim./ Difference Math. Other/ TYPE OF EMPLOYER Theory Analysis Topology Compo Sci. Probability Biostat. Math. mations Control Equations Educ. Unknown TOTAL

Group I (Public) 29 8 13 4 5 0 5 6 1 12 0 1 84 Group I (Private) 14 5 9 4 1 2 3 0 1 14 0 0 53 Group II 7 3 9 3 2 2 9 3 4 10 2 0 54 Group III 0 4 1 3 0 10 1 1 1 3 1 0 25 Group IV 0 0 1 0 2 36 0 0 0 0 0 0 39 Group Va 0 0 0 2 1 0 4 2 0 0 0 0 9 Master's 8 9 8 3 3 6 4 2 0 2 3 2 50 Bachelor's 30 11 16 8 1 7 6 7 4 13 5 0 108 Two-Year College 0 0 0 0 0 0 1 1 0 1 0 0 3 Other Academic Dept. 7 3 2 5 3 56 15 3 1 6 6 0 107 Research Institute/ 3 0 0 3 0 11 0 2 0 0 0 0 19 Other Nonprofit Government' 6 0 1 2 2 11 3 4 0 3 0 0 32 Business and Industry 9 6 2 8 4 50 11 5 3 1 0 0 99 Non-U.S. Academic 22 5 10 8 0 14 6 14 2 15 1 0 97 Non-U.S. Nonacademic 2 1 1 0 0 4 1 2 0 2 0 0 13 Not Seeking Employment 1 1 0 0 0 5 0 2 0 1 0 0 10 Still Seeking Employment 8 1 3 2 0 8 7 4 3 6 0 0 42 Unknown (U.S.) 11 10 9 8 0 34 17 8 1 6 1 4 109 Unknown (non-U.S.)1 12 8 9 6 5 19 14 6 0 4 1 0 84 TOTAL 169 75 94 69 29 275 107 72 21 99 20 7 1037 Column I Male 133 61 69 52 21 161 75 55 18 73 6 5 729 Subtotals I Female 36 14 25 17 8 114 32 17 3 26 14 2 308

1 Includes those whose status is reported as "unknown" or "still seeking employment".

Table 28: 2002-2003 U.S. Doctoral Recipients: Type of Degree-Granting Department by Fall 2003 Employment Status, Updated April 2004

TYPE OF DOCTORAL DEGREE-GRANTING DEPARTMENT Group I Group I Row (Public) (Private) Group II Group III Group IV Group Va Subtotals TYPE OF EMPLOYER Math. Math. Math. Math. Statistics Applied Math. TOTAL Male Female

Group I (Public) 44 25 9 2 0 4 84 64 20 Group I (Private) 18 29 1 0 2 3 53 43 10 Group II 17 5 24 3 2 3 54 40 14 Group III 2 1 5 10 4 3 25 19 6 Group IV 1 2 1 0 35 0 39 20 19 Group Va 1 0 0 0 1 7 9 6 3 Master's 8 2 19 15 6 0 50 27 23 Bachelor's 20 11 45 19 7 6 108 73 35 Two-Year College 0 0 3 0 0 0 3 1 2 Other Academic Dept. 10 8 11 12 52 14 107 66 41 Research Institute/ 2 7 0 0 10 0 19 9 10 Other Nonprofit Government 5 2 8 2 8 7 32 19 13 Business and Industry 19 14 5 8 46 7 99 75 24 Non-U.S. Academic 33 21 11 14 13 5 97 76 21 Non-U.S. Nonacademic 2 2 3 2 2 2 13 12 1 Not Seeking Employment 2 0 2 2 4 0 10 7 3 Still Seeking Employment 7 8 8 11 4 4 42 27 15 Unknown (U.S.) 34 8 9 13 31 14 109 81 28 Unknown (non-U.S.)1 33 9 7 9 14 12 84 64 20 TOTAL 258 154 171 122 241 91 1037 729 308 Column I Male 201 126 121 68 141 72 729 Subtotals I Female 57 28 50 54 100 19 308

1 Includes those whose status is reported as "unknown" or "still seeking employment".

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Table 2C: 2002-2003 U.S. Doctoral Recipients: Field of Thesis by Type of Degree-Granting Department, Updated April 2004

FIELD OF THESIS TYPE OF DOCTORAL Real, Comp., Discr. Math./ Numerical Linear Differential, Algebra Analysis/ Nonlinear Integral, & DEGREE-GRANTING Funct., & Combin./ Number Harmonic Geometry/ Logic/ Statistics/ Applied Approxi- Optim./ Difference Math. Other/ DEPARTMENT Theory Analysis Topology Compo Sci. Probability Biostat. Math. mations Control Equations Educ. Unknown TOTAL Group I (Public) 72 33 41 21 11 4 16 15 5 38 1 1 258 Group I (Private) 43 12 25 15 5 7 19 8 1 17 0 2 154 Group II 39 17 23 10 5 6 21 18 7 22 2 1 171 Group III 13 13 4 11 2 17 11 19 1 14 17 0 122 Group IV 0 0 0 0 4 232 2 0 0 0 0 3 241 Group Va 2 0 1 12 2 9 38 12 7 8 0 0 91 TOTAL 169 75 94 69 29 275 107 72 21 99 20 7 1037

Table 2D: Percentage of Total Employed New Doctoral number ofnon-U.S. citizen new doctoral recipients Recipients by Type of Employer, Fall 1999 to Fall 2003 was 679 in 1992-1993. Table lC gives a breakdown ofthe 1,037 doctoral u.s. Employed Non-U.S. Employed TOTAL degrees awarded in the mathematical sciences NUMBER % Academic Nonacademic Academic Nonacademic EMPLOYED between July 1, 2002, and June 30, 2003, by type of degree-granting department. Fall 1999 64 23 11 2 955 Tables 2A, 2B, and 2C display updates of em­ Fall 2000 62 28 10 1 957 ployment data, found in these same tables in the Fall 2001 63 27 9 2 914 First Report, for the fall count of 2002-2003 Fall 2002 67 22 10 1 829 doctoral recipients plus twenty additional doctoral recipients reported late. These tables are parti­ Fall 2003 70 17 12 2 792 tioned by field of thesis research, by the survey group of their degree department, and by type of number ofnew doctoral recipients who are u.s. cit­ employer. At the time of this Second Report, the izens is 499, an increase of 71 over last year; last fall 2003 employment status of 844 of the 1,037 year's number of u.s. citizens was the lowest fig­ doctoral recipients was known. ure reported since 1989-1990. The all-time high The fall 2003 unemployment rate for new doc­ toral recipients, based on information gathered by

Figure 1: Percentage of New Doctoral Recipients Unemployed, . As Reported in the Respective Annual Survey Second Reports, 1979 to 2003

Year Percentage 12.0 1979 1.5 1980 0.9 11.0 1981 0.0 10.0 1982 1.8 1983 2.2 9.0 1984 2.1 1985 0.8 8.0 1986 2.3 1987 3.0 7.0 1988 1.4 1989 3.0 6.0 1990 2.2 1991 5.0 5.0 1992 6.7 1993 8.9 4.0 1994 10.7 1995 10.7 3.0 1996 8.1 1997 3.8 2.0 1998 4.9 1.0 1999 4.7 2000 3.3 0.0 2001 3.7 ,...... O"l N ..q- LI"'I 1..0 ,...... 00 O"l N ..q- LI"'I 1..0 00 O"l 0 N m 2002 2.9 ,...... 0 m 0 m 00 00 00 00 00 00 00 00 00 00 O"l O"l O'l O'l O'l O'l O'l O'l O'l O'l 0 0 0 0 2003 5.0 O"l O'l O'l O"l O"l O"l O"l O"l O"l O"l O"l O"l O"l O"l O"l O"l O'l O"l O"l O"l O'l 0 0 0 0 N NNN

792 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences the time of the Second Report, was 5.0%. This is the dividual new doctoral recipients who responded to highest unemployment rate since 1996, whenit was the follow-up questionnaire. The nextfew paragraphs 8.1%. Figure 1 presents the fall 1979 through fall discuss some ofthe informationpresentedin these 2003 trend in the final unemployment rate of new tables. doctoral recipients. The counts onwhich these rates are determined do not include those new doctoral recipients whose fall employment status was un­ Table 3B: Number of New Doctoral Recipients known at the time of the Second Report. Although Taking U.S. Academic Positions by the number of recipients whose employment sta­ Type of Degree-Granting Department, tus was reported as unknown had been declining, Fall 1999 to Fall 2003 from 150 in 1997 to a low of 94 in 2002, this year it spiked to a high of 193. For future reports, mea­ Group I (Pu) I (Pr) II III IV Va TOTAL sures are being taken which should reduce the Fall 1999 166 91 146 82 86 39 610 number of recipients whose employment status Fall 2000 144 82 126 79 131 28 590 gets reported as unknown. Note that prior to 1999 the new doctoral recipients from Group Vb are in­ Fall 2001 159 71 126 80 108 30 574 cluded in the total unemployment rate for each Fall 2002 133 86 107 91 102 34 553 year. Fall 2003 123 90 118 61 119 40 551 Of the 844 new doctoral recipients whose employment is known, 682 were employed in the Table 3A shows that the fall 2003 total U.S., 110 were employed outside the U.S., 42 were number of doctoral recipients taking positions in still seeking employment, and 10 were not seeking business or industry is 99; this number reflects a employment. continued decline, and a 56% decrease since fall Table 2D presents the trend in the percentage 2000's high of 223. While some groups have of employed new doctoral recipients by type of em­ shown a slight increase, Groups II and IV show ployer for the last five years. Academic employment the largest decreases over last year. includes those employed by research institutes and other nonprofits. The percentage of the total Table 3C: Number of New Doctoral Recipients employed new doctoral recipients that are in U.S. Taking U.S. Academic Positions by Type of academic positions is at a five-year high, while the Hiring Department, Fall 1999 to Fall 2003 percentage of the total employed in U.S. nonacad­ emic positions is at a five-year low. Group I-III IV Va M&B Other TOTAL Among new doctoralrecipients who are employed, Fall 1999 233 47 19 193 118 610 the percentage taking nonacademic employment Fall 2000 216 51 11 180 132 590 (U.S. government, U.S. business and industry, and non-U.S. nonacademic) varied significantly by field Fall 2001 214 49 11 178 122 574 of thesis. For those whose field of thesis is in the Fall 2002 222 45 10 148 128 553 first three columns in Table 2A, this percentage is Fall 2003 216 39 9 158 129 551 thelowest, at 11% (up from 9%), while the percentage

Table 3A: Number of New Doctoral Recipients Table 3C shows that the number of new Taking Positions in Business and Industry doctoral recipients taking U.S. academic posi­ in the u.s. by Type of Degree-Granting tions has continued to decline over each of the Department, Fall 1999 to Fall 2003 past five years, from 610 in 1999 to 551 in 2003. The number hired by Groups M and B has Group I (Pu) I (Pr) II III IV Va TOTAL dropped each of the years 1999-2002, but is Fall 1999 32 24 28 21 66 14 185 slightly up this year; there has been a 18% Fall 2000 33 28 37 24 83 18 223 Table 3D: Females as a Percentage of Fall 2001 28 15 27 26 75 23 194 2002-2003 U.S. Doctoral Recipients Produced by and Hired by Fall 2002 18 12 19 7 65 15 136 Doctoral-Granting Groups, Fall 2003 Fall 2003 19 14 5 8 46 7 99 % I (Pu) I (Pr) II III IV Va TOTAL for those with theses in probability or statistics is Produced 22 18 29 44 41 21 30 the highest, at 30% (down from 39%). Hired 24 19 26 24 49 33 27 Tables 3A through 3D first appeared in the First Report for 2000-2001, although they do not have decrease from fall 1999 to fall 2003. This decline the same table numbers inthatreport. They have all may reflect more hiring at these institutions of beenupdatedwithinformationobtainedfrom the in-

AUGUST 2004 NOTICES OF THE AMS 793 2003 Annual Survey of the Mathematical Sciences

Table 3E: 2002-2003 Male U.S. Doctoral Recipients: Type of Citizenship by Fall 2003 Employment Status

CITIZENSHIP TOTAL MALE NON-U.S. CITIZENS DOCTORAL U.S. CITIZENS TYPE OF EMPLOYER Permanent Visa Temporary Visa Unknown Visa RECIPIENTS U.S. Employer 255 16 173 18 462 U.S. Academic 205 14 137 12 368 Groups I, II, III, and Va 92 4 68 8 172 Group IV 12 1 7 0 20 Non-Ph.D. Department 99 9 55 4 167 Research Institute/Other Nonprofit 2 0 7 0 9 U.S. Nonacademic 50 2 36 6 94 Non-U.S. Employer 11 0 75 2 88 Non-U.S. Academic 9 0 66 1 76 Non-U.S. Nonacademic 2 0 9 1 12 Not Seeking Employment 6 0 1 0 7 Still Seeking Employment 13 2 12 0 27 Subtotal 285 18 261 20 584 Unknown (U.S.) 55 6 18 2 81 Unknown (non-U.S.)1 1 0 50 13 64

TOTAL 341 24 329 35 729

1 Includes those whose status is reported as "unknown" or "still seeking employment".

Table 3F: 2002-2003 Female U.S. Doctoral_ Recipients: Type of Citizenship by Fall 2003 Employment Status

CITIZENSHIP TOTAL FEMALE NON-U.S. CITIZENS DOCTORAL U.S. CITIZENS TYPE OF EMPLOYER Permanent Visa Temporary Visa Unknown Visa RECIPIENTS U.S. Employer 130 16 66 8 220 U.S. Academic 107 12 57 7 183 Groups I, II, III, and Va 23 3 24 3 53 Group IV 10 1 7 1 19 Non-Ph.D. Department 68 7 24 2 101 Research Institute/Other Nonprofit 6 1 2 1 10 U.S. Nonacademic 23 4 9 1 37 Non-U.S. Employer 3 1 18 0 22 Non-U.S. Academic 3 1 17 0 21 Non-U.S. Nonacademic 0 0 1 0 1 Not Seeking Employment 2 0 1 0 3 Still Seeking Employment 9 4 2 0 15 Subtotal 144 21 87 8 260 Unknown (U .S.) 14 5 7 2 28 Unknown (non-U.S.)1 0 0 15 5 20

TOTAL 158 26 109 15 308

1 Includes those whose status is reported as "unknown" or "still seeking employment".

individuals completing a postdoctoral appoint­ doctoral recipients known to have jobs in fall 2003. ment. Last year this percentage was 88%. Table 3D gives information about the production Sex and citizenship is known for all of the 1,037 and hiring offemale new doctoral recipients in the new doctoral recipients. The final count of new doctoral-granting departments ofthis survey. From doctoral recipients who are U.S. citizens is 499 Table 3D we see that the percentage of females (48%). For the last five years this figure has re­ hired ranges from a high of 49% in Group IV to a mained very close to 50%, the largest percentage low of 19% in Group I (private). reportedby the Annual Survey since the mid-1980s. Updated Information about 2002-2003 U.S. Pages 224-6 of the First Report present further Doctoral Recipients by Sex and Citizenship information related to the citizenship of the Tables 3£ and 3F show the sex and citizenship of 2002-2003 new doctoral recipients. the 1,037 new doctoral recipients and the fact Of the 499 U.S. citizen new doctoral recipients that 682 new doctoral recipients found jobs in the reported for 2002-2003, 158 are female and 341 U.S. this year. This is 86% of the 792 new are male. While females accounted for 32% of the

794 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences

Table 3G: Number of 2002-2003 New Table 4A: Number (and Percentage) of Annual EENDR Doctoral Recipients Employed in the u.s. by Respondents Taking U.S. Positions by Job Status, Citizenship and Type of Employer Fall 1999 to Fall 2003

CITIZENSHIP U.S. Employed Fall 1999 Fall 2000 Fall 2001 Fall 2002 Fall 2003 U.S. EMPLOYER U.S. Non-U.S. TOTAL TOTAL 512 536 473 510 469 Academic, Groups I-Va 137 127 264 Permanent 273 (53) 317 (59) 266 (56) 264 (52) 253 (54) Academic, Other 175 112 287 Temporary 237 (46) 218 (41) 205 (43) 245 (48) 216 (46) Nonacademic 73 58 131 Perm not avail. 101 (43) 92 (42) 107 (52) 90 (37) 87 (40) TOTAL 385 297 682 Postdoctorate 155 (65) 157 (72) 143 (70) 203 (83) 164 (76) Perm not avail. 58 (37) 55 (35) 42 (29) 69 (34) 53 (32) U.S. citizen total, both figures represent an Unknown 2 1 2 1 0 increase over last year's counts of 130 and 298, respectively. Table 48: Percentage of Annual EENDR Respondents Taking Table 3G shows that U.S. academic doctoral U.S. Positions by Employment Sector within Job Status, departments, Groups I through Va, hired 52% U.S. Fall 1999 to Fall 2003 citizens, while groups M, B, and all other academic departments hired 61% U.S. citizens. U.S. citizens U.S. Employed Fall 1999 Fall 2000 Fall 2001 Fall 2002 Fall 2003 represented 56% of those hired into nonacademic Permanent positions. Among the 682 new 2002-2003 doc­ Academia 59 59 62 70 76 toral recipients employed in the U.S., 19% took Government 4 4 6 6 4 nonacademic employment (government or busi­ Bu siness/Ind. 37 36 32 23 20 ness and industry.) This percentage is down from Temporary 24% in 2001-2002 and from 30% in 2000-2001. Academia 94 95 95 93 94 Government 5 2 4 6 3 New Information from the EENDR Survey Business/Ind. 0 2 0 1 3 Of the 1,017 new doctoral recipients reported in the First Report, the 910 whose addresses were known were sent the Employment Experiences of 87 (40%) reported taking temporary employment New Doctoral Recipients (EENDR) survey in Octo­ because a suitable permanent position was not ber 2003, and 551 (54%) responded. The response available and 164 (76%) classified their position as rates varied considerably among the various sub­ postdoctoral. Furthermore, among those in post­ groups of new doctoral recipients defined by their doctoral positions, 32% responded that they took employment status as reported by departments. the positionbecause a suitable permanent position Among those who were employed, the highest re­ was not available. Of particular note in Table 4A sponse rate, 74%, was from those in academia in is that after showing a 13% increase last year, this the U.S., while the lowest, 57%, was from those in year there is a decline in the percentage of tem­ U.S. nonacademic. porarily employed respondents who reported The EENDR gathered de- tails on employment experi­ ences not available through Figure 2: Age Distribution of 2002-2003 EENDR Respondents departments. The rest of this ~ section presents additional in­ 80 formation available on this ~I·~(i 70 subset of the 2002-2003 doc­ ~:)Id toral recipients. 60 :~l; .6. Table 4A provides the ;iii I:'; ,Ii >. 50 ::~ trend in EENDR respondents v s::::: ~~ I:~ Ii taking permanent and tem­ QI 40 porary positions in the U.S for ::::s ~ ::~;;~ '1 C" ~< .~ ~,1.' ~ t'I." ' 0"; fall 1999 through fall 2003. 30 ';~,,,< '\, u.. '~ I;~~ !:E ',d' ;\' :~{; This year we see that among i<,~:l : , ';i]: 20 the 469 employed in the U.S., bi, ~' oJ ~ 253 reported obtaining a per­ 10 ~~1 I~~~~ ti . 's,' manent position and 216 a F r., I'~F II~;~ .J:In. ~ temporary position. Of the o '" 20 25 30 35 40 45 50 55 60 65 216 in temporary positions, Age

AUGUST 2004 NOTICES OF THE AMS 795 2003 Annual Survey of the Mathematical Sciences

taking a postdoctoral position; last year the per­ Starting Salary Survey of the centage of temporarily employed respondents who were hired in postdoctoral positions was 83%, and 2002-2003 U.S. Doctoral this year it was 76%. The figures reported in this Recipients table for fall 2001 and fall 2002 have been corrected, as they were incorrectly reported onpage 807 of the The starting salary figures for 2003 were "2002 Annual Survey of the Mathematical Sciences compiled from information gathered on the Second Report" (Notices ofthe AMS, August 2003). EENDR questionnaires sent to indiViduals who received doctoral degrees in the mathematical Table 4B shows the employment trends of sciences during the 2002-2003 academic year permanent and temporary positions broken down from universities in the United States (see by sector for the last five years. There has been a previous section for more details). continuing increase in the proportion of EENDR The questionnaires were distributed to 910 respondents taking permanent employment in recipients of degrees using addresses provided academia and an offsetting decline in the propor­ by the departments granting the degrees; 551 tion taking permanent positions in business and individuals responded between late October and industry. April. Responses with insufficient data or from Among the 253 who reported obtaining a per­ individuals who indicated they had part-time or non-U.S. employment were excluded. Numbers manent position in the u.S. in fall 2003, 76% were of usable responses for each salary employed in academia (including 3% in research category are reported in the following tables. institutes and other nonprofits), 4% in government, Readers should be warned that the data in and 20% in business or industry. Women held 37% this report are obtained from a self-selected of the permanent positions. sample, and inferences from them may not be Among the 216 individuals with temporary representative of the population. employment in the u.S. this year, 94% were em­ Key to Tables. Salaries are listed in hundreds ployed in academia (including 5% in research of dollars. Nine-month salaries are based on institutes and other nonprofits), 3% in government, 9-10 months' teaching and/or research, not and 3% in business or industry. adding extra stipends for summer grants or summer teaching or the equivalent. Years listed Figure 2 gives the age distribution of the 551 denote the survey cycle in which the doctorate new doctoral recipients who responded to this was received. For example: survey cycle July 1, question. The median age of new doctoral recipi­ 2002-June 30, 2003 is designated as 2003. ents was 30 years, while the mean age was 32 Salaries are those reported for the fall years. The first and third quartiles were 28 and 34 immediately following the survey cycle. M and F years, respectively. These figures are the same as are male and female respectively. Some persons those reported last year and very similar to those receiving a doctoral degree had been employed reported in previous years. in their present position for several years, so those who had "one year or less experience" were analyzed separately from the total. Male Previous Annual Survey Reports and female figures are not provided when the The 2003 First Annual Survey Report was number of salaries available for analysis in a published in the Notices of the AMS in the particular category was five or fewer. Also, February 2004 issue. For the last full year of quartile figures are not available for 1970 reports, the 2002 First, Second, and Third through 1980. All categories of Annual Survey Reports were published in the "Teaching/Teaching and Research" and Notices of the AMS in the February, August, and "Research Only" contain those recipients September 2003 issues respectively. These employed at academic institutions only. The reports and earlier reports, as well as a wealth of "Academic Research Only, 9-10-Month Salaries" category was dropped from the published other information from these surveys, are analyses in 1998 because so few recipients available on the AMS website at www. ams . 0 rgj respond in this category that the data were not employmentjsurveyreports.html. considered meaningful. Starting salaries for those reporting a 9-10-month salary postdoctoral position are available for a sixth year. These salaries are also included within the "Academic Teaching/Teaching and Research, 9-10-Month Salaries" table and boxplot on page 797.

796 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences

Academic Teaching/Teaching and Research Academic Postdoctorates 9-10-Month Salaries 9-10-Month Salaries (in hundreds of dollars) (in hundreds of dollars) Reported Reported Ph.D. Median in Ph.D. Median in Year Min Q 1 Median Q 3 Max 2003 $ Year Min Q1 Median Q3 Max 2003 $ 1970 85 110 195 422 1997 180 350 385 410 450 399 1975 90 120 128 135 173 356 1998 290 350 390 420 500 399 1980 105 155 171 185 250 334 1999 130 365 400 418 540 404 1985 170 230 250 270 380 379 2000 300 385 420 450 550 415 1990 230 305 320 350 710 414 2001 250 400 425 450 566 410 1995 220 320 350 382 640 402 2002 230 425 450 487 595 430 1996 240 333 360 400 636 405 2003 240 420 450 480 600 450 1997 180 340 366 400 840 405 1999 M 220 373 400 428 540 1998 140 340 370 410 700 405 1999 F 130 350 390 410 475 1999 180 360 400 430 700 432 2000 M 300 390 420 450 550 2000 250 380 415 450 650 439 2000 F 360 389 448 458 544 2001 259 400 420 461 660 434 2002 230 400 450 500 840 457 2001 M 250 400 430 454 566 2003 220 415 450 510 920 450 2001 F 310 395 421 438 490 1999 M 220 370 400 430 700 2002M 230 425 450 488 595 1999 F 180 350 390 420 540 2002 F 380 430 450 485 589 2000 M 250 380 415 450 650 Total (67 male/16 female) 2000 F 321 380 413 450 620 2003 M 240 420 450 485 600 2001 M 259 400 430 475 660 2003 F 359 408 449 459 510 2001 F 310 390 413 443 620 2002 M 230 420 450 500 840 2002 F 300 400 441 498 610 Total (168 male/72 female) 2003 M 220 420 450 509 855 2003 F 359 414 444 512 920 One year or less experience (143 male/54 female) 2003 M 220 420 450 505 855 2003 F 359 411 434 501 700

1500 1500

1400 1400

1300 1300

1200 ~ 1200 ~ V) V) ~ ~ ~ 1100 ns 1100 0 (5 "'C 1000 "'C 1000 m m 0 0 0 900 0 900 N * N "I- "I- 0 800 * 0 800 V) 0 V) "'C * "'C Q) Q) ~ 700 * * ~ 700 "'C * "'C s::: *§ s::: ::::s 600 § ::::s 600 @ .s: .s: 0 0 ~ * s:: s:: 0 "-'" 500 "-'" 500 >. >. ~ ~ ns 400 ns 400 ~ ~ tU tU ~ ~ ~ ~ ~ ~ ~ 0 300 300 0 0 0 0 I 200 200 0 *

100 100 *

1996 1997 1998 1999 2000 2001 2002 2003 1997 1998 1999 2000 2001 2002 2003

AUGUST 2004 NOTICES OF THE AMS 797 2003 Annual Survey of the Mathematical Sciences

Academic Teaching/Teaching and Research Academic Research Only 11-1 2-Month Salaries 11-1 2-Month Salaries (in hundreds of dollars) (in hundreds of dollars) Reported Reported Ph.D. Median in Ph.D. Median in Year Min Q1 Median Q3 Max 2003 $ Year Min Q1 Median Q3 Max 2003 $ 1970 95 128 200 491 1970 90 120 205 461 1975 87 145 204 403 1975 90 119 180 331 1980 143 195 350 381 1980 120 180 321 352 1985 220 230 273 300 470 414 1985 190 295 342 400 520 518 1990 225 318 365 404 670 473 1990 180 280 300 365 546 389 1995 300 354 410 478 600 470 1995 196 280 340 370 587 390 1996 150 302 340 390 720 383 1996 192 270 330 400 585 372 1997 260 370 400 497 650 443 1997 190 300 350 400 600 388 1998 275 405 480 575 700 526 1998 200 333 360 428 617 394 1999 200 374 420 469 650 453 1999 270 390 440 500 720 475 2000 300 400 485 600 1170 512 2000 300 384 400 555 1000 423 2001 350 420 465 615 870 480 2001 300 367 420 625 800 434 2002 310 439 500 597 840 508 2002 270 380 440 500 700 447 2003 345 438 475 550 780 475 2003 300 415 470 613 900 470 1999 M 280 370 420 458 650 1999 M 270 383 400 493 600 1999 F 200 393 435 590 630 1999 F 340 468 530 581 720 2000 M 300 390 460 650 1170 2000 M 300 390 400 486 1000 2000 F 395 465 500 570 750 2000 F 300 360 410 580 630 2001 M 350 420 443 498 870 2001 M 300 348 425 655 800 2001 F 380 465 588 658 750 2001 F 342 400 420 588 700 2002 M 310 420 485 595 840 2002 M 270 388 440 500 650 2002 F 400 453 500 558 700 2002 F 310 350 440 505 700 Total (28 malel1 2 female) Total (29 malel1 9 female) 2003 M 397 440 490 555 780 2003 M 300 420 450 510 820 2003 F 345 400 440 513 620 2003 F 310 390 480 650 900 One year or less experience (21 malel10 female) One year or less experience (24 male/16 female) 2003 M 397 440 470 520 700 2003 M 300 400 440 496 770 2003 F 345 400 433 455 620 2003 F 330 405 470 613 720

1500 1500

1400 1400

1300 1300

1200 1200 Vi' Vi' ~ ~ ~ 1100 ~ 1100 0 (5 "'C 1000 "'C m m 1000 0 0 0 900 0 N N 900 ~ ~ 0 800 0 800 "'C'" "'C QJ '"QJ ~ 700 ~ 700 "'C "'C C C ::::s 600 ::::s 600 ..c ..c c c ~ 500 ~ 500 ~ ~ ~ ~ nsn:s 400 nsn:s 400 ~ ~ 300 300

200 200

100 100

1996 1997 1998 1999 2000 2001 2002 2003 1996 1997 1998 1999 2000 2001 2002 2003

798 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences

Government Business and Industry 11-12-Month Salaries 11-1 2-Month Salaries (in hundreds of dollars) (in hundreds of dollars) Reported Reported Ph.D. Median in Ph.D. Median in Year Min Q1 Median Q3 Max 2003 $ Year Min Q1 Median Q3 Max 2003 $ 1970 100 150 223 576 1970 96 --- 170 --- 235 653 1975 78 182 247 506 1975 114 --- 187 --- 240 520 1980 156 244 501 477 1980 190 --- 284 --- 400 555 1985 263 294 325 381 440 493 1985 260 360 400 420 493 606 1990 320 345 378 430 587 490 1990 320 438 495 533 700 641 1995 370 440 494 507 650 567 1995 288 480 568 690 1250 652 1996 360 420 427 504 650 481 1996 250 510 580 610 1000 653 1997 350 454 573 600 750 635 1997 300 483 600 658 1000 665 1998 320 475 540 736 1250 591 1998 240 550 650 750 2250 712 1999 400 495 550 651 720 594 1999 360 600 680 761 2450 734 2000 440 540 600 640 830 634 2000 200 640 720 800 1500 761 2001 400 580 644 758 920 665 2001 475 716 770 865 1850 795 2002 450 551 650 775 1005 661 2002 325 734 780 850 1400 793 2003 290 668 705 763 1008 705 2003 300 700 800 900 1250 800 1999 M 400 495 540 587 720 1999 M 360 626 700 763 2450 1999 F 1999 F 440 580 644 676 1100 2000 M 440 563 620 649 830 2000 M 200 640 730 800 1500 2000 F 530 545 566 593 650 2000 F 200 645 690 788 980 2001 M 400 590 647 780 920 2001 M 520 717 788 875 1700 2001 F 450 550 630 670 896 2001 F 475 710 750 850 1850 2002 M 450 551 642 725 1005 2002 M 325 738 782 858 1100 2002 F 540 600 700 850 880 2002 F 600 713 768 838 1400 Total (l0 male/6 female) Total (32 male/1 5 female) 2003 M 290 648 710 788 830 2003 M 550 725 840 920 1250 2003 F 600 683 695 723 1008 2003 F 300 628 780 816 900 One year or less experience (9 male/5 female) One year or less experience (24 male/11 female) 2003 M 290 630 710 750 830 2003 M 550 719 815 905 1250 2003 F 600 680 690 700 730 2003 F 300 610 702 805 880 (Note: Three salaries above $1 50,000 are not shown.) 1500 1500 1400 1400 * * 1300 1300

~ ~ 1200 1200 '"~ ~'" ~ 1100 ~ 1100 (5 o "'C 1000 "'C m m 1000 o o o o N 900 N 900 ~ ~ o o 800 800 "'C'" "'C'" cu cu ~ 700 ~ 700 "'C "'C C C ~ 600 ~ ..c ..c 600 c c 500 500

400 300 * 200 200

100 100

1996 1997 1998 1999 2000 2001 2002 2003 1996 1997 1998 1999 2000 2001 2002 2003

AUGUST 2004 NOTICES OF THE AMS 799 2003 Annual Survey of the Mathematical Sciences

Graphs. The graphs show standard boxplots Definitions ofthe Groups summarizing salary distribution information As has been the case for a number of years, much of the data for the years 1996 through 2003. Values plotted in these reports is presented for departments divided into for 1996 through 2002 are converted to 2003 groups according to several characteristics, the principal one dollars using the implicit price deflator prepared being the highest degree offered in the mathematical sci­ annually by the Bureau of Economic Analysis, U.S. ences. Doctoral-granting departments of mathematics are fur­ Department of Commerce. ther subdivided ac~ording to their ranking of "scholarly qual­ For each boxplot the box shows the first ity of program faculty" as reported in the 1995 publication quartile (Q1), the median (M), and the third Research-Doctorate Programs in the United States: Continu­ quartile (Q3). The interquartile range (IQR) is ity and Change. 1 These rankings update those reported in a defined as Q3-Q1. Think of constructing invisible previous study published in 1982.2 Consequently, the de­ fences 1.5xIQR below Q1 and 1.5xIQR above Q3. partments which. now comprise Groups I, II, and III differ Whiskers are drawn from Q3 to the largest significantly from those used prior to the 1996 survey. observation that falls below the upper invisible The subdivision of the Group I institutions into Group 1Pub­ fence and from Q1 to the smallest observation lic and Group 1Private was new for the 1996 survey. With the that falls above the lower invisible fence. Think increase in number of the Group 1departments from 39 to 48, of constructing two more invisible fences, each the Data Committee judged that a further subdivision of pub­ falling 1.5xIQR above or below the existing invisi­ lic and private would provide more meaningful reporting of ble fences. Any observation that falls between the data for these departments. the fences on each end of the boxplots is called Brief descriptions of the groupings are as follows: an outlier and is plotted as 0 in the boxplots. Group 1 is composed of 48 departments with scores in the Any observation that falls outside of both 3.00-5.00 range. Group I Public and Group I Private are fences either above or below the box in the box­ Group I departments at public institutions and private plot is called an extreme outlier and is marked institutions respectively. as * in the boxplot. Group II is composed of 56 departments with scores in the Acknowledgments 2.00-2.99 range. The Annual Survey attempts to provide an accurate Group III contains the remaining U.S. departments .reporting appraisal and analysis of various aspects of the a doctoral program, including a number of departrDents not academic mathematical sciences scene for the use included in the 1995 ranking of program faculty. and benefit of the community and for filling the Group IV contains U.S. departments (or programs) of statis­ information needs of the professional organiza­ tics, biostatistics, and biometrics reporting a doctoral tions. Every year, college and university depart­ program. ments in the United States are invited to respond. Group V contains U.S. departments (or programs) in applied The Annual Survey relies heavily on the conscien­ mathematics/applied science, operations research, and tious efforts of the dedicated staff members of management science which report a doctoral program. these departments for the quality of its informa­ Group Va is applied mathematics/applied science; Group Vb, tion. On behalf of the Annual Survey Data Com­ which is no longer surveyed as of 1998-99, was operations mittee and the Annual Survey Staff, we thank the research and management science. many secretarial and administrative staff mem­ Group M contains U.S. departments granting a master's de­ bers in the mathematical sciences departments gree as the highest graduate degree. for their cooperation and assistance in responding Group B contains U.S. departments granting a baccalaureate to the survey questionnaires. degree only. Listings of the actual departments which comprise these Other Data Sources groups are available on the AMS Website at AmericanAssociation ofUniversity Professors, Don'tBlame www.ams.org/outreach. the Faculty for High Tuition: The AnnualReport on the Economic Status ofthe Profession 2003-2004, Academe: 1Research-Doctorate Programs in the United States: Continuity Bull. AAUP (March/April 2004), Washington, DC. and Change, edited byMarvin L. Goldberger, Brendan A. Maher, American Statistical Association, 2003-2004 Salary Re­ andPamela EbertFlattau, National AcademyPress, Washington, portofAcademicStatisticians, AmStat News (December DC, 1995. 2003), Alexandria, VA. 2 These findings were published in An Assessment ofResearch­ Commission onProfessionals in Science and Technology, Doctorate Programs in the United States: Mathematical and Professional Women andMinorities, 14thed., epST, Wash­ Physical Sciences, edited byLyle Jones, GardnerLindzey, and v: ington' DC, 2002. Porter E. Coggeshall, National Academy Press, Washington, DC, 1982. The information on mathematics, statistics, and com­ __, Salaries ofScientists, Engineers, and Technicians: A puterscience was presented in digest form in the April 1983 is­ SummaryofSalarySurveys, 20thed., CPST, Washington, sue ofthe Notices, pages 257-67, and an analysis ofthe clas­ DC, 2003. sifications was given in the June 1983 Notices, pages 392-3. Conference Board ofthe Mathematical Sciences, Statistical AbstractofUndergraduate Programs in the Mathematical

800 NOTICES OF THE AMS VOLUME 51, NUMBER 7 2003 Annual Survey of the Mathematical Sciences

Sciences in the United States: Fall 2000 CBMS Survey, American Mathematical Society, 2002. NEW JERSEY __, Statistical Abstract of Undergraduate Programs in Princeton University (16) the Mathematical Sciences in the United States: Fall 1995 CBMS Survey, MAAReports No.2, 1997. MATHEMATICS National Opinion Research Center, Doctorate Recipients Askshay, Venkatesh, Limiting forms of the trace formula. from United States Universities: SummaryReport2001, Banner, Adrian, Restriction of the Fourier transforms to Survey ofEarned Doctorates, Chicago, IL, 2002. quadratic submanifolds. National Research Council, Strengthening the Linkages Booker, Andrew, Numerical tests of modularity. between the Sciences and the Mathematical Sciences, Chudnovsky, Maria, Berge trigraphs and their applications. National Academy Press, Washington, DC, 2000. Doran, Brent, Intersectionhomologyhypergeometric func- __, U.S. Research Institutes in the Mathematical Sciences: tions, and moduli spaces as ball quotients. AssessmentandPerspectives, National Academy Press, Hall, Christopher, L-functions oftwisted legendre curves. Washington, DC, 1999. Harcos, Gergely, New bounds for automorphic L-functions. __, Research-Doctorate Programs in the United States: Helfgott, Harald, Root numbers and the parity problem. Continuity and Change, National AcademyPress, Wash­ Kerr, Matthew, Geometric constructionofregular currents ington, DC, 1995. with applications to algebraic cycles. National Science Board, Science and Engineering Indica­ Krieger, joachim, Global regularity ofwave maps in 2 and tors-2004. Two Volumes (volume 1, NSB 04-01; 3 spatial dimensions. volume 2, NSB 04-1A), National Science Foundation, McKee, Mark, On the finite order of Whittaker functions, Arlington, VA, 2004. Eisenstein series, and automorphic L-functions. National Science Foundation, Characteristics ofDoctoral Milley, Peter, Tube volumes and small hyperbolic 3-mani­ Scientists andEngineers in the United States: 2001 (NSF folds. 03-310), Detailed StatisticalTables, Arlington, VA, 2003. Parson, james, Level-raising congruences inthe represen­ __, Graduate Students andPostdoctorates in Science and tation theory ofreductive groups over large fields. Engineering: Fall 2001 (NSF 03-320), Arlington, VA, Sepanski, Peter, ASeiberg-Wittenproductformula for cer­ 2003. tain circle-bundles over surfaces. __, Science and Engineering Degrees: 196,6-2001 (NSF Spinu, Florin, The L4 norms of the Eisenstein series. 04-311), Detailed StatisticalTables, Arlington, VA, 2004. Tymoczko, julianna, Decomposing Hessenberg varieties over classical groups. __, Science and Engineering Degrees, byRace/Ethnicity ofRecipients: 1992-2001 (NSF 04-318), Detailed Statistical Tables, Arlington, VA, 2004. OHIO __, Science and Engineering Doctorate Awards: 2002 Case Western Reserve University (2) (NSF 04-303), Detailed Statistical Tables, Arlington, VA, 2003. STATISTICS __, Statistical Profiles ofForeign DoctoralRecipients in Sci­ Subramanian, Neepa, Monte Carlo methods for large que­ ence andEngineering: Plans to Stay in the United States ing networks. (NSF 99-304), Arlington, VA, 1998. Yan, Guofen, Evaluation ofBayesian diagnostic methods for __, Women, Minorities, and Persons with Disabilities in hierarchial data. Science andEngineering: 2002(NSF 03-312), Arlington, VA, 2003. OREGON

Portland State University (1) MATHEMATICS Rosson, john, Multiplicative invariants of speciaI2-com­ Doctoral Degrees plexes. Conferred 2002-2003 Supple111entary List The following list supplements the list of thesis titles published in the February 2004 Notices, pages 246-263. CALIFORNIA

University of California, Davis (1) MATHEMATICS Scott, Michael, General relativistic shock-waves propagat­ ing at the speed of light.

AUGUST 2004 NOTICES OF THE AMS 801 MathematicsPeople

were nominated by the National Science Foundation. Joyce Wins Adams Prize EDMOND CHOW of Lawrence Livermore National Laboratory DOMINIC JOYCE ofUncoln College, Oxford University, has been was nominated by the Department of Energy. awarded the Adams Prize for 2004 by the University of The recipients were selected from nominations made Cambridge. The Adams Prize is awarded each year by the by nine participating federal agencies. Each awardee re­ Faculty of Mathematics and St. John's College to a young ceives a five-year grant of up to $750,000 to further his researcher based in the United Kingdom who is doing or her research and educational efforts. first-class international research in the mathematical sci­ ences. Joyce's research interests are chiefly in differential -From an NSF announcement geometry, as well as in areas of theoretical phys.ics. He is particularly concerned with the study of geometrical struc­ tures on manifolds and special holonomy groups. Ferran Sunyer i Balaguer Prize The Adams Prize is named after the ,mathematician John Couch Adams and was endowed by members of Awarded St. John's College. It is currently worth £15,000 (approxi­ mately US$24,000), of which one-third is awarded to the The Ferran Sunyer i Balaguer Foundation has awarded the prizewinner on announcement of the prize; one-third is 2004 Ferran Sunyer i Balaguer Prize to GUY DAVID of Uni­ provided to the prizewinner's institution (for research ex­ versite Paris-Sud for his monograph Singular Sets ofMin­ penses of the prizewinner); and one-third is awarded to imizers for the Mumford-Shah Functional. the prizewinner on acceptance for publication in an in­ The monograph is a complete exposition of the known ternationally recognized journal of a substantial (normally results for the Mumford-Shah conjecture. Although this at least twenty-five printed pages) original survey article conjecture has its origins inimage segmentation, David sees of which the prizewinner is an author. it as a model for a whole class of problems with free boundaries and a length of area term. The work is an ex­ -From a University of Oxford announcement tended study of the underlying variational problem. The Ferran Sunyer i Balaguer Foundation PECASE Awards Announced (http://www.crm.es/FerranSunyerBalaguer/ffsb.htm) of the Institut d'Estudis Catalans awards this international Fifty-seven young researchers were chosen to receive the prize every year to honor the memory of Ferran Sunyer i 2002 Presidential Early Career Awards for Scientists and Balaguer (1912-1967), a self-taught Catalan mathematician Engineers (PECASE). This award is the highest honor be- who gained international recognition for his research in stowed by the u.s. government on outstanding young sci­ mathematical analysis. His achievements are the more im­ entists, mathematicians, and engineers who are in the early stages of establishing their independent research pressive given the serious physical disabilities with which careers. he was born. Among the awardees are three who work in the math­ ematical sciences. GEORGE G. PAPPAS ofthe University ofPenn­ -From a news release ofthe Ferran Sunyer i Balaguer sylvania and ROBERT W. GHRIST of the University of Illinois Foundation

802 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics People

Mathe Receives 2004 Prize for Szpiro Receives Media Prize Achievement inInformation­ GEORGE SZPIRO has received the 2003 Prix Media from the Swiss Academy of Natural Sciences for his column about Based Complexity mathematics, "Kleines Einmaleins" ("Little [Multiplication] Table"), which appears monthly in the Swiss newspaper PETER MATHE of the Weierstrass Institute for Applied Analy­ Neue Ziircher Zeitung. The prize carries a cash aw'ard of sis and Stochastics, Berlin, Germany, has been awarded the 10,000 Swiss francs (about US$7,800). 2004 Prize for Achievement in Information-Based Com­ Trained as a mathematician, Szpiro is based in plexity. His research interests are in the efficiency of nu­ Jerusalem, Israel, as a political correspondent for the Neue merical methods, including the complexity ofMonte Carlo Ziircher Zeitung. His book Kepler's Conjecture: How Some methods, the efficiency ofMarkov chain Monte Carlo meth­ of the Greatest Minds in History Helped Solve One of the ods, approximation theory, and the complexity ofill-posed Oldest Math Problems in the World was published by John problems. Wiley and Sons in 2003. The prize consists of $3,000 and a plaque. The award will be presented at the 2004 Workshop on Continuous -Allyn Jackson Algorithms and Complexity at Schloss Dagstuhl, Germany, in September 2004.

-Joseph F. Traub, Columbia University Putnam Prizes Awarded The winners of the 64th William Lowell Putnam Competi­ tion have been announced. The Putnam Competition is ad­ Allgower Wins Leibniz Prize ministeredby the Mathematical Association ofAmerica and consists of an examination containing mathematical prob­ FRANK ALLGOWER of the Universitat Stuttgart has been lems that are designed to test both originality and tech­ awarded a 2004 Leibniz Prize by the Deutscheforschungs­ nical competence. Prizes are awarded to both individuals gemeinschaft, the main scientific research funding agency and teams. of the German government. The prize provides a research The five highest ranking individuals, listed in alpha­ grant of 1.55 million euros (approximately US$1.8 mil­ betical order, were: REID W. BARTON, Massachusetts Insti­ lion) over five years. tute of Technology; ANA CARAIANI, Princeton University; Allgbwer specializes in nonlinear systems and control GABRIEL D. CARROLL, Harvard University; RALPH C. FURMANIAK, theory. His work focuses on the control of technical sys­ University of W~terloo; and DANIEL M. KANE, Massachusetts tems such as energy supply networks, the Internet, and Institute of Technology. transport systems; and he has developed methods to Institutions with at least three registered participants analyze and influence these systems. His work links obtain a team ranking in the competitionbased on the rank­ fundamental research and practical solutions to technical ings of three designated individual participants. The five problems. He received his Ph.D. from the Universitat top-ranked teams (with team members listed in alpha­ Stuttgart and is currently head of the Institute for Systems betical order) were: Massachusetts Institute of Technology Theory in Engineering at the university. (Reid W. Barton, Daniel M. Kane, Yevgeny K. Zaytman); Harvard University (Gabriel D. Carroll, George Lee, Jr., -Elaine Kehoe Alexander B. Schwartz); Duke University (David G. Arthur, Nikifor C. Bliznashki, Oaz Nir); California Institute ofTech­ nology (Zhihao Liu, Po-Ru Loh, Po-Shen Loh); and Harvey Mudd College (David J. Gaebler, Jason Murcko, Andrew G. Prizes of the Mathematical Niedermah~r). Sodety ofJapan The top five individuals in the competition received cash awards of $2,500; the next ten received $1,000. The The Mathematical Society of Japan (MSJ) has awarded its first-place team was awarded $25,000, with each team 2004 Spring Prize to TAKASHI KUMAGAI of Kyoto University member receiving $1,000. The team awards for second and the Algebra Prize to TOMOHIDE TERASOMA of Tokyo place were $20,000 and $800; for third place, $15,000 University. and $600; for fourth place, $10,000 and $400; and for fifth The Spring Prize is awarded each year to a mathematician place, $5,000 and $200. who is not older than forty and who has made an The Elizabeth Lowell Putnam Prize is awarded period­ outstanding contribution to mathematics. The Algebra ically to a woman whose participation in the Putnam Com­ Prize is awarded every year to a maximum of two petition is deemed particularly meritorious. In the recent algebraists in recognition of outstanding contributions in competition, this prize went to ANA CARAIANI of Princeton algebra. University. The prize carries a cash award of $1,000.

-From an MSJ announcement -Elaine Kehoe

AUGUST 2004 NOTICES OF THE AMS 803 Mathematics People

USA Mathematical Olympiad The thirty-third annual USA Mathematical Olympiad (USAMO) was held April 27 and 28,2004. The students par­ ticipating in the Olympiad were selected on the basis of their performances on the American High School and American Invitational Mathematics Examinations, which in­ volved hundreds of thousands of students. The twelve highest scorers in the USAMO, listed in al­ phabetical order, were: JAE BAE, Hackensack, New Jersey; JONGMIN BAEK, Cupertino, California; OLEG GOLBERG, Exeter, New Hampshire; MATT INCE, Chesterfield, Missouri; JANOS KRAMAR, Toronto, Ontario, Canada; TIANKAI LIU, Exeter, New Hampshire; ALISON MILLER, Niskayuna, New York; AARON PIXTON, Vestal, New York; BRIAN RICE, Dublin, Virginia; JACOB TSIMERMAN, Toronto, Ontario, Canada; AMEYA VEUNGKER, Al­ lentown, Pennsylvania; TONY ZHANG, Exeter, New Hamp­ shire. Tiankai Liu received a perfect score. The twelve USAMO winners will attend the Mathemat­ ical Olympiad Summer Program (MOSP) from June 13 through July 3. Then six of the twelve students will be se­ lected as the United States team to compete in the Inter­ national Mathematical Olympiad (IMO) to be held inAthens, Greece, July 6-18, 2004.

-Elaine Kehoe

AmericanAcademy ofArts and Sdences Elections Eight mathematical scientists have been elected to mem­ bership in the American Academy ofArts and Sciences for 2004. They are: DAVID ALDOUS, University of California, Berkeley; LEONARD GROSS, Cornell University; ANATOLE KATOK, Pennsylvania State University; FANG-HuA LIN, Courant In­ stitute of Mathematical Sciences, New York University; YURI I. MANIN, Northwestern University; GANG TIAN, Massa­ chusetts Institute of Technology; NOLAN WALLACH, Univer­ sity of California, San Diego; and lAI-SANG YOUNG, Courant Institute of Mathematical Sciences, New York University. YVES COLIN DE VERDIERE, Universite de Grenoble, France, was elected as a foreign honorary member. The American Academy of Arts and Sciences was founded in 1780 to foster the development of knowledge as a means of promoting the public interest and social progress. The membership of the academy is elected and represents distinction and achievement in a range of in­ tellectual disciplines: mathematical and physical sciences, biological sciences, social arts and sciences, and humani­ ties and fine arts.

-From an AAAS announcement

804 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics Opportunities

proposals is August 20, 2004. The deadline date for full Enhancing the Mathematical proposals is September 17, 2004. The FRG solicitation may be found on the Web at Sdences Workforce in the 21st http://www.nsf.gov/pubs/2002/nsf02129/nsf02129. Century htrn.

The National Science Foundation has launched a new pro­ -From an NSF announcement gram called Enhancing the Mathematical Sciences Work­ force in the 21st Century (EMSW21). The long-range goal of the program is to increase the number of u.S. citizens, NSF Mathematical Sdences nationals, and permanent residents who are well prepared in the mathematical sciences and who pursue careers in Postdoctoral Research the mathematical sciences and in other NSF-supported disciplines. EMSW21 builds on the VIGRE (Vertical Fellowships Integration of Research and Education in the Mathematical The Mathematical Sciences Postdoctoral Research Fellow­ Sciences) program and now includes a broadened VIGRE ship program of the Division of Mathematical Sciences activity, a component for Research Training Groups (RTG) (DMS) of the National Science Foundation (NSF) awards fel­ in the Mathematical Sciences, and a component for Men­ lowships each year for research in pure mathematics, ap­ toring through Critical Transition Points (MCTP) in the plied mathematics and operations research, and statistics. Mathematical Sciences. The deadline for this year's applications is October 15, The program solicitation is available at 2004. Applications must be submitted via FastLane on http://www.nsf.gov/pubsys/ods/getpub.cfrn?nsf03575. the World Wide Web. Go to http: j jwww. fastl ane. It replaces NSF 02-120 (VIGRE). The proposal deadline is nsf. gOY/ and click on "Postdoctoral Fellowships". Infor­ September 16 each year. mation can be found there for the Mathematical Sciences Postdoctoral Research Fellowships, as well as other NSF -From an NSF announcement fellowship opportunities. For more information, telephone the DMS at 703-306-1870 or send email to rnsprf@nsf. gOY. NSF Focused Research Groups -From an NSF announcement The Focused Research Groups (FRG) activity of the Division of Mathematical Sciences (DMS) of the National AMS Scholarships for "Math in Science Foundation (NSF) supports small groups of researchers in the mathematical sciences. Moscow" The DMS has announced deadline dates for the fiscal The Independent University of Moscow has created a pro­ year 2004 competition for FRG grants. The deadline for gram called "Math in Moscow", which offers foreign stu­ receipt of the required letters of intent to submit FRG dents (undergraduate or graduate students specializing in

AUGUST 2004 NOTICES OF THE AMS 805 Mathematics Opportunities mathematics and/or computer science) the chance to should be written in English and should be at least 150 spend a semester in Moscow studying mathematics. The pages long. AMS provides a small number of scholarships to students The prize, amounting to 10,000 euros (about US$8,600), to attend the program. is provided by the Ferran Sunyer i Balaguer Foundation. Math in Moscow provides students with a fifteen-week The winning monograph will be published in Birkhauser program similar to the Research Experiences for Under­ Verlag's series Progress in Mathematics, subject to the graduates programs that are held each summer across the usual regulations concerning copyright and author's rights. United States. Math in Moscow draws on the Russian tra­ Submissions should be sent before December 1, 2004, dition of teaching mathematics, which emphasizes creative to: Fundaci6 Ferran Sunyer i Balaguer, Carrer del Carme, approaches to problem solving. The focus is on develop­ 47, E-08001 Barcelona, Spain. For further informationvisit the ing in-depth understanding of carefully selected material websttehttp://www.crm.es/FerranSunyerBalaguer/ffsb.htm rather than broad surveys of large quantities of material. or send email [email protected]. Discovering mathematics under the guidance of an expe­ rienced teacher is the central principle ofMath in Moscow. -From an Insitut d'Estudis Catalans announcement Most of the program's teachers are internationally recog­ nized research mathematicians, and all of them have con­ siderable teaching experience in English, typically in the United States or Canada. All instruction is in English. With funding from the National Science Foundation (NSF), the AMS awards each semester five $5,000 scholar­ ships to U.S. students to attend the Math in Moscow program. To be eligible for the scholarships, students must submit applications to both the Math in Moscow program and to the AMS. An applicant should be an undergraduate mathematics or computer science major enrolled at a U.S. institution. April 15 is the deadline for scholarship applications for the fall semester immediately following that date; September 30 is the deadline for the spring semester. Information and application forms for Math inMoscow are available on the Web at http://wvwv.mccme.ru/mathi nmoscow or by writing to: Math in Moscow, P.O. Box 524, Wynnewood, PA 19096; fax +7095-291-65-01; email: mi m@mccme . ru. Information and application forms for the AMS scholarships are available on the Web at http://www.ams.org/employment/mimoscow.html or by writing to: Math in Moscow Program, Membership and Programs Department, American Mathematical Society, 201 Charles Street, Providence, RI 02904; email: [email protected].

-Allyn Jackson Call for Submissions for Sunyer i Balaguer Prize Ferran Sunyer i Balaguer (1912-1967) was a self-taught Catalan mathematician who despite a serious physical disability was very active in research in classical analysis, an area in which he acquired international recognition. Each year in honor of the memory of Ferran Sunyer i Balaguer the Institut d'Estudis Catalans awards an international re­ search prize bearing his name. The prize is awarded for a mathematical monograph of an expository nature presenting the latest developments in an active area of research in mathematics in which the author has made important contributions. The monograph

806 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Inside theAMS

Kevin Clancey Named MR Executive Editor Kevin Clancey of the University of Louisville has been named executive editor of Mathematical Reviews. He suc­ ceeds Jane Kister, who is retiring from the position after twenty-five years of service at MR. Born in Halifax, Nova Scotia, Canada, Clancey received his B.S. degree from Saint Mary's University in Halifax (1965) and his M.S. (1967) and Ph.D. (1969) from Purdue University. He held positions at Carleton University and the University of California, Los Angeles, before moving in 1971 to the University of Georgia, where he spent most ofhis career. He was head of the mathematics department there from 1995 to 2001, when he accepted a position as chair of the mathematics department at the University of Louisville. From 1993 to 1995 he was a program director in the modern analysis program of the Division of Math­ ematical Sciences at the National Science Foundation (NSF). He received the Outstanding Performance Award for Pro­ gram Management for his service at the NSF. Kevin Clancey, MR Executive Editor. Clancey's area of research is operator theory. He is the author or coauthor ofnearly fifty research papers and two dedicated scientific and production staff at MR, interact­ books in the subject. He has presented many lectures at ing with the community of reviewers, viewing the broad universities and conferences in the United States and Eu­ mathematical landscape, and communicating with the rope. Since 1978 he has been on the editorial board of In­ publishing world. The position involves significant man­ tegral Equations and Operator Theory. agerial duties, and I believe my previous administrative ex­ What follows is the text of an email interview with perience can be put to good use at MR. My goal is to ef­ Clancey, in which he reflects on the importance of main­ fectively guide the publication in a manner consistent taining and enhancing MR as the premier bibliographic tool with the excellent stewardship that has been offered by for mathematicians. past executive editors such as Jane Kister. I am quite hon­ Notices: Why did you want to become the executive ed­ ored to have this opportunity. itor ofMR? Notices: How do you see the role of MR in the mathe­ Clancey: Uke almost all mathematical scientists, I've de­ matical enterprise? rived great benefit from Mathematical Reviews over the Clancey: The readers of the Notices know the value of years. Thus, when the opportunity to serve as executive Mathematical Reviews as both a research tool and a his­ editor popped up, I eagerly explored the matter. A visit to torical record of our discipline. In an age of copious pub­ the MR offices made it clear that serving as executive ed­ lication, MR, aka MathSciNet, allows us to stay current itor would be an interesting and satisfying occupation. with research activity "close to home" and then to gain per­ There are many facets of the duties of executive editor that spective on new directions. From personal experience I appeal to me. I'm looking forward to working with the know that the best place to start learning about a new

AUGUST 2004 NOTICES OF THE AMS 807 Inside the AMS mathematical topic is a text search of MR. Similarly, an au­ thor search in MR defines an individual researcher in Fan China Exchange Program greater depth than a CV. The links provided in MR yield Names Awardees important connections to past research that could easily have slipped off our radar screens. MR is a pretty amaz­ The Society's Fan China Exchange Program awards grants ing product and plays a central role in the mathematical to support collaborations between Chinese and U.S. or enterprise. Canadian researchers. Institutions in the United States or Notices: What are your thoughts about the future of Canada apply for the funds to support a visitor from China MR? How will MR be influenced by, or influence, the elec­ or vice versa. This funding is made possible through a gen­ tronic revolution in how mathematical research is com­ erous gift to the AMS from Ky and Yu-Fen Fan in 1999. municated? The names of the awardees for 2004 follow. The first Clancey: In the age of voluminous and electronic pub­ name given is that of the host; the second is the name of lication of mathematical research, the information con­ the visitor. RUSSELL THOMPSON, Utah State University, U.S.; tained in Mathematical Reviews is more important than YONGQING LI, Fujian Normal University, China, two months. ever. The significance of many electronic research publi­ DAVID GRIFFEATH, University ofWisconsin, Madison, U.S.; JIAN cations can be easily lost in the ether. MR must be there ZHONG PAN, Academia Sinica, China, one month. BERNARD A. to put things in context and develop connecting links. Es­ MArR, North Carolina State University, U.S.; SHAO-MING FEI, tablishing appropriate relationships with electronic jour­ Capital Normal University, China, two months. BERNARD nals and publications is an essential future task for MR. Russo, University of California, Irvine, U.S.; Bo DAr, Chinese It is going to take considerable effort to keep on top of elec­ Academy of Sciences, China, six months. tronic publishing developments; however, MathSciNet is For information about the Fan China Exchange Pro­ already an electronic publication and has demonstrated gram, visit the website http://www . ams .0 rg/ its ability to keep pace with publishing developments. employment/chi naexchange. html, or contact the AMS Notices: Do you have specific plans for things you want Membership and Programs Department, email: prof­ to accomplish at MR? [email protected], telephone 401-455-4058 (within the U.S. Clancey: MR seems to be quite healthy, and I don't call 800-321-4267, ext. 4058). think it is wise to change successful strategies. Accordingly, I'm not going into the position of executive editor with a -Elaine Kehoe heavy agenda. My objective will be to work hard to main­ tain the many qualities and value of MR for the commu­ nity. Naturally, issues related to electronic publishing and the volume of mathematical publications will require our attention at MR. There will be ways to enrich the publica­ tion via the electronic media. For these and other issues, we'll look to the community and advisoryboard for input. In general I'm a firm believer in the merits of collective wisdom and will welcome input onways the publication canimprove its service to the com­ munity. At the end of the day, the quality of this service is the measure of accomplishment at MR.

-Allyn Jackson AMS-AAAS Media Fellow Named Each year the AMS sponsors a fellow to participate in the Mass Media Fellowship program of the American Associ­ ation for the Advancement of Science (AAAS). This program places science and mathematics graduate students in sum­ mer internships in media outlets. The 2004 AMS-AAAS Mass Media Fellowship has been awarded to LISA DEKEUKELAERE, a mathematics graduat~ stu­ dent at Brown University. She will be based at Scientific American for her summer fellowship upon completion of the program orientation in Washington, DC.

-AMS Washington office

808 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Reference and Book List

The Reference section of the Notices August 1, 2004: Applications for brochu re. shtml, or write to Sloan is intended to provide the reader with National Research Council Research Research Fellowships, Alfred P. Sloan frequently sought information in Associateships. See http://www4 . Foundation, 630 Fifth Avenue, Suite an easily accessible manner. New nationalacademies.org/pga/rap. 2550, New York, NY 10111. information is printed as it becomes nsf, or contact Research Associate­ September 16, 2004: Proposals for available and is referenced after the ship Programs, Keck Center of the Na­ NSF Enhancing the Mathematical Sci­ first printing. As soon as information tional Academies, 500 Fifth Street, ences Workforce in the Twenty-First is updated or otherwise changed, it NW, GR322A, Washington, DC 20001; Century program (including VIGRE). will be noted in this section. telephone 202-334-2760; fax 202-334­ See "Mathematics Opportunities" in 2759; email: rap@nas. edu. this issue. Contacting the Notices August 20, 2004: Letters of intent September 17, 2004: Full propos­ The preferred method for contacting for NSF Focused Research Groups. See als for NSF Focused Research Groups. the Notices is electronic mail. The "Mathematics Opportunities" in this See "Mathematics Opportunities" in editor is the person to whom to send issue. this issue. articles and letters for consideration. September 15, 2004: Nominations September 30, 2004: AMS schol­ Articles include feature articles, for Alfred P. Sloan Foundation fel­ arships for Math in Moscow. See memorial articles, communications, lowships. See http://www.sloan. "Mathematics Opportunities" in this opinion pieces, andbook reviews. The org/programs/fellowship_ issue. editor is also the person to whom to send news of unusual interest about other people's mathematics research. Where to Find It The managing editor is the person A brief index to information that appears in this and previous issues of the to whom to send items for "Mathe­ Notices. f matics People", "Mathematics Op­ AMS Bylaws-November 2003, p. 1283 portunities", "For Your Information", AMS E-mail Addresses-November 2003, p. 1266 "Reference and Book List", and "Math­ ematics Calendar". Requests for AMS Ethical Guidelines-June/July 2004, p. 673 permissions, as well as all other AMS Officers 2000 and 2001 (Council, Executive Committee, inquiries, go to the managing editor. Publications Committees, Board of Trustees)-May 2004, p. 566 The electronic-mail addresses are AMS Officers and Committee Members-October 2003, p. 1115 noti ces@math .ou . edu in the case of Backlog of Mathematics Research Journals-September 2003, p. 961 the editor and noti ces@ams. org in Conference Board of the Mathematical Sciences-September 2003, p. 945 the case of the managing editor. The Information for Notices Authors-June/July 2004, p. 668 fax numbers are 405-325-7484 for the editor and 401-331-3842 for the Mathematics Research Institutes Contacf Information-August 2004, managing editor. Postal addresses p.810 may be found in the masthead. National Science Board-January 2004, p. 54 New Journals for 2003-June/July 2004, p. 670 Upcoming Deadlines NRC Board on Mathematical Sciences and Their Applications-March 2004, July 22, 2004: Proposals for NSF p. 350 CAREER Program. See http: // www.nsf.gov/pubs/2002/ NRC Mathematical Sciences Education Board-April 2004, p. 446 nsf02111/nsf02111.htm. NSF Mathematical and Physical Sciences Advisory Committee-February August 1, 2004: Nominations for 2004,p.242 Clay Senior Scholars Program. See Program Officers for Federal Funding Agencies-October 2003, p. 1107 http://www.claymath.org/ (DoD, DoE); December 2003, p. 1429 (DMS Program Officers); December senior_scholars, or call 617-995­ 2003, p. 1430 (NSF Education Program Officers) 2600.

AUGUST 2004 NOTICES OF THE AMS 809 Reference and Book List

September 30, 2004: Nominations Banff International Research Centre for Mathematical Physics for Information-Based Complexity Station and Stochastics (MaPhySto) Young Researcher Award. Contact c/o PIMS Central Office Department of Mathematical joseph F. Traub at traub@cs. University of British Columbia Sciences co1umbi a. edu. 1933 West Mall University of Aarhus October 1, 2004: Nominations for Vancouver, BC V6T 1Z2, Canada Ny Munkegade the Louise Hay Award and the Telephone: 403-762-6100 DK-8000 Aarhus C, Denmark Alice T. Schafer Mathematics Prize. Fax: 403-763-6990 Telephone: (+45) 8942 3515 Call the AWM at 301-405-7892 or send email: [email protected] Fax: (+45) 8613 1769 email to awm@math . umd . edu. World Wide Web: email:[email protected] October 1, 2004: Nominations for http://www.pims.math.ca/birs/ World Wide Web: CRM-Fields Prize. See http://www. http://maphysto.dk fields.utoronto.ca/ Center for Discrete Mathematics proposals/crm-fields_prize. and Theoretical Computer Science Centre for Mathematics and Its html, or contact the Director, The (DIMACS) Applications Fields Institute, 222 College Street, CoRE Building, 4th Floor Building 27 Toronto, Ontario M5T 3j1, Canada. Rutgers University Australian National University October 15, 2004: Applications for 96 Frelinghuysen Road Canberra ACT 0200, Australia NSF Postdoctoral Research Fellow­ Piscataway, Nj 08854-8018 Telephone: (+61) 2 6125 2897 ships. See "Mathematics Opportuni­ Telephone: 732-445-5930 Fax: (+61) 2 6125 5549 ties" in this issue. Fax: 732-445-5932 email:[email protected]. December 1, 2004: Submissions email: center-admin@dimacs. edu.au for Ferran Sunyer i Balaguer Prize. rutgers.edu World Wide Web: See "Mathematics Opportunities" in World Wide Web: http://www.maths.anu.edu. this issue. http://dimacs.rutgers.edu au/CMA/ January 1, 2005: Entries for Cryp­ tologia undergraduate paper compe­ Center for Scientific Computation Centre de Recherches Mathema­ titions. See http://www. dean. usma. and Mathematical Modeling tiques (CRM) edu/math/pubs/cryptologia/, or (CSCAMM) Universite de Montreal contact Cryptologia, Department of University of Maryland C.P. 6128, Succ. Centre-ville Mathematical Sciences, United States 4146 CSIC Building #406 Montreal, Quebec, Canada H3C 3j7 Military Academy, West Point, NY Paint Branch Drive Telephone: 514-343-7501 10996; email: Crypto1ogi a@usma. College Park, MD 20742-3289 Fax: 514-343-2254 edu. Telephone: 301-405-0662 email: Fax: 301-314-6674 [email protected] Contact Information for email: i nfo@cscamm. umd. edu World Wide Web: Mathematics Institutes World Wide Web: http://www.crm.umontreal.ca AmericanInstitute ofMathematics (AIM) http://www.cscamm.umd.edu AIM Research Conference Center Centro de Investigacion en (ARCC) Centre International de Rencontres Matematicas (CIMAT) 360 Portage Avenue Mathematiques (CIRM) A. P. 402, Guanajuato, Gto. Palo Alto, CA 94306-2244 163, avenue de Luminy Case 916 C.P. 36000, Mexico Telephone: 650-845-2071 F-13288 Marseille Cedex 09, France Telephone: 473 73 271-55 Fax: 650-845-2074 Telephone: (+33) 04 91 83 30 00 Fax: 473 73 257-49 email: [email protected] Fax: (+33) 04 91 83 30 05 email: cimat@cimat~mx World Wide Web: email: cirm@cirm.. univ-mrs.fr World Wide Web: http://www.aimath.org World Wide Web: http://www.cimat.mx http://www.cirm.univ-mrs.fr/ Stefan'Banach International Mathe­ indexE.html Chennai Mathematical Institute matical Center 92 G. N. Chetty Road 8 Sniadeckich str., P.O. Box 21 Centre de Recerca Matematica Chennai 600 017, India 00-950 Warszawa, Poland (CRM) Telephone: (+91) 44-28157854, Telephone: (+48-22) 522-82-32; 628­ Apartat 50 28157855 01-92 E-08193 Bellaterra, Spain Fax: (+91) 44-28157671 Fax: (+48-22) 622-57-50; 629-39-97 Telephone: (+34) 935 811 081 email: [email protected] email: [email protected] Fax: (+34) 935 812 202 World Wide Web: World Wide Web: email: [email protected] http://www.cmi.ac.in http://www.impan.gov.pl/BC World Wide Web: http://crm . es

810 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Reference and Book List

Erwin Schrodinger International Fax: 609-924-8399 Academic Building NO.1 Institute for Mathematical Physics email:[email protected] Shatin, Hong Kong Boltzmanngasse 9 World Wide Web: Telephone: 852 2609-8038 A-1090 Vienna, Austria http://www.math.ias.edu/ Fax: 852 2603-7636 Telephone (+43) 1 317 20 47/11 [email protected] Fax: (+43) 1 3172047/30 Institute for Mathematical Sciences World Wide Web: email: [email protected] National University of Singapore http://www.ims.cuhk.edu.hk World Wide Web: 3 Prince George's Park http://www.esi.ac.at/ Singapore 118402, Republic of International Center for Singapore Theoretical Physics (ICTP) Euler International Mathematical Telephone: 65 6874 1897 Strada Costiera 11, P.O. Box 586 Institute Fax: 65 6873 8292 34100 Trieste, Italy nab. Fontanka, 27 email: [email protected] Telephone: (+39) 040 2240111 St. Petersburg 197023, Russia World Wide Web: Fax: (+39) 040 224163 Telephone: 7 812 312-40-58 http://www.ims.nus.edu.sg email: [email protected] Fax: 7 812 310-53-77 World Wide Web: email: [email protected] Institute for Mathematics and its http://www.ictp.trieste.it World Wide Web: Applications (IMA) http://www.pdmi.ras.ru/EIMI/ University of Minnesota International Centre for index.html 400 Lind Hall Mathematical Sciences (ICMS) 207 Church Street, SE 14 India Street Fields Institute for Research in Minneapolis, MN 55455-0436 Edinburgh EH3 6EZ, UK Mathematical Sciences Telephone: 612-624-6066 Telephone: +44 (0)131 220 1777 222 College Street, 2nd Floor Fax: 612-626-7370 Fax: +44 (0)131 220 1053 Toronto, Ontario, Canada M5T 3j1 email: i ma-staff@i rna. umn. edu email: [email protected] . uk Telephone: 416-348-9710 World Wide Web: World Wide Web: Fax: 416-348-9714 http://www.ima.umn.edu/ http://www.ma.hw.ac.uk/icms email: [email protected] Institut Henri Poincare Isaac Newton Institute for World Wide Web: 11, rue Pierre et Marie Curie Mathematical Sciences http://www.fields.utoronto.ca 75231 Paris Cedex OS, France 20 Clarkson Road Telephone: 01 44 27 67 89 Cambridge CB3 OEH, UK Forschungsinstitut fur Mathematik Fax: 01 43 25 40 67 Telephone: (+44) 1223-335999 (FIM) http://www.ihp.jussieu.fr/ Fax: (+44) 1223-330508 Eidgenbssische Technische email: [email protected] . uk Hochschule Zentrum Institut Mittag-Leffler World Wide Web: http://www . Ramistrasse 101 Auravagen 17 newton.cam.ac.uk/ 8092 Zurich, Switzerland S-182 60 Djursholm, Sweden Telephone: (+41) 1-632-3475 Telephone: (+46) 8 622 05 60 Instituto Nacional de Matematica email: Fax: (+46) 8 622 05 89 Pura e Aplicada (IMPA) [email protected] [email protected] Estrada Dona Castorina, 110 World Wide Web: World Wide Web: jardim Botanico http://www.fim.math.ethz.ch/ http://www.ml.kva.se/ 22460-320 Rio de janeiro, Rj, Brazil Telephone: (+55) 21-2529-5000 Institut des Hautes Etudes Scien­ Institute for Pure and Applied Fax: (+55) 21-2512-4115 tifiques (IHES) Mathematics (IPAM) World Wide Web: Le Bois-Marie IPAM Building http://www.impa.br 35, route de Chartres 460 Portola. Plaza F-91440 Bures-sur-Yvette, France Box 957121 Istituto Nazionale di Alta Matematica Telephone: (+33) 1 60 92 66 00 Los Angeles, CA 90095-7121 "F. Severi" (INDAM) Fax: (+33) 1 60 92 66 69 Telephone: 310-825-4755 Citta Universitaria World Wide Web: Fax: 310-825-4756 P. Ie AIdo Moro 5 http://www.ihes.fr email:[email protected] 00185 Rome, Italy World Wide Web: Telephone: 39 06490320 Institute for Advanced Study (lAS) http://www.ipam.ucla.edu Fax: 39 064462293 School of Mathematics email:[email protected] Einstein Drive Institute of Mathematical Sciences World Wide Web: Princeton, Nj 08540 Chinese University of Hong Kong http://www.altamatematica.it Telephone: 609-734-8000 Unit 601, 6/F

AUGUST 2004 NOTICES OF THE AMS 811 Reference and Book List

Korea Institute for Advanced Fax: (+49) 7834 979 55 Alfred Renyi Institute of Mathe­ Study (KIAS) email: [email protected] matics 207-43 Cheongnyangni 2-dong World Wide Web: Hungarian Academy of Sciences Dongdaemun-gu http://www.mfo.de Realtanoda utca 13-15 Seoul 130-722, Korea H-1053 Budapest, Hungary Telephone: (+82) 2-958-3711 Max-Planck-Institut fur Telephone: +361-483-8302 Fax: (+82) 2-958-3770 Mathematik Fax: +361-483-8333 World Wide Web: P.O. Box 7280 email: [email protected] http://www.kias.re.kr D-53072 Bonn, Germany World Wide Web: Telephone: (+49) 228 402 0 http://www.renyi.hu Mathematical Biosciences Fax: (+49) 228 402277 Institute email: [email protected] Research Institute for The Ohio State University World Wide Web: Mathematical Sciences (RIMS) 231 West 18th Avenue http://www.mpim-bonn.mpg.de Kyoto University Columbus, OH 43210 Kyoto 606-8502, Japan Telephone: 614-292-3648 Max-Planck-Institut fur Fax: (+81) 75 753 7276 Fax: 614-247-6643 Mathematik in den World Wide Web: http: // email: rebecca@mbi .osu.edu Naturwissenschaften www.kurims.kyoto-u.ac.jp/ World Wide Web: Inselstrasse 22 http://www.mbi .osu.edu D-04103 Leipzig, Germany Statistical and Applied Telephone: (+49) 341 9959 50 Mathematical Sciences Institute Mathematical Research Centre Fax: (+49) 341 9959 658 (SAMSI) (MCAA) World Wide Web: 19 T. W. Alexander Drive University of Aarhus http://www.mis.mpg.de P.O. Box 14006 Department of Mathematical Research Triangle Park, NC 27709­ Sciences Nankai Institute of Mathematics 4006 Ny Munkegade Nankai University Telephone: 919-685-9350 8000 Aarhus C, Denmark Weijin Road 94 Fax: 919-685-9360 Telephone: (+45) 8942 1111 Tianjin 300071, China email: [email protected] Fax: (+45) 8613 1769 _ Fax: (+86) 22-23501532 World Wide Web: email: [email protected] World Wide Web: http://www.samsi.info/ World Wide Web: http://202.113.29.3/english/ http://www.imf.au.dk/ 2.htm Steklov Institute of Mathematics ResearchC/MCAA/index.html Russian Academy of Sciences New Zealand Institute of 8 Gubkina str., 119991 Mathematical Sciences Research Mathematics and its Applications Moscow, Russia Institute (MSRI) (NZIMA) Telephone: (+7) 095 135-22-91 17 Gauss Way University of Auckland Fax: (+7) 095 135-05-55 Berkeley, CA 94720-5070 Private Bag 92019 World Wide Web: http://www.mi . Telephone: 510-642-0143 Auckland, New Zealand ras.ru/index_e.html Fax: 510-642-8609 Telephone: (+64) 9-373-7599 email: i nqui ri es@msri .org Fax: (+64) 9-373-7457 Steklov Institute of Mathematics World Wide Web: email: nzi ma-admi n@nzi ma. 27, Fontanka http://www.msri .org/ auckland.ac.nz St. Petersburg 191023, Russia World Wide Web: Telephone: (+7) 812-312-40-58 Mathematics Institute http://nzima.auckland.ac.nz Fax: (+7) 812-310-53-77 University of Warwick email: [email protected] Coventry CV4 7AL, UK Pacific Institute for the World Wide Web: Telephone: +44 (0)24 7652 4661 Mathematical Sciences (PIMS) http://www.pdmi.ras.ru Fax: +44 (0)24 7652 4182 University of British Columbia World Wide Web: 1933 West Mall Tata Institute of Fundamental http://www.maths.warwick.ac.uk/ Vancouver, BC V6T 1Z2, Canada Research Telephone: 604-822-3922 School of Mathematics Mathematisches Forschungsinsti­ Fax: 604-822-0883 Dr. Homi Bhabha Road tut (Oberwolfach) email: [email protected] Bombay 400 005, India Schwarzwaldstr. 9-11 (Lorenzenhof) World Wide Web: Telephone: (+91) 22 22804545 D-77709 Oberwolfach-Walke, http://www.pims.math.ca Fax: (+91) 22 22804610,22804611 Germany email: [email protected] Telephone: (+49) 7834 979 50 World Wide Web:

812 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Reference and Book List http://www.math.tifr.res.in by Larry Wos and Gail Pieper. Rinton W. W. Norton & Company, August Press, December 2003. ISBN 1-58949­ 2003. ISBN 0-393-02001-0. Weierstrass Institute for Applied 023-1. Everything and More: A Compact Analysis and Stochastics Beyond the Limit: The Dream of History of Infinity, by David Foster Mohrenstrasse 39 Sofya Kovalevskaya, by Joan Spicci. Wallace. W. W. Norton & Company, 10117 Berlin, Germany Forge, August 2002. ISBN 0-765­ October 2003. ISBN 0-393-00338-8. Telephone: (+49) 30-203720 30233-0. (Reviewed January 2004.) (Reviewed June/July 2004.) Fax: (+49) 30-2044975 The Book ofMy Life, by Girolamo ;'~ The Fabric ofthe Cosmos, by Brian email: [email protected] Cardano. New York Review of Books Greene. Knopf, February 2004. ISBN 0­ World Wide Web: Classics Series/Granta. ISBN 1-590­ 375-41288-3. http://www.wias-berlin.de/ 17016-4. Fields Medalists' Lectures, edited Calculated Risks: How to Know by Sir Michael Atiyah and Daniel Iagol­ Book List When Numbers Deceive You, by Gerd nitzer. World Scientific, 2nd edition, Gigerenzer. Simon & Schuster, March The Book List highlights books that December 2003. ISBN 9-812-38259-3. have mathematical themes and are 2003. ISBN 0-743-25423-6. Four Colors Suffice: How the Map aimed at a broad audience potentially California Dreaming: Reforming Problem Was Solved, by Robin Wilson. Mathematics Education, including mathematicians, students, by Suzanne M. Princeton University Press, March and the general public. When a book Wilson. Yale University Press, January 2003. ISBN 0-691-11533-8. (Reviewed has been reviewed in the Notices, a 2003. ISBN 0-300-09432-9. (Reviewed February 2004.) reference is given to the review. Gen­ November 2003.) The Fractal Murders, by Mark erally the list will contain only books The Changing Shape ofGeometry: Cohen. Mysterious Press, Warner published within the last two years, Celebrating a Century of Geometry Books, May 2004. ISBN 0-89296-799-4. though exceptions may be made in and Geometry Teaching, edited by (Reviewed October 2003.) Chris Pritchard. Cambridge Univer­ cases where current events (e.g., the -k From Newton to Hawking: A death ofa prominent mathematician, sityPress, January 2003. ISBN 0-521­ History of Cambridge University's coverage ofa certain piece of mathe­ 53162-4. Lucasian Professors of Mathematics, matics in the news) warrant drawing Codebreakers: Arne Beurling and edited by Kevin C. Knox and Richard readers' attention to older books. Sug­ the Swedish Crypto Program during Noakes. Cambridge University Press, World War II, by Bengt Beckman, gestions for books to include on the list November 2003. ISBN 0-521-66310-5. translated by Kjell-Ove Widman. AMS, may be sent to noti ces-bookl i st@ Galois' Theory of Algebraic February 2003. ISBN 0-8218-2889-4. ams.org. Equations, by Jean-Pierre Tignol. (Reviewed September 2003.) ;'~Added to "Book List" since the World Scientific Publishing. ISBN 981­ ~', Cogwheels ofthe Mind: The Story list's last appearance. 02-4541-6 Abel's Proof: An Essay on the of Venn Diagrams, by A. W. F. Ed­ wards. Johns Hopkins University Gamma: Exploring Euler's Constant, Sources andMeaning ofMathematical by Julian Havil. Princeton University Unsolvabiltty, by Peter Pesic. l\1lT Press, Press, April 2004. ISBN 0-801-87434-3. The Constants of Nature: From Press, May 2003. ISBN 0-691-09983-9. May 2003. ISBN 0-262-16216-4. (Re­ (Reviewed in this issue.) viewed March 2004.) Alpha to Omega-The Numbers That Encode the Deepest Secrets ofthe Uni­ Geometry: Our Cultural Heritage, Across the Board: The Mathematics by Audun Holme. Springer, April of Chessboard Problems, by John ]. verse, by John D. Barrow. Jonathan 2002. ISBN 3-540-41949-7. (Reviewed Watkins. Princeton University Press, Cape, September 2002. Pantheon May 2004.) April 2004. ISBN 0-691-11503-6. Books, January 2003. ISBN 0-375­ G6del's Proof, by Ernest Nagel and Adam Spencer's Book ofNumbers, 42221-8. James R. Newman. New York Univer­ by Adam Spencer. Four Walls Eight Correspondance Grothendieck­ Windows, January 2004. ISBN 1-568­ Serre, Pierre Colmez and Jean-Pierre sity Press, revised edition, February 58289-7. Serre, editors. Societe Mathematique 2002. ISBN 0-8147-5816-9. (Reviewed After Math, by Miriam Webster. de France, 2001. ISBN 2-85629-104-X. March 2004.) Zinka Press, June 1997. ISBN 0-9647­ (Reviewed October 2003.) The Golden Ratio: The Story ofPhi, 1711-5. (Reviewed October 2003.) ;'~ CountDown: Six Kids Vie for Glory the World's MostAstonishing Number, Alan Turing: Life and Legacy of a at the World's Toughest Math Compe­ by Mario Livio. Broadway Books, Great Thinker, edited by Christof tition, by Steve Olson. Houghton Mif­ October 2002. ISBN 0-767-90815-5. Teuscher. Springer, 2004. ISBN 3-540­ flin, April 2004. ISBN 0-618-25141-3. A Handbook ofMathematical Dis­ 20020-7. (Reviewed in this issue.) course, by Charles Wells. Infinity Pub­ Alpha & Omega: The Search for the The Curious Life ofRobert Hooke, lishing Company, 2003. ISBN 0-7414­ Beginning andEnd ofthe Universe, by the Man Who Measured London, by 1685-9. Charles Seife. Viking, July 2003. ISBN Lisa Jardine. HarperCollins, February How Economics Became a Mathe­ 0-670-03179-8. 2004. ISBN 0-060-53897-X. matical Science, by E. Roy Weintraub. AutomatedReasoning and the Dis­ Einstein's Clocks, Poincare's Maps: Duke University Press, June 2002. covery ofMissing and Elegant Proofs, Empires of Time, by Peter Galison. ISBN 0-822-32856-9.

AUGUST 2004 NOTICES OF THE AMS 813 Reference and Book List

Imagining Numbers (particularly the Mathematics and Culture I, edited On the Nature ofHuman Romantic square root ofminus fifteen), by Barry by Michele Emmer. Springer, January Interaction, by Karl Iagnemma. Dial Mazur. Farrar, Straus and Giroux, 2004. ISBN 3-540-01770-4. Press, April 2003. ISBN 0-385-33593-8. February 2003. ISBN 0-374-17469-5. Mathematics and War, edited by -k Phase Change: The Computer (Reviewed November 2003.) BernheIm Booss-Bavnbek and Jens Revolution in Science and Mathemat­ Infinity:·The Quest to Think the Un­ H0yrup. Birkhauser,December 2003. ics, by Douglas S. Robertson. Oxford thinkable, byBrian Clegg. Carroll &Graf, ISBN 3-764-31634-9. University Press, March 2003. ISBN 0­ December 2003. ISBN 0-786-71285-6. Mathematics, Art, Technology, and 195-15748-6. Information: The New Language of Cinema, editedbyMichele Emmer and Portraits ofthe Earth·A Mathematidan Science, byHans ChristianvonBaeyer. Mirella Manaresi. Springer, 2003. ISBN Looks atMaps, byTimothy G. Freeman. Weidenfeld&Nicolson, October 2003. 3-540-00601-X. AMS, September 2002. ISBN 0-8218­ ISBN 0-297-60725-1 (hardcover), 0-753­ Mathematics by Experiment: Plau­ 3255-7. 81782-9 (paperback). sible Reasoning in the 21st Century, by Predicting Presidential Elections and Isaac Newton, byJames Gleick. Pan­ David Bailey and Jonathan Borwein. A Other Things, by Ray C. Fair. Stanford theon Books, May 2003. ISBN 0-375­ KPeters, September 2003. ISBN 1-568­ University Press, August 2002. ISBN 0­ 42233-1. (Reviewed December 2003.) 81136-5. 804-74509-9. Kepler's Conjecture: How Some ofthe Mathematics for the Imagination, Prime Obsession: BernhardRiemann GreatestMinds in HistoryHelped Solve by Peter M. Higgins. Oxford University and the Greatest Unsolved Problem, by One ofthe OldestMath Problems in the Press, November 2002. ISBN 0-198­ John Derbyshire. Joseph Henry Press, World, byGeorge G. Szpiro. JohnWiley 60460-2. March 2003. ISBN 0-309-08549-7. & Sons, January 2003. ISBN 0-471­ Mathematics in Nature: Modeling -k Probability Theory: The Logic ofSci­ 08601-0. Patterns in the Natural World, byJohn ence, by E. T. Jaynes, edited by G. Larry Linked: The New Science of Net­ Adam. Princeton University Press, No­ Bretthorst. Cambridge University Press, works, by Albert-Laszl6 Barabasi. vember 2003. ISBN 0-691-11429-3. April 2003. ISBN 0-521-59271-2. Perseus Publishing, May 2002. ISBN The Mathematics of Juggling, by Proofs from the Book, by Martin 0-738-20667-9. (Reviewed February Burkard Polster. Springer, November Aigner and GUnter M. Ziegler. Springer­ 2004.) 2002. ISBN 0-387-95513-5. (Reviewed Verlag, third edition, December 2003. ,:- Masters of Theory: Cambridge January 2004.) ISBN 3-540-40460-0. and the Rise ofMathematical Physics, Memoirs ofa ProofTheorist: Godel The Riemann Hypothesis: The by Andrew Warwick. University of and OtherLogicians, by Gaisi Takeuti, Greatest Unsolved Problem in Mathe­ Chicago Press, July 2003. ISBN 0-226­ translated by Mariko Yasugi and matics, by Karl Sabbagh. Farrar Straus 87375-7. & Giroux, April 2003. ISBN 0-374­ Math through the Ages: A Gentle Nicholas Passell. World Scientific, Feb­ 25007-3. History for Teachers and Others, by ruary 2003. ISBN 981-238-279-8. William P. Berlinghoff and Meta Math! The Quest for Omega, The Saga of Mathematics: A Brief Fernando Q. Gouvea. Oxton House, by Gregory J. Chaitin. April 2004. History, byMarty Lewinter andWilliam 2002. ISBN 1-881929-21-3. ,Available at http://www.cs. Widulski. Prentice Hall, January 2002. Mathematical Constants, by umaine.edu/~chaitin/omega.html. ISBN 0-130-34079-0. Steven R. Finch. Cambridge Univer­ The Millennium Problems: The Science in the Looking Glass, by sity Press, August 2003. ISBN 0-521­ Seven Greatest Unsolved Mathemati­ E. Brian Davies. Oxford University 81805-2. cal Puzzles of Our Time, by Keith J. Press, August 2003~ ISBN 0-19­ Mathematical Journeys, byPeter D. Devlin. Basic Books, October 2002. 852543-5. Schumer. Wiley-Interscience, Febru­ ISBN 0-465-01729-0. (Reviewed Sep­ Shooting the Sun, byMax Byrd. Ban­ ary 2004. ISBN 0-471-22066-3. tember 2003.) tam, December 2003. ISBN 0-553­ A Mathematician's Survival Guide: More Mathematical Astronomy 80208-9. Graduate School and Early Career De­ Morsels, byJean Meeus. Willmann-Bell ,'~ Signs ofthe Inka Khipu: Binary Cod­ velopment, by Steven G. Krantz. AMS, Inc., 2002. ISBN 0-943396-743. ing in the Andean Knotted-String August 2003. ISBN 0-8218-3455-X. The Music ofthe Primes: Searching Records, by Gary Urton. University of (Reviewed April 2004.) to Solve the Greatest Mystery in Texas Press, August 2003. ISBN 0-292­ -k Mathematicians As Enquirers: Mathematics, by Marcus Du Sautoy. 78540-2. Learning about Learning Mathemat­ HarperCollins, April 2003. ISBN 0-066­ Six Degrees: The Science of a Con­ ics, edited by Leone Burton. Kluwer, 21070-4. nected Age, by Duncan]. Watts. W. W. April 2004. Hardbound, ISBN 1­ Newton's Apple: Isaac Newton and Norton & Company, February 2003. 4020-7853-6; paperback, ISBN 1­ the English Scientific Renaissance, by ISBN 0-393-04142-5. (Reviewed Febru­ 4020-7859-5; eBook, ISBN 1-4020­ Peter Aughton. Weidenfeld &Nicolson, ary 2004.) 7908-7. October 2003. ISBN 0-297-84321-4. Strange Curves, Counting Rabbits, Mathematicians under the Nazis, by The Number IT, by Pierre Eymard and Other Mathematical Explorations, Sanford L. Segal. Princeton University and Jean-Pierre Lafon. AMS, 2004. byKeith Ball. PrincetonUniversity Press, Press, July 2003. ISBN 0-691-00451-X. ISBN 0-8218-3246-8. November 2003. ISBN 0-691-11321-1.

814 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Reference and Book List About the Cover In preparing diagrams for Peter Sarnak's article in this issue on Sync: The Emerging Science of Ramanujan graphs, we decided that it would be an interesting exercise Spontaneous Order, bySteven Strogatz. to verify that its expansion constant h is 1/4. I ~,? ~,~{ Hyperion, February 2003. ISBN :i '1. recall that if the graph has N nodes, then this 0-786-86844-9. (Reviewed March 2004.) constant is the minimum value of IaXI / lXI, Travels in Four Dimensions: The ... where X varies over the subsets of nodes of size Enigmas of Space and Time, by at most N /2. Thus a priori one might expect to Robin Le Poidevin. Oxford University have to look at close to 280 subsets of nodes, and, Press, February 2003. ISBN 0-19­ "''' ~J!u ~ indeed, it has been shown by M. Blum et al. 875254-7. (Inform. Process. Lett. 13 (1981» that this is a very ""~ Turing (A Novel about Computa­ difficult problem. For the graph at hand, tion), by Christos H. Papadimitriou. ~~~ however, Sarnak was able to verify by hand that MIT Press, November 2003. ISBN 0­ .• h = 1/4, and it was also possible to verify the 262-16218-0. calculation with a computer program that might What Is Thought?, by Eric B. Baum. work as well for more general 3-regular graphs. The basic idea of the MIT Press, January 2004. ISBN 0-262­ program is to look at the possible cut sets separating X from its 02548-5. complement. There are two key observations that the program is based What the Numbers Say: A Field on. The first is that one need only look at connected subsets X, and in Guide to Mastering Our Numerical fact only at cut sets that are Jordan curves. The second observation is World, by Derrick Niederman and that in certain circumstances one need only look at cut sets that satisfy David Boyum. Broadway Books, April a kind of convexity condition at each vertex. The exact conditions ought 2003. ISBN 0-767-90998-4. to be clear from the accompanying diagrams, where the dashed lines When Least Is Best: How Mathe­ cannot be the cut sets for a candidate X, since adjusting them in a maticians Discovered Many Clever simple way increases IXI without decreasing IaXI. (The nodes in X are Ways to Make Things As Small (or As dark.) Large) As Possible, by Paul J. Nahin. PrincetonUniversity Press, November 2003. ISBN 0-691-07078-4.

It is straightforward and entirely practical to make up an algorithm that constructs all admissible cut sets. If IXI for all of these is not greater than half the number of nodes, the convexity argument above shows that h can be calculated by perusing the list. Because of the symmetries in the graph at hand, it is necessary to consider only two types of cut sets, and the cover illustration is, in effect, the program output for one of these types. It shows all convex cut sets passing through the top two gray faces (up to mirror symmetry). The minimum value 1/4 is achieved in the large diagram at lower left, where IXI is also the maximum value of 40. This graph is a Ramanujan graph. A result of Lipton and Tarjan (SIAM ]. AppI. Math. 36 (1979» implies that there are at most finitely many planar Ramanujan graphs. The largest ones known are 84:20, and 84:23 in the Atlas of Fullerenes by P. Fowler and D. Manolopoulos. I'd like to thank A. Gamburd for calling my attention to the graph used here, which he found in a paper by P. Frankin on the four-color problem, and also for telling me about Fullerenes. -Bill Casselman Covers/Graphics Editor (not;[email protected])

AUGUST 2004 NOTICES OF THE AMS 815 Visiting Mathematidans

Name and Home Country Host Institution Field ofSpecial Interest Period ofVisit AmericanandCanadianMathematiciansVisitingAbroad Ban, Dubravka (U.S.A.) UniversiUit Miinster, Germany Automorphic Forms 8/04- 7/05 Bose, Amitava (Canada) Institut Mittag-Leffler, Royal Queueing Theory and Teletraffic Theory 10/04-11/04 Swedish Academy of Sciences, Sweden Gehrke, Mai (U.S.A.) Danish Technical University, Denmark Lattice Theory, Logic 8/04 Gehrke, Mai (U.S.A.) University of Copenhagen, Denmark Lattice Theory, Logic 9/04- 6/05 Gehrke, Mai (U.S.A.) Oxford University, England Lattice Theory, Logic 7/05 Jing, Naihuan (U.S.A.) Bielefeld University, Germany Quantum Groups, Infinite Dimensional 12/04- 8/05 Lie Algebras Lukin, William D. (U.S.A.) University ofFlorence, Italy Fluid Dynamics, Biomechanics, Applied 9/04-10/04 and Computational Mathematics Weiss, Richard M. (U.S.A.) University ofWiirzburg, Germany Group Theory 5/04- 1/05 Wilson, J. Michael (U.S.A.) University of Seville, Spain Classical Fourier Analysis and Weighted 8/04- 5/05 Norm Inequalities Zirbel, Craig (U.S.A.) University of Salzburg, Austria Applied Probability and Stochastic Processes 9/04- 6/05

Visiting Foreign Mathematicians Agoston, Istvan (Hungary) CarletonUniversity Representation Theory 9/04-12/04 Ali, Jodayree Akbrafam (Iran) CarletonUniversity Differential Equations, Inverse 7/04- 9/04 Spectral Theory Belopolskaya, Y. (Russia) Case Western Reserve University Stochastic Differential Equations 8/05-12/05 Bernstein, Joseph (Israel) New York University, Courant Institute Representation Theory 9/04-12/04 deBievre, Stephen (France) McGill University Quantum Maps, Scarring, Quantum Chaos 1/05- 7/05 Biler, P. (Poland) Case Western Reserve University Partial Differential Equations 7/04- 8/04 Bueno Cachadina, Maria College ofWilliam and Mary Numerical Linear Algebra 8/04- 7/05 Isbel (Spain) Dai, Bo (China) University of California, Irvine Integrable Systems, Symplectic Geometry 1/05- 6/05 Ding, Xuanhao (People's Vanderbilt University Analysis 9/04-11/04 Republic of China) Fei, Shao-ming (China) North Carolina State University Quantum Computation 10/04-11/04 Gonzalez, Alfredo (Argentina) Auburn University Differential Equations, Numerical Analysis, 8/04-12/04 Approximation Theory, Harmonic Analysis Hsu, Nan-Jung (Taiwan) Colorado State University Statistics 8/04-11/04 Hu, Naihong (People's York University Representation Theory ofLie Algebras 9/03- 9/04 Republic of China) and Quantum Groups Huang, Hsin-Cheng (Taiwan) Colorado State University Statistics 8/04-11/04 Hulanicki, Andrej (Poland) Cornell University Harmonic Analysis on Groups and 1/05- 5/05 Applications Kang, Chang Wook (Korea) University of South Carolina, Columbia Quality Management, Industrial Statistics 7/04- 8/04

The list of visiting mathematicians includes both foreign mathematicians visiting in the United States and Canada, and Americans and Canadians visiting abroad. Note that there are two separate lists.

816 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Visiting Mathematicians

Name and Home Country Host Institution Field of Special Interest Period ofVisit Kang, Chang Wook (Korea) University of South Carolina, Columbia Quality Management, Industrial Statistics 12/04- 2/05 Karageorghis, Andreas (Cyprus) Colorado School of Mines Numerical Analysis 8/04- 6/05 Karch, G. (Poland) Case Western Reserve University Partial Differential Equations 7/04- 8/04 Kroo, Andras (Hungary) Vanderbilt University Approximation Theory 1/05- 5/06 Kuttler, Jochen (Switzerland) Northeastern University Singularity 4/04- 3/06 Levin, Eli (Russia) Vanderbilt University Constructive Approximation 1/05- 5/05 Lin, Furong (China) West Virginia University Computational Mathematics 1/05- 4/05 Lindner, Alexander M. (Germany) Colorado State University Applied Probability and Statistics 8/04-12/04 Logomasino, G. T. Lopez (Spain) Vanderbilt University Approximation Theory 9/04-12/04 Lu, Guo-Fu (China) Washington State University Partial Differential Equations 7/04- 1/05 Michler, Gerhard (Germany) Cornell University Representation Theory and the 8/04- 5/05 Theory of Finite Groups Mierczynski, Janusz (Poland) Auburn University Differential Equations and Dynamical 8/04-10/04 Systems Morozova, Nadezhda (Russia) Northeastern University Physics and Biology 9/04 Mueller, Peter (Germany) University of California, Irvine Mathematical Physics 7/04- 6/05 Novikov, Dmitry (Israel) Cornell University Core Analysis, Differential Equations 8/04- 5/05 and Geometry Pankov, Alexander (Russia) College ofWilliam and Mary Mathematical Physics 8/04- 8/05 Paoli, Simona (Italy) University at Buffalo Algebra Cohomology, Category Theory 6/04- 8/05 Park, Jong Sea (South Korea) University ofAlabama Fixed Point Theory, Nonlinear Analysis, 8/04- 8/05 Control Systems Governed by Integro­ Differential Equations Park, Sangue (Korea) University ofMassachusetts, Amherst Biostatistics 9/03-12/04 Rahgozar, Mehdi (Iran) Memorial University of Newfoundland Biostatistics 4/04- 9/04 Rattanakul, Chontita (Thailand) West Virginia University Biometrics, Partial Differential Equations 4/04- 3/05 Rosas, Mercedes (Venezuela) York University Algebraic Combinatorics 9/04- 8/05 Saal, Jiirgen (Germany) Vanderbilt University Analysis 8/04- 8/05 Sadaka, Hanai (Switzerland) Northeastern University Singularity 4/04- 3/06 Safoui, Abdessamad (Morocco) Universite Laval Equations aux derivees partielles 6/04- 8/04 Satoh, Shin (Japan) University of South Florida Topology, Knot Theory 4/03- 3/05 Schoenenberger, Johanna Universite Laval Enseignement des mathematiques 12/04-10/05 (Switzerland) Schmidt, Rafael (Germany) Universite Laval Processes 9/04-12/04 Siegwart, Martin (Germany) Vanderbilt University Analysis 11/04- 8/05 Solynin, Alexandre (Russia) University ofArkansas ComplexAnalysis 8/03- 8/04 Tableman, Mara (Switzerland) Portland State University Models and Methods 7/04- 9/04 Tian, Qingchun (China) McGill University Number Theory and Arithmetic Geometry 8/04- 8/05 Wang, Qin (People's Vanderbilt University Non-commutative Geometry 9/04- 8/06 Republic of China) Weerakoon, Sunethra (Sri Lanka) Cornell University Numerical Analysis and Differential 8/04- 5/05 Equations Uchiyama, Koichi (Japan) Massachusetts Institute of Asymptotic Analysis and Differential 4/05- 5/05 Technology Equations Yang, Danping (China) California Institute of Technology Applied Mathematics 9/03- 8/04 Zhang, Chaowen (People's Auburn University Representation Theory of Lie Algebras 8/04-12/04 Republic of China)

AUGUST 2004 NOTICES OF THE AMS 817 FromtheAMS Secretary

Overview Report of the Executive The Society's membership grew to over 28,000 in 2003; dur­ Director, State ofAMS, 2004 ing the currentyear we expect membership to exceed 29,000. Most of that growth is in nominee members (graduate stu­ The American Mathematical Society shows different faces dents who become members because their universities are to different groups of mathematicians. Many young math­ institutional members). Ordinary membership has been ematicians focus on the Society's employment services slightly declining, as has reciprocity and Category-S (in the and surveys, which help them to make career decisions. developing world). Life and emeritus memberships are up, More established mathematicians sometimes associate reflecting our aging population. the AMS with meetings, where they not only do mathe­ The Societywill carry out planningin specific areas each matics but connect with friends and acquaintances as year for the next several years. The first of these focused well. Mathematicians abroad oftenidentify with the Notices, planning efforts tookplace in 2003, and as a result the AMS which keeps them informed about mathematics and the will consider making various change'S in membership: the profession. principles by which we set dues, our delivery of member benefits, new categories of membership, etc. Most impor­ But for many mathematicians-especially those most ac­ tantly, the planning process has helped us to understand tively engaged in research-the most apparent face of the some of the major issues surrounding membership. Mem­ Society is MathSciNet, the entryway to the Mathematical bership development is now a separate function in the Reviews database. Math Reviews has always been impor­ Society, viewed as an important ongoing activity. tant to mathematicians, but MathSciNet has changed the Our meetings continue to be healthy and robust. The way in which people do research (and many other things recent Joint Meeting in Phoenix was the fourth largest as well). It is an enormously successful part of the AMS ever held by the Society, and the program was widely publication program, and this year's annual report fo­ praised by those attending. The sectional meetings con­ cuses on Mathematical Reviews and its great success. tinue to be one of the key ways inwhich the Society reaches First, here is an overview of the Society early in 2004. mathematicians throughout the country. The Society held a joint international meeting in Bangalore, India, during December, and attendance exceeded 500 mathematicians 2003 Membership (from the U.S., Europe, and India). The summer research conferences, which are now held at Snowbird in Utah, Life seem to be vigorous and healthy with a resurgence of pro­ 1% posals for future conferences. Student/Unemp Ordinary-High 3% The Washington office of the Society has represented 10% the interests of the mathematical community in Wash­ Emeritus ington for more than 7% Ordinary-Low ten years now. This 16% is increasingly diffi­ cult in the present budgetary environ­ Nominee ment, but we have 28% been successful Ordinary-Entry nonetheless. The key 11% to success in Wash­ ington is making sure that mathematics has a presence, that policymakers think AMS president David Eisenbud ofmathematics when testifying in Washington, D.C.

818 NOTICES OF THE AMS VOLUME 51, NUMBER 7 From the AMS Secretary they think about science and technology. We have ac­ careers, allowing them to spend a year fully engaged in complished this through such things as congressionallun­ research. cheons, public service awards, and special receptions. But • We award annual Ky Fan travel grants to promote the it is the day-to-day activities that ultimately bliild con­ exchange ofmathematicians between North America and nections with all parts ofWashington. The Washington of­ China. fice also oversees special projects, including such things • We 'conduct the Math in Moscow program, providing as the Media Fellows Program, the chairs workshops, and funds for approximately ten students each year to study mentoring workshops done jointlywith the Mathematicians for a semester at the Independent University in Moscow and Education Reform network (MER). in a demanding mathematics program that builds fu­ Many of the programs and services of the Society are ture connections between our communities. ongoing, and for that reason it is easy to overlook their It is important to note that, with the exception of the quality and value to the community; new and elaborate pro­ last item, everyone of these programs is ongoing and jects are easier to tout, even when they are short-lived. But does not rely on grant support of any kind. providing sure and steady service to the community has One of our greatest services been a hallmark of the Society. For example: to the mathematics community • We continue to conduct the annual Survey, l which pro­ is the publication of our mem­ vides essential information about the profession. ber journals, the Notices and 2 • We plan and run the Employment Center at the Joint the Bulletin. While only mem­ Meetings. bers receive these journals in • We administer the Mathlobs service, which connects job paper form, the electronic ver­ applicants and participating departments. sions are available to all math­ • We publish various professional publications for the ematicians with access to the mathematics community, including Employment Infor­ Internet. They are widely ac­ mation in the Mathematical Sciences,3 Assistantships cessed throughout the world. and Fellowships,4 the Professional Directory, and the Public awareness is a special kind of service, and it has Combined Membership List.S accomplished a great deal in the few years since the cre­ • We provide annual Ep­ ation of our Public Awareness silon awards to encour­ Office. We are building contacts age young scholars pro­ with the press, a slow process grams (summer programs that requires patient diligence to for talented high school accomplish. Mathematical Mo­ students) totaling ap­ ments (one-page fliers with the proximately $80,000. simple message that mathe­ • We sponsor an Arnold matical research is important) Ross Lecture each year, have been extremely popular, aimed at high school students and normally held at a especially among high school science museum. teachers, who post them on • We hold 4-5 "Who Wants to Be a Mathematician" games classroom walls. The "Who each year, attracting the best mathematics students in Wants to Be a Mathematician" some geographical area to contests mentioned above have participate in mathemat­ drawn enthusiastic crowds of ics. shouting students (which isn't something mathematics • We award annual Tr­ does 'very often). The Public Awareness Office has helped jitzinsky scholarships of to highlight the Society, both inside and outside the math­ $4,000 each to eight ematics community. mathematics undergrad­ Of course, we could accomplish few of these services uates. if the Society did not have a source of funding, and mainly • We award Centennial Fel­ that source is our publications program. We derive about lowships each year to 75 percent of our revenues from publications. (Members young mathematicians in Who Wants To Be A are almost always surprised to discover that individual dues the early stages of their Mathematician? account for only a small percentage of revenues-6.2 per­ cent in 2003.) The health of our publications are therefore 1 Under the auspices of the AMS-ASA-IMS-MAA Data Committee. directly connected to the health of the Society overall. 2 With the advice of the AMS-MAA-SIAM joint Committee on Em­ Our research journals continue to do extremely well in ployment Opportunities (jCEO). almost every respect. Submissions to the journals are ro­ 3 With the advice ofJCEO. bust, backlogs are under control, and mathematicians view 4 Compiled under the direction of the AMS-ASA-IMS-MAA Data the journals with respect. While citation indices should be Committee. viewed cautiously in mathematics, it is interesting to note SA joint project ofthe AMS, MAA, and SIAM. that the Journal ofthe American Mathematical Society now

AUGUST 2004 NOTICES OF THE AMS 819 From the AMS Secretary

Percent References Impact Factors 6.00% (2002)

1. Journal of AMS 2.533 5.00% 2. Comm. Pure and Appl. Math. 2.022 3. Annals of Mathematics 1.905 4.00% 4., Bulletin of the AMS 1.824 5. Memoirs of the AMS 1.661 3.00% 6. Acta Mathematica ' 1.621 7. Inventiones 1.616 2.00% (All others are below 1.100) 1.00% III ranks first as measuredby the lSI impact factor. Like 0.00% IIIIII11111111111111111 most journals, ours experience slight attrition in 2000 1995 1990 1985 1980 1975 1970 1965 1960 1955 1950 1945 1940 subscriptions each year, but the attrition remains slight and the journals remain extremely healthy. The Society has devoted a great deal of time and energy editor for Zentralblatt fur Mathematik und ihre Grenzge­ to its book program over the past ten years. We now pub­ biete (ZBL). When ZBL began to enforce German racial poli­ lish approximately 100 new titles each year, and because cies in 1938, Neugebauer resigned his position and im­ of our policy to keep all mono­ migrated to the United States. American mathematicians graphs in print indefinitely, we discussed how to save ZBL, but eventually were led to list more than 3,000 titles in start a new reviewing journal, which they called Mathe­ print. We have extended distri­ matical Reviews. Neugebauer and J. D. Tamarkin were its bution agreements throughout first editors. the world and increasingly dis­ In its first year of operation (1940) Math Reviews pub­ tribute books for other pub­ lished 2,120 reviews in 400 pages. The staff consisted of lishers. We have recently added four people: the two editors, a technical assistant (Willy several benefits for authors of Feller), and a secretary (Evelyn Spencer). Expenses for the books, including websites for year totaled $14,356.77, and the journal actually showed each book (to post corrections a small profit. and additional material) and Over the years Math Reviews slowly evolved into an ever­ author discounts (50% for 5 larger operation, growing from its initial staff of four to a years on all AMS books). We staff of more than seventy. The operation was originally have evaluated all aspects of based inProvidence (where Neugebauer and Tamarkin held the program, and we plan to expand our book program fur­ appointments at Brown), and that was part of the motiva­ ther in the future. tion for the Society moving its headquarters to Providence in 1951. For the next fourteen years, the Math Reviews staff Mathematical Reviews shared space with the rest of the AMS staff. It was not al­ The third part of our publications program is Mathemat­ ways an easy sharing, however, because Math Reviews op­ ical Reviews, which plays an increasingly important role erated as a semiautonomous unit of the Society, with its ex­ both in the mathematics community and in the Society ecutive editor reporting directly to the Board. For this reason itself. Most mathematicians see only the front face ofMath (and others) Math Reviews moved its offices to Ann Arbor, Reviews-either the Web version onMathSciNet or (less fre­ Michigan, in 1965. That year it had a staff of thirty-five and quently) the large orange published about 15,000 reviews. volumes. Sitting behind In the intervening years Math Reviews has had its ups that facade, however, is an and downs. In the late enormously complicated 1970s Math Reviews fell set of interlocking compo­ behind in reviewing, cre­ nents that produce a high­ ating a giant backlog of quality product and make material that was subse­ it widely available. Mathe- quently processed in a matical Reviews has be­ short period of time in come invaluable to math­ order to catch up. (The ematicians around the number of published re­ world. views rose by over 50 per­ It wasn't always that cent in a single year!) For way. Math Reviews was manyyears Math Reviews founded by Otto Neugebauer, who had been the Feller showed a financial loss

820 NOTICES OF THE AMS VOLUME 51, NUMBER 7 From the AMS Secretary

chooses from over 100,000 items in about Math Reviews 2003 1,800 journals and many hundreds of books. They deal with nearly 3,000 pub- Number of items in the MRDB lishing entities, tracking down material 1,894,000 when it is lost and sometimes download­ Number of reviews 'in the MRDB 1,661,000 ing material from the Web. They uniquely Number of individuals (authors) in the MRDB 400,000 identify each of the more than 400,000 au­ Number of new items added-to the MRDB in 2003 77,493 thors included in the database, which Number of new reviews added to the MRDB in 2003 57,438 makes it possible to locate all papers by Number of new items processed each working day 325 specific authors (even when names change Number of journals currently covered by MR 1,799 or are transliterated differently!). They clas­ Number of classifications in MSC2000 sify, they evaluate, and they annotate arti­ 5,529 cles for further processing. They do all this Number of currently active reviewers 10,843 day after day and week after week, at the rate of about 350 items per day. for the Society, and there was talk of selling it to commercial Reviewers are one of the key assets of publishers. Repeatedly over the years people discussed Math Reviews, of course, and there are more than 10,000 merging Math Reviews and ZBL in the hope that this would of them today. Editors decide which reviewers should con­ save costs, both for the Society and for subscribers. Early sider which papers, and the staff must track (and gently in the development of the Internet, some predicted the im­ remind) reviewers to complete their work. Items are lost in transit, reviewers go on leave (sometimes unexpect­ minent demise ofMath Reviews, which would "become use­ edly), occasionally items are returned after long periods less once all mathematics was available on the Web." Of of time. Mainly, however, reviews arrive and are edited, not course exactly the opposite has happened. just for style and grammar (that's the easy part), but to Today Math Reviews is thriving. Each year it adds more add detailed references and their corresponding Math Re­ than 75,000 items to the database, which now includes views numbers to make certain that the database is con­ more than 1.9 million items. Assembling that database re­ sistent and interconnected. quires more effort than most people imagine. The staff Today, of course, there is even more data to be captured:

Percent References there are about 120 possible fields underlying each item in the database. The Math Reviews database now has more than 360,000 links to original articles, allowing users to access the articles under review with the click of a button. The staff adds more links each year and keeps the old ones up to date. Math Reviews includes a richer and richer col­ lection of internal links showing users which reviews refer to which others, both backwards and forwards. And the most recent additional data is the collection of references for selected journals. It has the potential to add an entirely new way in which to use Math Reviews to understand the mathematical literature. Beginning two years ago, Math Reviews began captur­

0_00%-fA-rLrA~~'-"""""~"""""""""""'~~~'-"""""~"""""""""""""""'~~~~~ ing reference lists for approximately 100 journals, going 2000 1995 1990 1985 1980 1975 1970 1965 1960 1955 1950 1945 back to 1997 for most. Using a sophisticated application, we are able to match those references to the correspond­ # Refs/# Items in MR ing item in the MR database about 80 percent of the time.

0.800 (Many items do not correspond to anything in the data­ base, either because they were never published or were 0.700 never included. The matching rate for items in the data­

0.600 base is about 95 percent.) Like links to original articles, the links from references to the corresponding items in Math 0.500 ,..-- Reviews provide a wonderfully efficient way to navigate 0.400 r- the literature. The entire collection of data can do much

0.300 f- more, however. With nearly 800,000 references collected so far, we can 0.200 begin to study and to understand the mathematical liter­ 0.100 1111,1 I ill ature far better than we ever have before. Mathematicians have always claimed that mathematical literature is valu­ 0.000 111111111 2000 1995 1990 1985 1980 1975 1970 1965 1960 1955 1950 1945 1940 able for many years after publication (unlike some other areas); we now can make that evident by graphing the

AUGUST 2004 NOTICES OF THE AMS 821 From the AMS Secretary

dividing the 11umber of citations to a particular Journal # Items # Refs Ratio journal by the total number of articles published in that journal for the past sixty years, we get a 1. Inst. Hawes Etudes Sci. Publ. Math. 343 3327 9.70 much broader understanding of the frequency of 2. ]. Arner. Math. Soc. 486 3333 6.86 citations. And we can choose the time period to re­ 3. Invent. Math. 3206 15526 4.84 fine the information. 4. Ann. Sci. Ecole Norm. Sup. (4) 778 3289 4.23 Of course, citation data can be misused (in some 5. ]. Differential Geom. 1198 4778 3.99 disciplines, self-citations are a major problem), and 6. Comm. Pure Appl. Math. 2025 7144 3.53 one should be careful to understand the limita­ 7. Mem. Arner. Math. Soc. 732 2441 3.33 tions of citation data (at the moment, the MR data 8. Geom. Funct. Anal. 470 1565 3.33 is too limited to be reliable). But over time, as we 9. Internat. Math. Res. Notices 553 1757 3.18 double the number of journals and build up a ci­ 10. ]. Algebraic Geom. 347 1089 3.14 tation database, users of Math Reviews will be able 11. Ann. of Math. (2) 4606 13569 2.95 to study the literature in ways they never could have 12. Advances in Math. 379 1030 2.72 before. 13. Ann. Inst. H. Poincare Anal. Non Lin. 537 1356 2.53 The Mathematical Reviews database is spectac­ 14. Math. Res. Lett. 698 1740 2.49 ular, but Math Reviews today is much more than 15. Asterisque 954 2348 2.46 just the database. The Web interface, MathSciNet, 16. ]. Funct. Anal. 3308 7838 2.37 is what has revolutionized the way in which math­ 17. Comm. Math. Phys. 6280 13334 2.12 ematicians use the database. MathSciNet is not a 18. Bull. Amer. Math. Soc. (N.S.) 911 1899 2.08 collection of Web pages, however; it is a sophisti­ 19. Ergodic Theory Dynam. Systems 1313 2710 2.06 cated piece of software that undergoes extensive 20. K-Theory 521 1010 1.94 development on an annual cycle. In late winter of each year, staff in the Ann Arbor and Providence offices consider a list of year of publication for all references from recent papers. potential improvements and en­ Better yet, since the number of items in Math Reviews hancements for the next cycle. each year measures the approximate size of the mathe­ They review comments made to matical literature, we can measure how often the past the customer support personnel mathematical literature is still cited. It is a convincing ar­ in Providence, they consider sug­ gument that mathematics has a very long life! gestions made directly to the ex­ For many years scientists in other fields have used the ecutive editor in Ann Arbor, and "impact factor" to measure the quality ofjournals. The im­ they generate ideas from every­ pact factor is measured by Thompson lSI, which compiles one involved in the publication citation data for a large collection ofjournals. For any given program.The list is narrowed, journal, the impact factor for a specific year is the num­ modified, and (sometimes) ex- ber of citations to the previous two years of that journal panded. Development takes place Jane Kister, MR divided by the total number of articles published in the over the next six months, and the Executive Editor journal in those two years. While this provides some in- new version of MathSciNet is re- retiring in July 2004.

Al'I~RICAN ,.,ATHEHATICAL 50CI.TY leased to all sites in September. I.IJ.IIIIJI' FOI SNnd

822 NOTICES OF THE AMS VOLUME 51, NUMBER 7 From the AMS Secretary which Math Reviews was priced. Sub­ scribers were asked to pay a Data Ac­ cess Fee (the DAF, which this year is $5,467 for institutional members) and then to purchase whatever individual products they chose (in 2004 this is $526 for paper and $1,998 for Math­ SciNet or MathSciDisc). It was a sen­ sible way to price a database product, because it separated the various com­ ponents assembling the database and delivering it invarious formats. But it also made it possible to create a flexible scheme for pricing Math Reviews for consortia. Consortia ofinstitutions can now purchase Math Reviews products by joining together. The Data Access Fee for the consortiumis the sum of all pre­ vious subscribers (no savings there). But adding new subscribers does not increase the DAF. Each mem­ ber of the consortium can then purchase Math­ SciNet at a reduced price of $250-$1,000 depend­ ing on "mathematical activity" at the institution. This means that a small college, which never had access to Math Reviews in the past, can join with nearby large univer­ sities to access MathSciNet for as little as $250 per year. Large and small institutions gain by this arrangement, whether or not they previously subscribed. There are now more than one hundred consortia It is used bymathematicians everywhere, around the world involving well over it is widely admired, and it is financially one thousand institutions. secure. We have reached this state In addition to the regular consortia, through the efforts of people over many the Society has a National Data Access years (including the present executive Fee program that allows certain coun­ editor, Jane Kister, who will retire in tries in the developing world to obtain July of this year). access at greatly reduced prices. Coun­ Twenty-five years ago when Math tries currently participating are Algeria, Reviews was struggling, some people Bosnia, Bulgaria, Costa Rica, Croatia, Es­ had the foresight to make certain tonia, Fiji, Lebanon, Macedonia, Morocco, that the underlying database was Romania, Serbia, Vietnam, Yugoslavia, and computerized, long before they were Zimbabwe. certain about its usefulness. Math- Keeping track of all the institutions in SciNet was developed initially at considerable ex­ consortia as well as the single-institution pense, largely because people had the belief that putting subscribers is a large job, involving many staff. We have a database online was the right thing to do. When the So­ to track subscriptions, send invoices in the right amounts ciety decided to add the old reviews from prior years, it to the right institutions, deal with agents (not always easy), invested nearly $1 million keyboarding those reviews. It help institutions find a consortium to join, add and sub­ was an investment that paid off. We need to continue to tract Internet addresses for access, etc. This work is done invest in Math Reviews in similar ways in the future so that by our Marketing/Sales and Customer Services Depart­ it remains a vital part of the Society's publication program­ ments in the Providence headquarters. in many ways, the most important part. And we will. Because of this flexible pricing, the number of institu­ tions with access to the Math Reviews database has more -john Ewing than doubled during the past ten years, a remarkable Executive Director achievement at a time when journal subscriptions are under enormous pressure. By almost every measure, Mathematical Reviews is healthier now than at any time in the past sixty-four years.

AUGUST 2004 NOTICES OF THE AMS 823 From the AMS Secretary

liabilities of the Society. Finally, in the last part of the re­ Report of the Treasurer (2003) port, there are summary financial statements which pre­ sent the balance sheet and other financial statements of I. Introduction the Society. One of the most important duties of the treasurer is to lead The Society segregates its net assets and the activities the Board of Trustees in the oversight of financial activi­ that increase or decrease net assets into three types. Un­ ties of the Society. This is done through close contact with restricted net assets are those that have no requirements the executive staff of the Society, review of internally gen­ as to their use placed on them by donors outside the So­ erated financial reports, review of audited financial state­ ciety. A substantial majority of the Society's net assets and ments, and direct contact with the Society's independent activities are in this category. Temporarily restricted net auditors. Through these and other means, the trustees gain assets are those with donor-imposed restrictions or con­ ditions that will lapse upon the passage of time or the ac­ an understanding of the finances of the Society and the complishment of a specified purpose. Examples of the So­ important issues surrounding its financial reporting. The ciety's temporarily restricted net assets and related "Report of the Treasurer" is presented annually and dis­ activities include grant awards and the spendable income cusses the financial condition of the Society as of the im­ from prize and other income-restricted endowment funds. mediately preceding fiscal year end and the results of its Permanently restricted net assets are those that must be operations for the year then ended. It contains summary invested in perpetuity and are commonly referred to as en­ information regarding the operating results and financial dowment funds. The accompanying financial information condition of the Society for 2003, a "Review of 2003 Op­ principally relates to the unrestricted net assets, as this erations", containing more detailed information regarding category includes the operating activities of the Society. the Society's operations, and a discussion of the assets and Unrestricted revenues in excess of unre­ stricted expenses for the year ended December Key Operating Results 31, 2003, resulted in an increase inunrestricted 25,000 r------net assets of approximately $10,082,000. Of this amount, net income on the unrestricted 20,000 m2001 .2002 ~2003 portion of the long-term investment portfolio to­ taled $7,781,000 and net income from opera- 15,000 tions totaled $2,301,000. The recovery in the do­ mestic and international financial markets that 10,000 occurred in 2003 resulted in a return on the long­ term portfolio of approximately 23.9%. These 5,000 and other matters are discussed in more detail in the following sections. o The Society's net assets totaled $53,821,000 at December 31, 2003: $3,156,000 is perma­ (5,000) r------== nently restricted, consisting principally of the original amount of donor restricted gifts andbe­ (10,000) .L-- _ quests received by the Society: $1,591,000 is Operating Operating Operating Net Long-term Net Revenues Expenses Income Investment Income Income temporarily restricted by donor-imposed limi­ tations that will lapse upon the passage of time or the use of the asset for its intended purpose; $49,074,000 is unrestricted, of which Operating Income As % of Revenue $38,438,000 has been designated by the Board of Trustees as reserved for future expenditure, principally in the form of the Economic Stabi­ 20.0% r------;(:f(~o---:::(:f(SlO------~ lization Fund (ESF). This fund's purpose is to pro­ C\J 0 ~ ~ '# vide a source of cash in the event of a financial o crisis. The Society's Board of Trustees set the minimum level at which to maintain the ESF at the current estimate of the postretirement health benefit obligation plus 75% of annual operating expenses. As of the end of 2003, the value of the ESF exceeds the established minimum level. The -10.0% remaining unrestricted net assets consist of $4,316,000 invested in fixed assets and

-20.0% ....o..-- ~ $6,320,000 of undesignated net assets.

80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

824 NOTICES OF THE AMS VOLUME 51, NUMBER 7 From the AMS Secretary

page shows operating income as a percentage of op­ Sales Trends - Historical Dollars erating revenues. Two observations are noteworthy. 10,000 ..------, First, the margins achieved from 1997 to 2003 are 9,000 +------l somewhat higher than the average of all the years pre­

8,000 ~ sented. Second, the variation inmargin over the more ~"""""'------'------_____t recent years is smaller than the variation in the ear­ 7,000 .. lier years. Taken together, these are positive finan­ 1::::~o~~~T~ated I 6,000 +------lI:...... Books ,II------l cial indicators. =X--Xlndividualdues ---Institutional dues If the Board of Trustees had not appropriated in­ 5'OOOI~ vestment income to support operations in 2002 and 4,000 "'- "-~------"""-_I-----=------2003, the operating income margin percentage above 3,000 +------=:::;:::;;;;;::;;::;;;;;;;;;;;;;;;<~~~==~~~;;;;;;;;;;;=~;;;:;;;;;;;;;;d would have been approximately 5.5% in 2002 and 7.1% in 2003. These are both above the average for the pe­ 2,000 -I-,------l riod shown above and consistent with the results for

1 ,000 t:=:;:=:::;;::=:;::=::;==~==~==~~:::::::+==~ the five prior years. O+----____.___------,.--~------.----.,....__-____.___-______r--~-----l Sales Trends 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 The graphs to the left show sales trends from 1994 through 2003, first in historical dollars and second in constant dollars (using 2003 as the base year and Sales Trends - Constant Dollars adjusting other years for inflation). 10,000 .-.... The trends shown in "Sales Trends-Historical 9,000 ...-.- ~ --...... Dollars" are mildly upward, and this is partly due to ~ --...... -....I ...... pricing strategies that counter the effects ofinflation. 8,000 Below, the chart is repeated with the underlying data 7,000 converted to constant dollars. I~MR &related I 6,000 Mathematical Reviews. Total revenue from MR in .'-- - , ...... JOUrnaIS I --Books its various forms increased in 2003. This is due to 5,000 "- - price increases effective in 2003, net of attrition 4,000 -- - -... - (which was minor). Also, the value of the dollar in " ------....--- ~ 3,000 many overseas markets continued to be favorable l?-""'- - --.., from the perspective of the overseas markets, thus 2,000 maintaining or lowering the effective cost of the

1,000 products in many other countries. The Society con­ tinues to concentrate its marketing efforts on work- o 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 ing with consortia, where costs can be spread over a larger number of institutions. This has the effect of providing the MR product line to a much wider au­ dience than could afford it as individual institutions, as II. Review of 2003 Operations well as protecting the current revenue stream for future As indicated in the top graph on the prior page, the past years. MR is currently financially healthy; however, it is three years have been very good years financially for the probably unrealistic to expect significant increases in sales Society, apart from investment losses incurred in two of from additional subscribers. these years. Journals. Journal revenues are doing well, with perhaps Although the Society experienced investment losses slight improvements in the last three years, as attrition of from 2000-02, a significant portion of those losses was subscribers has been less than expected. A portion of the recouped in 2003. Further, in spite of these losses, long­ 2003 increase in revenues is due to the publication of terminvestments have generated high returns over a long both the 2002 and 2003 volumes of Trudy Moscow. If rev­ period (an average annual return of 9.2% over the last ten enues had been recorded in the years expected, the jour­ years), and that income has helped the endowment funds nal revenue increase in 2003 would have been reduced to (and the income they produce) to keep pace with inflation. approximately $100,000. However, in 2003 a subscription In 2002 and 2003 the Board of Trustees appropriated agent defaulted on payments to publishers and subse­ investment income from those endowment funds with in­ quently declared bankruptcy. The Society gave gratis sub­ come whose use is unrestricted and from a portion of the scriptions to the affected institutions for 2003. Therefore, Economic Stabilization Fund to support operations. The the fact that the Society experienced any increase in jour­ amount of such appropriations included in operating rev­ nal revenue in,2003 demonstrates the strength of the So­ enue is $760,811 in 2002 and $865,696 in 2003. ciety's journal program. When reflecting on years with good operating results, There continue to be financial pressures on libraries it is instructive to review the Society's record for a some­ everywhere in the world. The decline in the value of the what longer period. The bottom chart on the previous dollar compared to many other currencies has helped the

AUGUST 2004 NOTICES OF THE AMS 825 From the AMS Secretary

Major Expense Categories 2001 2002 2003 Personnel Costs $12,801 65% $12,945 64% $13,388 67% Building and equipment related 1,541 8% 1,436 7% 1,387 7% Postage 838 4% 844 4% 815 4% Outside printing and binding 817 4% 848 4% 691 3% Travel, staff and volunteers 462 3% 488 3% 778 5% All other expenses 3,213 16% 3,602 18% 2,887 14% TOTAL $19,672 100% $20,163 100% $19,946 100%

Society's retention efforts with respect to non-U.S. sub­ and Board-designated (quasi-endowment) fund, including scribers. Hopefully the domestic economy will continue to any reinvested earnings. recover, which should help these retention efforts. The Society's fiscal year coincides with the period cov­ The drop in 1996 resulted from decisions made by ered by subscriptions and dues. Since dues and subscrip­ those in control of four Russian journals (Izvestiya, Sbornik, tions are generally received in advance, the Society re­ Steklov, and Doklady) to use sources other than the AMS ports a large balance of cash and short-term investments for translation into English and distribution of the re­ on its financial statements at year-end. This amounted to sulting translation journals. approximately $15,893,000 and $13,985,000 at December Books. Book revenues decreased in 2003 inhistorical and 31, 2003, and 2002, respectively. The recorded liability for constant dollars, due primarily to the shortfall in new ti­ the revenues received in advance was approximately tles as compared to prior years. There continues to be an $10,797,000 and $11,155,000 at December 31, 2003, and overall sluggish market in scholarly book sales worldwide, 2002, respectively. consistent with the overall economic conditions. The So­ The Society's property and equipment include land, ciety continues to work with distributors and has re­ buildings and improvements, office furniture and equip­ vamped marketing efforts in order to keep the book pro­ ment, and software. The Society also owns a small amount gram as healthy as possible in a difficult market. of transportation equipment. The land, buildings, and im­ Dues. Dues, the sum of individual and institutional, provements include the Society's Rhode Island headquar­ have shown a slight upward slope on the historical dol­ ters, with buildings in Providence and Pawtucket, and the lars chart and a flat or slightly decreasing line in constant Mathematical Reviews offices inAnnArbor. The largest part dollars. A flat constant.dollar line is expected for institu­ of the Society's office equipment is its investment in com­ tional dues, as the number of members varies little from year to year and the dues rates have been set so that dues puter facilities. will increase at about the same level as inflation. There has The Society's endowment is managed under the "total been a slight decline in individual dues from their high in return concept". Under this management policy, income 1998. in excess of a reasonable amount (set by the Board of Trustees) is reinvested and increases the value of the fund. Major Expense Categories This allows for growth in income over time. As discussed The table above shows the major expense for 2001,2002, and 2003 in thousands of dollars. In terms ofhow expense previously, in 2002 the Board ofTrustees appropriated in­ dollars are allocated, there is not much change from year vestment income from the true endowment funds whose to year. use of income is unrestricted and from a portion of the Economic Stabilization Fund to support operations. The III. Assets and Liabilities amount of such appropriations included in operating rev­ So far, this report has dealt with revenues and expendi­ enue is $865,696 and $760,811 in 2003 and 2002, re­ tures that affect unrestricted net assets. Another aspect spectively. of the Society's finances is what it owns and owes, or its assets and liabilities, which are reported in the Balance IV. Summary Financial Information Sheets. As discussed previously, the Society's net assets The following are summaries of the audited annual fi­ and activities that increase or decrease net assets are clas­ nancial statements of the Society. Each year the Audit sified as unrestricted, temporarily restricted, or perma­ Committee of the Board of Trustees meets with the Soci­ nently restricted. A majority of the assets and liabilities ety's auditors to review the conduct of the audit, the So­ detailed on the accompanying Balance Sheets constitute ciety's financial statements, and the auditors' report on the the unrestricted net assets. The permanently restricted net financial statements. Pursuant to the recommendation of assets are supported by investments in the long-term in­ the Audit Committee, the Board of Trustees has accepted vestment portfolio, and the temporarily reStricted net the audited financial statements. A copy of the Society's assets are supportedby investments in the long-term and audited financial statements, as submitted to the trustees short-term investment portfolios. The Market Value of In­ and the Council, will be sent from the Providence office vested Funds shows the market value of each endowment to any member who requests it from the treasurer. The

826 NOTICES OF THE AMS VOLUME 51, NUMBER 7 From the AMS Secretary treasurer will be happy to answer any questions members Investment earnings available may have regarding the financial affairs of the Society. for spending 790,700 649,500 Meetings 944,433 872,541 -Respectfully submitted, Total membership and professional services revenue 5,849,044 5,997,357

John M. Franks Short-term investment income 452,613 262,141 Treasurer Other 154,919 94,434

BALANCE SHEETS Total operating revenue $22,409,547 $22,094,586 December 37, 2003 and 2002

Assets 2003 2002 Operating Expenses

Cash and cash equivalents $ 678,795 $ 447,334 Publication: Short-term investments 13,537,923 13,537,923 Mathematical Reviews and Receivables, less allowances 1,223,912 1,135,479 related activities $ 5,488,300 $ 5,374,976 Deferred prepublication costs 686,279 614,291 Journals (excluding MR) 1,267,824 1,038,486 Com pleted books 1,165,507 1,285,692 Books 2,480,675 2,512,238 Prepaid expenses and Divisional indirect 689,493 1,083,229 deposits 1,044,717 1,041,481 Warehousing and distribution 704,464 689,277 Land, bldgs., and equipment, less Marketing and sales 104,653 188,187 accumulated depreciation 4,316,071 4,466,363 Sale of services 224,353 252,722 Long-term investments 47,292,301 38,282,201 Total publication expense 10,959,762 11,241,636 Total assets $ 71,621,398 $60,81 0,764 Membership and professional services: Dues, services, and outreach 2,851,239 2,772,368 Liabilities and Net Assets Grants, prizes, and awards 844,852 1,029,662 Liabilities: Meetings 856,032 803,132 Accounts payable $ 1,271,481 $ 1,181,860 Governance 464,816 384,256 Accrued expenses 2,623,354 2,683,431 Divisional indirect 438,360 51 5,949 Deferred revenue 10,796,619 11,153,592 Total membership and professional Postretirement benefit services expense 5,455,299 5,505,367 obligation 3,108,747 2,748,747 Other 64,965 71,075 Total liabilities 17,800,201 17,767,630 Member and customer services 759,530 733,987 Net assets: General and administrative 2,868,969 2,610,586 Unrestricted 49,074,025 38,991,709 Temporarily restricted 1,591,000 1,361,037 Total operating expenses 20,108,525 20,162,651 Permanently restricted 3,156,172 2,690,388 Excess of operating revenue Total net assets 53,821,197 43,043,134 over operating expenses 2,301,022 1,931 ,935 Total liabilities and Long-term investment return (loss) net assets $71,621,398 $60,810,764 in excess of investment earnings available for spending 7,781,294 (6,247,209) Change in unrestricted net assets 10,082,316 (4,315,274) Changes in temporarily STATEMENTS OF ACTIVITIES restricted net assets: Years Ended December 37, 2003 and 2002 Contributions and grants 86,158 58,069 Long-term investment Changes in unrestricted net assets: income (loss) 439,822 (l09,647) Net assets released from Operating Revenue 2003 2002 restrictions (296,017) (373,01 5) Publication: Change in temporarily Mathematical Reviews and restricted net assets (229,963) (424,593) related activities $ 8,658,388 $ 8,361,089 Change in permanently Journals (excluding MR) 4,043,300 3,891,416 restricted net assets- Books 2,797,201 2,936,959 Contributions 465,784 337,198 Sale of services 312,760 420,552 Other 141,322 130,638 Change in net assets 10,778,063 (4,402,669) Total publication revenue 15,952,971 15,740,654 Net assets, beginning of year 43,043,134 47,445,803

Membership and professional services: Net assets, end of year $ 53,821,197 $43,043,134 Dues, services, and outreach 3,323,900 3,514,799 Grants, prizes, and awards 790,011 960,517

AUGUST 2004 NOTICES OF T"~E AMS 827 From the AMS Secretary

MARKET VALUE OF INVESTED FUNDS December 31 2003 2002 2001

Endowment Funds:

Prize Funds: Steele $547,113 $ 466,213 $ 570,297 Birkhoff 33,098 28,204 34,500 Veblen 11,178 9,525 11,651 Wiener 11,178 9,525 11,651 Bacher 8,129 6,927 8,474 Conant 36,488 31,092 38,034 Cole 19,196 16,358 18,787 Satter 29,059 24,763 30,291 Morgan 39,707 33,836 41,390 Albert Whiteman 37,438 35,438 30,438 Arnold Ross Lectures 50,000 50,000 50,000 Trjitzinksky 439,892 374,719 458,254 C. V. Newsom 204,702 174,433 213,376 Centennial 104,958 89,438 107,570 Menger 10,272 9,250 10,602 Ky Fan (China) 366,757 366,757 366,757 Epsilon 550,335 448,808 340,350 Total (Income Restricted) 2,509,850 2,175,286 2,342,422

Endowment 661,856 563,358 679,815 Morita 118,107 100,000 109,011 Henderson 3,547,938 3,019,927 3,644,206 Schoenfeld/Mitchell 665,628 221,139 Laha 224,498 189,309 189,309 Ritt 211,380 179,923 217,116 Moore 19,925 16,960 20,466 Total (I ncome 5,449,332 4,290,616 4,859,923 Unrestricted)

Total Endowment Funds 7,959,182 6,465,902 7,202,345

Quasi-Endowment Funds:

Friends of Math 123,572 123,572 123,572 Russian Royalties 17,829 17,829 17,829 Jou rnal Archive Fu nd 334,714 237,078 225,750 Economic Stabilization Fund 37,476,366 30,815,123 32,493,104 You ng Scholars 485,162 391,485 450,787 Charitable Gift Annuities 42,854 Total Quasi-Endowment Funds 38,437,643 31,585,087 33,353,896

828 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics Calendar

The most comprehensive andup-to-date Mathematics Calendar information is available on e-MATH at http://www.ams.org/mathcal/.

August 2004 in Honour of Professor Andrzej Granas on the Occasion of His 75th Birthday, Centre de Recherches Mathematiques, Universite ;', 3-1 3 Frontiers of Mathematical Physics: Summer School on de Montreal, Montreal, Quebec, Canada. Strings, Gravity & Cosmology, PIMS at the University of British Themes: Fixed point theory and its applications to problems Columbia, Vancouver, British Columbia, Canada. arising in nonlinear analysis, differential equations, and dynamical Speakers and Titles: IanAffleck (Univ. ofBritishColumbia), Confor­ systems. m~l Field Theory; Robert Brandenburger (Brown Univ.), Cosmology; Clifford Johnson (Univ. Southern California and Durham Univ.), Organizer and Information: Marlene Frigon, email: frigonString Theory?; Washington Taylor (MIT), Nonpertur­ important and versatile tool coming from representation theory bative Formulations of String Theory; Rob Leigh (Univ. Illinois at whichhas found numerous applications, from explicit construction Urbana-Champaign), SUSY (to be confirmed). ofexpandergraphsthroughtheuniquenessofinvariantmeanstothe Contact: email: fmp

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

AUGUST 2004 NOTICES OF THE AMS 829 Mathematics Calendar

September 2004 Organizers: Robert Gulliver, Nai-Chung Leung, Tian-Jun Li, and Jiaping Wang. -I( 7-1 0 International Conference on PDE Methods in Applied Fi nancial Su pport: We have funding to defray workshop expenses Mathematics and Image Processing, Sunny Beach, Bulgaria. for a number of participants, with highest preference given to Purpose: To bring together people interested in partial differential younger scientists (grad students, postdocs, young faculty at most equations and their applications, with a special emphasis on their five years after Ph.D.), although all active people are eligible. Women novel applications in image processing. and minorities are especially encouraged to apply. Apply to the Organizers: Ognyan Kounchev, email: [email protected]. organizers at email: yamabe©math. umn. edu. http://www.math.bas.bg/-kounchev; Svetozar Margenov, email: Deadlines: July 15, 2004, for full consideration for funding and [email protected], http://parallel.bas.bg/ for guaranteed hotel rooms. -margenov. Information: http://www .math. umn. edu/yamabe/. Deadline: For submissions of abstracts is July 31,2004. Information: http://www.math.bas.bg/-kounchevI ~.( 18-20 Workshop on Harmonic Analysis and Number Theory, 2004_conference/HomePage2004.html. University of Exeter, Exeter, United Kingdom. Organizers: Nigel Byott and Anton Deitmar. ~.( 7-11 International Workshop on Analysis and Its Application, Speakers: D. Bump, K. Buzzard, S. deBacker, M. Harris, G. Henniart, Mersin University, Mersin, Turkey. W. Hoffmann, R. Langlands, C. Moeglin, W. Mueller, F. Murnaghan. Workshop Fields: ApproximationTheory, ComplexAnalysis, Func­ Information: email: [email protected] . uk. tional Analysis, OrdinaryDifferential Equations, Partial Differential Equations, Theoretical Physics, Theory of Functions. ~.( 19-25 7th Volterra-CIRM International School Quantum Prob­ Invited Speakers: V. V. Andrievskii (Kent Univ., USA); H. P. Blatt ability and Spectral Analysis on Large Graphs, Grand Hotel (Kath. Univ. of Eichstatt, Germany); Z. Ditzian (Univ. of Alberta, Bellavista, Levico Terme, Trento, Italy. Canada); V. V. Goryainov (Volgograd Univ., Russia); N. Kerimov Scientific Organizers: L. Accardi (Roma II), A. Hora (Okayama), N. (Baku State Univ., Azerbaijan); K. Kopotun (Univ. of Manitoba, Obata (Tohoku). Canada); D. Leviatan (Tel-Aviv Univ., Israel); R. Petrishin (Chernivtsi Information: email: michelet©science. unitn . it; http://www . Nath. Univ., Ukraine); R. Rzayev (Baku Econ. Univ., Azerbaijan); A. science.unitn.it/cirm/. M. Samoilenko (Math. Inst., Ukraine); L A. Shevchuk (Kyiv Nath.

Univ., Ukraine); A. L Stepanets (Math. Inst., Ukraine). -I( 22 DIMACS Working Group on Reticulated Evolution, DIMACS Information: http://mathworkshop .mersin. edu. tr. Center, Rutgers University, Piscataway, New Jersey. Sponsor: DIMACS. ~.( 9-1 2 CabriWorld 2004: Third CabriGeometry International Con­ Organizers: Mel Janowitz, DIMACS, email: melj@dimacs . rutgers. ference, University of Rome "La Sapienza", Rome, Italy. edu; Randy Linder, University of Texas, email: rlinder@mail. Description: The conference is dedicated to the teaching and utexas . edu; Bernard Moret, University of New Mexico, email: learning ofgeometrythroughthe use ofdynamic geometrysoftware [email protected]. and is addressed to the needs of mathematics teachers at all levels, Short Description: Species evolution has long been modeled as researchers, mathematicians, and others interested in the use of a branching process that can uniquely be represented by a tree new technologies in the teaching of mathematics. topology. In such a topology, each species can only be linked to its Information: http://italia2004.cabriworld. com. closest ancestor, while interspecies relationships such as species hybridization or lateral gene transfer in bacteria are not allowed. ~.( 12-17 CR Geometry and Partial Differential Equations, Grand With the advent of phyloge'netic analysis at the molecular level, Hotel Bellavista, Levico Terme, Trento, Italy. there is increasing evidence that such a model is inadequate. This Scientific Organizers: Marco Peloso (Polit. Torino), Dmitri Zaitsev workshop will explore the history and latest status of these new (Trinity ColI. Dublin), Giuseppe Zampieri (Univ. Padova). models of "reticulate evolution" and will be coupled with a smaller Information: email: michelet@science. unitn. it; http://www. working group meeting designed to explore promising avenues for science.unitn.it/cirm/; http://www.science.unitn.it/ future research. cirm/listCRGeometry.html. Deadlines: Main speakers are by invitation only. Workshop partic­ ipants may submit papers by contacting one of the organizers no ~.( 16-18 ADG 2004 (5th International Workshop on Automated later than August 1, 2004. Deduction in Geometry), University ofFlorida, Gainesville, Florida. Information: http://dimacs . rutgers. edu/Workshopsl Information: http://www .math. ufl. edu/-whitelADG2004. html. Reticulated/.

~.( 16-1 8 Fluid Mechanics: A Workshop in Honor of Amable Liiian, -I( 27-0ctober 1 Recent Advances in Complex and Real Geometry, Fundacion Euroarabe, Granada, Spain. Grand Hotel Bellavista, Levico Terme, Trento, Italy. Topics: Fluid mechanics, combustion, turbulence, electrohydrody­ Scientific Organizers: V. Ancona (Firenze), P. de Bartolomeis namics, applied mathematics. (Firenze), and A. Silva (Roma I). Invited Speakers and Participants: A. Barrero, A. Bermudez de Information: email: michelet@science. unitn . it; http://www . Castro, L. L. Bonilla, P. Clavin, A. Crespo, L Diaz, J. Dold, C. Dopazo, science.unitn.it/cirm/. ]. L. Fernandez de la Mora, A. C. Fernandez Pello, P. L. Garcia Ybarra, M. A. Herrero, J. Jimenez, G. Joulin, J. C. Lasheras, M. Martinez ~.( 28-29 DIMACS Workshop on Applications of Order Theory Sanchez, P. Moin, N. Peters, E. Sanchez Palencia, J. R. Sanmartin, G. to Homeland Defense and Computer Security, DIMACS Center, Sivashinsky, C. Trevino, M. G. Velarde, F. A. Williams. Rutgers University, Piscataway, New Jersey. Information: http://www. ugr. es/local/kinetic/amable; email: Short Description: The importance of the problem of terrorism [email protected]. . canhardlybe overstated. Since the Second World War it has become clear that mathematics has an important role to play in securing ~.( 17-19Yamabe Symposium on "Geometry and Physics", Univer­ victory in any global conflict. This workshop will draw renowned sity of Minnesota, Minneapolis, Minnesota. international researchers who bring two important fields to bear Invited Speakers: Robert Bryant, Duke Univ.; Kefeng Liu, U.C.L.A.; in the current war on terror: Order Theory and Reflexive Theory. Duong Phong, Columbia Univ.; Yongbin Ruan, Univ. of Wisconsin; Organizers:JonathanFarley,MIT; AnthonyA. Harkin, HarvardUniv., Isadore M. Singer, M.LT.; Karen Uhlenbeck, Univ. of Texas; Shing­ email: harkin©deas . harvard. edu; Mel Janowitz, DIMACSjRutgers Tung Yau, Harvard Univ. Univ., email: melj @dimacs . rutgers. edu; Hector Rosario, Univ.

830 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics Calendar

of Puerto Rico, email: hrosario©math. uprm. edu; Stefan Schmidt, Goal: The main goal of the conference is mathematical hydrody­ Physical Science Lab., email: schmidt©psl. nmsu. edu. namics in a broad context. The scope of the conference includes local Arrangements: Maria Mercado, DIMACS Center, email: but is not limited to the following topics. [email protected], 732-445-5928. Topics: Mathematical theory of fluid dynamics: solvability and Information: http://dimacs . rutgers. edu/Workshops/Defensel uniqueness; Analytical dynamics and geometric-differential meth­ ods inhydrodynamics; Stability offlows for ideal andviscous fluids; '~29-30 CLIMA V: Fifth International Workshop on Computational Convective flows; Asymptotics in hydrodynamics, Vibrodynamics, logic in Multi-agent Systems, Lisbon, Portugal. Parametric resonance, Boundary layers; Spectral theory in the Purpose: To discuss techniques, based on computational logic, stability problems of hydrodynamics; Qualitative and numerical for representing, programming, and reasoning about multi-agent methods; Bifurcation analysis, Transitions, Systems with symmetry systems in a formal way. and cosymmetry; Computer experiment methods and results. Call for Papers: We solicit unpublished papers that address formal Organization Committee: Rostov State Univ.: V. N. Govorukhin, approaches to multi-agent systems. The approaches as well as L. G. Kurakin, A. B. Morgulis, M. V. Norkin, V. G. Tsybulin, O. A. being formal must make a significant contribution to the practice Tsyvenkova. of multi-agent systems. Information: Secretary: S. V. Revina; Rostov State University, Organizers: ]. Leite, New Univ. of Lisbon, Portugal (jleite©di . Mechanical Mathematical Department, Zorge St., 5, Rostov-on-Don, fct. unl. pt); P. Torroni, Univ. of Bologna, Italy (ptorroni©deis. 344090, Russia; email: kvm©math.rsu.ru; phone: (8632)-221312; unibo . it). Please send program suggestions and inquiries to either http://kvm.math.rsu.ru/conf2004/ENGLISH/. of the organizers. Submission Instructions: We welcome and encourage the submis­ ~'~ 18-20 DIMACSjDIMATIAjRenyi Working Group on Extremal sion of high-quality, original papers which are not simultaneously Combinatorics II, DIMACS Center, Rutgers University, Piscataway, submitted for publication elsewhere. Please refer to the work­ New Jersey. shop webpages for further instructions concerning the submission Short Description: This meeting will continue the work of the procedures. working group on two general topics: extremal graph theory and Important Dates: Submission of Abstracts: June 20, 2004. Sub­ extremal problems arising from combinatorial search and testing. mission of Papers: June 25, 2004. Notification of Acceptance: Participation: The working group will be by invitation only. If you July 30, 2004. Final version due: September 6, 2004. CLIMA IV: are interested in participating, please contact the organizers. September 29-30, 2004. Sponsors: DIMACSjDIMATIAjRenyi. Information: http://centria.di.fct.unl. pt/~j leitelclimaVI Organizers: Janos Komlos, Rutgers Univ., komlos©math. rutgers. index.htm. edu; Endre Szemeredi, Rutgers Univ., szemered@cs. rutgers. edu. {~30 DIMACS Working Group on Applications of Order Theory local Arrangements: Maria Mercado, DIMACS Center, mercado© to Homeland Defense and Computer Security, DIMACS Center, dimacs. rutgers. edu, 732-445-5928. Rutgers University, Piscataway, New Jersey. Information: http://dimacs.rutgers . edu/Workshopsl Short Description: The importance of the problem of terrorism Extrema12/. canhardlybe overstated. Since the Second World War ithas become clear that mathematics has an important role to play in securing * 22-23 Twenty-Fourth Annual Southeastern-Atlantic Regional victory in any global conflict. This workshop will draw renowned Conference on Differential Equations, University of Tennessee at international researchers who bring two important fields to bear Chattanooga, Chattanooga, Tennessee. in the current war on terror: Order Theory and Reflexive Theory. Principal Speakers: Ravi P. Agarwal (Florida Institute of Tech­ Organizers:JonathanFarley, MIT; AnthonyA.Harkin, HarvardUniv., nology), Recent Trends in Singular Boundary Value Problems for email: harkin©deas .harvard. edu; Mel Janowitz, DIMACSjRutgers Ordinary Differential Equations; Johnny Henderson (Baylor Univ., Univ., email: melj ©dimacs . rutgers. edu; Hector Rosario, Univ. Texas), Topological Transversality and Boundary Value Problems of Puerto Rico, email: hrosario©math. uprm. edu; Stefan Schmidt, on Time Scales; Peter Kuchment (Texas A&M Univ.), Differential Physical Science Lab., email: schmidt©psl. nmsu. edu. Operators on Graphs and Their Applications; David R. Russell (Vir­ Local Arrangements: Maria Mercado, DIMACS Center, email: ginia Tech), Spline Interpolation and Approximation as a Discrete mercado©dimacs.rutgers.edu, 732-445-5928. Dynamical System. Participation: This working group is by invitation only. Invitations Information: In addition to the principal speakers, there will also may be obtained by contacting Mel Janowitz, email: melj©dimacs . be sessions of twenty-minute contributed talks. Pending funding rutgers. from the National Science Foundation, travel support funds will be edu or Hector Rosario, email: hrosario©math. uprm. edu. available for advanced graduate students and recent Ph.D. recipi­ Information:http://dimacs.rutgers . edu/Workshops/DefenseWGI ents. Women and minority participants are especially encouraged to participate in this conference and to apply for support. Please go to the conference website, http://www . utc . edulAcademicl Mathematicslsearcde-041index. html, to get instructions on reg­ October 2004 istration, lodging, submission of abstracts, and application for -)~ 2-6 Workshop on Algebraic K-Theory 2004, Centre de Recherches support; or send email to Boris-Belinskiy©utc. edu; phone: 423­ Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. 425-4748. Organizers: Eric Friedlander, Dan Grayson, Rick Jardine, Manfred Kolster. ~'~ 24-30 Partial Differential Equations in Mathematical Physics, in Memory of Olga A. Ladyzhenskaya, Grand Hotel Bellavista, Topics: The topics covered at this meeting include the most recent Levico Terme, Trento, Italy. developments in algebraic K-theory and the closely allied areas of motivic homotopytheory, algebraic cycles, andmotivic cohomology Scientific Committee: H. Beirao da Veiga (Pisa), G. Seregin (St. theory, along with applications in other areas of mathematics. Petersburg), V. Solonnikov (Ferrara), N. Uraltseva (St. Petersburg), A. Valli (Trento). -)~ 4-8 Mathematical .Hydrodynamics: Models and Methods, Dedi­ Information: email: michelet©science. unitn . it; http://www . cated to the 70th Anniversary of Professor V. Yudovich, Rostov science.unitn.it/cirm/;http://www.science.unitn.it/cirml State University, Rostov-on-Don, Russia. Ladylecture.html.

AUGUST 2004 NOTICES OF THE AMS 831 Mathematics Calendar

November 2004 Institute Membership: Membership application for visiting the institute under the program is also available from the website -1~ 8-1 0 Models of Financial Market Microstructure, MIT, Cambridge, above. Members of the Institute do not need to register for specific Massachusetts. activities. Program: This is a special session at the 2nd lASTED International Contacts: For general enquiries, please email ims

832 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics Calendar

www.hicstatistics.org/cfp\_stats05.htm. effects due to the media or the advective flow randomness. Sponsors: American Statistical Association, Hawaii Chapter; East This workshop will be an opportunity for interaction between West Council for Education; Center of Asian Pacific Studies of mathematicians at the forefront of this area and scientists involved Peking Univ. in the design of models and numerical methods for various information: email: statistics@hicstatistics. org; http: // applications, in particular, turbulent combustion. www.hicstatistics.org/index05.htm. March 2005 ~'12-14 Second Joint IMSjlSBA International Conference, Bormio, Italy (Italian Alps). -,', 2-5 Representing Unresolved Degrees of Freedom for the Program: A central theme of the conference will be Markov chain Atmosphere and Ocean, Centre de Recherches Mathematiques, Monte Carlo (MCMC) andrelatedmethods and applications inthe 15 Universite de Montreal, Montreal, Quebec, Canada. years since the publication of Gelfand and Smith (1990, JASA), the Organizer: A. J. Majda (Courant). paper that introduced these methods to mainstream statisticians. Description: A central problem in attempts to understand and The conference will also feature 3 plenary speakers and 6 invited predict the evolution of atmospheric or oceanic flows is how best sessions from internationally known experts covering a broad to represent the unresolved scales in these flows. In the jargon of array of current and developing statistical practice: molecular biol­ dynamic meteorology or physical oceanography, this is called the ogy, spatial and spatiotemporalmethods, bio-informatics/genetics, parameterization problem, while in the jargon of turbulence it is MCMC algorithms/software, statistical data mining, modern non­ called the closure problem. The most pertinent areas of analysis parametrics. and applied mathematics are homogenization theory, probability, Information: http://www . eco.uninsubria. i t/IMS-ISBA-05/. and nonlinear stochastic PDEs. The purpose of this workshop is to explore two complementary issues that arise in the context

-J, 24-July 22 Developments in Quantitative Finance, Isaac Newton of the parameterization problem: (1) the extent to which modern Institute for Mathematical Sciences, Cambridge, UK. techniques in applied mathematics can be brought to bear on Description: The field of mathematical finance is comparatively its formulation and partial solution, and (2) the extent to which young, and the modern theory can be traced back to the Black­ problems in the representation of atmospheric and oceanic flows Scholes-Merton solutionofthe problemofhowto price a call option, create fertile new areas of mathematical inquiry. a financial security whose payoff is contingent on the behaviour of an underlying asset. Over the past three decades the explosive -,', 24-27 Geometric Representation Theory, University of Arizona, growth in trading of financial derivatives has been reflected in Tucson, Arizona. a commensurate growth in the study of financial mathematics, Goal: To gather leading specialists in representation theory as well which in turn has helped to support the increasing sophistication as beginning researchers and advanced graduate students to create of financial markets. a forum where participants can exchange new ideas, communicate As a branch of mathematics, finance is extremely diverse, and recent advances, and assist younger participants in developing the subject has attracted the interest of, and generated research successful research strategies. Women and minority participants problems for, researchers from a broad spectrum of mathematical are especially encouraged to apply. disciplines. The theory is based on stochastic models, and there are Principal Speakers: S. Evens (Notre Dame), D. Gaitsgory (Chicago), obvious applications from statistical analysis, but there have also V. Ginzburg (Chicago, TBC), S. Kumar (North Carolina), J. Millson been significant contributions from functional and convex analysis. (Maryland), 1. Mirkovic (Amherst), K. Vilonen (Northwestern), D. There are also strong connections with numerical analysis and Vogan (MIT). computational methods, not least because many of the equations Organizing Committee: P. Bressler, P. Foth, and K. Joshi (all from which arise have long been studied by applied mathematicians. Arizona). The healthy development of the subject also needs input from Information: http://math. arizona. edu/-foth/grt.html. economists and industry professionals. The major themes ofthis programme are assetpricemodelling April 2005 and inference for financial models, market imperfections and derivative pricing in incomplete markets, insurance applications -,', 6-1 0 Extracting Macroscopic Information from Molecular Dy­ andthemodellingandquantificationofcreditevents, computational namics, Centre de Recherches Mathematiques, Universite de finance, and financial economics and agent interactions. The aim Montreal, Montreal, Quebec, Canada. is that researchers from all related disciplines-from economics, Organizers: P. F. Tupper (McGill), A. M. Stuart (Warwick). physics, and finance, as well as pure and applied'mathematics and Description: Models used in molecular dynamics are high-dimen­ statistics-should meet and interact to share their knowledge and sional dynamical systems (or stochastic dynamical systems) with advance their understanding. multiple time-scales. A major challenge for computational math­ Organizers: D. Duffie (Stanford), D. Hobson (Bath), C. Rogers ematics is the extraction of accurate macroscopic information at (Cambridge), J. Scheinkman (Princeton). minimal cost. This workshop will concentrate on two topics: (1) the Information: Isaac Newton Institute for Mathematical Sciences, 20 analysis and development of standard time-stepping algorithms in Clarkson Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, the context of molecular dynamics, with the purpose of the indirect fax.: +44 1223 330508; email: info@newton. cam. ac. uk; http: // calculation of macroscopic information; and (2) the design of new www.newton.cam.ac.uk/programmes/DQF/. algorithms aimed at extracting macroscopic information directly.

-J, 26-30 Front Propagation and Nonlinear Stochastic PDEs for -:'18-July 13 Time at Work, Institut Hemi Poincare, Paris, France. Combustion and Other Applications, Centre de Recherches Description: A trimester on asymptotic properties of dynamical Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. systems courses, minicourses, lectures throughout the trimester; Organizers: A. Bourlioux (Montreal) and P. Souganidis (Texas). also a one-week workshop on each of the main topics (details Description: The development of efficient large-scale models for on http://www.math.jussieu.fr/-baladi/ihp.html): extended the numerical simulation of turbulent premixed flames requires a systems, Hamiltonian systems, SRB measures and their asymptotic good understanding of the mathematical principles governing the properties, dynamical zeta functions, and quantum chaos. dynamics of self-propagating fronts. One of the most challenging Organizers: V. Baladi, J. Bricmont, P. Collet, F. Ledrappier, and C. issues is the analysis of the complex interactions, at small scales, Liverani. between advection, reaction and diffusion, including stochastic Information: http://www.ihp.jussieu.fr.

AUGUST 2004 NOTICES OF THE AMS 833 Mathematics Calendar

;'; 27-May 1 Multiscale Modeling in Solids, Centre de Recherches market uncertainties, theoretical and numerical approximations to Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. pricing, hedging and portfolio optimization control problems, and Organizers: Weinan E (Princeton), E. Vanden-Eijnden (Courant). data estimation issues. The goal is to bring together researchers Description: This workshop will focus on energetic and kinetic in a variety of disciplines (mathematics, engineering, operations issues associatedwithdefects, cross-slip, grainboundarymigration, research, and economics, for example) to emphasize different and phase boundary dynamics in solids. The objective is to develop techniques and approaches. mathematical models for complex multiscale phenomena such as crystal plasticity, nucleation and reconstruction of stepped ;', 13-25 CIMPA Summer School AGAHF 200S-Arithmetic and surfaces, and the behaviour of nano-materials in general. Geometry around Hypergeometric Functions, Galatasaray Uni­ versity, Ortakoy, Istanbul, Turkey. May 2005 Objectives: The aim of the school is the presentation of hot topics in the field in a form accessible to research students, and * 11-1 5 Integrative Multiscale Modeling and Simulation in Mate­ revival of the interest in the field by highlighting possible new rials Science, Fluids and Environmental Science, Centre de Re­ research directions. There will be minicourses· on hypergeometric cherches Mathematiques, Universite de Montreal, Montreal, Quebec, differential equations and related topics, such as discrete groups in Canada. the automorphismgroups ofcomplexballs,ballquotients, orbifolds Organizer: T. Y. Hou (Caltech). and corresponding moduli problems of algebraic geometry. Description: Multiscale modeling and simulation have already Organizers: Ceyhan (MPIfM, Bonn), L. Chaumard (GSU, Istanbul), impacted many scientific and engineering disciplines. Numerous Ozgur Kisisel (METU, Ankara) A. M. Uludag (GSU, Istanbul), A. Ulus developments have been scattered in various disciplines, and (GSU, Istanbul). there is a great need to integrate isolated efforts. This workshop ScientificAdvisory Board: F. Hirzebruch, R.P. Holzapfel, M. Yoshida, will recapitulate previous activirties, focus on the interdisciplinary E. Looijenga, M. Jambu, L. D. Trang, P. Cohen, I. Dolgachev, S. Kondo. interaction among these related fields, and try to develop new Registration: October 2004-March 2005. tools that combine mathematical analysis, multiscale modeling, Information: Details will soon be available on the website of the and computational analysis in an integrative way. school, which is under preparation.

-k 14-1 5 Conference in Honor of Heydar Radjavi's 70th Birthday, ;'; 20-24 Second Conference on Self-Similarity and Applications, Hotel Golf, Bled, Slovenia. INSA Toulouse, Toulouse, France. Motivation: The conference will consist of invited and contributed talks related to Heydar Radjavi's work. Radjavi's many important Scientific Committee: M. Ledoux, A. Benassi, A. Estrade, P. Flandrin, contributions to linear algebra and to operator theory include ]. Istas, S. Jaffard, J. Levy-Vehel, M. Taqqu. his seminal characterization of self-commutators of operators on Information: http://www.lsp.ups-tlse.fr/Autosim05/indexa. Hilbert space and his definitive trace condition for simultaneous html. triangularizability of semigroups of matrices, whichwas the culmi­ nation ofwork on this topic by several generations of distinguished July 2005 algebraists. Heydar has obtained numerous other results of broad interest on invariant subspaces, simultaneous triangularizability, * 9-11 Joint Meeting of the Chinese Society of Probability and products of involutions, semigroups of matrices, and many other Statistics (CSPS) and the Institute of Mathematical Statistics topics. As he approaches 70, his research productivityis increasing OMS), Beijing, China. with his age. It is hoped that this conference will reflect the breadth Deadline: Submissions of contributed papers are invited to the and influence of his research. conference website until January 20, 2005. Deadline: Those interested in attending should register by Jan­ Information:http://math.bnu.edu. cn/statprob/CSPS-IMS2005, uary 15,2005. index.html. Organizers: M. Bresar, L. Grunenfelder, T. Kosir, M. Omladic, P. Rosenthal, P. Semrl. ;'; 10-23 Cornell Summer School in Probability, Cornell University, Invited Speaker: P. Rosenthal. Ithaca, New York. Invited Participants: E. A. Azoff (USA), R. Bhatia (India), P. Binding Primary Speakers: R. Durrett (Cornell), J.-F. Le Gall (DMA-Ecole (Canada), L. Grunenfelder (Canada), R. Guralnick (USA), D. Hadwin Normale Superieure de Paris), R. Lyons (Indiana). (USA), J. Holbrook (Canada)" T. J. Laffey (Ireland), C. K. Li (USA), Organizer: G. Lawler, email: lawler@math. cornell. edu. L. Livshits (USA), R. Loewy (Israel), V. Lomonosov (Canada), G. Information: http://www .math. cornell. edu/-Iawler/ sum2005 . MacDonald (Canada), B. Mathes (USA), M. Mathieu (Germany), R. html. Meshulam (Israel), V. Muller (Czech Republic), J. Okninski (Poland), M. Radjabalipour (Iran), H. Radjavi (Canada), L. Rodman (USA), B. A. Sethuraman (USA), V. Shulman (Russia), A. Sourour (Canada), Y. Turovskii (Azerbaijan), J. Zemanek (Poland). The following new announcements will not be repeated until the criteria in the next to the last paragraph at the bottom of Information: http://www.law05.si/hrc/. the first page of this section are met. Secretary ofthe Conference: DamjanaKokol Bukovsek, Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, August 2005 Slovenia; phone: +386-1-476-65-50, fax: +386-1-251-72-81, email: [email protected]. ok 1-December 23 Pattern Formation in large Domains, Isaac Newton Institute for Mathematical Sciences, Cambridge, England. June 2005 Organizers: ].H.P. Dawes (Cambridge), M. Golubitsky (Houston), P.C. Matthews (Nottingham), A.M. Rucklidge (Leeds). ;'; 1-5 Stochastic Modeling in Financial Mathematics (joint with Information: http://www.newton.cam.ac .uk/programmes/PFD/; SAMSI), CentredeRecherchesMathematiques,UniversitedeMontreal, Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Montreal, Quebec, Canada. Road, Cambridge, CB3 OER, U.K.; tel.: +44 1223 335999, fax.: +44 Organizers: R. Sircar (Princeton), J.-P. Fouque (North Carolina 1223 330508; email: [email protected] .uk. State).

Description: The theme of this workshop is emerging directions -I, 8-December 23 Global Problems in Mathematical RelativitY,Isaac in financial mathematics, with emphasis on stochastic modeling of Newton Institute for Mathematical Sciences, Cambridge, England.

834 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Mathematics Calendar

Description: General relativity has been around for a long time as Organizers: M.R. Evans (Edinburgh), S. Franz (ICTP, Trieste), C. a physical theory and an object of mathematical study. It was a Godreche (SPEC, Saclay), D. Mukamel (Weizmann Inst.). subject of intense interest in the 1960s and 1970s, when advances Information: http://www.newton.cam.ac . uk/programmes/PDS/; included the discovery of the Kerr solution, the study of black Isaac Newton Institute for Mathematical Sciences, 20 Clarkson holes and singularity theorems, and the introduction of asymp­ Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, fax.: +44 topia as a framework for studying asymptotic properties, including 1223 330508; email: [email protected] .uk. gravitational radiation. At the same time there were many math­ ematical problems that resisted mathematical analysis. In recent ?', 16-July 7 logic and Algorithms, Isaac Newton Institute for Math­ years there have been significant advances in our understanding of ematical Sciences, Cambridge, England. the topological, geometrical, and PDE aspects of general relativity; Description: Theoretical computer science is broadly divided into and progress is once again becoming rapid. New results are being disciplines dealing with logic, semantics and formal methods on obtained, and older results re-proved in greater generality. the one hand, and algorithmics and computational complexity on Themes: This programme will be structured around four themes: the other. The programme will focus on active areas of research Elliptic aspects of general relativity: new methods of solving that cut across this divide, dealing with algorithmic and complexity the constraint equations, developments from the solution of the aspects of logic as well as logical methods in complexity. Among Riemannian Penrose inequality, the study of static and stationary the areas of focus are computer-aided verification, specifically solutions including black holes. Hyperbolic aspects of general dealing with algorithms and structures for verifying properties of relativity: local and global evolution problems, Cosmic Censorship computing system and the logical, combinatorial, and algebraic conjecture, and the nature of singularities. Global Lorentzian methods deployed in their study. Finite Model Theory: This draws geometry: global techniques and asymptotic structure, splitting on logic and combinatorial methods to study the expressive power theorems and extendibility. New methods in general relativity: of logical languages in the finite. Along with connections with inverse scattering and boundary-value problems, scattering theory complexity, the programme will explore applications in database for linear field equations, newmethods from Riemannian geometry. theory, constraint satisfaction, proofcomplexity andprocess logics. While allfour themeswillbeworkedonthroughouttheprogramme­ Proof Complexity: At the interface of logic and complexity theory, and indeed it would be neither possible nor desirable to keep them the study of proof complexity, both in terms of lengths of proofs rigidly separate-there will be periods of more focus on each. and complexity of inference steps, provides powerful methods for The overall emphasis will be on mathematical results and global complexity lower bounds. Constraint Satisfaction: This describes properties of solutions of the Einstein equations, but it is worth a class of combinatorial search problems that arise in a wide noting that there is a clear motivation from physics to deepen our variety of areas of computer science and that have been the focus understanding of general relativity at a time when gravitational of sustained research, drawing on a rich variety of techniques from wave detectors around the world have started collecting data. algebra, logic, and graph theory. Organizers: P.T. Chrusciel (Tours), H. Friedrich (Golm), P. Tad Games: While two-player games are used as a tool in many of the (Oxford). areas mentioned above, an emerging theory combines games with automata and logic into a powerful tool for the analysis of sytems. Information: http://www.newton.cam.ac .uk/programmes/GMR/; Fundamental questions concern the algorithmic complexity of Isaac Newton Institute for Mathematical Sciences, 20 Clarkson determining a winner or constructing a winning strategy, given a Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, fax.: +44 game and a winning condition. 1223 330508; email: [email protected] . uk. Organizers: A. Dawar (Cambridge), M.Y. Vardi (Rice). Information: http://www.newton.cam.ac .uk/programmes/LAA/; January 2006 Isaac Newton Institute for Mathematical Sciences, 20 Clarkson -,', 9-June 30 Principles of the Dynamics of Non-Equilibrium Sys­ Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, fax.: +44 tems, Isaac NewtonInstitute for Mathematical Sciences, Cambridge, 1223 330508; email: [email protected] . uk. England. Description: The collective behaviour of nonequilibrium systems July 2006 is poorly understood compared to systems in thermal equilibrium, -,', 17-August 11 Spectral Theory and Partial Differential Equations, for which statistical mechanics provides a well-established theory. Isaac Newton Institute for Mathematical Sciences, Cambridge, By nonequilibrium systems we refer both to systems held far England. from thermal equilibrium by an external driving force and the Organizers: M. van den Berg (Bristol), B. Helffer (Orsay), A. Laptev complimentary situation of systems relaxing towards thermal (Stockholm), A.V. Sobolev (Sussex). equilibrium. Such systems display a broad range of phenomena, Information: http://www.newton.cam.ac .uk/programmes/STP/; such as phase transitions and slow collective dynamics, which Isaac Newton Institute for Mathematical Sciences, 20 Clarkson one would like to understand at a deeper level. The study of Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, fax.: +44 nonequilibrium systems has arisen in many different contexts, 1223 330508; email: [email protected] .uk. such as reaction-diffusion processes, interacting particle systems, drivendiffusive systems, andthe slowdynamicsofglassysystems.In ?', 24-December 22 Noncommutative Geometry, Isaac Newton In­ recent years progress has beenmade towards better understanding stitute for Mathematical Sciences, Cambridge, England. these systems. Mathematical tools have been developed, and some Organizers: A. Connes (IHES), S. Majid (Queen Mary), A. Schwarz exact results pertaining to specific systems have been derived. (UC Davis). These developments bring us closer to the point where one can Information: http://www.newton.cam.ac .uk/programmes/NeG/; address fundamental questions of generality, both of techniques Isaac Newton Institute for Mathematical Sciences, 20 Clarkson and results. It is anticipated that bringing together the different Road, Cambridge, CB3 OEH, U.K.; tel.: +44 1223 335999, fax.: +44 communities of physicists and mathematicians working in this 1223 330508; email: [email protected] .uk. diverse field will foster the emergence of new directions and outlooks. Focus: Driven diffusive systems ofinteracting particles; coarsening andpersistence; glassy, constraineddynamics and ageing. Although all three of these areas will be explored throughout the programme, it is intended that there will be periods of focus on each, centered around topical workshops.

AUGUST 2004 NOTICES OF THE AMS 835 New Publications Offered by the AMS

autonomy, and quantum groupoids; j. W. Duskin, Algebra and Algebraic R. W. Kieboom, and E. M. Vitale, Morphisms of 2-groupoids and low-dimensional cohomology of crossed modules; Geometry M. Gran, Applications of categorical Galois theory in universal algebra; C. Hermida, Fibrations for abstract multicategories; j. Huebschmann, Lie-Rinehart algebras, descent, and quantiza­ Galois Theory, tion; P. johnstone, A note on the semiabelian variety of Heyting semilattices; G. M. Kelly and S. Lack, Monoidal func­ Hopf Algebras, tors generated by adjunctions, with applications to transport and Semiabelian of structure; M. Khalkhali and B. Rangipour, On the cyclic homology of Hopf crossed products; G. Lukacs, On sequen­ Galois Theory, Categories tially h-complete groups; j. L. MacDonald, Embeddings of HopfAJgebras, and algebras; A. R. Magid, Universal covers and category theory in Semiabelian Categories George Janelidze, Razmadze polynomial and differential Galois theory; N. Martins-Ferreira, George Janelidze Bodo Pareigis Mathematical Institute of the Weak categories in additive 2-categories with kernels; T. Palm, Walter Tholen Editors Georgian Academy of Sciences, Dendrotopic sets; A. H. Roque, On factorization systems and .A/!A. ~ Tbilisi, Republic of Georgia, admissible Galois structures; P. Schauenburg, Hopf-Galois and Bodo Pareigis, University of bi-Galois extensions; j. D. H. Smith, Extension theory in Munich, Germany, and Walter Mal'tsev varieties; L. Sousa, On projective generators relative to coreflective classes; j. j. Xarez, The monotone-light factor­ Tholen, York University, ization for categories via preorders; j. J. Xarez, Separable Toronto, ON, Canada, Editors morphisms of categories via preordered sets; S. Yamagami, Frobenius algebras in tensor categories and bimodule exten­ This volume is based on talks given at the Workshop on Cate­ sions. gorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Insti­ Fields Institute Communications, Volume 43 tute for Research in Mathematical Sciences (Toronto, ON, August 2004, 570 pages, Hardcover, ISBN 0-8218-3290-5, Canada). The meeting brought together researchers working in LC 2004050271, 2000 Mathematics Subject Classification: these interrelated areas. 08Bxx, 12Hxx, 13Bxx, 14Lxx, 16Dxx, 17Bxx, 18-XX, 19Dxx, This collection of survey and research papers gives an up-to­ 22Axx, All AMS members $103, List $129, Order code FIC/43N date account of the many current connections among Galois theories, Hopf algebras, and semiabelian categories. The book features articles by leading researchers on a wide range of Analysis themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. In the Tradition of Articles are suitable for graduate students and researchers, specifically those interested in Galois theory and Hopf alge­ Ahlfors and Bers, III bras and their categorical unification. In the Tradition of William Abikoff and Contents: M. Barr, Algebraic cohomology: The early days; Ahlfors and Bers, III Andrew Haas, University of F. Borceux, A survey of semi-abelian categories; D. Bourn, William Abikoff Connecticut, Storrs, Editors Commutator theory in regular Mal'cev categories; D. Bourn Andrew Haas and M. Gran, Categorical aspects of modularity; R. Brown, Editors This proceedings volume reflects the Crossed complexes and homotopy groupoids as non commu­ 2001 Ahlfors-Bers Colloquium held at tative tools for higher dimensionallocal-to-global problems; the University of Connecticut (Storrs). M. Bunge, Galois groupoids and covering morphisms in topos This conference began nearly a half theory; S. Caenepeel, Galois corings from the descent theory century ago with a tradition based on point of view; B. Day and R. Street, Quantum categories, star profound mathematics, wide-ranging

836 NOTICES OF THE AMS VOLUME 51, NUMBER 7 New Publications Offered by the AMS interests, personal involvement, and scholarship. Once led by tant contributors are L. Caffarelli, M. Loss, and C. Villani. The Lipman Bers and Lars Ahlfors, the core of this tradition material is suitable for graduate students and research mathe­ unfolded around geometric function theory. maticians interested in methods of mass transport. Talks at the colloquium were devoted to various aspects of This item will also be of interest to those working in differential complex analysis, including Teichmuller spaces, quasicon­ equations. formal mappings, and geometric function theory. The book is Contents: J.-D. Benamou, Y. Bremer, and K. Guittet, Numer­ suitable for graduate students and researchers interested in ical analysis of a multi-phasic mass transport problem; complex analysis. Y. Brenier, Extension of the Monge-Kantorovich theory to clas­ Contents: W. Abikoff, C. j. Earle, and S. Mitra, Barycentric sical electrodynamics; L. A. Caffarelli, The Monge Ampere extensions of monotone maps of the circle; H. Akiyoshi, equation and optimal transportation; E. Carlen and M. Loss, H. Miyachi, and M. Sakuma, A refinement of McShane's iden­ Logarithmic Sobolev inequalities and spectral gaps; tity for quasifuchsian punctured torus groups; C. j. Bishop, D. Cordero-Erausquin, Non-smooth differential properties of An explicit constant for Sullivan's convex hull theorem; G. Ble, optimal transport; D. Cordero-Erausquin, W. Gangbo, and A. Douady, and C. Henriksen, Round annuli; M. Bonk, C. Houdre, Inequalities for generalized entropy and optimal j. Heinonen, and E. Saksman, The quasiconformal jacobian transportation; C. Villani, Trend to equilibrium for dissipative problem; M. Bridgeman, Random geodesics; R. D. Canary, equations, functional inequalities and mass transportation. Pushing the boundary; R. Chamanara, Affine automorphism Contemporary Mathematics, Volume 353 groups of surfaces of infinite type; G. Cui, F. P. Gardiner, and Y. jiang, Scaling functions for degree 2 circle endomorphisms; july 2004, 109 pages, Softcover, ISBN 0-8218-3278-6, E. de Faria, F. P. Gardiner, and W. j. Harvey, Thompson's LC 2004047695, 2000 Mathematics Subject Classification: group as a Teichmuller mapping class group; C. j. Earle, 35j60, 58j20, 49Q20, 35Q35, All AMS members $31, F. P. Gardiner, and N. Lakic, Asymptotic Teichmuller space, List $39, Order code CONM/353N part II: The metric structure; A. Eremenko, Geometric theory of meromorphic functions; H. M. Farkas and I. Kra, Identities CD5~ in the theory of theta constants; E. Fujikawa, Modular groups Computational ..oom acting on infinite dimensional Teichmuller spaces; D. M. Gallo, Some infinitesimal properties of the grafting map; Complexity Theory P. M. Gauthier and M. R. Pouryayevali, Failure of Landau's Steven Rudich, Carnegie theorem for quasiconformal mappings of the disc; j. Holt, Mellon University, Pittsburgh, Bumping and self-bumping of deformation spaces; j. Hu, PA, and Avi Wigderson, Earthquake measure and cross-ratio distortion; H. A. Masur and Y. N. Minsky, Quasiconvexity in the curve complex; Institute for Advanced Study, K. Matsuzaki, Indecomposable continua and the limit sets of Princeton, Nj, Editors Kleinian groups; I. Petrovic, A Teichmuller model for period doubling. Computational complexity theory is a major research area in mathematics Contemporary Mathematics, Volume 355 and computer science, the goal of july 2004, 351 pages, Softcover, ISBN 0-8218-3607-2, which is to set the formal mathematical foundations for effi­ LC 2004049691, 2000 Mathematics Subject Classification: cient computation. 30-06; 30C62, 30F30, 30F40, 30F60, 32G15, 57M50,AlIAMS There has been significant development in the nature and members $71, List $89, Order code CONM/355N scope of the field in the last thirty years. It has evolved to encompass a broad variety of computational tasks by a diverse set of computational models, such as randomized, Applications interactive, distributed, and parallel computations. These models can include many computers, which may behave coop­ eratively or adversarially. Recent Advances in Each summer the lAS/Park City Mathematics Institute Grad­ uate Summer School gathers some of the best researchers and the Theory and educators in the field to present diverse sets of lectures. This Recent Advances in Applications of Mass volume presents three weeks of lectures given at the Summer the Theory and School on Computational Complexity Theory. Topics are struc­ Applications of tured as follows: Mass Transport Transport M, C. Carvalho M. C. Carvalho, Georgia Week One: Complexity Theory: From Codel to Feynman. This J, F. Rodrigues section of the book gives a general introduction to the field, Editors Institute of Technology, with the main set of lectures describing basic models, tech­ Atlanta, and University of niques, results, and open problems. Lisbon, and J. F. Rodrigues, Week Two: Lower Bounds on Concrete Models. Topics University ofLisbon, Editors discussed in this section include communication and circuit This volume is the result of the Summer School on Mass complexity, arithmetic and algebraic complexity, and proof Transportation Methods in Kinetic Theory and Hydrodynamics complexity. held in Ponta Delgada (Azores, Portugal). It contains both Week Three: Randomness in Computation. Lectures are survey and research articles on methods of optimal mass devoted to different notions of pseudorandomness, interactive transport and applications in physics. Among the many impor-

AUGUST 2004 NOTICES OF THE AMS 837 New Publications Offered by the AMS proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). Logic and Foundations The volume is recommended for independent study and is suitable for graduate students and researchers interested in Ctl5!~syoom computational complexity. UmversIty The LECTURE This item will also be of interest to those working in logic and Series Stationary foundations. Tower Members of the Mathematical Association of America (MAA) and the The Stationary Tower National Council of Teachers of Mathematics (NCTM) receive a 20% Notes on a Course byW. Hugh Woodin Notes on a Course by discount from list price. w. Hugh Woodin Contents: Introduction; Week One. Complexity Theory: From Paul B. Larson G6del to Feynman: Steven Rudich, Complexity Theory: From Paul B. Larson, Miami G6del to Feynman: History and basic concepts; Resources, University, Oxford, OH reductions and P vs. NP; Probabilistic and quantum computa­ :\llIllillT1:\lllhtlll1lltIISIlIItI\ tion; Complexity classes; Space complexity and circuit The stationary tower is an important complexity; Oracles and the polynomial time hierarchy; Circuit method in modern set theory, invented lower bounds; "Natural" proofs of lower bounds; Bibliography; by Hugh Woodin in the 1980s. It is a means of constructing Avi Wigderson, Average Case Complexity: Average case generic elementary embeddings and can be applied to produce complexity; Bibliography; Sanjeev Arora, Exploring Complexity a variety of useful forcing effects. through Reductions: Introduction; PCP theorem and hardness Hugh Woodin is a leading figure in modern set theory, having of computing approximate solutions; Which problems have made many deep and lasting contributions to the field, in ~ strongly exponential complexity?; Toda's theorem: PH p#P; particular to descriptive set theory and large cardinals. This Bibliography; Ran Raz, Quantum Computation: Introduction; book is the first detailed treatment of his method of the Bipartite quantum systems; Quantum circuits and Shor's stationary tower that is generally accessible to graduate factoring algorithm; Bibliography; Week Two. Lower Bounds: students in mathematical logic. By giving complete proofs of Ran Raz, Circuit and Communication Complexity: Communica­ all the main theorems and discussing them in context, it is tion complexity; Lower bounds for probabilistic intended that the book will become the standard reference on communication complexity; Communication complexity and the stationary tower and its applications to descriptive set circuit depth; Lower bound for directed st-connectivity; Lower theory. bound for FORK (continued); Bibliography; Paul Beame, Proof Complexity: An introduction to proof complexity; Lower The first two chapters are taken from a graduate course bounds in proof complexity; Automatizability and interpola­ Woodin taught at Berkeley. The concluding theorem in the tion; The restriction method; Other research and open course was that large cardinals imply that all sets of reals in problems; Bibliography; Week Three. Randomness in Computa­ the smaliest model of set theory (without choice) containing tion: Preface; Oded Goldreich, Pseudorandomness-Part I: the reals are Lebesgue measurable. Additional sections include Preface; Computational indistinguishability; Pseudorandom a proof (using the stationary tower) of Woodin's theorem that, generators; Pseudorandom functions and concluding remarks; with large cardinals, the Continuum Hypothesis settles all Appendix; Bibliography; Luca Trevisan, questions of the same complexity as well as some of Woodin's Pseudorandomness-Part II: Introduction; Deterministic simula­ applications of the stationary tower to the studies of absolute­ tion of randomized algorithms; The Nisan-Wigderson ness and determinacy. generator; Analysis of the Nisan-Wigderson generator; The book is suitable for a graduate course that assumes some Randomness extractors; Bibliography; Saltl Vadhan, Proba­ familiarity with forcing, constructibility, and ultrapowers. It is bilistic Proof Systems-Part I: Interactive proofs; Zero-knowledge also recommended for researchers interested in logic, set proofs; Suggestions for further reading; Bibliography; Madhu theory, and forcing. Sudan, Probabiltstically Checkable Proofs: Introduction to PCPs; Contents: Elementary embeddings; The stationary tower; NP-hardness of PCS; A couple of digressions; Proof composi­ Applications; Appendix: Forcing prerequisites; Bibliography; tion and the PCP theorem; Bibliography. Index. lAS/Park City Mathematics Series, Volume 10 University Lecture Series, Volume 32 September 2004, 389 pages, Hardcover, ISBN 0-8218-2872-X, August 2004, 132 pages, Softcover, ISBN 0-8218-3604-8, 2000 Mathematics Subject Classification: 68Qxx; 03D15, All LC 2004047666, 2000 Mathematics Subject Classification: AMS members $55, List $69, Order code PCMS/10N 03E40, 03E15, 03E35, 03E55, 03E60, All AMS members $23, List $29, Order code ULECT/32N

838 NOTICES OF THE AMS VOLUME 51, NUMBER 7 New Publications Offered by the AMS

Mathematical Physics More Publications Available from the AMS Supersymmetry for Mathematicians: Algebra and Algebraic An Introduction Geometry Supersymmetry for Mathematicians: v. s. Varadarajan, University of An Introduction California, Los Angeles Seminaire Bourbaki Supersymmetry has been studied by theoretical physicists since the early VolulDe 2001/2002 1970s. Nowadays, because of its Exposes 894/908 novelty and significance- in both mathematics and physics - the issues As in the preceding volumes of this it raises attract the interest of mathematicians. seminar, one finds here fifteen survey lectures on topics of current interest: Written by the well-known mathematician, V. S. Varadarajan, three lectures on algebraic geometry, this book presents a cogent and self-contained exposition of two on dynamical systems, one on the foundations of supersymmetry for the mathematically­ actions of finite groups, one on combi­ minded reader. It begins with a brief introduction to the natorics and algebraic geometry, one physical foundations of the theory, in particular, to the classi­ on theoretical computer science, one fication of relativistic particles and their wave equations, such on p-adic monodromy, one on Iwasawa algebras, one on the as those of Dirac and Weyl. It then continues with the develop­ Kato conjecture, one on renormalization and Feynman ment of the theory of supermanifolds, stressing the analogy diagrams, one on dualities in string theory, one on the with the Grothendieck theory of schemes. Here, Varadarajan geometric Langlands correspondence, and one on "dessins develops all the super linear algebra needed for the book and d'enfants". establishes the basic theorems: differential and integral calculus in supermanifolds, Frobenius theorem, foundations of The book is suitable for graduate students and research math­ the theory of super Lie groups, and so on. A special feature is ematicians interested in recent progress in algebraic geometry, the in-depth treatment of the theory of spinors in all dimen­ number theory, differential equations, mathematical physics, sions and signatures, which is the basis of all supergeometry discrete mathematics, combinatorics, and applications to developments in both physics and mathematics, especially in computer science. quantum field theory and supergravity. This item will also be of interest to those working in number The material is suitable for graduate students and mathemati­ theory. cians interested in the mathematical theory of supersymmetry. A publication of the Societe Mathematique de France, Marseilles (SMF), The book is recommended for independent study. distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive Titles in this series are copublished with the Courant Institute of Math­ a 30% discount from list. ematical Sciences at New York University. Contents: Novembre 2001: A. Adem, Finite group actions on Contents: Introduction; The concept of a supermanifold; Super 'acyclic 2-complexes; B. Chazelle, The PCP theorem; J. Coates, linear algebra; Elementary theory of supermanifolds; Clifford Iwasawa algebras and arithmetic; P. Colmez, Les conjectures algebras, spin groups, and spin representations; Fine structure de monodromie p-adiques; C. Procesi, On the n!-conjecture; of spin modules; Superspacetimes and super Poincare groups. Mars 2002: D. Bennequin, Dualites de champs et de cordes; Courant Lecture Notes, Volume 11 L. Boutet de Monvel, Algebre de Hopf des diagrammes de Feynman, renormalisation et factorisation de Wiener-Hopf; August 2004, 300 pages, Softcover, ISBN 0-8218-3574-2, F. Loeser, Cobordisme des varietes algebriques; Y. Meyer, La LC 2004052349, 2000 Mathematics Subject Classification: conjecture de Kato; M. Rapoport, On the Newton stratification; 58A50, 58C50, 17B70, 22E99; 14A22, 14M30, 32C11, juin 2002: C. Bonatti, Dynamiques generiques: hyperbolicite et All AMS members $31, List $39, Order code CLN/11N transitivite; O. Debarre, Varietes rationnellement connexes; G. Laumon, Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands; J. Oesterle, Dessins d'enfants; R. Perez-Marco, KAM techniques in PDE. Asterisque, Number 290 April 2004, 234 pages, Softcover, ISBN 2-85629-149-X, 2000 Mathematics Subject Classification: 00B25, Individual member $74, List $82, Order code AST/290N

AUGUST 2004 NOTICES OF THE AMS 839 More Publications Available from the AMS

of almost complex structures; j. j. Kohn, Ideals of multipliers; Analysis G. Komatsu, The Bergman kernel of Hartogs domains and transformation laws for Sobolev-Bergman kernels; M. Kuran­ ishi, An approach to the Cartan geometry II: CR manifolds; Complex Analysis in L. Lempert, The aequation in N variables, as N varies; K. Matsumoto, Levi form of logarithmic distance to complex Several Variables submanifolds and its application to developability; Y. Miyaoka, Numerical characterisations of hyperquadrics; Mem.orial Conference of S. Mori, Meromorphic mappings and deficiencies; J. Noguchi, Kiyoshi Oka's Centennial Intersection multiplicities of holomorphic and algebraic curves Birthda}; Kyoto/Nara with divisors; T. Ohsawa, Generalization of a precise L2 divi­ sion theorem; M. Passare, Amoebas, convexity and the volume 2001 of integer polytopes; R. M. Range, On the decomposition of Kimio Miyajima, Kagoshima holomorphic functions by integrals and the local CR extension University, japan, theorem; o. Riemenschneider, The monodromy covering of the versal deformation of cyclic quotient surface singularities; Mikio Furushima, Kumamoto G. Schumacher, Moduli as algebraic spaces; S. Shimizu, University, japan, Hideaki Prolongation of holomorphic vector fields on a tube domain Kazama, Kyushu University, Fukuoka, japan, Akio and its applications; M. Shirosaki, Hypersurfaces and unique­ Kodama, Kanazawa University, Japan, Junjiro ness of holomorphic mappings; S. Takayama, Seshadri Noguchi, University of Tokyo, Japan, Takeo constants and a criterion for bigness of pseudo-effective line bundles; H. Tsuji, Subadjunction theorem; T. Ueda, Fixed Ohsawa, Nagoya University, japan, Hajime Tsuji, points of polynomial automorphisms of cn; K. Yamanoi, On Sophia University, Tokyo, japan, and Tetsuo Nevanlinna theory for holomorphic curves in abelian varieties; Ueda, Kyoto University, japan, Editors S.-T. Yau, Numerical characterization for affine varieties be a cone over nonsingular projective varieties; K.-I. Yoshikawa, This volume resulted from a conference held at Kyoto Univer­ Nikulin's K3 surfaces, adiabatic limit of equivariant analytic sity and Nara Women's University (Japan) in commemoration torsion, and the Borcherds -function. of the late Professor Kiyoshi aka, one of the most famous japanese mathematicians. Included are 34 research and survey Advanced Studies in Pure Mathematics, Volume 42 papers contributed by the invited lecturers and a letter written june 2004, 360 pages, Hardcover, ISBN 4-931469-27-2, for the occasion by H. Cartan. 2000 Mathematics Subject Classification: 00B20; 32Axx, 32Exx, Among the leading mathematicians who contributed to the 32Fxx, 32Hxx, 32Qxx, 32Sxx, 32Vxx, 32Wxx, 37Fxx, 14Cxx, volume are E. Bedford, S. Kobayashi, ]. j. Kohn, and M. Kuran­ 14jxx, 14Rxx, 58jxx, All AMS members $74, List $92, Order ishi. Topics discussed include pseudoconvex domains, a code ASPM/42N analysis (including L2 theory), the Bergman kernel, value distribution theory, hyperbolic manifolds, dynamical systems, infinite dimensional complex analysis, algebraic analysis, CR Number Theory structure, singularity theory, algebraic geometry, and others. The book is suitable for advanced graduate students and research mathematicians interested in complex analysis, alge­ Varietes de braic geometry, and complex geometry. Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, Shimura, espaces de and distributed worldwide, except in Japan, by the AMS. Rapoport-Zink et Contents: Part I: Photos of Kiyoshi aka; aka, Kiyoshi; Memo­ rial conference of Kiyoshi aka's centennial birthday on correspondances de complex analysis in several variables, Kyoto/Nara 2001; Langlands locales Message from Professor Henri Cartan; Part II: T. Nishino, Mathematics of Professor Oka-a landscape in his mind; Laurent Fargues, Universite Y. Aihara, Uniqueness problem for meromorphic mappings Paris/CNRS, Orsay, France, under conditions on the preimages of divisors; T. Akahori, On and Elena Mantovan, the middle dimension cohomology of Al singularity; T. Aoki, University of California, T. Kawai, and Y. Takei, The exact steepest descent method-a Berkeley new steepest descent method based on the exact WKB analysis; E. Bedford, Excursions of a complex analyst into the This volume contains two articles. Both deal with generaliza­ realm of dynamical systems; j. El Goul, Demailly's 2-jet nega­ tions of Michael Harris' and Richard Taylor's work on the tivity of certain hyperbolic fibrations; j. E. Forn

840 NOTICES OF THE AMS VOLUME 51, NUMBER 7 More Publications Available from the AMS p-divisible groups realizes some cases of local Langlands correspondences. For this the author establishes a formula linking the cohomology of those spaces to the one of the "supersingular" locus of a Shimura variety. Then he proves that the supercuspidal part of the cohomology of those vari­ eties is completely contained in the one of the "supersingular" locus. The second article links the cohomology of a Newton stratum of the Shimura variety, for example the "supersingular" stratum, to the cohomology of the attached local moduli space of p-divisible groups and to the cohomology of some global varieties in positive characteristic named 19usa varieties that generalize the classical Igusa curves attached to modular curves. The book is suitable for graduate students and research math­ ematicians interested in number theory and algebraic geometry. This item will also be of interest to those working in algebra and algebraic geometry. A publication of the Societe Mathematique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Contents: L. Fargues. Cohomologie des espaces de modules de groupes p -divisibles et correspondances de Langlands locales: Introduction; Varietes de Shimura de type P.E.L. non ramifiees; Espaces de Rapoport-Zink; Uniformisation des varietes de Shimura de type P.E.L.; Une suite spectrale de Hochschild-Serre pour l'uniformisation de Rapoport-Zink; Formule de Lefschetz sur la fibre speciale; Formule de Lefschetz sur la fibre generique; Contribution de la cohomologie de la strate basique; Application ala cohomologie des espaces Rapoport­ Zink de type E.L. et P.E.L.; Appendices; References; E. Mantovan. On Certain Unitary Shimura Varieties: Introduc­ tion; Preliminaries; Igusa varieties; A system of covers of the Newton polygon strata; Group action on cohomology; Formally lifting to characteristic zero; Shimura varieties with level structure at p; The cohomology of Shimura varieties; Refer­ ences. Asterisque, Number 291 April 2004, 331 pages, Softcover, ISBN 2-85629-150-3, 2000 Mathematics Subject Classification: 11G18, 11Fxx, 14G35, 14L05, 14G22, Individual member $86, List $95, Order code AST/291N

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To ensure full consideration, all mate­ Excellent recreational, entertainment and ALABAMA rials should be received by November 1, cultural facilities, modern shopping com­ 2004. Late applications will be reviewed plexes and professional sports organiza­ UNIVERSITY OF ALABAMA IN until the position is closed. Women and mi­ tions are available in Arlington and sur­ HUNTSVILLE norities are encouraged to apply. The Uni­ rounding areas. UTA offers baccalaureate Department ofMathematical Sciences versity of Alabama in Huntsville is an Af­ degrees in 90 fields, master's in 74, and firmative Action, Equal Opportunity doctorates in 34. More information onUTA The Department of Mathematical Sciences Institution. can be found at http://www. uta. edu/. at the University ofAlabama in Huntsville 000044 The College of Science (COS) has expe­ invites applications for a tenure track po­ rienced rapid growth in recent years reach­ sition at the rank of assistant professor, ing a fall 2003 enrollment in excess of beginning spring semester 2005 or fall se­ TEXAS 2,600. COS is home to the departments of mester 2005. In exceptional cases, more biology, chemistry and biochemistry, ge­ advanced appointments may be consid­ UNIVERSITY OF TEXAS AT ARLINGTON ology, mathematics, physics, and psy­ ered. A Ph.D. degree in mathematics or College ofScience chology. The college houses two newly es­ tablished centers; Converging Bio­ applied mathematics is required. Appli­ Dean cants must have a strong commitment to technology Center and Center for Nanos­ teaching and show evidence of excellent The University ofTexas at Arlington invites tructured Materials; as well as numerous teaching ability. Applicants should also nominations and applications for the po­ other centers of excellence. Ground was show evidence of outstanding research sition of Dean of the College of Science. broken in November of 2003 for a new potential in an area that matches the in­ The University of Texas at Arlington $40M state-of-the-art chemistry and terests of the department. Preference will (UTA) is a public research university serv­ physics building. Information on the out­ be given to applicants whose research ing over 25,000 students in eleven col­ standing research capabilities of the col­ areas are probability/ stochastic processes leges and schools on campuses in Arling­ lege can be found at http://www . uta. or numerical analysis. ton and Fort Worth. UTA is classified by edu/cos/. Applicants should send a curriculum the Carnegie Foundation as a Doctoral/Re­ As the chief academic officer of the col­ vita with the AMS standard cover sheet, search-Extensive university and is the sec­ lege, the dean has overall responsibility for transcripts, and three letters of recom­ ond largest institution in the University of leadership of the college including strate­ mendation (with at least one letter ad­ Texas System. The university is located in gic planning and analysis of college oper­ dressing teaching) to: the center of the rapidly growing Dal­ ations; program development for depart­ ments and schools in the college; budget Chairman las/Fort Worth metropolitan area, one of the nation's leading regions in the devel­ development; faculty recruitment; liaison Department of Mathematical Sci­ opment, production, and application of with department chairs, center directors, ences high technology. This environment pro­ and college staff; representing the college University of Alabama in Huntsville vides excellent opportunity for indus­ to the university administration; alumni re­ Huntsville, Ai 35899 try/university R&D collaborations as well lations; and fundraising. The dean reports For more information about the depart­ as with other institutions of higher edu­ to the provost. ment, visit our web site at http://www. cation including The University of Texas Minimum qualifications include an math. uah . edu. Southwestern Medical Center in Dallas. earned doctorate, a distinguished record

Suggested uses for classified advertising are positions available, books or November 2004 issue-August 27,2004; December 2004 issue-September lecture notes for sale, books being sought, exchange or rental of houses, 28, 2Q04; January 2005 issue-October 28, 2004; February 2005 issue­ and typing services. November 22, 2004 The 2004 rate is $100 per inch or fraction thereof on a single column (one­ U.S. laws prohibit discrimination in employment on the basis of color, age, inch minimum), calculated from top of headline. Any fractional text of 1/2 sex, race, religion, or national origin. "Positions Available" inch or more will be charged at the next inch rate. No discounts for mul­ advertisements from institutions outside the U.S. cannot be published tiple ads or the same ad in consecutive issues. For an additional $10 unless they are accompanied by a statement that the institution does not charge, announcements canbe placed anonymously. Correspondence will discriminate on these grounds whether or not it is subject to U.S. laws. De­ be forwarded. tails and specific wording may be found on page 1373 (vol. 44). Advertisements in the "Positions Available" classified section will be set Situations wanted advertisements from involuntarily unemployed math­ with a minimum one-line headline, consisting of the institution name ematicians are accepted under certain conditions for free publication. Call above body copy, unless additional headline copy is specified by the ad­ toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or vertiser. Headlines will be centered inboldface at no extra charge. Ads will 401-455-4084 worldwide for further information. appear in the language in which they are submitted. Submission: Promotions Department, AMS, P.O. Box 6248, Providence, There are no member discounts for classified ads. Dictation over the Rhode Island 02940; or via fax: 401-331-3842; or send email to telephone will not be accepted for classified ads. [email protected] . AMS location for express delivery packages is Upcoming deadlines for classified advertising are as follows: September 201 Charles Street, Providence, Rhode Island 20904. Advertisers willbe billed 2004 issue-June 28, 2004; October 2004 issue-July 28, 2004; upon publication.

842 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Classified Advertisements of scholarship and teaching that would Kansas State University and teaching interests; and should arrange qualify the candidate for tenure at the pro­ Manhattan, KS 66506 for three letters of recommendation to be fessor level in one of the academic units The department also requires that the sent directly to: of the college and administrative experi­ candidate arrange for letters to be sub­ Chair, Departmental Committee on ence at or above a department chair in a mitted evaluating teaching accom­ Appointments college or university setting. plishments and potential. Offers may Department of Mathematics, Qualifications include an outstanding begin by December 1, 2004, but appli­ #121-19a4 Mathematics Road record of academic leadership and funded cations for positions will be reviewed University of British Columbia scholarly activity, fiscal and human re­ until February 1, 2005, or until posi­ Vancouver, BC, Canada, V6T lZ2 source management, as well as community tions are closed. AA/EOE Applications and supporting material, in­ engagement and fundraising. A demon­ 000047 cluding letters of recommendations, must strated record of commitment to excel­ be received by September 1, 2004. lence in teaching and research, significant 000045 achievement in the promotion of diver­ MICHIGAN sity and affirmative action, and outstand­ ing communication and interpersonal MICHIGAN STATE UNIVERSITY skills are also preferred. East Lansing, MI48824 Please send letters of application or nom- proMSc Program in ination electronically to: Industrial Mathematics Dr. Bill D. Carroll Chair, College of Science Search Direct your students toward one of the Committee professional M.Sc. programs. Industry Dean, College of Engineering needs business-savvy mathematicians. See University of Texas at Arlington http://www.sciencemasters.com/. Phone: (817) 272-5725 000001 email: [email protected] Applications should be submitted elec­ tronically to the above email address and CANADA should include a current curriculum vita including a complete list of refereed pub­ UNIVERSITY OF BRITISH COLUMBIA lications and extramural grant support, a Mathematics Department list of five references, and a detailed let­ University Faculty Awards (UFA) ter describing the candidate's relevant ex­ perience, qualifications and leadership The Mathematics Department at the Uni­ philosophy. References will not be con­ versity of British Columbia is seeking to tacted until advanced stages of screening, nominate at least one woman and/or Abo­ and candidates will receive prior notifica­ riginal candidate for an NSERC University tion. Screening of applicants will begin Faculty Award in the Fall 2004 competition. July 12, 2004, and continue until the po­ A departmental nominee will have a strong sition is filled. research record in a field that fits in with The University of Texas at Arlington is the research interests of the department, an Equal Opportunity, Affirmative Action as well as strong teaching qualifications. Employer. A successful nominee would assume a 000046 tenure-track faculty position at the ap­ propriate rank to begin July 1,2005, with a reduced teaching load for the duration of the award. The University Faculty Award KANSAS was created by the Natural Science and Engineering Research Council to encour­ KANSAS STATE UNIVERSITY age Canadian universities to appoint very Department of Mathematics promising women and/or Aboriginal re­ searchers to tenure-track positions in sci­ Subject to budgetary approval, applica­ ence and engineering. tions are invited for tenure-track and vis­ Further information on the program can iting positions commencing August 14, be found at the NSERC web page: 2005; rank and salary commensurate with http://www.nserc.ca/sf_e.asp?nav= qualifications. The departmen.t seeks can­ sfnav&lbi=c7. didates whose research interests mesh In accordance with the NSERC regula­ well with current faculty. The department tions for the awards, an applicant must be has research groups in the areas of analy­ female and/or Aboriginal and must be a sis, algebra, geometry/topology, and dif­ Canadian citizen or a permanent resident ferential equations. Applicants must have of Canada. An applicant must either hold strong research credentials as well as a doctorate degree, or have completed all strong accomplishment or promise in the requirements for such a degree by the teaching. Letter of application, current proposed appointment date. Applicants vita, description of research, and at least are strongly encouraged to apply online as three letters of reference evaluating re­ described at: http://www.math. ubc. cal search should be sent to: Dept/jobs.htm\#Apply. Louis Pigno Alternatively, applicants may send a cur­ Department of Mathematics rent CV including a list of publications, Cardwell Hall 138 preprints, reprints, statement of research

AUGUST 2004 NOTICES OF THE AMS 843 I CAMBRIDGE

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Understanding The Covering Property Axiom, CPA Direct Methods in Soliton Theory Probability A Combinatorial Core of the Ryogo Hirota Chance Rules in Iterated Perfect Set Model Edited by Jon Nimmo and Claire Gilson Everyday Life Krzysztof Ciesielski and Translated by Atsushi Nagai Henk Tijms Janusz Pawlikowski The bilinear, or Hirota's, direct method was Mastering the concepts This book explores a new axiom of set theory­ invented in the early 1970s as an elementary of probability can cast a CPA, the Covering Property Axiom. CPA is means of constructing soliton solutions that new light on situations consistent with the usual ZFC axioms. It is avoided the use of the complex calculations in which randomness true in the iterated Sacks model and actually of the inverse scattering transform. This and chance appear to captures the combinatorial core of this model. analysis is essentially concerned with the rule. With an emphasis A plethora of results known to be true in the more modern version of the method. Still on why probability works and how it can Sacks model easily follow from CPA. Replacing maintaining the original philosophy of using be applied, Henk Tijms introduces the reader iterated forcing arguments with deductions relatively simple mathematics, the method to the world of probability in an informal from CPAs simplifies proofs, provides deeper has, nevertheless, been influenced by the way. Lotteries and casino games provide a insight, and leads to new results. work of the Kyoto school, and will be natural source of motivation, and he carefully Cambridge Tracts in Mathematics 164 essential for all working in soliton theory. discusses these with many worked examples $70.00*: Hardback: 0-521-83920-3: 272pp Cambridge Tracts in Mathematics 155 to illustrate the key concepts from probability $65.00: Hardback: 0-521-83660-3: 200pp theory. $75.00*: Hardback: 0-521-83329-9: 394pp Logic in $29.99*: Paperback: 0-521-54036-4 Computer Exploratory Galois Theory Science John Swallow Integer Partitions Modelling and Combining a concrete perspective with an Reasoning about exploration-based approach, this analysis George E. Andrews and Kimmo Eriksson Systems develops Galois theory at an entirely under­ Provides an accessible and wide-ranging Michael Huth and graduate level. The text grounds the introduction to partitions, without requiring Mark Ryan presentation in the concept of algebraic anything more than some familiarity with The second edition numbers with complex approximations and polynomials and infinite series. of this successful only requires knowledge of a first course $70.00*: Hardback: 0-521-84118-6: 132pp textbook continues to provide a clear in abstract algebra. It introduces tools for $24.99*: Paperback: 0-521-60090-1 introduction to formal reasoning relevant to hands-on experimentation with finite exten­ the needs of modern computer science and sions of the rational numbers for readers sufficiently exacting for practical applications. with Maple or Mathematica. Levy Processes Improvements have been made throughout . $75.00*: Hardback: 0-521-83650-6: 200pp $39.99*: Paperback: 0-521-54499-8 and Stochastic with many new and expanded text sections. Calculus The coverage of model-checking has been David Applebaum substantially updated and additional exercises are included. Topology, Geometry Levy processes form a and Quantum Field wide and rich class of $55.00: Paperback: 0-521-5431 O-X: 450pp random process, and Theory have many applications Proceedings of the ranging from physics Leibniz and 2002 Oxford Symposium in to finance. Stochastic His Correspondents Honour of the 60th Birthday of Graeme Segal calculus is the mathematics of systems inter­ Edited by Paul Lodge acting with random noise. David Applebaum Edited by U.L. Tillmann Unlike most of the other great philosophers, connects the two subjects together in this This volume covers Leibniz never wrote a magnum opus, so his monograph. After an introduction to the the proceedings of an philosophical correspondence is essential for general theory of Levy processes, he accessi­ international conference held in Oxford in an understanding of his views. This collec­ bly develops the stochastic calculus for June 2002. In addition to articles arising tion of new essays by preeminent figures in Levy processes. All the tools needed for from the conference, the book also contains the field of Leibniz scholarship is the most the stochastic approach to option pricing, the famous, as yet unpublished, article thorough account of Leibniz's philosophical including It6's formula, Girsanov's theorem /I correspondence available. It illuminates his by Graeme Segal on the Definition of and the martingale representation theorem, Conformal Field Theories. /I philosophical views and pays due attention to are described. the dialectical context in which the relevant London Mathematical Society Lecture Note Series 308 Cambridge Studies in Advanced Mathematics 93 passages from the letters occur. $90.00: Paperback: 0-521-54049-6: 590pp $75.00: Hardback: 0-521-83263-2: 408pp $70.00: Hardback: 0-521-83410-4: 324pp

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Editors Website: http://imrn.hindawi.com Morris Weisfeld IMRN provides very fast publication of research articles of high current interest in all Managing Editor areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics. Issues are published as frequently as Dan Abramovich necessary. IMRN is expected to publish 80± issues in volume 2004. The articles of IMRN are Enrico Arbarello reviewed/indexed in COMPUMATH Citation Index, Current Contents, lSI Alerting Services, Joseph Bernstein Mathematical Reviews, Science Citation Index, SciSearch, and Zentralblatt fur Mathematik. Enrico Bombieri There are no page charges. Submissions are made bye-mail to [email protected]. Richard E. Borcherds New and current online subscribers shall receive "perpetual" online access to volumes Alexei Borodin 1991-2004. Contact orders@hindawLcom for more details. Jean Bourgain FORTHCOMING ARTICLES Marc Burger Tobias Colding •A Note on Gaps of Hill's Equation, B. Grebert, T. Kappeler, and J. Poschel Corrado DeConcini • Analysis of Geometric Stability, Sean T. Paul and Gang Tian Percy Deift • Combinatorial Classes on Mg,n Are Tautological, Gabriele Mondello Robbert Dijkgraaf • Correlations between Zeros of Non-Gaussian Random Polynomials, P. Bleher and X. Di S. K. Donaldson • Hypergeometric Solutions to the q-Painleve Equations, Yakov Eliashberg K. Kajiwara, T. Masuda, M. Noumi, Y: Ohta, and Y: Yamada Edward Frenkel • Irreducibility of the Siegel Moduli Spaces with Parahoric Level Structure, Chia-Fu Yu Emmanuel Hebey • Janossy Densities, Multimatrix Spacing Distributions and Fredholm Resolvents, Dennis Hejhal J.Harnad Helmut Hofer • New Lagrangian Submanifolds of CIfD, River Chiang Gerhard Huisken • 04: Integrable Master Equation Related to an Elliptic Curve, ~ E. Adler and Yu. B. Suris Yasutaka Ihara • Semiclassical Analysis of Generic Codimension 3 Crossings, C. F. Kammerer Michio Jimbo • Some Estimates for Non-microstates Free Entropy Dimension with Applications to Kurt Johansson q-Semicircular Families, Dimitri Shlyakhtenko Masaki Kashiwara Kiran S. Kedlaya Carlos Kenig Sergiu Klainerman Applied Mathentatics Toshiyuki Kobayashi Maxim Kontsevich Research eXpress Igor Krichever Shigeo Kusuoka Gilles Lebeau Website: http://amrx.hindawLcom Joachim Lohkamp AMRX provides very fast publication of research articles of high current interest Nikolai Makarov dealing with the use of mathematics in all other areas of knowledge usually in Yu.1. Manin the form of mathematical/computational models and algorithms. There are no Barry Mazur restrictions on length and illustrations can be in color. Submissions are made by Haynes Miller e-mail to [email protected]. Shinichi Mochizuki Editors: Morris Weisfeld (Editor-in-Chief), A. Bertozzi, K. Bhattacharya, D. Cai, Fabien Morel R. Carmona, T. T. Chen, T. C. E. Cheng, J.-M. Coron, J. Demmel, A. DeSimone, Q. Du, Michael Rapoport Weinan E, L. C. Evans, G. Friesecke, J. Glimm, A. Guionnet, P. Holmes, S. Hosten, N. Yu. Reshetikhin T. Y. Hou, J. Jost, J. Keating, C. Le Bris, M. Ledoux, P. Maini, B. Mishra, S. Miiller, Igor Rodnianski T. Nizar, S. Osher, J. Pach, R. L. Pego, J. Satsuma, B. Sturmfels, X.-P. Wang, Peter Sarnak H. Weitzner, H.-T. Yau, P. Zhang. Freydoon Shahidi Forthcoming Articles Stanislav Smirnov • Simulation of Low Reynolds Number Flow Control over a Backward-Facing Michael Struwe Step Using Pulsed Inlet Velocities, Emmanuel Creuse and Iraj Mortazavi G. Tian • Quartets and Parameter Recovery for the General Markov Model ofSequence John Toth Mutation, Elizabeth S. Allman and John A. Rhodes Takeshi Tsuji David Vogan Dan Voiculescu Hindawi Publishing Corporation, 410 Park Avenue, 15thFloor, #287 pmb, New York, Andrei Zelevinsky HINDAWI NY 10022, USA; Fax 1-866-446-3294 (USA, toll-free) Maciej Zworski MATHEMATICS from Birkhauser

Dynamics of Foliations, Rings, Modules, Poincare Seminar 2003 Groups and Pseudogroups and, the Total Bose-Einstein Condensation - Entropy PAWEL WALCZAK, University ofLodz, Poland FRIEDRICH KASCH, Icking, Germany; and ADOLF MADER, JEAN DALIBARD, Paris, France; University ofHawaii, Honolulu BERTRAND DUPLANTIER, CEA-Saclay, France; and Foliations, groups, and pseudogroups are objects, which VINCENT RIVASSEAU, Polytechnique, Paris, France (eds.) are closely related via the notion of holonomy. This book The book deals with direct decompositions of modules deals with their dynamics, concentrating on notions and and associated concepts. The central notion of 'partially This volume contains the lectures of the third and results related to different ways of measuring complexi­ invertible homomorphisms', namely those that are fourth Poincare Seminar, both held in 2003. The third ty of the systems under consideration. It details differ­ factors of a non-zero idempotent, is introduced in avery seminar is devoted to Bose-Einstein condensation: it ent types of growth, entropies, and dimensions of accessible fashion. covers the physics of superfluid liquid helium as well as limiting objects. Invented in the 1980s (by E. Ghys, R. the recently discovered atomic Bose-Einstein conden­ 2004/APPROX 138 PP'/SOFTCOVER/$115.00 sates. The fourth seminar concentrates on entropy, Langevin, and the author), geometric entropy of a ISBN 3-7643-7071-8 foliation is the principal object of interest among all of FRONTIERS IN MATHEMATICS giving a comprehensive account of the history and them. Throughout the book, the reader will find a good various realizations of this concept, from thermodynam­ number of inspiring problems related to the Reconstructive ics to black holes, and including theoretical and experi­ topics covered. mental discussions of the corresponding fluctuations for mesoscopic systems near equilibrium. 2004/225 PP'/HARDCOVER/S 119.00 Integral Geometry ISBN 3-7643-7091-2 VICTOR PALAMODOV, Tel Aviv University, Israel 2004/264 PP'/HARDCOVER/$125.00 MONOGRAFIE MATEMATYCZNE ISBN 3-7643-71 06-4 This book covers facts and methods for reconstruction PROGRESS IN MATHEMATICAL PHYSICS of a function in a real affine or projective space from Applied Laplace data of integrals over lines, planes, and spheres. It Travelling Waves in Transforms and collects recent results on explicit analytic methods. Another focus consists of the relations between Nonlinear Diffusion­ z-Transforms for algebraic integral geometry and partial differential Convection-Reaction Scientists and Engineers equations. Aconcise introduction to harmonic analysis and distribution theory is provided in the first chapter. BRIAN H. GILDING, Sultan Qaboos University, Oman; and A Computational Approach Using a The first half of the book includes the ray, the spherical ROBERT KERSNER, Hungarian Academy ofScience, Mathematica Package mean transforms in the plane or in 3-space, and inver­ Budapest, Hungary URS GRAF, Hochschule till' Technik; and ARCHITEKTUR, sion from incomplete data. Later chapters are devoted Traveling waves are observed in many natural process­ Bief, Switzerland to the Funk-Radon transform on algebraic varieties of es, ranging from the spread of diseases to the combus­ This book presents the theory and applications of arbitrary dimension. tion of fuels. This book examines such waves in Laplace and z-transforms together with a Mathematica 2004/APPROX. 192 PP'/HARDCOVER/$89. 95 phenomena described by a nonlinear diffusion-convec­ package developed by the author. The package substan­ ISBN 3-7643-7129-3 tion-reaction equation. The technique employed is new tially enhances the built-in facilities of Mathematica and MONOGRAPHS IN MATHEMATICS and can be briefly described as an integral or integrated includes algorithms for the numerical inversion of approach to phase-plane analysis. It leads to results that Laplace transforms. Many examples from applied The Topology of were otherwise unobtainable, including the effects of sciences and engineering illustrate applications of the degenerate diffusion and nonlinear convection, or are theory and the usage of the package. Torus Actions on much sharper than those obtained previously. Symplectic Manifolds Applications are taken from: the evolution of biological 2004/500 PP'/HARDCOVER/S89. 95 populations, diffusion of oxygen in tissue, heat transfer, ISBN 3-7643-2427-9 Second Edition thermal convection, combustion, flow in porous media, MICHELE AUDIN, Universite Louis Pasteur et CNRS, hydrology, soil-moisture flow, boundary layer theory, Limit Operators and Strasbourg, France foam drainage, viscous fluid flow, turbulent fluid flow, Their Applications in In this is second edition, the material and references the motion of plasma particles in a magnetic field, solar have been updated. Symplectic manifolds and torus prominences, crystal growth, reaction chemistry, Operator Theory actions are investigated, with numerous examples of quenching, and other fields. VLADIMIR RABINOVICH, National Politechnic Institute torus actions, for instance on some moduli spaces. 2004/210 PP'/HARDCOVER/$115.00 of Mexico, Mexico; STEFFEN ROCH; and Although the book is still centered on convexity results, ISBN ·3-7643-7071-8 BERND SILBERMANN it contains more theorems, more (and better) proofs, PROGRESS IN NONLINEAR DIFFERENTIAL EQUATIONS AND THEIR This book examines a class of operators that occur in more exercises, and many figures. APPLICATIONS almost every part of mathematics: band and band­ 2004/APPROX. 332 PP'/HARDCOVER/$ 109.00 dominated operators on spaces of vector-valued ISBN 3-7643-2176-8 <·PRE.E.;jOURNAL SAMPLE COPIES! PROGRESS IN MATHEMATICS sequences. 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8/04 Promotion #Y9672 AMERICAN MATHEMATICAL SOCIETY Praise for AMS Publications These comments from scholarly journals in the mathematical community reflect the quality, timeliness, and significance of books in the AMS Publishing Program. To learn more about these and other AMS publications or to browse our full selection of titles, visit the online AMS Bookstore at www.ams.org/bookstore. Take advantage of substantial meeting discounts!

fX>;'«For Elennentary Algebraic A Mathennatician's ";~t~ •<' Classroom @j., .... Use Geol11etry Survival Guide Klaus Hulek, Universitot Hannover, Germany Graduate School and 'Early Career Development "The book does an excellent job ofexplaining what alge­ Steven G. Krantz, Washington Universit~ St. Louis, MO braic geometry is about ... invites the reader to continue exploring the subject "Very valuable for the prospective de~nitely ... recommend it as reading student ... de~nitely good advice ... material to a bright undergraduate ..." recommend every mathematics depart­ -MAAOnline ment keep a copy ofthis book ..." Student Mathematical Library,Volume -MAAOnline 20; 2003; 213 pages; Softcover; ISBN 0­ 2003; 222 pages; Softcover; ISBN 0-8218­ 8218-2952-1; List $35;AII AMS members 3455-X; List $28; All AMS members $22; $28; Order code STML/20N048 Order code GSCMN048

Mathel11atics of ." ..... For Lecture Notes in pi,·;,:. Classroom (t]',,,,", Use Inforl11ation and Algebraic Topology Coding James F. Davis and Paul Kirk, Indiana Te Sun Han and Kingo Kobayashi, The University of Universit~ Bloomington Electro-Communications, Tokyo, Japan "The book might well have been titled 'What Every Young "This volume is a very nice introduaion to information theory Topologist Should Know' ... recommend this book as a valu­ and to source coding ... plenty ofexercises able tool for everybody teaching ... strongly recommended for teaching and graduate courses as well as a self­ learning information theory." contained introduction ... for -Zentralblatt MATH independent reading." -Mathematica Bohemica Translations of Mathematical Monographs,Volume 203; 2002; 286 pages; Graduate Studies in Mathematics, Hardcover; ISBN 0-8218-0534-7; List $99; Volume 35; 200 I; 367 pages; Hardcover; Individual member $59; Order code ISBN 0-8218-2160-1; List $55;AII AMS MMONO/203N048 members $44; Order code GSM/35N048 F~~ :I:I·.·111 many more publications of interest, III.".' VISit the AMS Bookstore lAMS BOOKSTORE I www.amsbookstore.org

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8/04 See You There! Meetings &Conferences oftheAMS

IMPORTANTINFORMATION ~GARDING MEETINGS PROGRAMS: AMS SectionalMeetingprograms do notappear inthe printversionofthe Notices.IHowever, comprehensive and continuallyupdatedmeeting andprograminformation withlinkstotheabstractfor eachtalkcanbefoundon theAMS.website. See http://www. ams . 0 rg/meet; ng 5/.Programs andabstracts will continue tobe displayed ontheAMS website inthe Meetings and Conferences sectionuntil about six weeks afterthemeetingis over. Finalprogramsfor SectionalMeetingswillbearchivedontheAMS websiteinanelectronic issue ofthe Notices as notedbelowfor eachmeeting.

Geometry ofHyperbolic Manifolds (Code: SS lOA), John G. Nashville, Tennessee Ratcliffe and Steven T. Tschantz, Vanderbilt University. Vanderbilt University Index Theory and the Topology ofManifolds (Code: 5S 3A), Bruce Hughes and Guoliang Yu, Vanderbilt University. October 16-1 7, 2004 Inverse Problems (Code: SS 9A), Maeve L. McCarthy, Saturday - Sunday Murray State University, and Rudi Weikard, University of Alabama at Birmingham. Meeting #999 Local and Homological Algebra (Code: SS 6A), Florian Southeastern Section Enescu, University of Utah, and Adela N. Vraciu, Univer­ Associate secretary: John L. Bryant sity of South Carolina. Announcement issue of Notices: August 2004 Nonlinear Partial Differential Equations and Applications Program first available on AMS website: September 2, 2004 (Code: SS 11A), Gieri Simonett, Vanderbilt University. Program issue of electronic Notices: October 2004 Operator Theory on Function Spaces (Code: SS 7A), Dechao Issue of Abstracts: Volume 25, Issue 4 Zheng, Vanderbilt University. Deadlines Semigroup Theory (Code: SS 13A), Matthew I. Gould, Van­ For organizers: Expired derbilt University, and Karen Ann Linton, California State For consideration of contributed papers in Special Sessions: Polytechnic University, Pomona. Expired Topological Aspects of Group Theory (Code: SS SA), For abstracts: August 24, 2004 Michael L. Mihalik and Mark V. Sapir, Vanderbilt Univer­ sity. Invited Addresses Universal Algebra and Lattice Theory (Code: SS 4A), Ruth M. Charney, Brandeis University, Title to be an­ Ralph N. McKenzie, Vanderbilt University, and George F. nounced. McNulty, University of South Carolina. Peter S. Ozsvath, Columbia University, Title to be an­ Von Neumann Algebras andNoncommutative Ergodic The­ nounced. ory (Code: SS 1A), Dietmar Bisch, Vanderbilt University, and Sorin T. Popa, University of California Los Angeles. Sorin T. Popa, University of California Los Angeles, Title to be announced. Wavelets, Frames, and Sampling (Code: 5S 2A), Akram Al­ droubi and Douglas P. Hardin, Vanderbilt University, and Rudi Weikard, University ofAlabama at Birmingham, Title Qiyu Sun, University of Central Florida. to be announced. Accommodations Special Sessions Participants should make their own arrangements directly Algebraic Geometry and Commutative Algebra (Code: SS with a hotel of their choice as early as possible. Special rates SA), Juan C. Migliore, University of Notre Dame, and Uwe have been negotiated with the hotels listed below. Rates Nagel, University of Kentucky. quoted do not include sales tax of 14.25%. The AMS is not Biomathematics (Code: SS 12A), Laurent Pujo-Menjouet and responsible for rate changes or for the quality of the ac­ Glenn F. Webb, Vanderbilt University. commodations. When making a reservation, participants

AUGUST 2004 NOTICES OF THE AMS 849 Meetings & Conferences should state that they are with the American Mathemati­ lie parking spaces are 52-170. Cost at the time of this print­ cal Society (AMS) Fall Southeastern Sectional Meeting group. ing is $5/day. Cancellation and early checkout policies vary; be sure to check when you make your reservation. Registration and Meeting Information Days Inn-Vanderbilt, 1800 West End Avenue, Nashville, The meeting is on the main campus of Vanderbilt Univer­ TN 37203; 615-327-0922 (phone), 407-206-3001 (fax); sity in Nashville, Tennessee. Sessions and Invited Ad­ $65/single or double, includes complimentary continen­ dresses will take place in Wilson and Furman Halls. tal breakfast, exercise room on premises; about 1/3 mile The registration desk will be in the lobby of Wilson to the meeting site. Deadline for reservations is Hall and will be open Saturday, October 16, 7:30 a.m.­ September 15, 2004. Be sure to check cancellation and early 4:00 p.m. and Sunday, October 17, 8:00 a.m.-noon. Fees checkout policies. are $40 for AMS or CMS members; $60 for nonmembers; Hampton Inn & Suites at the University, 2330 Elliston and $5 for students/unemployed/emeritus, payable on Place, Nashville, TN 37203; 615-320-6060 or 888-880-5395 site by cash, check, or credit card. (phone), 615-327-4723 (fax); $79/single or double, com­ plimentaryhot breakfast daily; about 1/3 mile to the meet­ Travel and Campus Map ing site. Deadline for reservations is September 15, 2004. Vanderbilt University is located a mile and a half south­ Be sure to check cancellation and early checkout policies. west of downtown Nashville and is approximately 4 hours Holiday Inn Select Vanderbilt, 2613 West End Avenue, from Atlanta, 3 hours from Birmingham, 2 1/2 hours from Nashville, TN; 615-327-4707 (phone), 615-320-4850 (fax); Knoxville, 3 hours from Louisville, and 3 hours from Mem­ $80/single or double, full-service restaurant, cocktail phis. lounge, as well as coffee shop on premises; about 1/2 mile to the meeting site. Deadline for reservations is Sep­ A campus map is found at http://www. vande rbi 1t . tember 15,2004. Be sure to check cancellation and early edu/map/entry.html. checkout policies. The closest airport is Nashville International Airport (BNA) and is servicedby all major airlines. Delta Air Lines Food Service has been selected as the official airline for this meeting There are many restaurants along 21 stAvenue that are con­ because of economical rates and convenient schedules. The venient for lunch. following specially negotiated rates are available for this meeting and exclusively for mathematicians and their fam­ Local Information ilies for the period October 13, 2004, through October 20, 2004. Other restrictions/discounts may apply and seats The math department's webpage is located at are limited. http://si temason. vanderbi 1t. edu/math; the univer­ sity's homepage is http://www. vanderbi 1t. edu/. Delta Air Lines is offering: •A 5% discount off Delta's published round-trip fares Other Activities within the continental United States, excluding A, D, I, U, Book Sales: Examine the newest titles from the AMS! Many and T classes of service. of the AMS books will be available at a special discount •A 10% discount off Delta's domestic published unre­ available only at the meeting. Complimentary coffee will stricted round-trip coach fares (Y06/YR06) rates. No ad­ be served courtesy of AMS Membership Services. vance reservations or ticketing is required. • An additibnal 5% bonus discount if you purchase AMS Editorial Activity: An acquisitions editor from the your ticket 60 days or more prior to your departure through AMS book program will be present to speak with prospec­ tive authors. If you have a book project that you would like Meeting Network Reservations or your travelagent; online not applicable. to discuss with the AMS, please stop by the book exhibit. To take advantage of these discounts and make imme­ Parking diate reservations, call Delta Meeting Network at 800-241­ The closest public parking to the meeting site is found at 6760 between 8:00 a.m. and 11:00 p.m. Eastern Standard Wesley Place parking garage. From the airport take 1-40 Time, Monday through Sunday, referencing File Number West toward Nashville. Follow the signs for 1-24 205778A. West/I-40 West, then 1-65 South/I-40 West. Exit at Broad­ Ground transportation from the airport: The 10-mile way. At the base of the ramp, go straight through the traf­ cab ride to campus/hotels will take about 20 minutes and fic light at McGavock Street and turn left onto Broadway. cost about $25. A Gray Line shuttle van (www. Stay left. Continue on Broadway to 16th Avenue, where the gray1i nenashvi 11 e. com) to downtown hotels makes the road splits. Veer left onto Broadway. Broadway will become trip in about 30 minutes and costs $11 one way/$17 round 21st Avenue South. As you pass the traffic light at Grand trip; be sure to take the "West End/Vandy Area" shuttle. Avenue, the Vanderbilt main campus will be onyour right. The public bus to downtown leaves the airport 14 times Turn left at Scarritt Place (first traffic light after Grand Av­ a day, Monday-Friday, and four times a day on Saturdays enue). The entrance to the Wesley Place parking garage will and Sundays. The fare is $1.40, and schedules are avail­ be on your right one-halfblock past the intersection. Pub- able at the airport Welcome Center on the baggage claim

850 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences level. If you need information before you arrive, call the Arithmetic Geometry (Code: SS 6A), Alexandru Buium and center at 615-275-1674. Michael J. Nakamaye, University of New Mexico. Driving: Getting to Vanderbilt from Nashville Inter­ Braids andKnots (Code: SS 14A), Theodore Stanford, New national Airport: Take 1-40 west to Exit 209A. Turn left Mexico State University. onto Broadway. Follow Broadway and veer right to West Categories and Operads in Topology, Geometry, Physics and End Avenue. Allow approximately thirty minutes, driving OtherApplications (Code: SS SA), Hanna Ewa Makaruk and time from the airport to Vanderbilt University. If traffic is Robert Michal Owczarek, Los Alamos National Laboratory, light, it will take less time. and Zbigniew Oziewicz, Universidad Nacional Aut6noma Getting to Vanderbilt via the Interstates: From the de Mexico. north, take 1-65 to 1-40 West and then look for 1-40 East to Exit 209B. Turn right on Broadway. Financial Mathematics: The Mathematics of Derivative From the east or south, take 1-40 West to Exit 209A. Turn Securities (Code: SS 4A), Maria Cristina Mariani, New left on Broadway. Mexico State University, and Osvaldo Mendez, University From the west, take 1-40 East to Exit 209B. Turn right of Texas at El Paso. on Broadway (US 70S); follow Broadway and veer right to Interactions in Riemannian Geometry (Code: SS 8A), West End Avenue. Charles P. Boyer and Krzysztof Galicki, University of New Mexico. Mathematical Methods in Turbulence (Code: SS 9A), Monika Albuquerque, Nitsche and Vachtang Poutkaradze, University of New Mexico. New Mexico Mathematics for Secondary Teachers: Curriculum and University ofNew Mexico Assessment (Code: SS 16A), Adriana Aceves and Kristin Umland, University of New Mexico. October 16-1 7,2004 Multiscale Methods and Sampling in Time-Frequency Analy­ Saturday - Sunday sis (Code: SS lOA), Jeffrey Andrew Hogan, University of Arkansas, and Joseph·D. Lakey, New Mexico State Uni­ Meeting #1 000 versity. Western Section Nonlinear Partial Differential Equations Applied to Mate­ Associate secretary: Michel L. Lapidus rials Science (Code: SS lIA), Patricia Bauman, Purdue Uni­ Announcement issue of Notices: August 2004 versity, and Tiziana Giorgi, New Mexico State University. Program first available on AMS website: September 3, 2004 Program issue of electronic Notices: October 2004 Probabilistic and Geometric Methods in Learning Theory Issue of Abstracts: Volume 25, Issue 4 (Code: SS 15A), Vladimir Koltchinskii, University of New Mexico. Deadlines Random Matrix Theory and Growth Processes (Code: SS lA), For organizers: Expired Craig A. Tracy, University of California Davis. For consideration of contributed papers in Special Sessions: Regularity in PDEs and Harmonic Analysis (Code: SS 12A), Expired Marianne Korten and Charles Nelson Moore, Kansas State For abstracts: August 24, 2004 University, and Maria C. Pereyra, University of New Mex­ Invited Addresses ico. Several Complex Variables and CR Geometry (Code: SS Sara C. Billey, University of Washington, Seattle, Title to 2A), Peter Ebenfelt, University of California San Diego, and be announced. Marshall A. Whittlesey, California State University, San Mar­ Peter Ebenfelt, University of California San Diego, Analytic cos. and geometric properties of CR manifolds and their map­ Spectral Geometry (Code: SS 7A), Ivan G. Avramidi, New pings. Mexico Institute of Mining and Technology, and Thomas Theodore Stanford, New Mexico State University, Knots Patrick Branson, University of Iowa. modulo braids. Craig A. Tracy, University of California Davis, Title to be Accommodations announced. Participants should make their own arrangements directly with the hotel. The rates quoted for the DoubleTree Hotel Special Sessions are available until September 15. When making a reser­ Algebraic Geometry (Code: SS 3A), Hirotachi Abo and vation, identify yourself as being with the UNM Math. and Chris Peterson, Colorado State University. Stat. group attending the AMS Meeting. After September 15 Analysis and Geometry in Carnot-Caratheodory Spaces rooms may not be available at the quoted price. The AMS (Code: SS 13A), Luca Capogna, University of Arkansas, is not responsible for rate changes or for the quality of the and Robert Smits, New Mexico State University. accommodations. Hotels have varying cancellation or early

AUGUST 2004 NOTICES OF THE AMS 851 Meetings & Conferences checkout penalties; be sure to ask details when making your Travel reservation. By Air: Albuquerque Sunport International Airport : Al­ Double Tree Hotel, 201 Marquette NW, Albuquerque, buquerque is served by many of the major commercial car­ NM 87102; 866-224-9330, 888-223-4113; $79/single or riers and several commuter airlines. The Sunport is located double plus tax (currently 11%). Approximately 1 mile to 2 miles south of UNM and 4 miles southeast of the Dou­ campus. Deadline for reservations is September 15. Be bletree Hotel. Driving time to either location is approxi­ sure to check the cancellation policy. The website is mately 10 minutes. http://www.doubletreealbuquerque.com. Announcing Delta Air Lines as the official airline for Food Service the sectional meeting in Albuquerque, New Mexico! A list of Albuquerque restaurants will be available at the The following specially negotiated rates are available for registration desk. this meeting and exclusively to mathematicians and their families for the period October 13, 2004, through Local Information and Campus Map October 20, 2004. For further information please consult the website main­ Other restrictions/discounts may apply, and seats are tained by the Department of Mathematics at the Univer­ limited. sity of New Mexico: http://www . math. unm. edu/ Delta Air Lines is offering: conferences/ams/. •A 5% discount off Delta's published round-trip fares To view a campus map please visit: http://www . within the continental United States, excluding A, D, I, U, unm. edu/campusmap. html . Dane Smith Hall is building and T classes of service. 48 at F-G-3. For travel information please visit: •A 10% discount off Delta's domestic published unre­ http://www.newmexico.org/place/loc/transportation/ stricted round-trip coach fares (Y06/YR06) rates. No ad­ page/DB-place/category/1 05/place/5 73.html. vance reservations or ticketing is required. • An additional 5% bonus discount if you purchase Other Activities your ticket 60 days or more prior to your departure through AMS Book Sale: Examine the newest titles from AMS! Com­ Meeting Network Reservations or your travel agent; online plimentary coffee will be served, courtesy of AMS Mem­ not applicable. bership Services. The AMS Book Sale will operate during To take advantage of these discounts and make imme­ the same hours as registration and will be held in Dane diate reservations, call Delta Meeting Network at 800-241­ Smith Hall. 6760, between 8:00 a.m. and 11:00 p.m. Eastern Standard AMS Editorial Activity: An acquisitions editor from Time, Monday through Sunday, referencing File Number the AMS book program will be present to speak with 205778A. prospective authors. If you have a book project that you Traveling to Albuquerque by car: Albuquerque is would like to discuss with the AMS, please stop by the book served by two major interstates, 1-25 (north-south) and exhibit. 1-40 (east-west). The Martin Luther King exit one mile south of the 1-25-1-40 interchange allows access to the Double­ Parking tree Hotel (go approx. one mile west to downtown; hotel Private parking lots are within walking distance to the is on right between 2nd and 3rd Streets) and UNM (go east meeting and are located on Cornell Avenue between Cen­ to Univers~ty Blvd., enter campus, turn right on Redondo tral Avenue and Silver Avenue and on Silver Avenue be­ Rd., and follow it to visitor parking or residence halls). tween Cornell Avenue and Stanford Avenue. These lots Driving Directions from Sunport to Doubletree Hotel: have coin-operated machines at a cost of $2.25 per day. Take 1-25 north to Martin Luther King exit, turn left onMLK, Exact change must be deposited in the machine. go approximately one mile; Doubletree Hotel is on right Registration and Meeting Information between 2nd and 3rd Streets. Registration will take place in Dane Smith Hall, located on Driving Directions from Sunport to UNM: Take Yale Las Lomas Blvd., across from University House onYale Av­ Blvd. exit north until you dead-end at Central Ave. Cross enue, from 7:30 a.m. to 4:00 p.m. on Saturday, October 16, Central Ave. to enter campus. and from 8:00 a.m. to noon on Sunday, October 17. Reg­ Transportation from/to the Sunport: Shuttle and taxi istration fees are $40 for AMS or CMS members; $60 for service available after all flights outside at baggage claim nonmembers; and $5 for students/unemployed/emeritus, level. 24/7 service available from Airport Shuttle (505­ payable on site by cash, check, or credit card. All talks will 765-1234; typical charge $16) and Albuquerque Cab (505­ take place in Dane Smith Hall. 883-4888; typical charge $15). Sunport provides a shuttle to Rental Car Center. Doubletree Hotel does not have a pri­ Social Event vate shuttle. The Department ofMath will host a reception in the atrium Getting around Albuquerque is easiest by car, but the of Dane Smith Hall, Saturday, October 16, at 6:00 p.m. city has regular bus service along Central Ave. from down-

852 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences town to UNM for participants who wish to stay downtown. Algebraic Topology: Interactions with Representation The­ Taxi service is available but best arranged beforehand. ory and Algebraic Geometry (Code: SS 13A), Paul G. Car Rental: Special rates have been negotiated with Goerss, Northwestern University, Jesper Kragh Grodal, Upi­ Avis Rent A Car for the period October 9 to October 24, versity of Chicago, and Brooke E. Shipley, University of 2004, beginning at $24.99/day for a subcompact car at the Illinois at Chicago. weekend rate (the weekend rate is available from noon Applications of Motives (Code: SS 18A), Eric M. Friedlan­ Thursday until Monday at 11:S9 p.m.). All rates include der, Northwestern University, Alexander Goncharov, unlimited free mileage. Rates do not include state or local Brown University, Mikhail Kapranov, Yale University, and surcharges, tax, optional coverages, or gas refueling Yuri Manin, Max Planck Institute for Mathematics. charges. Renter must meet Avis's age, driver, and credit Codes andApplications (Code: SS SA), William C. Huffman, requirements, and return to the same renting location. Make Loyola University of Chicago, and Vera S. Pless, Univer­ reservations by calling 800-331-1600 or online at sity of Illinois at Chicago. www.avis.com. Please quote Avis Discount Number Computability Theory and Applications (Code: S5 8A), B159266 when making reservations. Robert I. Soare and Denis R. Hirschfeldt, University of Chicago. Weather Differential Geometry (Code: SS lOA), Anders Ingemar October weather is generally pleasant, with daytime tem­ Linner and Hongyou Wu, Northern Illinois University. peratures in the 60° F. range, and nighttime temperatures Extremal Combinatorics (Code: SS 2A), Dhruv Mubayi and in the 30-4So F. range. For up-to-the- minute weather, Yi Zhao, University of Illinois at Chicago. please visit http://www.weather.com/outl ook/ driving/local/U5NM0004. Fluid Dynamics, Diffusion and Reaction (Code: SS 4A), Peter S. Constantin and Leonid V. Ryzhik, University of Chicago. Evanston, Illinois Geometric Aspects of the Langlands Program (Code: SS 17A), Edward Frenkel, University of California Berkeley, Northwestern University Dennis Gaitsgory, University of Chicago, Mark Goresky, Institute for Advanced Study, and Kari Vilonen, North­ October 23-24, 2004 western University. Saturday - Sunday Geometric Partial Differential Equations (Code: SS 7A), Gui­ Meeting #1001 Qiang Chen and Jared Wunsch, Northwestern University. Central Section HopfAlgebras at the Crossroads ofAlgebra, Category The­ Associate secretary: Susan ]. Friedlander ory, and Topology (Code: SS 24A), Louis H. Kauffman and Announcement issue of Notices: August 2004 David E. Radford, University ofIllinois at Chicago, and Fer­ Program first available on AMS website: September 9, 2004 nando J. o. Souza, University of Iowa. Program issue of electronic Notices: October 2004 Index Theory, Morse Theory, and the Witten Deformation Issue of Abstracts: Volume 25, Issue 4 Method (Code: SS 3A), Igor Prokhorenkov and Ken Richard­ son, Texas Christian University. Deadlines Iterated Function Systems and Analysis on Fractals (Code: For organizers: Expired SS 12A), Ka-Sing Lau, Chinese University of Hong Kong, For consideration of contributed papers in Special Sessions: and Stephen S.-T. Yau, University of Illinois at Chicago. Expired Low-Dimensional Topology and Kleinian Groups (Code: SS For abstracts: August 31, 2004 21A), Ian Agol, John Holt, and Saul Schleimer, University Invited Addresses of Illinois at Chicago. Mathematical Problems in Robotics (Code: SS lSA), Ian Agol, University of Illinois at Chicago, Title to be an­ Robert W. Ghrist, University of Illinois at Urbana-Cham­ nounced. paign. Robert W. Ghrist, University of Illinois, Title to be an­ Mathematical Techniques in Musical Analysis (Code: 55 nounced. 23A), Judith Baxter, University of Illinois at Chicago, Yuri Manin, Northwestern University, Title to be announced. Richard Cohn, University of Chicago, and Robert Peck, Paul Seidel, Imperial College-London and University of Louisiana State University. Chicago, Title to be announced. Modern Schubert Calculus (Code: SS 1A), Ezra Miller, Uni­ versity ofMinnesota, and Frank Sottile, University ofMass­ Special Sessions achusetts. Algebraic Representations and Deformations (Code: SS Nonlinear Partial Differential Equations and Applications 19A), Stephen R. Doty and Anthony Giaquinto, Loyola Uni­ (Code: SS 6A), Gui-Qiang Chen, Northwestern University, versity of Chicago. and Mikhail Feldman, University ofWisconsin at Madison.

AUGUST 2004 NOTICES OF THE AMS 853 Meetings & Conferences

Nonlinear Waves (Code: SS 14A), Jerry L. Bona, University The Homestead, 1625 Hinman Avenue, Evanston, IL of Illinois at Chicago, Shuming Sun, Virginia Polytechnic 60201; 847-475-3300, fax: 847-570-8100; $100 single or Institute and State University, and Bingyu Zhang, Univer­ double plus tax (currently 13.5%). Two blocks to campus. sity of Cincinnati. Deadline for reservations is October 8, 2004. Representation Theory ofReductive Groups (Code: SS 20A), Jeffrey D. Adler, University ofAkron, and Ju-Lee Kim, Uni­ Food Service versity of Illinois at Chicago. There are a number of restaurants adjacent to the cam­ Solving Polynomial Systems (Code: SS 9A), Anton Leykin pus. A list of restaurants will be available at the registra­ and Jan Verschelde, University of Illinois at Chicago. tion desk. Special Functions, Orthogonal Polynomials, and Their Applications (Code: SS 22A), George Gasper, Northwestern Local Information University, and Ahmed Ia Zayed, DePaul University. Please visit the website maintained by the Department of Spectral Problems ofDifferential Operators (Code: SS 16A), Mathematics at http://www . math. northwestern. edu/, Qingkai Kong, Hongyou Wu, and Anton Zettl, Northern the Northwestern University website http://www . Illinois University. no rthwestern . edu/, and for a campus map: http: // Stability Issues in Fluid Dynamics (Code: SS lIA), Susan J. www.northwestern.edu/campus/maps/evanston_ Friedlander and Roman Shvydkoy, University of Illinois campus_map. html. at Chicago. Other Activities Accommodations AMS Book Sale: Examine the newest titles from AMS! Com­ Participants should make their own arrangements directly plimentary coffee will be served courtesy of AMS Mem­ with the hotel of their choice and state that they will be bership Services. The AMS Book Sale will operate during attending the American Mathematical Society/NU Math. the same hours as registration. The location of the Book Dept. meeting. The AMS is not responsible for rate changes Sale will be announced at a later date. or for the quality of the accommodations. Rates quoted AMS Editorial Activity: An acquisitions editor from do not include taxes. Hotels have varying cancellation or the AMS book program will be present to speak with early checkout penalties; be sure to ask for details when prospective authors. If you have a book project that you making your reservation. would like to discuss with the AMS, please stop by the book Best Western University Plaza, 1501 Sherman Ave., exhibit. Evanston, IL; 847-491-6400, toll-free reservations: 800-328-6786, fax: 847-328-3090; $89 single or double plus Parking tax (currently 13.5%). Four blocks to campus. Deadline for Vehicles may be parked without a permit and free of reservations is September 22,2004. charge (except in designated lots listed below) after 4:00 Comfort Inn, 9333 Skokie Blvd., Skokie, IL 60007; p.m. on Friday and before 7:30 a.m. on Monday. 847-679-4200, fax: 847-679-4218; $89 single/double plus tax (currently 9.5%). Approximately 4 miles to campus. Do not park in the following lots: Limited shuttle service to campus will be provided by the • Student Health (FH Permits Only) hotel. A complimentary hot buffet breakfast served daily. • University Police Deadline for reservations is September 22,2004. • Central Utility Plant Doubletree Hotel & Executive Meeting Center North Shore, 9599 Skokie Blvd., Skokie, IL 60077; 847-679-7000, • Tech Institute Lots and Drives toll-free reservations: 800-222-8733, fax: 847-679-9841; $79 • Foster-Walker Complex (R Spaces only)1' single or double plus tax (currently 12%). Approximately • Handicapped and Reserved Spaces 4 miles to camptts. Deadline for reservations is October 8,2004. 1'Not enforced Friday 4:00 p.m. to Monday 7:30 a.m. Hampton Inn and Suites, 5201 Old Orchard Road, Registration and Meeting Information Skokie, IL 60077; 847-583-1111, fax: 847-583-0300; $99 sin­ gle or double plus tax (currently 9.5%). Approximately 4 The registration desk will be open 7:30 a.m. to 4:30 p.m. miles to campus. Deadline for reservations is October 1, on Saturday and 8:00 a.m. to noon on Sunday. The loca­ 2004. tion of the registration desk and the location of the talks will be announced at a later date. Hilton Garden Inn Evanston, 1818 Maple Avenue, Evanston, IL 60201; 847-475-6400, fax: 847-475-6460; Registration fees: (payable on site only) $40/AMS mem­ $104 single or double plus tax (currently 13.5 %). Three bers; $60/nonmembers; $5/emeritus members, students, blocks to campus. Deadline for reservations is Septem­ or unemployed mathematicians. Fees are payable by cash, ber 24, 2004. check, VISA, MasterCard, Discover, or American Express.

854 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences

Social Event Sheridan Road. Refer to the campus maps at http://www.northwestern.edu/campus/maps/evanston_ The Department of Mathematics will host a reception on campus_map. html to find specific locations. Saturday evening (time and place to be announced). Taxi to Evanston Campus from O'Hare: Taxis are avail­ Travel able on a first-come, first-served basis from the lower-level curb front of all terminals. Shared ride service is available. By Air: Evanston is served by both Chicago's O'Hare Expect to spend approximately $35 to $40 for a taxi to International Airport and Midway Airport. Evanston. Announcing Delta Air Lines as the official airline for the Airport Shuttle from Midway: To take the Airport Shut­ sectional meeting in Evanston, Illinois! tle from Midway, you MUST make reservations one day in The following specially negotiated rates are available for advance by calling 312-454-7799. Check in with the Con­ this meeting and exclusively to mathematicians and their tinental Air Transport ticket agent located near the bag­ families for the period October 20, 2004, through gage claim area. Continental Airport Express offers daily October 27, 2004. shuttle service between Midway and north suburban lo­ Other restrictions/discounts may apply, and seats are cations from 6:00 a.m. to 11:30 p.m. Departures are about limited. every 15 minutes. The fare to the Evanston campus is Delta Air Lines is offering: about $28. The pickup location is located in front of the •A 5% discount off Delta's published round-trip fares main terminal building. within the continental United States, excluding A, D, I, U, Subway (the "L") from Midway: The Chicago Transit and T classes of service. Authority (CTA/elevated/"L") train provides transit service •A 10% discount off Delta's domestic published unre­ to downtown. There is an "L" stop at the airport. stricted round-trip coach fares (Y06/YR06) rates. No ad­ 1. Take the CTA Orange Line to the State/Lake stop in vance reservations or ticketing is required. Chicago's Loop. You willbe on the elevated level above State • An additional 5% bonus discount if you purchase Street and Lake Street at that location. your ticket 60 days or more prior to your departure through 2. Walk down the stairs to State Street. Meeting Network Reservations or your travel agent; online not applicable. 3. Walkabout a halfblock south and take the stairs down from State Street to the subway station. Take the Red Line To take advantage of these discounts and make imme­ north to Howard Street. diate reservations, call Delta Meeting Network at 800-241­ 6760 between 8:00 a.m. and 11:00 p.m. Eastern Standard 4. Howard Street is as far as the Red Line goes in the Time, Monday through Sunday, referencing File Number direction of Evanston, so you will need to get off the Red 205778A. Line train and transfer to a Purple Line train. Take the Pur­ ple Line north to the Foster Street stop in Evanston. Airport Shuttle from O'Hare: Continental Airport Ex­ press offers daily shuttle service from O'Hare Interna­ 5. Walk east, passing Sherman Avenue and Orrington tional Airport between downtown Chicago and north sub­ Avenue (about two and a half blocks), until you arrive at urban locations from 6:00 a.m. to 11:30 p.m. Departures Sheridan Road. Refer to the website http://www . are about every 5 to 10 minutes. The fare to Evanston is northwestern.edu/campus/maps/evanston_ approximately $20 for a standard passenger or $17 for stu­ campus_map. html to find specific locations. dents. You can reserve a seat in advance by calling 312­ Taxi: The fare from Midway to Evanston is approxi­ 454-7799, or you can make your reservation at the Conti­ mately $40. nental Air Transport ticket counter after you have collected Driving: From O'Hare International Airport: your baggage. 1. Take the main exit road from the airport, which turns Subway (the "L") from O'Hare: The Chicago Transit Au­ into 1-190, east for about a half mile. thority (CTA/elevated/"L") train provides rapid transit service from O'Hare to downtown and from downtown to 2. Exit right and go north on 1-294, the Tri-State Toll­ Evanston. way. You will be headed north toward Wisconsin, and shortly after entering you will need to pay a 50<1: toll. 1. Take the CTA Blue Line train from the O'Hare station to Washington Street in Chicago's Loop. 3. Go north about three miles to the Dempster Street eastbound exit. 2. Get off at Washington Street and walk through the pedestrian tunnel to the Washington Street stop on the Red 4. Take Dempster Street east approximately 10 miles Line. Take the Red Line north to Howard Street. through Park Ridge, Niles, Morton Grove, and Skokie into Evanston. 3. Howard Street is as far as the Red Line goes in the direction of Evanston, so you will need to get off the Red 5. At Chicago Avenue in Evanston, turn left and go Line train and transfer to a Purple Line train. Take the Pur­ north. ple Line north to the Foster Street stop in Evanston. 6. Main campus is approximately three-fourths of a 4. Walk east, passing Sherman Avenue and Orrington mile north at Chicago Avenue and Sheridan Road. Avenue (about two and a half blocks), until you arrive at

AUGUST 2004 NOTICES OF THE AMS 855 Meetings & Conferences

From Midway Airport: 7. Main campus is approximately three-fourths of a 1. From Midway take Cicero Avenue north to I-55 (the mile north at Chicago Avenue and Sheridan Road. Stevenson Expressway). To Evanston from the West or Northwest, via 1-88, 2. Turn right (northeast) on I-55 toward downtown 1-90, or 1-190: (Note: These directions also apply if travel­ Chicago. ing from O'Hare International Airport.) 3. Follow I-55 to Lake Shore Drive northbound. 1. Take 1-88,1-90, or 1-190 eastbound until you reach 4. Take Lake Shore Drive north through downtown northbound 1-294 (heading toward Wisconsin). Chicago and continue until Lake Shore Drive ends. Lake 2. Take 1-294 north to Dempster Street east. Shore Drive splits; you will need to be in one of the right 3. Follow Dempster Street approximately 10 miles two lanes to turn north on Sheridan Road. through the suburbs of Des Plaines, Park Ridge, Niles, 5. Turn right and continue north on Sheridan Road Morton Grove, Skokie, and Evanston. through Chicago into Evanston. 4. At Chicago Avenue in Evanston, turn left and go 6. Follow Sheridan Road north, which makes several north. right-hand turns to keep going north, until you reach the 5. Main campus is approximately three-fourths of a Northwestern campus. Refer to the campus maps to find mile north at Chicago Avenue and Sheridan Road. specific locations. To Evanston from the South or Southeast, via 1-94, To Evanston from the /north, via 1-94: (Note: Highway 1-90, 1-80, or I-57: is called 1-94 East, but runs north-south in Chicago met­ ropolitan area.) Once you reach the Chicago area, you want to go north on 1-94 West (Note: Highway is called 1-94 West, but runs 1. Take 1-94 East south to Skokie Highway. north-south in Chicago metro area.) If you are coming 2. Go south on Skokie Highway approximately 2 miles from: to Golf Road. 1-80/1-94: 1-94 splits from 1-80 south of Chicago and 3. Turn left (east) at Golf Road. heads north. 4. Follow Golf Road approximately 4 miles east into I-57: I-57 ends at and merges into 1-94 on the south side Evanston (the road name changes to Emerson Street in of Chicago. Evanston). 1-90: 1-90 merges with1-94 on the south side of Chicago. 5. Just past Maple Street inEvanston is a fork in the road; bear to the right to go onto Clark Street. Once you are northbo·und on 1-94 W: 6. Follow Clark Street through downtown Evanston. 1. When 1-94 splits from 1-90, bear to the right, staying You'll know you're going the correct way when you note on 1-94 (Edens Expressway). some landmarks: 2. Take 1-94 W north to Dempster Street east. • The Rebecca Crown Center, which houses the offices 3. Follow Dempster Street approximately 5 miles through of the Central Administration of Northwestern, on the the suburbs of Skokie and Evanston. north side of Clark Street (to your left). 4. At Chicago Avenue in Evanston, turn left and go •A Burger King restaurant on the south side of Clark north. Street (to your right). 5. Main campus is approximately three-fourths of a 7. The main campus is located at the corner of Chicago mile north at Chicago Avenue and Sheridan Road. Avenue and Sheridan Road. Car Rental: Special rates have been negotiated with To Evanston from the West or Southwest, via I-55 or Avis Rent A Car for the period October 16-0ctober 31, 1-80 to 1-94 West: (Note: Highway is called 1-94 West, but 2004. All rates include unlimited free mileage; beginning runs north-south in Chicago metro area.) at $24.99/day for a subcompact car at the weekend rate 1. If coming in on eastbound 1-80, follow 1-80 until the (the weekend rate is available from noon Thursday until interchange with I-55; take northbound I-55. Monday at 11:59 p.m.). Rates do not include state or local 2. Follow northbound I-55 until the interchange with surcharges, tax, optional coverages, or gas refueling 1-90/1-94; take northbound 1-90/1-94 (Kennedy Express­ charges. Renter must meet Avis's age, driver, and credit way). requirements. Make reservations by calling 800-331-1600 or online at www.avis.com. Nonweekend and weekly rates 3. When 1-94 splits from 1-90, bear to the right, staying are also available. Please quote Avis Discount Number on 1-94 W (Edens Expressway). B159266 when making reservations. 4. Take 1-94 W north to Dempster Street east. 5. Follow Dempster Street east approximately 5 miles Weather through the suburbs of Skokie and Evanston. October weather is generally pleasant and mild, with tem­ 6. At Chicago Avenue in Evanston, turn left and go peratures in the upper fifties and low sixties. Please visit north. http://wwwa.accuweather.com/adcbin/public/

856 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences

1oca1_index. asp?zi pcode=60201 for current and fu­ Mathematical Biology (Code: SS 13A), Jonathan E. Rubin ture weather conditions in Evanston. and Bard Ermentrout, University of Pittsburgh. Mathematical Finance (Code: SS 11A), David Saunders and John Chadam, University of Pittsburgh. Pittsburgh, Mathematical Modeling ofNonlinearPhenomena in Biology and Mechanics (Code: SS 6A), Anna Vainchtein and Pennsylvania William C. Troy, University of Pittsburgh. Universi.ty ofPittsburgh Modularity of Galois Representations and Serre's Conjec­ ture (Code: SS 14A), Mark E. T. Dickinson, University of November 6-7, 2004 Pittsburgh. Saturday - Sunday Multiscale Algorithms in Computational Fluid Dynamics Meeting #1002 (Code: SS SA), William J. Layton, University of Pittsburgh, and Anastasios Liakos, U.S. Naval Academy. Eastern Section Associate secretary: Lesley M. Sibner Multivariate Hypergeometric Functions: Combinatorial and Announcement issue of Notices: September 2004 Algebro-Geometric Aspects (Code: SS 9A), Eduardo Cat­ Program first available on AMS website: September 23, tani, University ofMassachusetts, Amherst, Alicia M. Dick­ 2004 enstein, Universidad de Buenos, Aires, and Laura Felicia Program issue of electronic Notices: November 2004 Matusevich, Harvard University. Issue of Abstracts: Volume 25, Issue 4 Partial DifferentialEquations andApplications (Code: SS 4A), Xinfu Chen and Dehua Wang, University of Pittsburgh. Deadlines PDE-Based Methods in Imaging and Vision (Code: SS 15A), For organizers: Expired Stacey E. Levine, Duquesne University, and Yunmei Chen, For consideration of contributed papers in Special Sessions: University of Florida. July 20, 2004 For abstracts: September 14, 2004 Trends in Operator Theory and Banach Spaces (Code: SS lOA), Christopher J. Lennard and Thomas A. Metzger, Uni­ Invited Addresses versity of Pittsburgh. Jeffrey F. Brock, Brown Univers~ty, Title to be announced. Der-Chen Chang, Georgetown University, Title to be an­ nounced. Atlanta, Georgia Robert Schapire, Princeton University, Title to be an­ Atlanta Marriott Marquis andHyatt Regency nounced. Atlanta Ofer Zeitouni, University of Minnesota, Minneapolis, Title to be announced. January 5-8,2005 Wednesday - Saturday Special Sessions Convexity and Combinatorics (Code: SS 2A), James F. Meeting #1 003 Lawrence and Valeriu Soltan, George Mason University. Joint Mathematics Meetings, including the 111th Annual Geometric Analysis and Partial Differential Equations in Meeting of the AMS, 88th Annual Meeting of the Mathe­ Subelliptic Structures (Code: SS 12A), Cristian E. Gutierrez, matical Association ofAmerica (MAA), annual meetings of Temple University, Guozhen Lu, Wayne State University, the Association of Women in Mathematics (A VVl\1) and the and Juan J. Manfredi, University of Pittsburgh. National Association of Mathematicians (NAM), and the winter meeting of the Association ofSymbolic Logic (ASL). Graph Polynomials (Code: SS 8A), E. Glen Whitehead Jr., University of Pittsburgh. Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2004 The History of Mathematics (Code: SS 3A), Robert E. Program first available on AMS website: November 1, 2004 Bradley, Adelphi University, and Lawrence A. D'Antonio, Program issue of electronic Notices: January 2005 Ramapo College of New Jersey. Issue of Abstracts: Volume 26, Issue 1 Invariants of Knots and 3-Manifolds (Code: SS 1A), Marta M. Asaeda, University of Maryland, Jozef H. Przy­ Deadlines tycki, George Washington University, and Adam S. Sikora, For organizers: Expired SUNY at Buffalo. For consideration of contributed papers in Special Sessions: Knots and Macromolecules (Code: SS 7A), Kenneth C. Mil­ August 10, 2004 lett, University of California Santa Barbara, and Eric J. For abstracts: October 5, 2004 Rawdon, Duquesne University. For summaries ofpapers to MAA organizers: September 14, 2004

AUGUST 2004 NOTICES OF THE AMS 857 Meetings & Conferences

Joint Invited Addresses Analysis Problems in Modern Physics (Code: SS 30A), Steven M. Zelditch, Johns Hopkins University. Andrea L. Bertozzi, University of California Los Angeles, Title to be announced (AMS-MAA Invited Address). Arithmetic Algebraic Geometry (Code: SS 32A), Matthew H. Baker and Dino J. Lorenzini, University of University of California Berkeley, Title Bernd Sturmfels, Georgia. to be announced (AMS-MAA Invited Address). Commutative Algebra (Code: SS 20A), Srikanth B. Iyengar, AMS Invited Addresses University of Missouri, Sean M. Sather-Wagstaff, Univer­ sity of Illinois at Urbana-Champaign, Anurag K. Singh, Ingrid Daubechies, Princeton University, Title to be an­ Georgia Institute of Technology, and Carolyn A. Yackel, nounced (Josiah-' Willard Gibbs Lecture). Mercer University. Benylonel, University ofWisconsin, Title to be announced. Complex and Functional Analysis (Code: 5S 25A), Mihaly Bruce A. Kleiner, University of Michigan, Ann Arbor, Title Bakonyi, Georgia State University, and Imre Patyi, to be announced. University of California San Diego. Robert K. Lazarsfeld, University of Michigan, Title to be Current Events (Code: SS IA), David Eisenbud, announced (Colloquium Lectures). Mathematical Sciences Research Institute and University Gunther Uhlmann, University of Washington, Title to be of California Berkeley. announced. Design Theory and Graph Theory (Code: SS 24A), Mike Avi Wigderson, Institute for Advanced Study, Title to be Daven, Mount Saint Mary College, and Atif A. Abueida, Uni­ announced. versity of Dayton. Steven M. Zelditch, Johns Hopkins University, Title to be D-Modules (Code: SS 14A), Steven Sperber, University of announced. Minnesota, Minneapolis, and Uli Walther, Purdue Univer­ sity. MAA Invited Addresses Dynamic Equations on Time Scales; Integer Sequences and Georgia Benkart, University of Wisconsin, Madison, Title Rational Maps (Code: SS 26A), Martin J. Bohner, Univer­ to be announced. sity ofMissouri-Rolla, Marc A. Chamberland, Grinnell Col­ Erik D. Demaine, Massachusetts Institute of Technology, lege, Billur Kaymak<;alan, Georgia Southern University, Title to be announced. Allan C. Peterson, University of Nebraska-Lincoln, and Diana M. Thomas, Montclair State University. (AMS-SIAM) Fernando Q. Gouvea, Colby College, Title to be announced. Dynamics ofMapping Class Groups on Moduli Spaces (Code: Steven G. Krantz, Washington University, Title to be SS IDA), Richard J. Brown, American University. announced. History ofMathematics (Code: SS 3A), Joseph W. Dauben, Ravi D. Vakil, Stanford University, Title to be announced. Lehman College (CUNY), Patti Hunter, Westmont College, Robin j. Wilson, The Open University, Title to be and Karen H. Parshall, University of Virginia. (AMS-MAA) announced. Integrable Systems and Special Functions (Code: SS 3IA), Andras Balogh, University ofTexas-Pan American, Mourad Invited Addresses ofOther Organizations E. H. Ismail, University of Central Florida, Wen-Xiu Ma, Pavel Pevzner, University of California San Diego, Title to University of South Florida, and Zhijun Qiao, Los Alamos be announced (SIAM). National Laboratory. (AMS-SIAM) AMS Special Sessions Inverse Spectral Geometry (Code: SS 16A), Carolyn S. Gor­ don, DartmouthUniversity, and Ruth Gornet and Peter A. Some sessions are cosponsored with other organiza­ Perry, University of Kentucky. tions. These are noted within the parentheses at the In the Wake ofJacobi andHamilton 200 Years Later(Code: end of each listing, where applicable. SS 37A), Maria-Clara Nucci, University of Perugia, and Algebraic Geometry Codes and Quantum Codes (Code: SS Pavel Winternitz, Centre de Recherches Mathematiques, 13A), Shuhong Gao and Gretchen L. Matthews, Clemson Universite de Montreal. University. Mathematical Image Processing (Code: SS 36A), jianhong Algorithmic 'Algebraic and Analytic Geometry (Code: SS Shen, University of Minnesota, Minneapolis, and Tony F. 34A), Saugata Basu, Georgia Institute of Technology, Chan, University of California Los Angeles. (AMS-SIAM) Victoria A. Powers, Emory University, Mika K. Sepala, Mathematical Sciences Contributions to the Biomedical Sci­ Florida State University, Tanush T. Shaska, University of ences (Code: SS 29A), Peter D. March, Ohio State Univer­ Idaho, and Emil J. Volcheck, National Security Agency. sity, De Witt L. Sumners, Florida State University, and Analysis andApplications in NonlinearPartial Differential John Whitmarsh, The National Institutes of Health. (AMS­ Equations (Code: SS 27A), Michael T. Lacey, Jason L. Met­ MAA) calfe, Gerd Mockenhaupt, Ronghua Pan, and Andrzej j. Mathematical Sciences Research for the Department of Swiech, Georgia Institute of Technology. (AMS-SIAM) Energy's Computational Biology Needs (Code: SS 7A),

858 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences

Jennifer R. Slimowitz, Board on Mathematical Sciences and Riemannian Geometry (Code: SS llA), Igor Belegradek, Their Applications. Georgia Institute of Technology, and Mohammad Ghomi, Mathematicians' Work on Mathematics Education (Code: SS Georgia Institute ofTechnology and Pennsylvania State Uni­ 19A), William G. McCallum, University of Arizona. (AMS­ versity. MAA) Spaces of Vector-Valued Functions (Code: SS 22A), Terje Mathematics and Education Reform (Code: SS 2A), HDim, Florida Atlantic University, and David A. Robbins, William H. Barker, Bowdoin College, Jerry L. Bona and Trinity College. Naomi Fisher, University of Illinois at Chicago, Kenneth Stochastic, Large-Scale, and Hybrid Systems (Code: SS 12A), C. Millett, University of California Santa Barbara, and Bon­ A. S. Vatsala, University of Louisiana at Lafayette, and nie Saunders, University of Illinois at Chicago. (AMS-MAA­ G. S. Ladde, University of Texas at Arlington. (AMS-SIAM) MER) Theoretical and Computational Aspects ofInverse Problems Mathematics and Mathematics Education in Fiber Arts (Code: SS 4A), Gunther Uhlmann, University of Washing­ (Code: SS 21A), Sarah-Marie Belcastro, Xavier University, ton, and David L. Colton, University of Delaware. (AMS­ and Carolyn A. Yackel, Mercer University. SIAM) Modular Representation Theory of Finite and Algebraic Topics in Geometric Function Theory (Code: SS 33A), Groups (Code: SS 8A), David J. Hemmer, University of Abdelkrim Farouk Brania, Morehouse College, David A. Toledo, and Cornelius Pillen, University of SouthAlabama. Herron, University of Cincinnati, and Shanshuang Yang, Emory University. Nonsmooth Analysis in Variational and Imaging Problems Tropical Geometry (Code: SS 3SA), Michael Develin and (Code: SS 17A), M. Zuhair Nashed, University of Central Bernd Sturmfels, University of California Berkeley. (AMS­ Florida, and University of Innsbruck. Otmar Scherzer, MAA) (AMS-SIAM) Orthogonal Polynomials-Random Matrices-Integrable Sys­ tems: Interdisciplinary Aspects (Code: SS 38A), Jinho Baik, Bowling Green, University of Michigan, Ann Arbor, Steven B. Damelin, Georgia Southern University, and Peter D. Miller, Univer­ Kentucky sity of Michigan, Ann Arbor. (AMS-SIAM) Western Kentucky University Quantum Topology (Code: SS ISA), Stavros Garoufalidis and T. T. Q. Le, Georgia Institute of Technology. March 18-19,2005 Radon Transform and Inverse Problems (Code: SS SA), Friday - Saturday Adel Faridani, Oregon State University, Gestur Olafsson, Louisiana State University, and Todd Quinto, Tufts Uni­ Meeting #1 004 versity. Southeastern Section Associate secretary: John L. Bryant Reaction Diffusion Equations and Applications (Code: SS Announcement issue of Notices: To be announced 28A), Xu-Yan Chen, Georgia Institute of Technology, Program first available on AMS website: To be announced Yuanwei Qi, University of Central Florida, Junping Shi, The Program issue of electronic Notices: To be announced College of William and Mary, and Ratnasingham Shivaji, Issue of Abstracts: Volume 26, Issue 2 Mississippi State University. (AMS-SIAM) Recent Advances in Mathematical Ecology (Code: SS 18A), Deadlines Semen Koksal, Florida Institute of Technology, Sebastian For organizers: July 19, 2004 Schreiber, The College ofWilliam & Mary, and Robert van For consideration of contributed papers in Special Sessions: Woesik, Florida Institute of Technology. (AMS-SIAM) To be announced Representations of Lie Algebras (Code: SS 23A), Brian D. For abstracts: To be announced Boe, University of Georgia, Ben L. Cox, College of Charleston, Vyacheslav M. Futorny, Universidade de Sao Newark, Delaware Paulo, William A. Graham, University of Georgia, Duncan J. Melville, St Lawrence University, and Daniel K. Nakano, University ofDelaware University of Georgia. April 2-3, 2005 Research in Mathematics by Undergraduates (Code: SS Saturday - Sunday 9A), DarrenA. Narayan and Tamara A. Burton, Rochester Institute of Technology, Michael J. Fisher, California State Meeting #1 005 University, Fresno, and Carl V. Lutzer, Rochester Institute Eastern Section of Technology. (AMS-MAA-SIAM) Associate secretary: Lesley M. Sibner Reverse Mathematics (Code: SS 6A), Jeff L. Hirst, Ap­ Announcement issue of Notices: To be announced palachian State University, and Reed Solomon, University Program first available on AMS website: To be announced of Connecticut. (AMS-ASL) Program issue of electronic Notices: To be announced

AUGUST 2004 NOTICES OF THE AMS 859 Meetings & Conferences

Issue of Abstracts: Volume 26, Issue 2 Mattias Jonsson, University of Michigan, Title to be an­ nounced. Deadlines Nicolas Monod, University of Chicago, Title to be an­ For organizers: September 2, 2004 nounced. For consideration of contributed papers in Special Sessions: To be announced Hee Dh, California Institute ofTech, Title to be announced. For abstracts: To be announced Special Sessions Invited Addresses Classical and Differential Galois Theory (Code: SS 3A), Xiu Xiong Chen, University of Wisconsin, Title to be an­ Lourdes Juan and Arne Ledet, Texas Tech University, and nounced. Andy R. Magid, University of Oklahoma. Anna Gilbert, AT&T Labs Research, Title to be announced. Differential Geometry and Its Applications (Code: SS 2A), Alex Lubotzky, Hebrew University of Jerusalem, Title to Josef F. Dorfmeister, Munich University of Technology, be announced. Magdalena D. Toda, Texas Tech University, and Hongyou Lorenz Schwachhoefer, University of Dortmund, Title to Wu, Northern Illinois University. be announced. Homological Algebra and Its Applications (Code: 5S 4A), Alex Martsinkovsky, Northeastern University, and Special Sessions Mara D. Neusel, Texas Tech University. Asymptotic Behavior ofEvolution Equations (Code: SS 4A), Recent Advances in Complex Function Theory (Code: SS SA), Gaston M. N'Guerekaya, Morgan State University, and Brock Williams, Roger Barnard, and Kent Pearce, Texas Nguyen Van Minh, James Madison University. Tech University. Homotopy Theory (in Honor of Donald M. Davis's and Topology of Continua (Code: SS 1A), Wayne Lewis, Texas Martin Bendersky's 60th Birthdays) (Code: SS 1A), Ken­ Tech University. neth G. Monks, University of Scranton, and W. StephenWil­ son, Johns Hopkins University. Mathematical Methods in Electromagnetic Wave Propagation Santa Barbara, (Code: SS 3A), Fioralba Cakoni and Peter B. Monk, University of Delaware. California Singular Analysis and Spectral Theory ofPartial Differential University ofCalifornia Santa Barbara Equations (Code: SS 2A), Juan B. Gil, Pennsylvania State Uni­ versity, Altoona, and Gerardo A. Mendoza, Temple Uni­ April 16-17, 2005 versity. Saturday - Sunday Meeting #1 007 Lubbock, Texas Western Section Texas Tech University Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced April 8-1 0, 2005 Program'first available on AMS website: To be announced Friday - Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 26, Issue 3 Meeting #1 006 Central Section Deadlines Associate secretary: Susan J. Friedlander For organizers: September 16, 2004 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Sessions: Program first available on AMS website: To be announced To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: Volume 26, Issue 3 Invited Addresses Deadlines Mei-Chu Chang, University of California Riverside, To be For organizers: September 9, 2004 announced. For consideration of contributed papers in Special Sessions: Mischa Kapovich, University of California Davis, To be an­ To be announced For abstracts: To be announced nounced. Mihai Putinar, University of California Santa Barbara, To Invited Addresses be announced. Nikolai Ivanov, Michigan State University, Title to be an­ James Sethian, University of California Berkeley, To be an­ nounced. nounced.

860 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences

Special Sessions che Hochschule, and Gunter M. Ziegler, Technical Uni­ versity of Berlin. Dynamical Systems in Neuroscience (Code: SS lA), Eugene M. Izhikevich, The Neurosciences Institute. Functional Analytic and Complex Analytic Methods in Lin­ History ofMathematics (Code: SS 2A), Shawnee L. McMurran, ear Partial Differential Equations, R. Meise, University of California State University, San Bernardino, and James J. Dusseldorf, B. A. Taylor, University of Michigan, and Tattersall, Providence College. Dietmar Vogt, University of Wuppertal. RecentAdvances in Combinatorial Number Theory (Code: History ofMathematics: Mathematics and War, Thomas W. SS 3A), Mei-Chu Chang, University of California Riverside, Archibald, Acadia"University, John H. McCleary, Vassar and Van Ha Vu, University of California San Diego. College, Moritz Epple, University of Stuttgart, and Norbert Schappacher, Technische Universitat Darmstadt. Homotopy Theory, Paul G. Goerss, NorthwesternUniversity, Mainz, Germany Hans-Werner Henn, Institut de Recherche Mathematique Avancee, Strasbourg, and Stefan Schwede, Universitat June 16-19,2005 Bonn. Thursday - Sunday HopfAlgebras and Quantum Groups, Susan Montgomery, University of Southern California, and Hans-Jurgen Schnei­ Meeting #1 008 der, University of Munich. Joint International Meeting with the Deutsche Mathematiker­ Vereinigung (DMV) and the Oesterreichische Mathematis­ Mathematics Education, Gunter Torner, Universitat Duis­ che Gesellschaft (OMG) burg-Essen and Alan Schoenfeld, School of Education, Associate secretary: Susan J. Friedlander Berkeley. Announcement issue of Notices: To be announced Nonlinear Waves, Herbert Koch, University of Dortmund, Program first available on AMS website: To be announced and Daniel I. Tataru, University of California Berkeley. Program issue of electronic Notices: To be announced Stochastic Analysis on Metric Spaces, Laurent Saloff-Coste, Issue of Abstracts: To be announced Cornell University, Karl-Theodor Sturm, University of Deadlines Bonn, and Wolfgang Woess, Graz Technical University. For organizers: To be announced Topology ofManifolds, Matthias Kreck, University of Hei­ For consideration of contributed papers in Special Sessions: delberg, and Andrew Ranicki, University of Edinburgh. To be announced For abstracts: To be announced Annandale-on­ Invited Addresses Helene Esnault, University of Essen, Title to be announced. Hudson, New York Richard Hamilton, Columbia University, Title to be Bard College announced. Michael J. Hopkins, Massachusetts Institute ofTechnology, October 8-9, 2005 Title to be announced. Saturday - Sunday Christian Krattenthaler, University of Lyon, Title to be Meeting #1 009 announced. Eastern Section Frank Natterer, University of Muenster, Title to be Associate secretary: Lesley M. Sibner announced. Announcement issue of Notices: To be announced Horng-Tzer Yau, New York University and Stanford Program first available on AMS website: To be announced University, Title to be announced. Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 26, Issue 4 Special Sessions Algebraic Combinatorics, Patricia Hersh, University of Deadlines Michigan, Christian Krattenthaler, Universite Claude For organizers: March 8, 2005 Bernard Lyon-I, and Volkmar Welker, Philipps University For consideration of contributed papers in Special Sessions: Marburg. To be announced Algebraic Geometry, Yuri Tschinkel, Georg-August-Uni­ For abstracts: To be announced versitat Gottingen, and Brendan E. Hassett, Rice Univer­ sity. Invited Addresses Discrete Geometry, Jacob Eli Goodman, The City College Persi Diaconis, Stanford University, Title to be announced of New York, CUNY, Emo Welzl, Eidgenossiche Technis- (Erdos Memorial Lecture).

AUGUST 2004 NOTICES OF THE AMS 861 Meetings & Conferences Johnson City, Eugene, Oregon Tennessee University of Oregon East Tennessee State University November 12-1 3, 2005 Saturday - Sunday October 15-16, 2005 Meeting #1 012 Saturday - Sunday Western Section Meeting #1 01 0 Associate secretary: Michel L. Lapidus Southeastern Section Announcement issue of Notices: To be announced Associate secretary: John L. Bryant Program first available on AMS website: To be announced Announcement issue of Notices: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 26, Issue 4 Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: Volume 26, Issue 4 For organizers: April 12, 2005 Deadlines For consideration of contributed papers in Special Sessions: To be announced For organizers: March 15,2005 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced For abstracts: To be announced Taiwan December 14-18,2005 Lincoln, Nebraska Wednesday - Sunday University ofNebraska in Lincoln Meeting #1 013 October 21-22, 2005 First Joint International Meeting between the AMS and the Friday - Saturday Taiwanese Mathematical Society. Associate secretary: John L. Bryant Meeting #1011 Announcement issue of Notices: To be announced Central Section Program first available on AMS website: To be announced Associate secretary: Susan J. Friedlander Program issue of electronic Notices: To be announced Announcement issue of Notices: August 2005 Issue of Abstracts: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 26, Issue 4 For organizers: To be announced For consideration of contributed papers in Special Sessions: Deadlines To be announced For organizers: March 22,2005 For abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced San Antonio, Texas For abstracts: To be announced Henry B. Gonzalez Convention Center Invited Addresses January 12-1 5, 2006 Barry Masur, Universiy of Illinois at Chicago, Title to be Thursday - Sunday announced. Joint Mathematics Meetings, including the 112th Annual Alejandro Uribe, University of Michigan, Title to be an­ Meeting of the AMS, 89th Annual Meeting of the Mathe­ nounced. matical Association of America, annual meetings of the JudyWalker, University ofNebraska, Title to be announced. Association for Women in Mathematics (A VV1\1) and the Na­ tional Association ofMathematicians (NAM), and the win­ Jack Xin, University of Texas, Title to be announced. ter meeting ofthe Association for Symbolic Logic (ASL). Special Sessions Associate secretary: John L. Bryant Announcement issue of Notices: October 2005 Algebraic Geometry (Code: SS lA), Brian Harbourne, Uni­ Program first available on AMS website: To be announced versity of Nebraska-Lincoln, and Bangere P. Purnaprajna, Program issue of electronic Notices: January 2006 University of Kansas. Issue of Abstracts: To be announced

862 NOTICES OF THE AMS VOLUME 51, NUMBER 7 Meetings & Conferences

Deadlines For organizers: April 12, 2005 San Diego, California For consideration of contributed papers inSpecial Sessions: San Diego Convention Center To be announced For abstracts: To be announced January 6-9, 2008 For summaries of papers to MAA organizers: To be an­ Sunday - Wednesday nounced Joint Mathematics Meetings, including the 114th Annual Meeting of the AMS, 91st Annual Meeting of the Mathe­ San Francisco, matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A WM) and the California National Association of Mathematicians (NAM), and the winter meeting ofthe Association for Symbolic Logic (ASL). San Francisco State University Associate secretary: Michel L. Lapidus April 29-30, 2006 Announcement issue of Notices: October 2007 Program first available onAMS website: November 1, 2007 Saturday - Sunday Program issue of electronic Notices: January 2008 Western Section Associate secretary: Michel L. Lapidus Issue of Abstracts: Volume 29, Issue 1 Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced For organizers: April 6, 2007 Issue of Abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For summaries of papers to MAA organizers: To be an­ For consideration of contributed papers inSpecial Sessions: nounced To be announced For abstracts: To be announced Washington, District New Orleans, of Columbia Louisiana Marriott Wardman Park Hotel and Omni Shoreham Hotel New Orleans Marriott and Sheraton New Orleans Hotel January 7-10,2009 Wednesday - Saturday January 4-7,2007 Joint Mathematics Meetings, including the 115th Annual Thursday - Sunday Meeting of the AMS, 92nd Annual Meeting of the Mathe­ Joint Mathematics Meetings, including the 113th Annual maticalAssociation ofAmerica (MAA), annual meetings of Meeting of the AMS, 90th Annual Meeting of the Mathe­ the Association for Women in Mathematics (A WM) and the matical Association ofAmerica (MAA), annual meetings of National Association of Mathematicians (NAM), and the the Association for Women in Mathematics (AWM) and the winter meeting ofthe Association for Symbolic Logic (ASL). National Association of Mathematicians (NAM), and the Associate secretary: Lesley M. Sibner winter meeting ofthe Association for Symbolic Logic (ASL). Announcement issue of Notices: October 2008 Associate secretary: Susan J. Friedlander Program first available on AMS website: November 1, 2008 Announcement issue of Notices: October 2006 Program first available on AMS website: To be announced Program issue of electronic Notices: January 2009 Program issue of electronic Notices: January 2007 Issue of Abstracts: Volume 30, Issue 1 Issue of Abstracts: To be announced Deadlines Deadlines For organizers: April 7, 2008 For organizers: April 4, 2006 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced For summaries of papers to MAA organizers: To be an­ For summaries of papers to MAA organizers: To be an­ nounced nounced

AUGUST 2004 NOTICES OF THE AMS 863 Meetings and Conferences of the AMS

Associate Secretaries ofthe AMS Western Section: Michel L. 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: 1api dus@math. ucr. edu; telephone: 909­ e-mail: lsi bner@duke. pol y. edu; telephone: 718-260-3505. 787-3113. Southeastern Section: John L. Bryant, Department of Math­ Central Section: SusanJ. Friedlander, Department of Math­ ematics, Florida State University, Tallahassee, FL 32306-4510; ematics, University of Illinois at Chicago, 851 S. Morgan (M/C e-mail: bryant@math. fsu. edu; telephone: 850-644-5805. 249), Chicago, IT.. 60607-7045; e-mail: susan@math. nwu. edu; tele­ phone: 312-996-3041.

The Meetings and Conferences section of the Notices 2007 gives information on all AMS meetings and conferences January 4-7 New Orleans, Louisiana p.863 approved by press time for this issue. Please refer to the page Annual Meeting numbers cited in the table of contents on this page for more 2008 detailed information on each event. Invited Speakers and January 6-9 San Diego, California p.863 Special Sessions are listed as soon as they are approved by Annual Meeting the cognizant program committee; the codes listed are needed 2009 for electronic abstract submission. For some meetings the list January 7-10 Washington, DC p.863 may be incomplete. Information in this issue may be dated. Annual Meeting Up-to-date meeting and conference information can be found at WWW. ams. orgjmeeti ngsj .

Meetings: Important Information regarding AMS Meetings Potential organizers, speakers, and hosts should refer tc 2004 page 84 in the January 2004 issue of the Notices for genera] October 16-17 Nashville, Tennessee p.849 information regarding participation in AMS meetings and October 16-17 Albuquerque, New Mexico p.851 conferences. October 23-24 Evanston, Illinois p.853 November 6-7 Pittsburgh, Pennsylvania p.857 Abstracts Several options are available for speakers submitting 2005 abstracts, including an easy-to-use interactive Web form. Nc January 5-8 Atlanta, Georgia p.857 knowledge of0T£X is necessary to submit an electronic form, Annual Meeting although those who use 0T£X may submit abstracts with March 18-19 Bowling Green, Kentucky p.859 such coding, and all math displays and similarily coded ma­ April 2-3 Newark, Delaware p.859 terial (such as accent marks in text) must be typeset in 0I'£X, April 8-10 Lubbock, Texas p.860 To see descriptions of the forms available, visit http: j J April 16-17 Santa Barbara, California p.860 www. ams. orgjabstractsji nstructi ons. html, or send maiJ to abs-submi t@ams. org, typing he1p as the subject line; de· June 16-19 Mainz, Germany p.861 scriptions and instructions on how to get the template ofyow October 8-9 Annandale-on-Hudson, choice will be e-mailed to you. New York p.861 Completed email abstracts should be sent to abs-submi t~ October 15-16 Johnson City, Tennessee p.862 ams. org, typing submi ssi on as the subject line. Question~ October 21-22 Lincoln, Nebraska p.862 about abstracts may be sent to abs-i nfo@ams. org. November 12-13 Eugene, Oregon p.862 Paper abstract forms may be sent to Meetings & Confer­ ences Department, AMS, P.O. Box 6887, Providence, RI 02940, 2006 There is a $20 processing fee for each paper abstract. TherE January 12-15 San Antonio, Texas p.862 is no charge for electronic abstracts. Note that all abstract dead· Annual Meeting lines are strictly enforced. April 29-30 San Francisco, California p.863 Close attention should be paid to specified deadlines in thi~ issue. Unfortunately, late abstracts cannot be accommodated,

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