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of the American Mathematical Society May 2001 Volume 48, Number 5

A New Solution to the Three-Body Problem page 471 Thomas H. Wolff (1954-2000) page 482

Choreography for Six Suns (see page 481) It All Adds Up At Maple®

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Feature Articles

471 A New Solution to the Three-Body Problem Richard Montgomery Somewhere in the universe do three suns chase each other in a figure-eight pattern? This article explains a recent proof of the mathematical existence of such an orbit.

482 Thomas H. Wolff (1954-2000) Lennart Carleson, Sun-Yung Alice Chang, Peter W. ]ones, Markus Keel, Peter D. Lax, Nikolai Makarov, Donald Sarason, Wilhelm Schlag, and Barry Simon Colleagues commemorate an outstanding analyst's life and work.

Communications Commentary 491 The Growth of the Professional 469 Opinion Master's in Mathematics 4 70 Letters to the Editor Sheila Tobias, Charles MacCluer, and Ralph Svetic 498 The Universal Computer: The Road from Leibniz to Turing-A Book 502 Connes Receives 2001 Crafoord Review Prize Reviewed by Brian E. Blank 503 Arnold and ShelahReceive 2001 Wolf Prize 505 2001JPBMCommunicationsAward 507 MAA Awards Presented in New Orleans 509 AWMAwardsPresentedinNew Orleans Notices Departments of the American ll!athematical Societv

EDITOR: Harold P. Boas Mathematics People ...... 5 11 ASSOCIATE EDITORS: Nesterov Wins Dantzig Prize, Kleinberg Wins National Academy of Susanne C. Brenner, Bill Casselman (Covers Editor), Robert J. Daverman, Nathaniel Dean, Rick Durrett, Sciences Award, Trevisan Awarded Oberwolfach Prize, Yang Wins Susan Friedlander, Robion Kirby, Elliott H. Ueb, 2001 Faisallnternational Prize, Ziegler Awarded Leibniz Prize, Andy Magid, Judith Roitman, Mark Saul, Karen E. Smith, Audrey Terras, Usa Traynor Humboldt Foundation Research Awards, AIM Five- Year Fellow SENIOR WRITER and DEPUTY EDITOR: Announced, ONR Young Investigators Awards Announced, Allyn Jackson Correction, Deaths. MANAGING EDITOR: Sandra Frost CONTRIBUTING WRITER: Elaine Kehoe Mathematics Opportunities ...... 514 PRODUCTION ASSISTANT: Muriel Toupin AWM Workshops for Women Graduate Students and Postdocs, PRODUCTION: Marcia Almeida, Kyle Antonevich, Nominations Sought for Peter Gruber Prize in Cosmology. Siulam Fernandes, Thomas J. Kacvinsky, Lori Nero, Donna Salter, Deborah Smith, Peter Sykes Inside the AMS ...... 51 5 ADVERTISING SALES: Leonard Moorehead, AMS Epsilon Fund Makes Awards. Anne Newcomb Reference and Book List...... 516 SUBSCRIPTION INFORMATION: Subscription prices for Volume 48 (2001) are $430 list; $272 ...... 527 institutional member; $204 individual member. (The Mathematics Calendar subscription price for members is included in the annual dues.) A late charge of 10% of the New Publications Offered by the AMS ...... 532 subscription price will be imposed upon orders received from nonmembers after January 1 of the Publications of Continuing Interest ...... 544 subscription year. Add for postage: Surface delivery outside the United States and India-$17; in India­ $38; expedited delivery to destinations in North Classifieds ...... 546 America-$35; elsewhere-$70. Subscriptions and orders for AMS publications should be addressed to Meetings and Conferences Table of Contents ...... 560 the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904. All orders must be prepaid. ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be faxed to 401-331-3842 (indicate "Notices advertising" on fax cover sheet). SUBMISSIONS: Articles and letters may be sent to the editor by e-mail at noti ces@math. tamu. edu, by fax at 979-845-6028, or by postal mail at Department of Mathematics, Texas A&M University, College Station, TX 77843-3368. E-mail is preferred. Correspondence with the managing editor may be sent to noti ces@ams. org. For more information, see the section "Reference and Book Ust". NOTICES ON e-MATH: Most of this publication is available electronically through e-MATH, the Society's resource for delivering electronic products and services. To access the Notices on e-MATH, use the URL: http: I /www. ams. org/noti ces/. Those with VT100-type terminals or without WWW browsing software can connect toe-MATH via Telnet (tel net e-math. ams. org; login and password are "e-math"). From· the [NoticesoftheAmericanMathematicalSocietyispublished monthly except bimonthly in June/ July by the American AMS Sec:r~t,•ry . Mathematical Society at 201 Charles Street, Providence, Rl 02904-2294. Periodicals postage paid at Providence, Rl, and additional mailing offices. POSTMASTER: Send ad­ OfficersoftheSociety~OOOand 2001 ...... 520 dress change notices to Notices of the American Mathe­ matical Sodety, P.O. Box 6248, Providence, Rl 02940-6248.] Publication here of the Society's street address and the Call for Nominations for 2001 AMS-MAA-SIAM Frank and other information in brackets above is a technical re­ Brennie Morgan Prize ...... 521 quirement of the U.S. Postal Service. Tel: 401-45 5-4000; e-mail: noti ces@ams. org. ©Copyright 2001 by the AMS Standard Cover Sheet ...... 522 American Mathematical Society. All rights reserved. Printed in the United States of America. Call for Nominations for 2002 Bocher Memorial Prize, Frank The paper used in this journal is acid-free and Cole Prize in Number Theory, Levi L. Conant Prize, falls within the guidelines established Nelson to ensure permanence and durability. and Distinguished Public Service Award ...... 525 Opinion

In Praise of the Bull Historically, many mathematical definitions have been the outcome of significant discussion. The definition of Session an abstract group or a topology did not solidify into the pre­ sent form until the beginning of the last (twentieth) Once, I resolved to teach myself German. On page five of a text­ century. The definition of a continuous function emerged book I found the explanation, "German is a declined from more than 150 years of dialogue among mathemati­ language .. .. " The next page contained a set of paradigms for de­ cians who are remembered, partly, for just this dialogue. clension of German nouns and articles. On the bottom of the As a teacher I find that the "bull session" is among my most page the author had written, "So now there should be no ques­ useful tools. One student can formulate a question that an­ tion about the declension of the German noun." other has been struggling to ask. A student can give an ex­ But of course there were questions. Anyone who has struggled to learn a language will recognize that the matter of planation in slightly different words from the teacher, which grammatical form is not settled until it has been integrated on will reach some students who did not get the idea the first a very low cognitive level, until it has been grasped by both the time. Students can ask each other questions that they might mind and the heart. Some of this process can happen through be embarrassed to ask the teacher. Most importantly, students introspection, but a powerful tool, for many, is discourse with are exposed to a variety of ways to express the understand­ others. ing encapsulated in the definition. This column is about learning mathematics, not German. It The importance of this process is not limited to is prompted by an anonymous quote cited in these pages: I "A definitions. I find that dialogue is of great importance in help­ definition is not something that can be discovered in a bull ses­ ing students integrate axioms, theorems, conjectures, and sion." My years in the classroom have convinced me otherwise: proofs. Again, I am guided partly by the history of the sub­ that dialogue with other students (what the anonymous writer ject. The classic expositions and results did not spring from calls a "bull session") is in fact a powerful tool for learning math­ their creator's forehead full grown. They developed as a re­ ematics, and particularly mathematical definitions, and that sult of deep reflection and prolonged discourse. the quotation conveys a serious misunderstanding of the Why is this view of mathematics so rarely taken by math­ processes of teaching and learning. ematicians? I can think of two reasons. First, the use of dis­ A definition is logically arbitrary, a matter of convenience. course in the classroom is sometimes misunderstood to im­ On one level, "learning" it can be accomplished by simple ply that "anything goes". No: the aim of discourse, however wild memorization. But the meaning of a definition has not really and wide-ranging, is to come to an understanding of a state­ been learned until it has been integrated on a very low cogni­ ment that others have already arrived at. The role of the tive level, until it has been grasped by both the mind and the teacher is to channel the discussion so that it advances this heart. Some of this process can happen through introspection, process. A fundamental and hard-won skill of the master but a more important path, for many, is discourse with oth­ teacher is to manage the tension between this role and the need ers. to get students talking about their own, sometimes useless, Here are some examples of definitions, drawn from a va­ conceptions. riety of disciplines: The second reason that the social nature of mathematical -An ordered pair (a, b) is the set {a, {a, b}}. discovery is overlooked is that the process often does not -A function f: A~ B is a set of ordered pairs (a, b), feel like a social one. You struggle with a problem: you take it where a E A and b E B such that (i) for all a E A on the bus, into the shower, even into bed; you try out this or there exists an (a, b) E f, and (ii) if (a, b) E f and that approach, and, eventually, with luck you get some in­ (a, c) E f, then b =c. sight. Where is the social aspect to this process? -A buffer is a section of memory that loses its contents Well, where does the insight come from? It is likely that the when read. process of mathematical discovery is a deeply unconscious one. -A plant is an organism whose cells contain a cell wall. We may be able to explain how our discovery makes sense, These definitions can be memorized, but to understand how the definition works, how it separates out a useful class of ob­ but we often cannot explain how we thought of it. And it is jects from other objects, how it describes precisely something often the case, when we reflect on our thought, that the germ of which we have, a priori, an intuitive recognition, requires of the idea came from outside us. That is, the idea developed considerable thought. through some form of discourse, perhaps through reading or And definitions are not constant. The middle school stu­ writing, with others. dent learns that the sine of an angle is a certain ratio in a right As the reader might guess, my German is still primitive. My triangle. The high school student redefines the sine as the or­ rote memorization of verb forms has not been given meaning dinate of a point on the unit circle. The college student may through their use in dialogue. And my mind's hold on what I define the sine as the sum of a certain infinite series or as a have managed to memorize is weak, unsupported by the network certain rational function of eiz. Definitions grow with the stu­ of associations that such dialogue would create. m contrast, the dent's understanding. "bull session" is alive and well in my classroom and plays a significant role in helping my students reach a level of under­ 1 joan Ferrini-Mundy, Principles and Standards for School Mathe­ standing of mathematics that cannot be achieved by rote. matics: A Guide for Mathematicians, Notices, September 2000, page 814; Wayne Bishop, Another View of PSSM, Notices, january -Mark Saul 2001, page 6. Associate Editor

MAY 2001 NOTICES OF THE AMS 469 Letters to the Editor

letters to the Editor Listing Foreign Ph.D.'s "professionals" into high school teach­ I noted with some interest the listing in ing via $20,000 signing bonuses, the February Notices of Ph.D.'s awarded 2. Its purpose is the recruitment Smale and Politics by U.S. institutions in the 1999-2000 of high school teachers-straying Robion Kirby, in his review of Batter­ academic year. I was interested to see quite far from the mission of the No­ son's biography of Steve Smale (De­ who had graduated from certain tices as far as I can tell, and, cember Notices, pages 1408-1411), schools I have been associated with 3. It's an advertisement!! Our college displays magnanimous generosity to­ and in what areas their theses were. I (a Massachusetts public college) had to ward the faults of the American Right. noted a few graduate students I was pay hundreds of dollars to advertise After all, forty-odd years have passed personally acquainted with who had our open, tenure-track position for ElMS since the Red-hunt expelled us and graduated, and looked for a couple of Ph.D. mathematicians on Information in the others from American universities, names of former undergraduates who (Employment Mathematical Sciences) and would and threatened Smale and many more might have completed degrees. The list have had to pay several hundred dol­ with the same expulsion. Kirby has is also a good reference, in addition to lars more to have an ad in the Notices. institutional Web pages, for informa­ by now achieved the detachment to In the future I hope you would tion regarding areas of graduate study forgive the expellers: their motives filter political propaganda a bit for students contemplating graduate were so pure, and those they attacked more judiciously. so vile, that "means even as extreme school. Fmally, I can see the list playing as McCarthyism" become under­ a role for both individuals and institu­ -julian F. Fleron standable. tions in the employment market. Westfield State College Indeed. What does your avowed lib­ I wonder if it might not be a good (Received January 25, 2001) ertarianism mean, Professor Kirby? idea to extend the list somewhat. I That a little tyranny is a good thing, have similar interests in relation to provided it is applied against us? Canadian universities. I know some­ Moving to the 1960s, Kirby is char­ one pursuing a degree in Canada and itable toward the American war in would be interested to note when he Vietnam, saying it was possibly "cor­ finishes.Sometimesstudentswantto rect (perhaps necessary, perhaps even apply to graduate school there, or we honorable)"-though he concedes it get applications for faculty positions was harmful, and he might have noted from people who stUdied in Canada. Also, it would be good to get notice that it was certainly illegal, as the Con­ of doctoral dissertations of other U.S. stitution allows the executive to make citizens (who would typically be AMS war only under a declaration of war by The Notices invites readers to members) who recently completed Congress. Kirby goes on to snipe at submit letters and opinion pieces on dissertations in other countries. topics related to mathematics. Elec­ Smale as the wrong kind of anti-war In truth, it is partly the omission of protester. tronic submissions are preferred; my name many years ago when I com­ see the masthead for addresses. Well, we may be awfully slow for­ pleted my doctorate at the University Opinion pieces are usually one getters, but we still think the Vietnam of London that prompts these sug­ printed page in length (about 800 war was evil, and we still honor Steve gestions. words). Letters are normally less Smale for his labors to stop it. than one page long, and shorter - WalterS. Sizer letters are preferred. -Chandler Davis Minnesota State University, University of Toronto Moorhead -Lee Lorch (Received January 24, 2001) York University (Received January 16, 2001) Massachusetts Signing Bonus I was quite disturbed by your Response to Davis-Lorch Letter "Mathematical Opportunities" an­ The Davis-Larch letter, particularly in nouncement of the "Massachusetts its first paragraph, totally misrepre­ Signing Bonus Program" in the sents what I wrote, and I urge the February 2001 Notices. There are many readers to consult my book review reasons that this controversial pro­ and decide for themselves. gram should not have been included as such an "announcement", among - Rob Kirby them: University of California, Berkeley 1. It is a political ploy to legitimize (Received February 19, 2001) this hotly debated program to draw

470 NOTICES OF 1HE AMS VOLUME 48, NUMBER 5 ANew Solution to the Three-Body Problem Richard Montgomery

describe a new solution to the Newton solved the two-body problem. The hree-body problem (C. Moore difference vector x = X1 - x2 satisfies Kepler's 1993], Chenciner and Montgomery problem: Wl 2001]) and motivate its discovery. d 2x -kx (2) We also sketch its existence proof, dt2 = lxl 3 ' which is based on the direct method of the calcu­ of which are conics with one focus lus of variations. We begin with the statement of all solutions at the origin. The Kepler constant k is m 1 + m2. the N -body problem and some of its solutions. Correspondingly, if we fix the center of mass of Newton told us that two masses attract each be the origin, then the two other, the force of attraction being directed along our two bodies to conic sections with one focus the line joining them, proportional to the product move along similar periodic two-body motions are of the masses, and inversely proportional to the at this origin. The to them as Keplerian ellipses. square of the distance between them. If we have ellipses. We refer They include degenerate ellipses, sometimes called N masses, then the force on any one is the sum of orbits, which are line the forces exerted on it by all the others. This elliptic collision-ejection endpoint at the origin. They gives us the nonlinear system of second-order segments with one solutions to the two-body differential equations represent collision problem. . d2x; _ , m;mj(X; - Xj) It is impossible to describe all the solutions to m, dt2 - - L. 3 ' the three-body problem. Following Poincare, we (1) Jh rij focus on the periodic solutions x;(t) = x;(t + T). i = 1, ... ,N, Here Tis called the period. The simplest periodic solutions for the three-body problem were dis­ of the ith mass, m; being the numerical value covered by Euler [1765] and by Lagrange [1772]. E rij distance x;(t) JRd its position vector, and the Built out of Keplerian ellipses, they are the only ex­ j. are interested in the pla­ between it and mass We plicit solutions. To form the Lagrange solution, nar cased= 2. A solution to theN-body problem start by placing the three masses at the vertices x~, is then a solution x(t) = (x1 (t), ... , XN(t)) to these equations. We contrast this notion with that of x~, x~ of an equilateral triangle whose center of "solving theN-body problem", which we suppose mass m1x~ + m2x~ + m3x~ is the origin. Identify to mean finding an explicit expression for the gen­ the plane of the triangle with the complex eral solution. Poincare showed, in effect, that this plane C, so that x? E C. Take any solution ,\(t) E C is impossible for N > 2. to the planar Kepler problem (2) where the Kepler constant k is a certain rational expression in the Richard Montgomery is professor of mathematics at the three masses m;. The Lagrange solutions are University of California, Santa Cruz. His e-mail address x;(t) = ,\(t)x?. Each mass moves in an ellipse in is rmont@math. ucsc. edu. such a way that the triangle formed by the three

MAY 2001 NOTICES OF THE AMS 471 masses evolves by a composition of instantaneous dilations and rotations and hence is equilateral for all time. For the Euler solutions start by placing the three masses on the same line with their positions x? such that the ratios riJ I rik of their distances are the roots of a certain polynomial whose coeffi­ cients depend on the masses. Again, take any solution .\(t) E C to Kepler's equation (2) where the Kepler constant is a certain other rational expres­ sion in the masses mi. The Euler solutions are Xi(t) = .\(t)x?. At every instant the masses are collinear, and the ratios of their distances remain constant. There are three different families of Euler solutions, according to which mass remains be­ tween the other two. Together, the Euler and Lagrange solutions form the only solutions for which the similarity class of the triangle remains constant throughout the motion. Their beginning mass case. Figure 1. Euler's solution in the equal configurations x? are called central configurations. Most important to astronomy are Hill's peri­ odic solutions, also called tight binaries. These model the earth-moon-sun system. Two masses are close to each other while the third remains far away. The two move in nearly circular orbits about their common center of mass. This center of mass and the third body in turn move in nearly circular orbits about the total center of mass. Like the Euler and Lagrange solutions, these Hill's solu­ tions exist for all ratios of masses. Perhaps next in order of complication is the orbit which is the subject of this paper, the figure eight. Unlike the earlier orbits, it is particular to the case when all three masses are equal. The three equal masses chase each other around the same figure-eight-shaped curve in the plane. The eight was discovered numerically by Chris Moore [1993]. Alain Chenciner and the author [2001] rediscovered it and proved its existence. Description of the eight. The eight is a periodic Figure 2. lagrange's solution in the equal mass case. solution x =(XI (t), x2(t), x3(t)) to the equal-mass three-body problem. If T is the period, then X2(t) =XI (t - T /3) and X3(t) =XI (t- 2T /3). This says that the three bodies travel the same planar curve, phase shifted from each other by one-third of a period. This curve has the form of a figure 0.5 eight. There is an eight orbit of any period T, ac­ cording to a scaling synimetry of the equations (1) to be described below. Modulo this scaling sym­ metry and the other obvious synimetries of (1), 0 the eight is unique according to all numerical investigations. Its unicity has not been proved. The double point of the figure-eight curve is at the origin. This is also the center of mass. The eight curve has the reflectional synimetries of the x-y -0.5 axes. Each of its two lobes is star-shaped (proved), indeed convex (Unproved). The solution begins at t = 0 with mass 1 at the origin, forming the mid­ -1 0 1 point of masses 2 and 3. We call any such config­ uration an Euler configuration of type 1, since it is Figure 3. The figure-eight solution. an initial configuration for Euler's solution

472 NOTICES OF THE AMS VOLUME 48, NUMBER 5 described above when the masses are all equal. The orthogonal transformation, which we will refer to set of Euler configurations with mass i forming the as a "rotation"; c represents translation; and tv rep­ midpoint will be denoted by EULi. Every one-sixth resents changing to a moving frame, moving with of a period the eight solution returns to an Euler constant velocity v. Associated to symmetries configuration, doing so in the order 132132 in a are conserved quantities, which are functions of full period. (A different ordering occurs in position and velocity that are constant along any Chenciner-Montgomery [2001].) At the times solution. The total linear momentum vector half-way between Euler configurations j and k, P =I miXi is the conserved quantity associated to the triangle formed by the masses is isosceles, translation. Its constancy lets us form the GaWean with ru = rik· ijk being a permutation of 123. We transformation with velocity v = -PI I mi which denote the set of all such isosceles configurations transforms us to a frame where the center of mass by IS 0 S C;. Thus in time T /12 the curve x travels is constant. between EUL1 and ISOSC2. This is the key to The conserved quantity associated with rotation constructing the eight. is the angular momentum I miXi 1\ Xi. Time trans­ Stability. Carles Sim6 [2000b] showed numeri­ lation Xi(t) ,_. Xi(t - to) is also a symmetry. Its cally that the figure eight is stable. This is surpris­ conserved quantity is the total energy H = K I 2 - U, ing for two reasons. First, we know very few where stable periodic orbits for the three-body problem and even fewer for the equal-mass case. Hill's solutions are always stable. The Lagrange orbits and are only stable when one of the three masses is U =I mimJiru. much greater than the other two. The Euler i

MAY 2001 NOTICES OF THE AMS 473 Shape Space Because Newton's equations are invariant under isometries, we can push them down to the quotient space of configuration space by isometries. This quotient is the space of congruence classes of N-gons. It is more convenient to divide out in­ stead by the group of orientation-preserving isome­ tries, which means dividing by rotations, since we have fixed the center of mass. We call this quotient shape space and denote it by C. In the case of the planar three-body problem, the shape space C is three-dimensional, by the side-side-side theorem from high-school geometry. It is homeomorphic to Euclidean three-space. Triple collision (XI = x2 = x3) is represented by the origin. It lies in the equatorial plane, the points of which Lag represent the degenerate triangles whose masses are collinear. Reflection about this plane corre­ Figure 4. The shape sphere. sponds to replacing a triangle by its reflection. Distance from the origin in C is measured by J1, where I = I(x) is the moment of inertia of a trian­ (3) L(x, x) = ~ K(x) + U(x), gle whose center of mass is the origin: I= I mi Ixi 12• (Note that I is invariant under rota­ to be contrasted with the energy H = K I 2 - U. The tions, so it yields a well-defined function on C.) space in which x moves is called configuration Dilating a triangle by the factor ,\ induces the space. For the planar N -body problem it is N copies change I .... ,\ 2 I. The sphere I = 1 in C is topolog­ of the Euclidean plane, if we allow collisions. We ically a two-sphere. Its points represent oriented We call it the shape will, without loss of generality, restrict to the similarity classes of triangles. sphere and denote it by S. The equatorial plane subspace Q of configuration space consisting of intersects S in the equator whose points represent x whose center of mass is at those configurations the similarity classes of collinear triangles. The the origin. The action of a path x : [0, T] ~ Q is three types of binary collisions-1 with 2, 2 with 3, and 3 with 1-are represented by three points on A(x) := I: L(x(t), x(t)) dt. this equator. The function U in the Lagrangian is invariant A curve x(t) is called collision-free if it has no under isometries, hence descends to a function on collisions, meaning that rij(t) is never zero for any shape space which we give the same name. This of degree -1, so it pair i of. j. The principle of least action asserts that function is homogeneous can be expressed in the form UI J1 where U is if {! is an "appropriate class" of curves in Q, if x homogeneous of degree 0. We identify [J with the curve in {!, and if the derivative is a collision-free restriction of U to the shape sphere I = 1 . The of the restriction of A to {! is zero at x, then that function U blows up at the three binary collision path x is a solution to Newton's equations (1). An points. It has five critical points: the two Lagrange example of an appropriate class is the set of all configurations, one for each orientation of an equi­ curves joining two fixed submanifolds of Q in lateral triangle, and the three Euler configurations some specified time. To construct the figure eight, lying on the equator. The Euler points are saddle we take the starting submanifold to be EUL1 and points for U and the Lagrange points are its the ending manifold to be ISOSC2. minima. This picture of the shape sphere allows one to Theorem 1. (Chenciner and Montgomery [200 1]) describe succinctly much of what is known about Fix a time f. There is a collison-free curve x which the planar three-body problem and suggests many minimizes the action among all curves joining EU L1 directions and open problems. We refer the reader to IS 0 S C2 in the time f. This curve comprises one­ to the beautiful article of Moeckel [1988]. twelfth of the figure-eight solution. We have been viewing shape space C as simply a topological space. But it also has a metric which Our figure-eight solution x(t) is assembled out of we call the kinetic energy metric, since it arises from pieces congruent to the minimizer of the theorem. the kinetic term Kin the Lagrangian. K is the norm The assembly requires understanding the shape squared of the velocities for the kinetic energy space for the three-body problem. inner product on the configuration space:

474 NOTICES OF THE AMS VOLUME 48, NUMBER 5 the negative potential, viewed as a function on C. The reduced action is then the function The associated norm squared of positions is the Ac(c) = fc Lc dt on curves in C. moment of inertia just introduced: We relate the action principle to the reduced one. Write rr : Q ~ C for the quotient map which associates to a triangle its shape, so that c(t) = rr(x(t)). Let M1, M2 c Q be two submanifolds The group of orientation-preserving isometries invariant under the action of the group of rotations of the plane acts isometrically on configuration about the center of mass. Then M; = rr-1(£;), space when we use this metric. Shape space, being i = 1, 2, for submanifolds £; c C. Consider the the quotient of configuration space by this problem of minimizing the action A among all group, inherits a metric. The distance between two paths in Q which connect M1 to M2 in time T. points in shape space is defined to be the According to the above, L(x, x) ~ Lc(c, c) with distance in configuration space between the two equality if and only if the curve x(t) has zero corresponding orbits. angular momentum. By applying a time­ This metric on C is Riemannian away from the dependent family of rotations g(t) to x(t), we can origin. The induced Riemannian metric on the obtain a new curve g(t)x(t) whose angular mo­ sphere S is that of the standard Euclidean two­ mentum 1 is zero and whose shape curve is the sphere of radius 112. The act of dilating a trian­ same c(t). This proves that any minimizer x for our gle is a metric dilation on C. This information original problem must satisfy A(x) = Ac(c). Our completely specifies the metric structure of C. It original problem reduces to that of minimizing is called a cone over a two-sphere of radius one­ Ac among all curves inC joining £1 to E2 in timeT. half. The triple collision is the cone point, and the We can reverse this procedure. Imagine that we Riemannian metric is singular there. have a minimizer c for our Ac-problem. Suppose For general masses, the binary collision points that this curve is without triple collisions. Each and central configurations are asymmetrically curve inC without triple collision has a lift to Q, placed on the shape sphere, enjoying only reflec­ this being a curve x which projects to c and has tional symmetry about the equator. In our case angular momentum zero. The lift is unique up to of all equal masses, these eight points are placed rigid rotation. The lifts x of our minimizer c will as symmetrically as possible. The three Euler minimize the original variational problem for points and the three collison points are spaced curves on Q. We now apply this fact with out in equal intervals along the equator. The two £1 = EUL1 and £2 = ISOSC2. Lagrange points are at the north and south poles. When we join the north pole to the six marked Building the Eight Using the Discrete points on the equator by great circles, we obtain Symmetries three meridians. Each meridian passes through We build the figure-eight solution from the mini­ the south pole and intersects the equator in a mizing curve x of Theorem 1, beginning from its pair of antipodal marked points, one an Euler point shape curve c(t), 0 ~ t ~ Tl12 =f. and the other a binary collison. These meridians The shortest curve in Euclidean three-space represent the manifolds of isosceles triangles which joins a point to a plane is a line segment JSOSC1, ISOSC2, and JSOSC3 from above. orthogonal to that plane. More generally, if a path minimizes the action over all paths which connect Reduced Action one submanifold to another in a given time inter­ If x(t) is a curve in configuration space, then we val, then this path must hit the endpoint manifolds will write c(t) for its projection to the shape orthogonally at the endpoint times. Upon apply­ space C. If x(t) is a solution to the three-body ing this idea to our reduced action principle, we problem, then we will call c(t) a reduced solution. find that our shape curve c(t) intersects EULo We show how to pass back and forth between orthogonally at t = 0 and ISOSC2 orthogonally at solutions and reduced solutions, provided the t=TI12. total angular momentum 1 := 2: m;x; A X.; = 0. The equality of the masses implies that the re­ Let Kc I 2 denote the kinetic energy associated flections about the meridian planes ISOSCJ inC to the metric on C. Kc is the function on the are symmetries, meaning that they preserve the tangent bundle of C given by Kc(c, c)= (c, c)c, reduced Lagrangian and hence take reduced solu­ where c represents a tangent vector to shape space tions to reduced solutions. Reflect our minimizing at the particular shape c E C and ( ·, ·) c denotes shape curve c(t), 0 ~ t ~ T 112, aboutJSOSC2. The the Riemannian metric at c. One proves that resulting reduced solution connects EUL3 to K(x) = Kc(c, c)+ 11(x, x)l 2 I I. It follows that K ~ Kc ISOSC2, hitting ISOSC2 orthogonally with deriv­ with equality if and only if the angular momentum ative opposite to c there. Thus if we concatenate 1 is zero. Define the reduced Lagrangian to be c with its time-reversed reflection, the derivatives Lc(c, c)= Kc(c, c)l2 + U(c), where U(x) = U(c) is match up at Tl12, and the result is a smooth

MAY 2001 NOTICES OF TilE AMS 475 reduced solution, still called c, now parameter­ (132). The same can be checked for the velocities. ized by the doubled interval 0 ~ t ~ Tl6. It follows that the continuation of the solution Continuing in this manner, applying reflections past time t=TI3 satisfies X2(TI3+t)=xl(t), about meridian planes and the equator, which is a X3(T 13 + t) = X2(t), and XI(T 13 + t) = X3(t). Armed symmetry for all mass ratios, we march around the with this functional equation, we can now shape sphere, returning to where we started at continue x over the whole period T. The result time 6 T I 12. The derivatives at t = 0 and t = T I 2 are is periodic of period T, since p 3 = 1. This also not equal, but instead are related by reflection shows that all three masses trace out the same about the equatorial plane. Continuing once more planar curve q(t) = x1 (t) during the solution. around the equator, we obtain a periodic reduced It remains to show that this planar curve q(t) solution curve c(t) inC of period T, made up of has the qualitative form of the eight. Using the twelve pieces congruent to the initial shape curve proper choices of g in the realizations g o o-1 of coming from Theorem 1. It passes through the the half-twist about EUL1 and of the equatorial Euler configurations in the order 132132 at the reflection, we can check that q is odd in time t and specified times and the isosceles configurations at that its image enjoys the symmetries of the x-y the intermediate times. axes, where we take for the x -axis the symmetry The lift X(t) =(XI (t), X2(t), X3(t)) E Q of the axis of the isosceles triangle x(TI12). The shape curve just constructed is a solution to qualitative form of the eight now follows from Newton (1). It remains to show that it is periodic the form taken by its first quarter, q(t), of period T and has the form described. We do this 0 ~ t ~ T 14. This quarter lies in the first quad­ by seeing how our reflections act on arcs of x. rant, is star-shaped (proved), indeed convex One might first guess that the symmetry (unproved). Star-shapedness is achieved by using 0"3(X1, x2. x3) = (x2. x1, x3) of interchanging masses minimality of c and the symmetries to show that 1 and 2 induces the reflection about JSOSC3. This the angular momentum q A q = r 2iJ is positive, is not true, since this interchange, unlike the hence e is monotone over this interval. reflection, reverses the orientations of triangles We now turn to the proof of Theorem 1, which and so interchanges north and south hemispheres is all that remains to demonstrate the existence of of the shape sphere. Instead, the interchange 0"3 the figure-eight solution. induces the half-twist about EUL3, by which we mean the isometry of C which is the identity on Hilbert's Direct Method; a Sketch ofthe the ray EUL3 and which rotates planes orthogo­ Proof of Theorem 1 nal to it by 180 degrees. The half-twist is the Consider the class {!. of all paths inC joining EUL1 composition of the reflection about ISOSC3 with to ISOSC2 in the time f = T 112. Choose a se­ the reflection about the equator and is the sym­ quence Xn E {!. such that metry used to extend c([O, Tl6]) past T16. Now x(T 16) = (x, - x, 0) with x =f. 0, thus o-3x(T 16) lim A(Xn) = inf A(x). n-oo XEC! =f. x(T16), and so the concatenation of x([O, T 16]) with o-3(x(T 16- t)) yields a discontinuous curve, We call such a sequence a minimizing sequence. which is not right. To correct matters we instead Try to extract a subsequence of the Xn which concatenate x with go o-3(T 16- t), where g is an converges to some x* E {!.. If we can show that x* appropriate rigid rotation. (Any rotation g induces is collision-free, then we will be done, according the identity on shape space.) The unique choice of to the principle of least action. This method of es­ g making the concatenation continuously differ­ tablishing the existence of critical points is Hilbert's entiable at T 16 is rotation by 180 degrees, direct method in the calculus of variations. gxi = -Xi. Since the arcs of x on either side of T I 6 The existence of some limiting x* follows satisfy Newton (1), and since the derivatives now quickly from the Arzela-Ascoli theorem of real match, this concatenation is a solution over analysis. The crux of the matter is to show that x* 0 ~ t ~ T I 3. The concatenation is achieved by suffers no collisions. To do this we compute the substituting the time-reversed and translated infimum Aco/1. of the action A over all curves time T 16 + (T 16 - t) = T 13 - t for the old time. which suffer a collision during the specified time Hence our curve satisfies the functional equation f, regardless of whether or not these curves (XI(t),X2(t),X3(t)) = (- X2(TI3- t), -XI(TI3- t), satisfy the endpoint conditions defining {!.. Then -X3(TI3- t)) for Tl6 ~ t ~ Tl3. we find a collision-free "test path" Xtest E {!. with The initial configuration x(O) is in EUL1 and so A(Xtest) < Acoll. . This will complete the proof, has the form (0, a, -a) for some a E C. According since A(x*) ~ A(Xtesr) < Aco/1., and sox* must be to the functional equation, x(T 13) = (-a, 0, a), or collision-free. x(T 13) = p(x(O)) where pis the symmetry of per­ To compute Acoll. observe that we decrease the mutation of masses: p(x1, x2, x3) = (x3, x1, x2). The action if we replace a path x(t) = (XI (t), x2(t), x3(t)) configurations x(O) and x(T I 3) are identical, except by a path in which one of the masses is positioned that the masses have undergone the permutation far from the other two and does not move. In other

476 NOTICES OF TifE AMS VOLUME 48, NUMBER 5 words, replace x(t) by x(nl(t) =(XI (t), x2(t), x~n)) the circle.) The winding number of a single orbit where x~n) is a fixed point of the plane satisfying of a Keplerian ellipse is either 1 or -1. Gordon asserts, among other things, that these are the lx~n)- XI(t)l > lx3(t)- XI(t)l only winding numbers which can be achieved by minimization. If we try to minimize the action and over the winding number 2 component, then we lx~n ) - X2(t)l > lx3(t)- X2(t)l. will be led to a collision curve at the boundary be­ In this way, a minimizing sequence x(n) realizing tween this component and the winding number + 1 Acoll. is obtained by letting two of the masses component. collide, while the third remains stationary and All Keplerian ellipses of period T have the same infinitely distant from them. This reduces the action cTI/ 3 . (The constant c is 3(2rr)213k2!3 /2 .) computation of Acoll. to a two-body problem, one This action is a function of the semimajor axis of which had already been solved by Gordon [1977] the ellipse alone, as are both the energy and the and which will be described momentarily. period. Consequently we have a whole family of To construct our test path, note that the minimizers, all sharing the same infimal value of potential function U takes on the same value the action. At the boundary of this family lie the U(EUL) at all three of the Euler points EULi E S elliptic collision-ejection orbits representing the by symmetry. Consequently, its level curve planet crashing into the sun and being expelled U = U(E U L) passes through each Euler point. These along the segment it came in on. These also points are saddle points for U, and as we follow realize the infimum of the action over our class the level curve we alternate hemispheres, crossing despite the fact that they are not in the class but the equator at the Euler points and nowhere else. rather on its boundary. In other words, the topology of this level curve These two linked phenomena-that of the direct vis-a-vis the collisions and Euler points is identi­ method leading us to the boundary of a topologi­ cal to that of the shape curve of the eight. Follow cal component and that of collision orbits sharing one-twelfth of this equipotential curve, namely, the infimal value of action-illustrate the main the upper branch connecting EULI to ISOSC2. subtleties of using action principles for Newton­ Parameterize it at constant speed so that it ian N -body problems. Identical phenomena occur arrives at JSOSC2 in the desired time f = T 112. in the three-body problem. The solutions of Scale it so that it lies in the sphere I= Io, thus Lagrange and Euler of a given period also occur in obtaining a one-parameter family of test curves. families parameterized by Kepler, all having the Minimize the action with respect to the scaling same action and including triple collision-ejection parameter Io. There will be a unique minimizing orbits on their boundary. Solutions corresponding Io = Io(T), and this yields Xtest· Its action depends only on the period f and the length of the to central configurations occur for all N, so this starting equipotential curve on the sphere phenomenon persists in the Newtonian N-body problem for any N. S = {I = 1}. We estimate this length with enough tolerance to guarantee that A(Xtesd < Acoll. Still, the Keplerian ellipses do minimize, al­ though they are degenerate minima. The crucial Gordon and Kepler's Problem fact which allowed Gordon to prove this, which is We have just finished the sketch of our proof of to say, to exclude the possibility of more than one the existence of the eight, except for the loose end collision per period, is the concavity of the action concerning Aeon .. We now tie up this loose end by cTI/3 of a Keplerian orbit as a function of its following Gordon's work on action minimization period T. Suppose, for example, we have a mini­ for the Kepler problem. inizer with countably many collisions, with the The action for the two-body problem is pro­ times between successive collisions being t1. t2, ... , portional to the Keplerian action f i lxl2 + kl lx l dt tj, .... By standard arguments from the calculus of for the difference vector x = XI - x2 in the plane. variations, in between collisions the minimizer We insist on excluding the collision x = 0. must be a solution to Kepler and hence must be a collision-ejection orbit. Its total action is thus Theorem. (Gordon [1977]) Consider the class of c 2:(ti)I13 . But 2:(ti)I13 ~ (2: ti )I/ 3 = yi/ 3 with closed non-contractible curves of period T in the equality if and only if there is but a single collision. plane minus the origin. The infimum of the This proves there is at most one collision. This is Keplerian action over this class is realized by any the argument which allows us to compute Acoll .. Keplerian ellipse with period T. The act of excluding collisions breaks up the Poincare Minimizes over a Homology Class space of closed curves in the plane into countably In the remainder of the article we describe other many components, indexed by the winding num­ applications of the direct method to N-body ber n E Z of the curve about the origin. (This index problems, relating some of these to the eight's is the degree of the map xI IxI from the circle into construction.

MAY 2001 NOTICES OF TilE AMS 477 Poincare [1896] was apparently the first to apply with kab > 0. These are strong-force if and only if the direct method to find new periodic solutions v ~ 2. to three-body equations. Poincare was most inter­ Poincare observed that when the potential is ested in reduced periodic solutions, as described strong-force, the action of any curve tending to earlier. Specifically, he looked for solutions x(t) with a collision diverges. Consequently, there are no the property that there is some fixed rotation g and finite action curves with collision. This let Poincare some time T for which prove the following, using the direct method. (R) x(t + T) = g(x(t)). Theorem. (Poincare [1896]) Consider the class of collision-free reduced periodic curves specified, as Equivalently, these are solutions x to (1) which above, by the period T, the winding numbers are not necessarily periodic but whose shape (K2,K3), and the overall rotation g. If the potential curves are periodic. is strong-force, then there is a solution to Newton's not want collisions any more than Poincare did equations lying in this class and realizing the we do. Shape space minus collisions is homotopic infimum of the action over this class. to the shape sphere minus three points. Its first homology group is Z x Z. Poincare fixed a homol­ There is a small oversight in Poincare's proof. ogy class (K2,K3) as well as the overall rotation For the statement to be correct we must exclude g in (R). Concretely he considered a rotating those classes in which at least one of the Ki is zero orthogonal frame with one axis being the edge and g is the identity. To see what goes wrong with­ X3 - x 2 of the moving triangle. The integers K3 out this constraint, consider the case g = 1, and K2 are the winding numbers of the nonzero K2 = K3 = 0. The infimal action is zero. It is real­ vectors X3 - XI and x2 - XI about the origin, as ized by a minimizing sequence Xn in which the Xn(t) viewed in this moving coordinate system. The ro­ are constant, forming the vertices of an equilateral tation g is the overall rotation of the coordinate triangle (or any fixed noncollinear triangle ~ E S) axes in one period. Together (K2,K3;g) specify a of size JI = n. As n- oo, all three masses recede a class of collision-free curves over which to apply to infinity and there is no limiting curve. the direct method. lf, on the other hand, K2K3 =f. 0 or g =f. 1, then all Poincare knew that the Newtonian problem ad­ three masses must move in order that the curve mits finite action solutions. Indeed, all solutions lie in the class C. Indeed, the projection to the ending in collision and defined over a bounded time three-sphere {I = 1} must be a curve whose length interval have finite action. He thus faced the pos­ is bounded away from zero by a positive constant sibility of action-minimizing collision-free se­ depending only on K2, K3, and g. It follows that if quences pinching off to collision. This possibility Xn is a sequence of curves realizing this class and blocked him from achieving reduced periodic so­ if In = I(Xn(tn)) - oo for some times tn, then the lutions in the given class by the direct method. To lengths of the Xn must tend to infinity, which circumvent this difficulty, Poincare changed the implies that their actionA(xn) - oo. This excludes potential and thus the differential equation (1), a the possibility of points of a minimizing sequence trick which dozens, if not hundreds, of researchers wandering off to infinity in the plane. have used since then without knowing that Gordon [1977] formalized the above argument. Poincare had done this a hundred years earlier. He called a class {! of collision-free curves tied to Replace the function U in the Lagrangian by the collisions if whenever we have a sequence of one of the general form curves Xn E {! with llxn(tn)ll 2 := I(xn(tn)) - oo for some sequence of times tn, the actions A(xn) must U = fab(rab), L also tend to infinity. A minimizing sequence ex­ a 0, fab(O) = +oo, and fab(oo) = 0. The by receding to infinity. resulting Euler-Lagrange equations define a new Instead of fixing a homology class of reduced N -body problem whose differential equation periodic curves in shape space, let us fix a ho­ is found by replacing the right-hand side of mology class of truly periodic curves in the usual Newton's equations (1) by the gradient of U with configuration space minus collisions. Such a class respect to Xi. This N-body problem will be called is determined by the winding numbers ni , n2, n3 strong-force if there exist positive constants 8 and of the three edges Xi - x j. Gordon's tied condition c such that the two-body potential functions fab is that at least two of these integers are nonzero. making up U satisfy The direct method then yields a periodic collision­ c free solution with these winding numbers, fab(r) > 2 whenever r < 8. r provided the potential is a strong-force potential. Venturelli [2001] transcended the strong-force The prime examples come from the power laws restriction. kab Theorem. (Venturelli [200 1]) Consider the class of fab(r) = rv' all closed collision-free periodic curves of period T

478 NOTICES OF THE AMS VOLUME 48, NUMBER 5 in the configuration space for the planar Newton­ ian three-body problem whose winding numbers ni 1 satisfy n1 nz n3 =f 0. The infimum of the Newtonian action over this class is realized by any elliptic or circular Lagrange solution of this period. 0.5 The Lagrange solutions have winding numbers 1, 1, 1 or -1, -1, -1. Their homology classes are the only classes realized by action-minimizing solu­ tions. If we fix any other homology class with all 0 nonzero winding numbers, such as 2, 1, 1, and try to minimize the action over this class using the direct method, then we will be led out of that class -0.5 to the Lagrange collision-ejection solution. This collision solution has the same action as the noncollision Lagrange solutions. -1 Note the close parallel with Gordon's theorem. Gordon's theorem is one of the main -1 0 1 ingredients in Venturelli's proof. Another key ingredient is that the Lagrange configuration Figure 5. Gerver's solution. minimizes U. we decorate the letters with superscripts + and - Eclipse Sequences: Minimizing over a to indicate whether the corresponding ecliptic Homotopy Class arc is approached from the northern or southern hemisphere. The pluses and minuses must alter­ Instead of fixing the homology class of a reduced nate. A given eclipse sequence has exactly two periodic curve, we now fix its free homotopy class. such decorations, corresponding to a curve and its We explain how to encode such a class into an reflection. We have established a one-to-one eclipse sequence. A loop in C\ collisions radially correspondence between grammatically correct projects to a loop inS\ collisions. Perturbing the eclipse sequences and free homotopy classes in loop slightly makes it intersect the equator of C \ collisions, up to reflection. collinear triangles a finite number of times, each transversally. Each intersection represents an Open Problem. (Wu-Yi Hsiang) Is every grammat­ eclipse, a configuration in which the three masses ically correct eclipse sequence realized by a are collinear, one lying between the other two. We solution to the three-body problem? label these eclipses 1, 2, or 3 according to which body is in the middle. Consequently, our curve Wu-Yi Hsiang posed this question in the spring of 1995. It helped lead to the construction of the yields a finite sequence, or "word", abc ... , a, b, figure-eight solution. and c being 1, 2, or 3 written down in the order they appear. We call such a word the eclipse Cheating by assuming the strong-force sequence of the curve. For example, Hill's solution condition and applying the direct method just realizes the eclipse sequence 12 if 1 and 2 are as Poincare did, we get an answer. the close bodies, while the figure eight has eclipse sequence 13 213 2. Theorem. (Montgomery (1998)) LetT > 0 be given. If two consecutive eclipses a a of the same type For the three-body problem with a strong-force po­ occur, then they can be deleted by pulling the arc tential, any given eclipse sequence in which all between the two eclipses up into the opposite three eclipse letters 1, 2, and 3 appear is realized hemisphere. This deletion decreases the action, so by a reduced periodic solution of period T which minimizes the action over this eclipse sequence it is consistent with the direct method. This process allows us to delete all "stutters" ... aa ... occurring class. in our word. Since we are interested in periodic Gordon's tied condition is precisely that all three orbits, we should think of our eclipse sequence eclipse letters occur. as being written out along the circumference of a Poincare, the inventor of the homotopy group, circle, i.e., as a periodic word. For example, the class clearly had the tools to state and prove the above of the figure eight is 132132 = 321321 = 213213 . theorem. It is slightly surprising that he did not. We insist that each such cyclic permutation be Summary. The set of periodic curves inC breaks stutter-free. A cyclic word in the letters 1, 2, and 3 up into infinitely many components when we subject to this stutter-free rule will be called a gram­ exclude collisions. These components are indexed matically correct eclipse sequence. Such by eclipse sequences. The (reduced) action func­ an eclipse sequence uniquely specifies the free tion creates walls between them. When we take homotopy of the collision-free shape curve once the potential to be strong-force, these walls are

MAY 2001 NOTICES OF 1HE AMS 479 other. Indeed, it lies in the boundary of every component and has very small action, hence is a very attractive place to head. See the discussion of 1 Venturelli's theorem and also Montgomery [2000]. It could be that the direct method applied to any 0.5 component will always lead to XL in the Newton­ ian case. If so, the direct method would yield no solutions beyond Lagrange and thus no eclipse 0 sequences beyond the empty sequence, which is the sequence for Lagrange. -0.5 We were able to get around these difficulties in constructing the figure eight because of our assumption of equal masses, with its consequent -1 discrete symmetries. In effect, these allowed us to replace the specification of the eclipse sequence 132132 with the endpoint conditions appearing in -1 0 1 2 Theorem 1. New N-Body Solutions Figure 6. A choreography of nine bodies. We askN equal Newtonian masses to dance around a fixed curve. The eight is such a solution, as is 1 the circular Lagrange solution. For each N we can obtain such a solution where the curve is a circle by placing the N points at the vertices of a regu­ lar N -gon inscribed in the circle and then rotating 0.5 this N-gon at the proper frequency. Additional orbits like this were found numerically by Davies et al. (1983) and G. Hoynant (1999) in space and by Moore [1993] in the plane. Right after our 0 rediscovery of the eight, Gerver conjectured the existence of solutions of this type for all N, with the number of distinct topological types of solu­ -0.5 tions increasing rapidly with N. He numerically found one of his conjectured solutions when N = 4. In Gerver's solution the configuration formed by the four bodies is a parallelogram at every instant, -1 and the curve they move on is a "super-eight", a -1 0 1 figure eight with an extra twist. Sim6 [2000a] numerically verified Gerver's conjecture, finding hundreds of new equal mass planar N-body Figure 7. A floral arrangement with eleven bodies. solutions. See also Chenciner et al. [2001]. Of all infinitely high, so we can minimize separately in these solutions, apparently the only stable one is each component. But as we lower the exponent v the eight. And, except for the eight, we lack rigor­ of fab = mamblr"%.b coming from the power law ous existence proofs for any of these new New­ potential above and so try to approach the case tonian N -body solutions. of real interest (the Newtonian case of v = 1 ), the To make N equal bodies perform a desired walls descend in height, becoming finite in height dance, begin with the circle 51 = !RITZ of circum­ as soon as v < 2. Then we can cross over from one ference T. The cyclic group ZN of order N acts on component to the other. The walls are probably still this circle with its generator w acting by steep and sharp in most places, but crossing is now w(t) = t +TIN, which is to say by rotation by possible. The Hilbert direct method might even lead 2rriN. It acts on c_N by W(XI,Xz, ... ,XN)= us to "valleys" or "breaks" in these walls, forcing (XN, x1, xz, ... , XN-1). Then ZN acts on the space the limit x* to have collisions. Because of this we of all loops x: 51 ~ c_N by (wx)(t) = w(x(w-1(t)). can say little to nothing at present about whether A fixed point of this action on loops is a map or not the given eclipse sequence is realized by an x:S1 ~eN satisfyingXj+I(t)=XI(t-jTIN). We action minimizer. call such a map a choreography. In a choreogra­ The situation is worse. The Lagrange collision­ phy all N masses travel along the same closed ejection orbit XL is a kind of "universal" low pass planar curve q(t) = XI(t), staggered in phase allowing us to cross from any component to any from each other by TIN.

480 NOTICES OF THE AMS VOLUME 48, NUMBER 5 This action of ZN on loops x leaves the N -body in Oeuvres, Seria Secunda tome XXV Commenta­ action A(x) invariant when the masses are all equal. tiones Astronomicce (page 286). [1977), A minimizing property of It follows from a general principle, which Palais [4) W. B. GoRDON Amer.]. Math. 99, 961-971. of symmetric criticality, that if Keplerian orbits, calls the principle (5) jOSEPH-LOUIS LAGRANGE (1772), Essai sur le Probleme des we restrict A to the fixed points of the action­ Trois Corps, Prix de l'Academie Royale des Sciences the choreographies-and find a collision-free de Paris, tome IX, in val. 6 of Oeuvres (page 292). choreography which is a critical point for this [6) R. MoEcKEL [1988), Some qualitative features of the restricted A, then this choreography will be a three-body problem, Contemp. Math., vol. 81, Amer. solution to theN-body problem. Excluding colli­ Math. Soc., Providence, Rl, pp. 1-21. sions breaks up the set of choreographies into [7] C. MooRE [1993], Braids in classical gravity, Phys. countably many different components, which Rev. Lett. 70, 3675-3679. [1896], Surles solutions periodiques et we call choreography classes. By the argument [8] H. POINCARE de moindre action, C.R.A.S. Paris 123, if the potential is le principe which Poincare used above, 915-918. strong-force, then there is a solution realizing each [9] C. SIM6 [2000a], New families of solutions in choreography class. N-body problems, to appear in the Proceedings Sim6 has numerically implemented this equi­ of the ECM 2000 (Barcelona, july 10-14); see variant action-minimization procedure for the http://www.iec.es/3ecm/. Newtonian problem. He discretizes the gradient [10] __ [2000b], Dynamical properties of the eight so­ flow of the action, starting with an initial guess for lution of the three-body problem, to appear in the a choreography. The resulting iteration scheme Proceedings of the Chicago Conference Dedicated to . stays within the space of (discrete) choreographies Don Saari (December 15-19, 1999) [11] A. VENTURELU [2001), Une caracterisation variation­ will either converge to a choreography with and neUe des solutions de Lagrange du ProblE~me plan collisions or to a collision-free choreography. des trois corps, to appear in C.R.A.S. Paris. (In the strong-force case the first possibility is excluded.) number In this way Sim6 has produced a huge About the Cover of new solutions to the equal mass Newtonian cover shows several frames of an anima­ N -body problem. Apparently all choreographies are The tion constructed by Carles Sim6- redrawn unstable except the original figure eight for N = 3. slightly-that exhibits the periodic orbital mo­ Sim6's results suggest that for each N only finitely of six objects of equal mass on a single many choreography classes are realized by local tion orbit. This is what is called in Richard Mont­ action minimizers. For N = 3 it seems only two gomery's article a "simple choreography". The choreography classes are realized: that of the original animation is number 42 in a spectacu­ Lagrange solution and that of the eight. What lar series of animations by Sim6 which run in selects the classes which topological criterion gnuplot. The whole series can be obtained are realized by a Newtonian choreography? An from http://www.maia.ub.es/dsg/nbody. this question might lead to an existence answer to html. horde of new orbits and perhaps proof for this As Montgomery explains, almost all of these of the planar N -body a better understanding choreographies are known at the moment only problem. through computer calculation, primarily the them is a dif­ Acknowledgments work of Carles Sim6. Producing ficult but interesting process. To see what is in­ e-mail I thank Alain Chenciner for numerous volved one should look at various papers of for most conversations and critiques, Carles Sim6 Sim6. Many recent ones are available as and Phil of the pictures here, Robert MacKay preprints at http: I jwww. mai a. ub. es/dsg/. Holmes for alerting us (via A.C.) to Poincare's Sim6 is currently working on finding rigor­ variational work [1896] and to Chris Moore's [1993] ous computer-assisted proofs of the existence discovery of the eight, Wu-Yi Hsiang for starting of choreographies. me on these problems, and finally Gil Bor for his Casselman (covers@ams . org) strong criticism of an early version of this article. - Bill

Bibliography (1) A. CHENCINER and R. MONTGOMERY (2000), A remark­ able periodic solution of the three-body problem in the case of equal masses, Ann. ofMath., November. (2) A. CHENCINER, j. GERVER, R. MONTGOMERY, and C. SIM6 [2001), Simple choreographic motions of N-bodies, a preliminary study, to appear. [3) LEoNARD EULER [1 767], De motu rectilineo trium cor­ porum se mutua attrahentium, Novi commentarii academice scientarum Petropolitance 11, pp. 144-151

MAY 2001 NOTICES OF TilE AMS 481 Thomas H. Wolff (1954-2000)

Lennart Carleson, Sun-Yung Alice Chang, Peter W. ]ones, Markus Keel, Peter D. Lax, Nikolai Makarov, Donald Sarason, Wilhelm Schlag, and Barry Simon

homas H. Wolff, a leading analyst and a win­ Lennart Carleson Tner of the Salem and Bacher Prizes, was killed in an automobile accident on July 31, 2000, Harmonic analysis has a position in mathe­ when he was forty-six years old. matics comparable to that of the theory of the Tom was raised in a mathematical environment. atom in physics. By understanding what goes on His uncle, Clifford Gardner, was a professor at at the micro level, we can understand large-scale NYU's Courant Institute of Mathematics for many phenomena (even such as meteorology). By Fourier years, and Tom's mother, Lucile, was a technical expansions we can analyze and understand global editor of volume 1 of the English translation of the functions, arithmetic problems, or differential celebrated book Methods of Mathematical Physics equations. As physics has a miracle method in by Courant and Hilbert. quantum mechanics, we use the theory of analytic Tom was an undergraduate at Harvard, where, functions, and of course, in a deeper sense, all he once told me, he regularly played poker with a this comes together as one piece. These methods fellow student named Bill Gates. After graduating are efficient when the number of variables n is from Harvard in 1975, Tom went to Berkeley, where either small or very large (where the theories be­ he got his Ph.D. under Don Sarason in 1979. Tom come probability). In the intermediate range, say then spent one year at the University of Washing­ 10 < n < 100, physicists and chemists have been ton and two at the University of Chicago before more successful than we mathematicians, and it remains an important challenge for the future to coming to Caltech in the fall of 1982 as an assis­ develop relevant harmonic analysis in this range. tant professor. The problems in harmonic analysis (of few vari­ Tom spent most of the rest of his career at Cal­ ables) today, after two hundred years of research, tech, although, for personal reasons, he resigned are very combinatorial and very complicated. Tom twice, spending two years (1986-88) at Courant and Wolff had a unique talent and a profound knowl­ three (1992-95) at Berkeley. His promotion or edge in the area. The fundamental problems that appointment to a professorship at Caltech three he considered required long preparation, deep times is a record for our institution. concentration, and an ability to keep a very com­ Tom is survived by his widow, Carol Shubin, a plicated set of arguments simultaneously active mathematics professor at California State Univer­ and available in his mind. Two quotations come to sity, Northridge; two sons (aged three and five at mind. The first is Newton's answer to how he his death); his parents; and two sisters. -Barry Simon Lennart Carleson is professor emeritus of mathematics at the Royal Institute of Technology, Stockholm, Sweden and Photographs included in this article are courtesy of Carol at the University of California, Los Angeles. His e-mail ad­ Shubin. dress is carl eson@math. kth. se.

482 NOTICES OF THE AMS VOLUME 48, NUMBER 5 found the law of gravitation: "By constantly think­ ing about it," and the second is Tom's remark Ph.D. Students ofTom Wolff when he was awarded the Bacher Prize in 1999: "It's never been easy for me." Nobody else could have Ivo Klemes, Caltech, 1985 done what he did. Peter Holden, Caltech, 1987 Let me highlight some of Tom's most striking Gregory Hungerford, Caltech, 1988 results, all coming from combinatorial harmonic Stewart Gleason, NYU, 1990 analysis. Tom's thesis and early work were in the Dean Evasius, Caltech, 1992 direction of complex analysis. (His thesis is de­ Wensheng Wang, Caltech, 1993 scribed below by D. Sarason.) While working on his Lawrence Kolasa, Caltech, 1994 thesis Tom found a new proof of the corona the­ Wilhelm Schlag, Caltech, 1996 orem for H"" of the unit disk. Here he managed to David Alvarez, UC Berkeley, 1997 summarize the combinatorics in a beautiful lemma Themistoklis Mitsis, Caltech, 1998 Oleg Kovrijkine, Caltech, 2000 on bounded solutions, u of au = f. As first pointed out by Hormander, one should first solve the rel­ Burak Erdogan, Caltech, 2001 evant equations in a nonanalytic way and then modify the solution to make it analytic bounded, which leads to the above equation. To get the w > 0, but barely so. Consider, for example, n = boundedness of the solutions, the old combina­ complement of disjoint disks torics came back, and this is where Tom's lemma works; see [4]. y . J1 avJ.I=N+zN, This lemma has had a large impact, although Tom never published it. He did not return much 0 < Ivi, IJ1I ~ N, to function algebras. I can mention a joint paper with Alan Noell (1989) which gives an almost com­ and choose TvJ.I so that wWvJ.I) = ttk for all v and plete description of sets Eon lzl = 1 where an an­ Jl. The radii in the outer part of the square must alytic function f(z) in lzl < 1, satisfying be very small, but a proof is not easy and the lf(z)- f(w)l ~ Clz- wla, lf(z)l ~ 1,f(O) = 0, can precise size of T VJ.I is not known (to me). It was a be = 1 (so-called peak sets). This is surprisingly tour de force when Tom and Peter Jones in 1988 hard! proved the statement about dim(£) ~ 1 in full From function algebras Tom turned to potential generality [6], and Tom even proved (published theory, where his most important work in the 1980s in 1993 but from the same time) that E can be can be found. The first fundamental result [5] is chosen of a-finite length. In the above example, from 1983. L. I. Hedberg had proved that if n is an Tom's method proves that 2: Tv11 ~ C, lav 11 1~ 1/2. open set in ~n, then CQ' (0) is dense in all higher­ There is a natural conjecture for the analogous order Sobolev spaces w;·P(n) when p > 2- Iff. problem in higher dimension n (0ksendal 1981) The exceptions 1 < p ~ 2 - Iff had their origin in that E c an can be chosen of dimension n - 1. In potential theory and more precisely in the miss­ 1987 Tom produced a counterexample to this ing Kellogg lemma on the size of the set of thin conjecture for n = 3. This was a sensation not boundary points outside that range. Tom closed only because of its complexity but also because it this gap by a clever construction. Today more simultaneously disproved two other conjectures. streamlined approaches exist. It was published only in 1991 [11]. Tom next turned to harmonic measure. Let n The construction is based on the following be a domain in the plane containing oo, and sup­ lemma: For each unit vector e in JR3 , there is a har­ pose that we can solve the Dirichlet problem in n monic function u in X3 > 0, vanishing at oo, so for arbitrary continuous f on an. The value of the that solution at oo is given by integrating f against a positive probability measure w in the plane sup­ (*) ported on an. This is called harmonic measure and is also the hitting probability on an of a Brown­ This fails in two dimensions by subharmonicity.

ian particle in n I starting at 00. From this inter­ This entropy-type of integral is relevant because pretation, it is natural to surmise that most mea­ the counterexample domain n is obtained in a sure lies in the exposed parts of an. The harmonic dynamic way by a snowflake construction where measure should therefore be !-dimensional; more almost independent products of gradient vectors precisely: 3?£ of dimension ~ 1 on an with as above occur. (A result by Bourgain tells us that w(E) = 1. By using the Riemann map, N. G. dim(£)~ n- Dn is possible, but the correct Dn is Makarov proved this in 1985 in a very precise not known.) Tom's construction also solves (par­ sense for simply connected domains. For general tially) a problem of Bers: There is u harmonic in domains it is much more complicated. The main x3 > 0, c1+a up to X3 = 0, such that u = IY'ul = 0 difficulty comes from those parts of an where on a set of positive measure on X3 = 0. Here oc is

MAY 2001 NOTICES OF TIIE AMS 483 Charles Fefferman's solution of the "ball-multi­ plier-problem" (1971). If fc~), ~ E JRd, d ~ 2, is the Fourier transform off E LP, p > 1, is f, restricted to 1~1 :::; 1, the Fourier transform of some g E LP? This is true for d = 1 (Hilbert transform) and for p = 2 and all d, but Fefferman gave a counterex­ ample for all other p and d. The obstruction comes from thin layers 1 - 8 :::; I~ I :::; 1 which contain long ( ~ 8112) rectangles of width 8 of all directions. In this way the counterexample comes from Besi­ covitch's 1928 construction (as a solution of a problem of Kakeya's concerning "sets where you can turn a needle") of a compact set in JR 2 of zero measure that contains a line segment of length 1 in every direction. (For this classical theory see (3].) It is rather easy to see that the Hausdorff di­ mension of a Kakeya set is 2. The corresponding conjecture in JR" concerning needles in any direc­ Tom Wolff at his desk in luminy in the summer of 1994. tion is open. One can also study restriction prob­ lems off(~) to 1~1 = 1 (Stein 1976). Kakeya prob­ lems for circles (the dimension of a compact set a fixed number, but it is not known if we can take, in JR2 that contains a circle of every small radius) say, oc = 1 (most likely not!). The construction also are similarly related to these restriction problems. disproves a third conjecture which I omit. In more Bourgain introduced the idea of fattening the seg­ recent work, the explicit and painful construction ments and circles to have width 8 and then look­ ( *) has been simplified and extended to certain ing for the size of the constant C(8) in LP -estimates powers of le + 'V'ul, but a complete picture is still of the corresponding maximal functions (8- missing. method). During the early part of the 1990s, Tom's research was focused on "unique continuation" Here Tom contributed very important new ideas (UC). In its simplest form this concerns differen­ and incorporated methods from combinatorial tial inequalities such as l~ul:::; Vo(x)lu(x)l or geometry. The most striking is the proof (1997) 3 l~ul:::; V1(x)IV'ul. If we know that u vanishes to (using the 8-method; here an L -estimate is rele­ infinite order at a point (x = 0) (strong UC, SUC) or vant; see [12]) that a Kakeya set for circles in JR 2 in an open set (UC), under which conditions on V; has dimension 2. For the needle case in JR", the can we conclude that u =0? This has obvious im­ dimension estimate ";1 is easy, but Tom proved plications for uniqueness of solutions to PDE. All (1995) ";2 . A growth rate better than ! x n as work here originates in a paper by T. Carleman n - oo has been proved by Bourgain. from the 1930s. One studies norms of functions such as lxl - tlul or exp(k · x)lul as the parameter Much remains here, and we are still hoping for t or k grows. For Vo and SUC, Jerison-Kenig proved a deeper understanding of how lower-dimensional in dimension d that Vo E L d/2 suffices, which is the objects fit into higher dimension. There is little optimal exponent, and Tom proved the analogous doubt that this type of mathematics is of great im­ statement for V1 ELd, d=3,4. He also proved portance for the analysis of the 3-dimensional that 3d2- 4 suffices. For UC, Tom proved that expo­ structure of molecules and therefore of, for ex­ ample, nents d/2 for Vo and d for V1 suffice, but here it polymers. is not at all clear if these exponents are relevant: What I have summarized here is a sample of e.g., L 1 may suffice. His method is an impressively Tom's work. The common denominator is combi­ detailed analysis of the sets where real Laplace natorial harmonic analysis, and it was here that he transforms had his unique talent. Where most of us would give up because of the complexity, he would exp(k · x)df.i(x) organize J the facts and concentrate for a very long time until of positive measures are of maximal size [10]. the goal was reached. He had, however, wide in­ Tom's main interest during the last years of his terests in mathematics and was always eager to life was "Kakeya sets". The methods here had the understand and discuss. This resulted in papers right mix of classical harmonic analysis, geometry, on interpolation spaces, Hardy spaces, geometry, and combinatorics to fit his unique talents. As a and, let me finally mention, a nice joint paper with problem in harmonic analysis, it goes back to Barry Simon on spectra for self-adjoint operators.

484 NOTICES OF THE AMS VOLUME 48, NUMBER 5 function in QA. Since nonzero functions in QA Donald Sarason are in fact nonzero almost everywhere, it follows subset of the It was my good fortune that Tom Wolff became immediately that every measurable a function in QC, a result interested in my area and decided to write his dis­ circle is the zero set of surprised me. sertation with me. He worked on some questions that of the theorem involved an in one-variable function theory with which I was Tom's proof application of BMO techniques. The then involved. insightful was unexpected because Q C functions Tom's graduate school years (1975-79) came theorem possess a sort of average continuity that general toward the beginning of the BMO era, initiated L oo functions lack. The theorem says that, in some around 19 72 by Charles Fefferman and Elias Stein. sense, the set of discontinuities of a general L oo Fefferman's duality theorem for the space of func­ function is smaller than we had imagined. Tom tions of bounded mean oscillation, a high point in showed that a Banach algebra perspective enables the program to develop real-variable methods in one to make this statement precise. analysis, was having a large impact in the harmonic Everyone who knew Tom during his student complex realm as well. days in Berkeley recognized his intelligence. It of Toeplitz operators there had In the study was not until that day he came to my office with a certain function space known as Q C (for arisen his breakthrough that I appreciated his remarkable It consists of the bounded func­ quasicontinuous). talent. plane tions on the unit circle in the complex Tom's dissertation is somewhat removed from functions and writable as sums of continuous the main currents in harmonic and complex analy­ functions. The Hilbert transforms of continuous sis. It is thus not widely known, despite its elegance be space turns out, for nonobvious reasons, to and beauty. After completing it, but while still a closed both under the essential supremum norm Berkeley student, Tom made another breakthrough. and under multiplication, thus is a Banach subal­ Word of his work on the corona problem spread gebra of the algebra L oo (on the circle). From quickly and made him famous. Fefferman's theorem one can derive an alternative description of QC: it consists of the bounded Sun-Yung Alice Chang functions on the circle that have vanishing mean I first met Tom Wolff in April1979, the year he oscillation. A related algebra is QA, consisting of finished his Ph.D. at UC Berkeley. We both had in QC that are boundary functions the functions written our thesis under the supervision of Donald of bounded holomorphic functions in the unit Sarason. The topics of our theses-the study of the disk. behavior of functions defined on some subalgebra A basic problem about Q C which had stumped me between H 00 (bounded analytic functions) and L oo its zero sets. I had been was that of characterizing of the unit circle on the plane- are closely related. e a plausible conjecture. unable even to formulat I was an assistant professor at the University of this problem specifically, While I did not assign Tom Maryland, College Park, where Lennart Carleson he seems to have gravitated to it instinctively. He was visiting during the special year in harmonic and unex- eventually solved it in a very imaginative analysis. Tom came to give a lecture on his simple, pected way. . elegant re-proof of Carle son's result on the Corona rather dis­ Tom came into my office at one point problem. It was a striking experience to see this shy, couraged, feeling he wasn't getting anywhere. I humble young man talking about his work in front forget exactly what I said to him, but undoubtedly of the world's leading expert on the subject. The it consisted of the usual reassuring words about beauty of Tom's proof and the sharpness of his how nearly every mathematics Ph.D. student has mind left a deep impression. a similar experience, about how a mathematician Later on we had many occasions to meet in con­ working on a hard problem can expect to be ferences and to have joint seminars together while stuck for long periods, about how progress would he was at Caltech and I was at UCLA. Tom had a come if he just kept at it. It was not too many comprehensive and incisive understanding of many days later that he reappeared in my office, this time topics in analysis. Those of us around him all grea~y to announce his beautiful resolution of the zero benefited from the interactions with him. In partic­ set problem. He in fact simultaneously resolved ular, during the month of ~ay 1990, while Tom and several other questions about Q C. Needless to say, I both were visitors at IHES (Institut des Hautes such a moment is a dissertation supervisor's ulti­ Etudes Scientifiques), we spent long hours talking. mate gratification. I got to know and was deeply impressed Tom's basic theorem states that every function by his work on Bers' problem, his criticisms- in L oo can b e multiplied into QC by a nonzero Sun· Yung Alice Chang is professor of mathematics at Donald Sarason is professor of mathematics at the Princeton University and the University of California, University of California, Berkeley. His e-mail address is Los A ngeles. H er e-mail address is chang@math. [email protected]. princeton. edu.

MAY 2001 NOTICES OF THE AMS 485 unsparing of his own work-of contemporary that reigned in the department, the matter was mathematics, his heavy sense of responsibility discussed for a minute or two, and then Calderon toward his students, and also his point of view declared the situation to be obvious: Tom must on politics in the mathematical community and come. society in general. At the time Tom arrived the University of I always enjoyed talking to Tom about the Chicago was an exciting place for Fourier analysts. mathematics projects each of us was involved in. The full ramifications of H 1-BMO duality, proved Although our interests later diverged, Tom's a decade earlier by Charles Fefferman, were still comments were always insightful and honest. We being worked out. (Tom's new proof of the Corona also had two joint papers together; each grew out Theorem was very much a product of that program of discussions where his contribution was so and is now seen as a model application of those significant that I, together with my coauthor, L 2 methods.) Calderon had recently proven the L 2 invited him to join our work. In [2] we partially boundedness of the Cauchy Integral on small con­ answered a question posed byCharles Fefferman: stant Lipschitz curves, and it was clear that a whole Let v be a nonnegative integrable function on JRd; new area had opened around that result. There when is it true that were lots of young analysts who flocked to Chicago f lul 2 vdx~c f IY'ul 2 dx in those years, but virtually from the start it was JIRd JRd clear that Tom had a special brilliance. his for some constant c? When c = 1 , this is equiva­ Very rapidly after arriving, Tom broadened lent to the positivity of the associated Schrodinger scope and was working on conjectures in many different areas. His approach to mathematics was operator L = ~ - v when L is essentially self-adjoint on CQ'. We also answered a question raised by remarkable and obvious even to those who knew Elias Stein by showing that if a square function him only slightly. Tom's hallmark was to select a S(f)-which is a variant of Lusin's area function­ problem where the present tools of harmonic analy­ is bounded, then the function f is in the Orlicz sis were wholly inadequate for the task. The exact space exp(L2) and this is the best order. This area was not necessarily so important, but he had latter result has been useful in the work of Robert a knack for finding precisely the central problems. Fefferman, Carlos Kenig, and Jill Pipher; in the Beginning with almost no background, he would study of Hardy spaces; and also in the work of interview experts at length and devour the litera­ Chuck Moore and Michael Wilson. In [1] we gave ture. Within a month or two he knew basically examples of sequences on a compact d-dimen­ everything that was relevant to the problem and sional manifold (d ~ 3) in a fixed conformal class would then turn in earnest to the attack. satisfying a uniform L(d/2l bound on curvature This is the part I remember best: when Tom and a bound on volume that are not compact in a was in full gear. He would enter a period of extreme c0 topology, which indicated that the result in the concentration, several notches up from his usual thesis of Matt Gursky is sharp. intense state. During this period nothing could Our mathematical community has lost a leader distract him, and he would stay in Eckhart Hall until at the prime of his productivity, and many of us late in the night, when he returned the few who worked with Tom have lost a friend. hundred meters to his apartment. In the day he could be seen pacing the corridors with a cup of coffee in hand. Other frequent haunts were the coffee lounge or the front steps of Eckhart Hall, Peter W. ]ones where he could be found smoking a cigarette. He I was a Dickson Instructor at the University of had a very characteristic body language, with his Chicago when I first heard about Tom Wolff. He shoulders hunched slightly forward and a distant had written an outstanding thesis under the gaze in his eyes. All the time he was calculating direction of Donald Sarason and created quite a stir lemmas or trying out a different tack. Tom was not with his now celebrated method of proving Lennart always communicative about what he was work­ Carleson's Corona Theorem. At this time the analy­ ing on, even though it was clear that something was sis group at Chicago was led by Alberto Calderon, up. I would sneak into his office on these occasions with Bill Beckner and Bob Fefferman the junior and look at his voluminous notepads, trying to professors. Antoni Zygmund was old and infirm, divine the exact nature of the problem. Eventually, but made a point of coming in every day and never the mathematical door would open a crack as Tom missed a s eminar. I presented Tom's results to discovered a new technique, usually of astonish­ Calderon, Beckner, and Fefferman and urged them ing originality. The end would now be in sight, as to try to bring him to the university. In the spirit Tom unleashed his tremendous technical abilities and overcame the remaining difficulties. Tom Peter W. ]ones is professor and chairman of mathematics attributed his results solely to hard work, but I at Yale University. His e-mail address is jones@math . never found this a satisfactory explanation and yale .edu. believe the true answer to be a mystery.

486 NOTICES OF 1HE AMS VOLUME 48, NUMBER 5 It was my fortune to write two papers with Tom. One of these, also coauthored by Donald Marshall, showed that (e.g.) if given corona data in the disk algebra, one could find corona solutions with one of the functions invertible. This was not too hard, but the other problem we tackled was tough to crack. 0ksendal had conjectured that for an arbitrary domain in !Rn, the harmonic measure was supported on a set of (Hausdorff) dimension at most n - 1. N. G. Makarov first proved a very sharp version of this for simply connected planar domains. Tom's paper with me solved the problem for general planar domains, using a delicate geometric construction. It was during this time that I got to see first-hand all of Tom's amazing talents, both conceptual and technical. Later, Tom disproved 0ksendal's conjecture for n ~ 3, and this has deep physical significance. It is not much of an exaggeration to say that this explains why Wolff with his uncle Clifford Gardner (November 1994). lungs work! Perhaps the most astonishing thing about that particular paper is that Tom also love of the cello, and a joint project to build- with solved two other well-known, only philosophically their bare hands-a log cabin in the woods, a related, conjectures. It took quite some time for retreat from the madding crowd. All this gave Tom to write up that paper. I recall him explain­ ample chance for them to discuss mathematics. ing that the problem-solving phase was exciting When Tom's affairs brought him to settle on the but the writing was drudgery. East Coast, we at the Courant Institute jumped at Time after time, Tom Wolff would pick a cen­ the chance to appoint him to the faculty. We were tral problem in an area and solve it. After a few very enthusiastic about his research, but a little more results, the field would be changed forever. apprehensive about his teaching because of his Tom would move on to an entirely new domain of shyness. We needn't have worried; within a short research, and the rest of the analysts would spend time he became the most popular teacher in the years trying to catch up. In the mathematical com­ department because of the clarity of his lectures munity, the common and rapid response to these and the wholehearted support he offered his breakthroughs was that they were seen, not just students. We were very sorry to lose him when as watershed events, but as lightning strikes that another change in his life brought him back to his permanently altered the landscape. beloved California. When news of the accident that took Tom's life reached Loon Lake this past summer, it was a thunderbolt from a sunny sky. "As flies to wanton Peter D. Lax boys are we to the gods, they kill us for their I have known Tom's parents for over fifty years. sport." I met them through Tom's maternal uncle, Clifford Nikolai Makarov Gardner, an outstanding applied mathematician whose accomplishments have been recognized by The first time I heard of Tom Wolff was in the a Norbert Wiener Prize, awarded jointly by the early 1980s when the news of his amazing new AMS and SIAM (Society for Industrial and Applied proof of the Corona Theorem reached Leningrad. Mathematics). Clifford was the first to arouse I still remember the excitement we felt as the proof Tom's interest in mathematics. The second was was presented in the analysis seminar. Before long, Jiirgen Moser. Their paths crossed at Loon Lake, several other spectacular results by Tom followed. an isolated spot in Franklin County in northern­ For us working in Leningrad on problems in most New York State, where the Masers and the linear and complex analysis, his name became Wolffs have had vacation homes since 1970. almost legendary. Jiirgen and young Tom were drawn together by In the course of his career Wolff deeply influ­ their love of hiking (although Jiirgen liked well­ enced various fields of modern analysis. He mapped footpaths, while Tom loved to bushwhack, made groundbreaking discoveries in harmonic and which made my sons dub Tom "the Viking"), a complex analysis, potential theory, and differen­ tial equations. In the last, exceptionally productive,

Peter D. Lax is professor emeritus of mathematics at the Nikolai Makarov is professor of mathematics at the Courant Institute of Mathematical Sciences of New York California Institute of Technology. His e-mail address University. His e-mail address is 1 ax@ci ms. nyu. edu. is makarov@caltech. edu.

MAY 2001 NOTICES OF THE AMS 487 years of his life, the Kakeya problem techniques people. He had friends all over the world. Many became his main topic, and Wolff made great analysts are deeply indebted to him for his advice progress in some of the most fundamental prob­ and support. I had the privilege of working with lems of harmonic analysis related to geometric Tom at Caltech for several years. He was a fine measure theory. His work had remarkable depth person, invariably friendly, reliable, and helpful. and breadth. Talking mathematics with him was always thrilling One part of Wolff's legacy that I can appreciate and inspiring. the most was his work concerning harmonic Most people would agree that Tom Wolff was measure in the complex plane and in higher one of the greatest analysts of our time. Those of dimensions. I had been working on related prob­ us who were lucky enough to have known him lems, and when I first saw Tom's results, they personally feel a tremendous sense of loss of looked like a miracle to me. He proved (with Peter someone very special. Jones) a long-standing conjecture that harmonic measure in the plane lives on a set of dimension at most one. He also showed Barry Simon that a similar fact I was a colleague and admirer of Tom. Three does not hold in higher di­ aspects of Tom's personality were especially mensions. The latter is a noticeable. The first was his passion and intensity. stunning example of how Tom was not only passionate about mathematics, one person can change the he was passionate about mathematicians and went whole subject. By con­ out of his way to help young mathematicians all structing beautiful and over the world, both with their mathematics and rather surprising coun­ with making sure they got the recognition they terexamples, Tom dis­ deserved. As for his intensity, Tom was a fixture proved all conjectures that on campus pacing outside the department, existed in this area and lost in thought. showed that the most The most common comment I got after basic facts about harmonic Tom's death-from Caltech's provost to a delega­ functions in the plane fail tion of undergrads-was that they missed him in dimension three or when they came in in the morning and didn't see greater. At the same time, him pacing there. he indicated new direc­ The second was his honesty. Tom didn't wave Tom Wolff with family in Mammoth, tions in the study of the his hands and didn't let others get away with CA, on his 45th birthday Uuly 1999). higher-dimensional case. waving their hands. The third was his shyness, which was so strong that one could see him Tom never wasted time overcome it when dealing with others. on finding all possible applications of his ideas. Tom was not only a deep thinker After solving the problem, he would move on to a in mathe­ different subject, leaving it to other people to fully matics but also a technical master. My own joint work him implement his inventions. Some of his arguments with arose when I was thinking about the (such as his proof of the Corona Theorem) im­ implication of some ideas of Kotani for localiza­ press with elegant simplicity. But more often his tion in random quantum systems. I realized that argument would be of the "hard analysis" type, and one could base things on rank one perturbation Tom was a genius of "hard analysis". His typical generalities if only I could prove a certain fact proof would be based on some very complicated about measures. After a week of trying, I called in and incredibly clever combinatorial construction Tom. We spent several hours trying to crack it of a geometric nature. with no success, but the next morning Tom walked in with the needed result in hand. Our subsequent Reading Wolff's papers is a difficult task, talks led to a simple way of avoiding the problem though he put great effort into writing his papers totally and resulted in our localization meticulously. It usually takes a long time to fully criterion [9]. As understand the depth of his ideas, but the upshot a side benefit of our joint work, Tom got is very rewarding for the reader. Tom had great interested in problems of localization and returned to the subject success in raising graduate students. Part of the ten years later in a beautiful joint reason, I believe, was that by merely studying his paper with Shubin and Vakilian [8] and, just prior constructions, the students were able to discover to his death, a joint work with Klopp [7]. Typically, new ideas and master all the latest technologies. these papers exploited subtle ideas from harmonic analysis. In particular, Tom's insight on the role of Tom commanded unanimous respect and admiration. He was a perfect colleague: very Barry Simon is IBM Professor of Mathematics and Theo­ generous with his insights and ideas, uncompro­ retical Physics at the California Institute of Technology. misingly honest, extremely respectful of other His e-mail address is bsi mon@ca 1 tech. edu.

488 NOTICES OF THE AMS VOLUME 48, NUMBER 5 uncertainty principle inequalities in the subject is gave a little bit of a significant contribution. advice, but seemed Tom was a gentleman and a gentle man. I miss reluctant to speak him. over lunch or maybe hesitant to Markus Keel say anything that I first met Tom Wolff four years ago when he'd was less than ab­ visit the UCLA analysis seminar, but got to know solutely precise. him better while a postdoc at Caltech these past Five minutes after two years. From his research, to his teaching, I'd returned to my to his day-to-day scientific interactions, Tom's office I found out unflinching honesty had as much to do with his reticence had his impact on me and my generation as did his more to do with his incredible analytical strength. doubting my entire Tom's utter lack of vanity should be seen first premise: I an­ swered his light in the light of his work, the depth and breadth of knock, opened the Wolff "reading math" with son Ricky. which are outlined by the collection of authors here. Even at the time of his death, Tom's preprints door, and in were nourishing entire fields. In [13], for example, poured Tom to explain that it'd be a real chore to produce a counterexample, since the estimate was he answered (up to an endpoint) a conjecture of Klainerman and Machedon dealing, roughly, with actually true. He put the proof on the board, apol­ ogized for interrupting my work, set the chalk the way that two solutions to the wave equation down, and strode right back out again. can interact. Tom's paper surprised everyone in the Tom Wolff was one of the most potent human field, and it seems likely that the arguments he in­ beings I've ever met. It was a great thrill to work troduced there will seriously impact work on the in the vicinity of the man. regularity and stability of nonlinear wave equations. Tom's teaching had a similar effect on the students and young faculty at Caltech, from his undergraduate calculus courses to the advanced Wilhelm Schlag harmonic analysis class he taught here last spring. Twice a week Tom would speed into the room, I met Tom Wolff in the fall of 1992 at UC Berke­ looking for all the world like he'd just wrestled ley. I was a first-year graduate student and Tom about 300 wild cats, half of whom were wielding had just moved there from Cal tech. He taught an squirt guns loaded with coffee. The lecture that introductory class in harmonic analysis that same followed was both terrific and terrifying: he'd term that left a lasting impression on me. He lec­ cover intricate arguments in a way that made them tured with great care, paying particular attention appear almost inevitable, but a weird unease would to details that are frequently skipped in classes of creep over me towards the end of the term. At my this type. At the same time he would draw our very best, I realized, the mathematics classes I attention to the essential points, not allowing the teach are a lot like taxidermy-the stuffing of a details to obscure the main line of the argument. cadaver so that if you don't look too closely or too He thus made fundamental results of the subject long, you might be fooled into believing it's alive. completely transparent. Tom, on the other hand, put an intensity and care I think it was very clear even to us first-year into his lectures which made them nothing short students that he was a master in his field. His of reanimation. By the end of the course we really seriousness about mathematics and his high believed that the ideas he had presented were still standards made him an ideal, albeit demanding, vigorous, still steeped in potential. advisor. He insisted on seeing his students every week, and the discussions we had were always The way he listened and spoke in informal fruitful. It was clear that he also enjoyed discussing mathematical discussions was similarly unique. mathematics with us, and he was basically avail­ Tom would listen not just to your words, but for able at all times. the hidden biases and unacknowledged gaps I think that the most inspiring feature of his that color most arguments. For example, I once personality was his uncompromising and relentless kept Tom at a table in the Caltech cafeteria to see search for the key points of a problem. He would if he had any ideas for a counterexample to an assemble the main facts and methods and then estimate which I, and every senior mathematician combine them in an almost miraculous way that I could corner, thought was probably false. Tom

Markus Keel is Taussky-Todd Research Instructor Wilhelm Schlag is · assistant professor of mathematics Department of Mathematics, California Institute of at Princeton University. His e-mail address is Technology. His e-mail address is keel @its. cal tech. edu. [email protected].

MAY 2001 NOTICES OF TilE AMS 489 made everything appear simple. This was true of his teaching as well as of his research. His command of the subject as well as his somewhat serious appearance could make him intimidating at times. But this really belied his gentle personality. I am very lucky to have met Tom, and I regard it as a privilege to have been his student.

References (1) S.-Y. A. CHANG, M. GURSKY, and T. WOLFF, Lack of compactness in conformal metrics with L(d/2) curvature,]. Geom. Anal. 4 (1994), 143-153. [2) S.-Y. A. CHANG, M. WILSON, and T. WOLFF, Some weighted norm inequalities concerning the Schri:idinger operators, Comment. Math. Helv. 60 (1985), 217-246. [3) F. CUNNINGHAM ]R., The Kakeya problem for simply connected and for star·shaped sets, Amer. Math. PUZZLERS' TRIBUTE Monthly 78 (1971), 114-129. [4) ]. GARNETI, Bounded Analytic Functions, Academic David Wolfe, Tom Rodgers, editors Press, New York, 1981. 2001; hardcover; approx. 350 pp.; $34.00 (tent.) [5) L. HEDBERG and T. WOLFF, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 This collection, in tribute to Martin Gardner, (1983), 161-187. presents poems, puzzles, and personal [6) P. JoNES and T. WoLFF, Hausdorff dimension of recollections of the greatest puzzler of all, who harmonic measures in the plane, Acta Math. 161 introduced many to the joy ofm athematics. (1988), 131-144. [7) F. KLOPP and T. WOLFF, Internal Lifshitz tails for Here's to The Game Master! random Schri:idinger operators, preprint, 2000. [8) C. SHUBIN, R. VAKIIlAN, and T. WoLFF, Some harmonic analysis questions suggested by Anderson-Bernoulli models, Geom. Funct. Anal. 8 (1998), 932-964. A GARDNER'S WORKOUT: [9) B. SIMON and T. WoLFF, Singular continuous spectrum Training the Mind and Entertaining the Spirit under rank one perturbations and localization for Comm. Pure Appl. Math. 39 Martin Gardner random Hamiltonians, (1986), 75-90. 2001; hardcover; approx. 350 pp.; $30.00 (tent.) [10] T. WoLFF, A property of measures in JRN and an Truly a treat for Martin Gardner fans. application to unique continuation, Geom. Funct. Anal. 2 (1992), 225- 284. A collection ofproblems and puzzles, including games [11] __ , Counterexamples with harmonic gradients in ofchance, word ladders and mathematical word JR3, in Essays on Fourier Analysis in Honor of Elias play games, tiling puzzles, magic squares, M. Stein, Princeton Math. Ser., vol. 42, Princeton Univ. and computer and calculator "magic" triclrs. Press, Princeton, NJ, 1995, pp. 321-384. [12] __ ,A Kakeya-type problem for circles, A mer. ]. It to sweat! will cause your brain Math. 119 (1997), 985-1026. [1 3) __ ,A sharp bilinear cone restriction estimate, to appear in Ann. of Math., 2001.

great A K Peters tides at www.akpeters.com

A K Peters, Ltd. t 63 South Ave., Natick, MA 01760-4626 Tel: (508) 655-9933 Fax: (508) 655-5847 www.akpeters.com • ser [email protected] Publishers of the journal Experimental Mathematics

490 NOTICES OF THE AMS VOLUME 48, NUMBER 5 The Growth of the Professional Master's in Mathematics Sheila Tobias, Charles MacCluer, and Ralph Svetic

"Mathematics has an age-old mission in unifying the sciences and technology." -Avner Friedman, 1993

In terms of student enrollment, American students, that disarm the objections of the skep­ mathematics is in crisis. Although secondary school tics. Jay Walton, whose mathematics department students increasingly express an interest in at Texas A&M began offering the professional mathematics, the number of mathematics majors master's options in 1994, reports an increased is declining. Students choose, instead, majors visibility for the department both inside and out­ that lead more directly to jobs in today's techno­ side the university, an increase in collaborations logical workplace [AMS]. While industry needs with other departments, an increase in graduate mathematically trained people, it hires few math­ mathematics course enrollments by nonmathe­ ematicians-a consequence of our discipline's matics graduate students, and an increase in the century-long disengagement from industrial work. proportion of university resources targeted for But this bad news has been allayed in the past the mathematics graduate program. Moreover, decade by the flourishing of a professional none of these benefits have come at the expense master's degree in mathematics, one that promises of the Ph.D. program, which, at Texas A&M, has the six new careers for the serious student of mathemat­ steadily increased in size and quality over options have been ics. Skeptics argue that such programs are devoid years the professional master's offered []W]. of serious mathematical content, attract weaker In what follows we will sketch the growth of the students, take resources and students from the professional master's degree (the first new degree traditional Ph.D. program, and force departments since the MBA at Harvard in 1908), what niche it fewer courses in traditional mathematical to offer fills, how it is implemented at several institutions, point to many areas. Practitioners, however, and its potential benefits to the mathematics com­ the benefits, both for the providers and for munity as a whole. Sheila Tobias is a math/science education consultant, author of Overcoming Math Anxiety (1978 and 1994), The Origins Succeed with Math ~1987), and coauthor of Rethinking In 1995 the Committee on Science, Engineering, and Science As a Career (1995). She is currently the outreach Public Policy (COSEPUP) of the National Academy coordinator for the Sloan Science Master's Degree of Sciences suggested that graduate schools of Initiative. Her e-mail address is 102531.1746@ arts and sciences might consider a "different" kind compuserve. com. of graduate degree, less oriented toward research Charles MacCiuer is professor of mathematics at Michi­ and requiring less time to obtain [CU]. The diffi­ gan State University and directs the MSU Professional cult labor markets faced by many new Ph.D.'s and Mathematics. He is the Master's Program in Industrial postdocs and the growing number of part-time Mathematics, Prentice Hall, 2000. His author ofindustrial the view that e-mail address is maccl uer@math. msu. edu. and adjunct faculty contributed to the Ph.D. degree might not be the best terminal appointment Ralph Svetic holds a research postdoctoral degree for many science- or mathematics­ at Michigan State University and codirects in mathematics Report was the MSU Professional Master's Program in Industrial trained professionals. The COSEPUP Mathematics. His e·mail address is rsveti c@math. a long-delayed acknowledgment that graduate ed­ msu. edu. ucation in the sciences had become severely

MAY 2001 NOTICES OF THE AMS 491 decoupled both from the career needs of students For most of the sciences, reform was thought and from the supply needs of an ever more tech­ about in terms of doctoral education, for what nically based national economy [MG]. else is graduate education in the sciences but Even before the National Academies had broached doctoral education? the topic, there were stirrings in the mathematics community. At conference after conference, begin­ Reinventing the Master's Degree ningin the late 1980s, and in journals like this one, All the while, the "silent success" (see Table 1) of talks were given and articles appeared with titles the terminal (professional) master's degree in fields like: "Preparing for a job outside academia" [Ben94], other than science and mathematics was pointing "Preparing M.S. and Ph.D. students for nonacade­ in another direction. Beginning in the late 1980s, mic careers" [BLTSLD93], "The mathematical an increasing number and proportion of M.S. de­ training of the nonacademic mathematician" grees were being awarded in professional fields, [Boy75], "Graduate education in transition" [CBMS92], with M.S. degrees in arts and sciences in decline. and so on. At the helm of the nascent movement In their 1993 study of master's education in the was SIAM, the Society for Industrial and Applied U.S., commissioned by the Council of Graduate Mathematics, which recognized early that not only Schools and supported by the Pew Trust, Conrad were there jobs to be had and interesting careers to et al. posited an important distinction between mas­ be made by mathematicians in business and indus­ ter's degrees: the predoctoral (or, when terminal, try but that some of the more interesting problems the failed Ph.D.) versus the self-contained profes­ in mathematics could come from such settings. sional certification [CHM93, p. xiv]. As they describe

Table 1. Number of master's degrees conferred by fields of study

Year Professional* Arts and Sciences** Mathematics Total

1996-97 326,895 14,219 3,783 419,401

1995-96 317,559 14,229 4,031 406,301

1994-95 310,514 13,312 4,181 397,629

1993-94 301,454 13,164 4,100 387,070

1992-93 288,358 12,365 4,067 369,585

1991-92 272,004 12,403 4,011 352,838

•k** •1:*-!r

1990-91 259,724 13,369 3,615 328,645

1989-90 253,544 13,700 3,676 322,465

1988-89 243,268 13,929 3,447 309,770

1986-87 224,942 14,101 3,319 289,341

1984-85 218,285 14,586 2,831 280,421

Source: U.S. Department of Education, National Center for Education Statistics, Integrated Postsecondary Education Data System, "Completions" survey and "Consolidated" survey, for the quoted academic year. *Professional includes Education, Health Professions, Business Administration, Engineer­ ing, Psychology, Computer Science, Public Administration, Library Sciences, and Law. **Arts and Sciences includes Physical Sciences, Agricultural Sciences, and Biological Sciences. *''*Change in source definitions of fields of study.

492 NOTICES OF 1HE AMS VOLUME 48, NUMBER 5 it, this professional degree is " ...an important means about (1) the latest applications of mathematics, for enriching the knowledge base and skills of (2) the benefit to faculty and students from inter­ pre-professionals in an information-centered soci­ actions with industry, and (3) the design and im­ ety." Furthermore, " ... people with master's consti­ plementation of new programs [Online resource: tute the professionals upon which business, indus­ SIAM98]. The idea behind the workshop series, ac­ try, education, government, and the nation's health cording to one of its organizers, was that (1) and care systems are increasingly coming to depend for (2) would lead naturally to (3). The goal was to have expertise and leadership" [CHM93 p. xiv]. participants from other universities learn about "What kind of professional master's degrees these efforts and, if possible, implement their own might we invent for science?" asked the authors [Online resource: DAVIS]. To further propagate of Rethinking Science As a Career [TCA95], and the idea of the professional master's, a set of how, once these degrees were in place, could they "Guidelines for the Professional Master's in Math­ be marketed to faculty, students, and to the em­ ematics", prepared by SIAM, is scheduled provi­ ploying and subsidy-paying public? If one looks to sionally for publication in July 2001. The "Guide­ the master's degree more generally, one finds that lines" will recommend a core curriculum; course instead of training producers of scholarship-the work or training in presentation and writing skills; traditional purpose of graduate education­ and internships in business, government, or in­ master's educators aim to produce people who dustry. Appended to the brochure as a kind of ex­ are able to use the products of scholarship in their istence proof will be descriptions of several pro­ work and who are familiar with the practical as­ fessional master's programs from a variety of pects of emerging problem areas. So the general institutions [SIAMGuidelines01]. outlines of a professional science or mathematics Meanwhile, furthering a different but comple­ master's degree were available to the community mentary agenda, the American Mathematical Society in the mid-1990s. The next task was to begin the (AMS) and the Mathematicians and Education process of persuasion and develop the momentum Reform Forum (MER) organized two national necessary to launch the new degree. workshops, the first at New York University on November 5-7, 1998, and the second at Arizona The Role of AMS, MER, and SIAM State University on November 4-6, 1999. The There is no question that the mathematics com­ organizers' intention was to stimulate departments munity itself played a leading role in expanding the of mathematics to think creatively about their professional master's degree from a narrow band mission in graduate education and as part of that of programs in fields such as actuarial science and to consider the range of professional master's that teaching to a wider range of subject specialties were available to them. Twenty-seven departments and applications. First, there was industrial and were represented at the NYU workshop and thirty­ applied mathematics. In the 1980s Avner Friedman, three departments were represented at the ASU president of the Society of Industrial and Applied workshop. To further support departments, AMS Mathematics (SIAM) during 1993-1994, launched and MER are producing a directory of professional the Institute for Mathematics and its Applications master's programs, based on their survey of 426 (IMA) at the University of Minnesota, followed (in master's- and Ph.D.-granting mathematics 1993) by Friedman's and John Lavery's booklet, departments, which both interested faculty and How to Start an Industrial Mathematics Program prospective students will be able to access online in the University [FrLa]. SIAM's next influential [Online resource: MER] . publication, Mathematics inlndustry(1995) [MI95], While AMS, MER, and SIAM deserve credit for based on interviews with 500 mathematicians in encouraging the establishment of the professional industry and their managers, documented the need master's degree by way of publications and for graduates specifically trained in a combination numerous workshops, two contemporaneous of mathematics, applications, and computation. developments have given the professional master's By May 1998 SIAM had secured National Science a further boost: first, the emergence in the 1990s of Foundation (NSF) funding for a series of regional a Professional Master's in Financial Mathematics in workshops both to build bridges between indus­ some of the nation's most prestigious institutions; try and academia and to encourage the founding second, the commitment on the part of two national of professional master's degree programs.l foundations, beginning in 1997, to support the Typically, mathematicians from industry and establishment of new professional science master's their faculty counterparts were invited to talk programs in which mathematics would be an integral part. 18-20, 1998, Worces­ l SJAM Workshops: Northeast, May With the stock market gaining in visibility in ter Polytechnic; Midwest, October 2-3, 1998, Univ. ofilli ­ nois at Chicago; West, june 16-19, 1999, The Claremont the early 1990s and the increasing mathematical Colleges; Southeast, October 10-12, 1999, North Carolina complexity of financial i nstruments, a sizeable State Univ.; Northwest, October 12-14, 2000, Washington number of physicists and mathematicians began State; Southwest, A pril 2001, Houston. finding their way to Wall Street. This bloom in

MAY 2001 NOTICES OF THE AMS 493 Online Resources who were also knowledgeable about intellectual property rights, finance, and business manage­ [S~~8] http://www.siam.org/meetings/archives/ ment, particularly as applied to biotechnology. The index. htm#mll · resulting institution, Keck Graduate Institute, which enrolled its first class of twenty-eight students in [DAV1S] http://www.wpi.edu/~heinrich/MII.html August 2000, recruits graduates in mathematics ' along with graduates in the life sciences, engi­ [MER] http://www.math.uic.edu/MER/masters.html neering, physics, and chemistry [Online resource: KGI]. [KGI] http://www.kgi.edu/ The contribution of the Sloan Foundation was to embed mathematics in a broader concept of the [SLOAN] http://www. s 1oa:n. org/ (Sloan Science Master's "science master's" and to provide start-up funding Initiative) for program development [Online resource: SLOAN]. Unlike the Keck Graduate Institute, with its dedi­ [SCI] http://www.sciencemasters.com/ cated biotechnology program, Sloan was willing to entertain any program of study-ideally in an [Ga] http://www.qcf.gatech.edu/ emerging discipline or combination of fields-for which a faculty group could document that there [WISC] http://www.wisc.edu/computationalsciences/ were employment possibilities for graduates at the master's level. As of this writing, Sloan has [ARIZ] http://cos.arizona.edu/sloan/ funded twenty-four degree "tracks" in seventeen Ph.D.-granting institutions, of which ten tracks (in [MSIM] http://www.math .msu.edu/ six different institutions) are in mathematics. With a substantial degree of local initiative and local control, the mathematics tracks funded by Sloan opportunities certainly fueled the establishment are not of a single type, although all require busi­ of master's-level graduate programs in financial ness/industry interaction [Online resource: SCI]. mathematics. The oldest among them is Compu­ For example, in Michigan State's Professional tational Finance at Carnegie Mellon University Master's Degree Program in Industrial Mathematics, (1992), followed soon after by Financial Mathe­ students are expected to complete a ten-module matics at University of Chicago, Mathematics with business certificate program; in Arizona's Specialization in the Mathematics of Finance at Mathematical Sciences Master's program, students Columbia University, and Mathematics in Finance take two semester-long courses, one in basics for at New York University. business, one in the legal environment of busi­ Meanwhile, the NSF-funded workshops stimu­ ness. In contrast, students enrolled in the lated the expansion of master's programs in applied Quantitative and Computational Finance track at and industrial mathematics. Examples of programs Georgia Tech take all their courses in financial established during this period are at the University instrument development and usage, investment, of Massachusetts, Amherst (established in 1989) and risk analysis, with an emphasis on the con­ and Carnegie Mellon (established in 1992); the struction, implementation, and testing of models University of Illinois, Urbana-Champaign; Clemson; used in these areas of finance [Online resource: Ga]. Texas A&M; and Virginia Tech. Of these new degree Unlike many of the other professional mathemat­ programs, only Texas A&M employs the term "pro­ ics degree programs, which are creating new con­ fessional" as a descriptor. But because of the par­ figurations largely of existing mathematics and ticipation of industrial contacts and the effort to computational science courses, Georgia Tech's, attract the nonpredoctoral student, the programs with ten new courses, is building a new discipline. are de facto professional. Director Robert Kertz says students in the pro­ gram will be able to select different emphases and The Keck and Sloan Foundations' specializations. Currently, eighteen are enrolled Initiatives in the program's first year. In 1997both the WilliamM. Keck andAlfredP. Sloan Two of the authors (MacCluer and Svetic) direct Foundations, independently, began to explore the an Industrial Mathematics Professional Master's possibility of launching new professional science program sponsored by Sloan at Michigan State master's degrees. Keck was approached by Henry University. Fifteen students are currently enrolled Riggs, the outgoing president of Harvey Mudd, the in its second year of operation. The degree require­ science and engineering college in the Claremont ments are four courses in applied mathematics, four Colleges group. Riggs wanted to build an all-new, courses in engineering or economics, two courses master's-only graduate school to supply California's in statistics, a fall term survey of industrial mathe­ biotech industry with professionals skilled in the matics (taught from MacCluer's text [CRM]), and a life sciences, mathematics, and engineering, but spring term project course with local industry. The

494 NOTICES OF TilE AMS VOLUME 48, NUMBER 5 process of discovering the underlying mathematical careers in advanced computing applications such character of any interesting problem is a critical part as parallel computing, grid computing, and visu­ of the training for the students. The industrial alization/animation [Online resource: WISC]. Mean­ projects are managed by faculty from various while, Arizona's master of mathematical sciences departments and coordinated by MacCluer and features interdisciplinarity and the possibility of Svetic. Ralph Svetic has brought verisimilitude to tailoring programs to individual students' interests these projects from his fifteen years of industrial and needs. For example, among the first three en­ experience. Each project engages students in rollees at Arizona (2000-01) are a full-time student solving a single real-world problem provided by doing his M.S. thesis in bioinformatics; a full-time local industry (see below). The problems are solidted student working on image analysis after an in­ by Svetic and MacCluer, who mold them into ternship at Raytheon; and a part-time student, semester-long projects solvable by two- or three­ fully employed at Raytheon, working on signal person teams of master's students. Undergraduates analysis [Online resource: ARIZ]. serve as assistants to the graduate student teams, The newest family of Sloan-funded programs at exposing the undergraduates to industrial mathe­ the University of Pittsburgh is remarkable because matics and the graduate students to management of its four separate tracks within the Department responsibility. At the end of the semester the teams of Mathematics: Mathematical Modeling of generate a formal report and present their findings Complex Systems (coordinated by Carson Chow), both to fellow students and on site to their industry Industrial Mathematics (coordinated by Xinfu liaisons, a presentation to which a large number of Chen), Scientific Computing (coordinated by people from the client organization are invited. The William Layton), and Analytical and Computational on-site event increases the visibility of the Michigan Methods in Finance and Risk (coordinated by John State program and exhibits the quality of the stu­ Chadam). Yet all four are designed around a core dents being produced. The students are astonished of courses required of all students: Methods of by how many people are actually interested in what Mathematical Analysis, Scientific Computing they have done. Techniques, Statistics and Stochastic Methods, and In 1999 Sloan funded a professional master's in Dynamical Systems. Upon completion of the core, mathematics program at Worcester Polytechnic students enroll in "focus courses" in each of the Institute (WPI). It features two tracks, one in fi­ four tracks and, depending on individual interest, nancial mathematics, coordinated by Domokos a group of related courses offered by other Vermes, and the second in industrial mathemat­ departments, including engineering. ics, coordinated by Bogdan Vernescu. Students in both tracks, however, are able to participate in Incorporating Industrial Participation industrial projects and internships facilitated by How does a program in industrial mathematics the established connections of WPI's Center of actually operate? What kinds of interactions with Industrial Mathematics and Statistics. To give industry are feasible in practice? We discuss some students a competitive advantage in careers that examples from the program that we supervise in require the combination of sophisticated quanti­ the hope that our experience may be useful to tative skills and a thorough understanding of the others who plan to create similar programs. underlying industrial problem, the industrial math­ Our Professional Master's in Industrial Mathe­ ematics program features coordinated modules matics is, as we have said, built around real-world of mathematics and engineering/ science courses problems solicited from and then delivered back in areas like dynamics and control, materials to industrial clients. The staff begins a year in design, fluid dynamics, biomedical engineering, advance to solicit projects from local industry. machine learning, cryptography, etc. The antici­ This involves "cold calling", chasing down any pated job opportunities for graduates in financial leads, persistently calling back, and keeping a mathematics are in the money management and careful diary of all contacts. Whoever makes the insurance firms in the Boston-Hartford financial first call explains the program and asks the corridor near WPI. Therefore, the main thrust of contact to "help us better educate our students." the finandal mathematics degree program is to In any one year, forty companies are contacted, provide the skills and knowledge needed in asset­ with an eventual yield of 5-8 projects. Our liability management, portfolio selection, risk guiding principle is to take on any "interesting" control, and financial product development. problem, on the presumption that behind every Another way that mathematics is being recon­ interesting problem lurks interesting mathematics. figured in a professional master's degree program What follows are descriptions of the kinds of prob­ is demonstrated by the University of Wisconsin's lems provided by local industry and government Sloan-funded Professional Master's Degree in to the student teams. The problems can be exam­ Computational Sciences, chaired by Gregory Moses. ined in more detail at [Online resource: MSIM]. The degree is intended to prepare students with B. F. Goodrich Avionics asked for help in bachelor's degrees in science and engineering for correcting temperature-induced errors in a

MAY 2001 NOTICES OF THE AMS 495 solid-state roll sensor. Careful graphing of the ex­ Another team of students constructed for Delta perimental errors revealed a hysteresis pattern­ Dental of Michigan a clustering of states into the errors took different paths during cooling of similar fee-for-service groups. For some reason, the the sensor and heating of the sensor. This students did not choose the more interesting hysteresis turned out to be merely the thermal problem of predicting monthly claims given the delay as heat diffused in and out of the sensor contracts outstanding. case, easily corrected by inserting a. software block Instrumented Sensor Technologies manufac­ simulating heat diffusion. tures recording sensors that are embedded in solid In the current year McCleer Power Inc., a premier rocket fuel in order to record the thermal cycling electric motor design firm, has provided three that induces microcracks. The company would problems. One of these involves the most expen­ like the students to model the propagation of sive and failure-prone components in future cracking given temperature history for use in de­ 42-volt automobile power inverters (which trans­ termining the remaining shelf life of the propellant. form DC to three-phase AC), namely, the Our cold calling has been unexpectedly success­ electrolytic capacitors that are hung across the ful-we are embarrassed by a surplus of projects DC bus to smooth out the square-wave current for which we are unable to constitute teams. We are trains drawn by the pulse-width modulating presently experimenting with a process by which inverters. The students' job is to search for an we "farm out" surplus projects to other institutions alternate pulse-width modulation pattern that with similar master's programs. will reduce the required number of electrolytics. It is made clear to the client from the outset that This may involve a huge search via a genetic these are student projects, with faculty playing algorithm. Another team will determine the only a supervisory role. Indeed, the students take optimum size ratio of the electric motor and the away from these exercises invaluable lessons, the gasoline engine in hybrid vehicles. A third team will equivalent of experience in industry. At first they optimize wind generator design. are wary of taking on an industrial problem that Another student team helped the Air Taxies they may not be able to solve. They soon learn, how­ Division of the Michigan Department of Environ­ ever, that industrial problems are never "solved" mental Quality decide if it would be advantageous in the way end-of-chapter problems are solved; to switch from 6-day sampling to 12 -day sampling rather that only pieces of the problem can be re­ for noncriteria pollutants, using the savings to solved given the time allotted. Nevertheless, these invest in additional testing sites. The students found solutions have real value to the firm, because they 12~day sampling too dangerous because important permit the product development to continue. events will be missed. They also suggested an ingenious method for accounting for "no-detects" Conclusions (readings below the instrument's threshold). Current professional master's programs appear We outreach to our own university researchers. to be doing well, in that they enjoy selective fac­ In the current year a student team will model the ulty/ administrators' support and the willingness outbreak of a certain grape pest along the shore of students to enroll. The true test of their utility of Lake Michigan using degree-day data. Last year (and hence their viability) comes when their a team analyzed the forensic use of fiber evidence graduates go on the job market with their new by applying small-world network theory (of "Six degrees in hand. There is a clear tendency among Degrees of Separation" fame). hiring managers to "hire a degree". What once was Neogen, a Lansing firm that manufactures test descriptive (such as "hire me the kind of person kits for detecting pathogens, has proposed four who can solve a technical problem") has become interesting problems: One proposal is to model the prescriptive ("hire me an engineer"). It will take dispersion of E. Coli via groundwater, where significant marketing of the new degree to change E. Coli may adhere to clay particles, bloom, and corporate culture. On the one hand, certain busi­ thereby become a new point-source. Other prob­ ness and industry representatives have reported lems concern sampling strategies for detecting that "the master's is our missing degree" and pathogens or the presence of genetically modified that the traditional MBA is not as relevant in a high­ grains. tech economy as it once was. On the other hand, Artificial wetlands are being designed and built there are few professional master's graduates in within suburban neighborhoods to retain and management to smooth the way for others. A absorb storm water runoff. One student team critical test of the "promise" being made to stu­ studied Ingham County's demonstration wetland dents will be their career trajectories. The first and derived its water-level response to an arbitrary graduates will need to cut steps in stone as they rain event (via convolution with its impulse sell themselves and their degrees. Networking response, derived from the generalized derivative and tracking of graduates of the programs are of its response to a "unit rain event"). therefore essential.

496 NOTICES OF THE AMS VOLUME 48, NUMBER 5 A second potential problem might be that, as universities ... even higher decreases in the number of outside funding ends, professional master's degree first-year graduate students. Data from "Mathemati­ programs will be reabsorbed into existing depart­ callandscape in the U.S. at the end of the century", ments and will lose their spedal characteristics. Or, unpublished presentation by John Ewing, executive director of the American Mathematical Society, should the programs succeed, they might be taken at September 30-0ctober 2, 1999, Conference on over by colleges of business and/or engineering. Summer Undergraduate Mathematics Research Avner Friedman talks about a "total and enduring Programs, conveyed in summary form by e-mail to commitment" by administrators for such new pro­ the authors. grams. The same could be said for the faculty. [.JW] J, R. WALTON, Abstract of talk to SIAM minisympo­ Faculty are enthusiastic, but few can afford to sium, San Diego, CA, July 2001. make the professional master's their first priority. [CU] Reshaping the Graduate Education of Scientists and A third potential problem is quality control. Engineers, National Academy Press, Washington, Absent a thesis, national boards, or state bar exams, DC, 1995. [MG] W. F. MAssY and C. A. GoLDMAN, The Production and how will the wide variety of professional master's Utilization of Science and Engineering Doctorates in programs (many crossing traditional disciplinary the United States, Report to the Alfred P. Sloan Foun­ boundaries) be accredited? The engineering ac­ dation, 1995. See also, Massy and Goldman, The creditation organization ABET 2000 is facing the Ph.D. Factory: Training and Employment of Science same dilemma. Although SIAM's guidelines may not and Engineering Doctorates in the U.S., Anker, 2001, be intended to set standards, this may be their Ch. 7. practical effect. To some extent, mathematics has [Ben94] S. BENKOWSKI, Preparing for a job outside acade­ an advantage over the sciences-compared to mia, Notices Amer. Math. Soc. 41 (1994), 917-919. [BLTSID93] C. D. BosMAN, et al., Preparing M.S. and Ph.D. "human-computer interaction" or "environmental students for nonacademic careers, Proceedings ofthe risk management", the mathematics that master's Conference on Graduate Programs in the Applied candidates need to know is fairly well laid out. Mathematical Sciences II (R. Fennell, ed.), Clemson Despite uncertainties, the mid-term outlook for University, Clemson, SC, 1993, pp. 61-65. the professional mathematics master's is very [Boy75] W. E. BoYcE, The mathematical training of the promising. Not every company contacted by MSU's nonacademic mathematician, SIAM Rev. 17 (1975), program provides a problem for the students to 541-557. solve. But all are pleased to be asked, and all tell [CBMS92] Conference Board of the Mathematical Sciences, the program directors that the students being Graduate Education in Transition, Conference Board trained are precisely the kind of "knowledge of the Mathematical Sciences, Washington, DC, 1992. [CHM93] C. F. CONRAD, ]. G. HAWOR1H, and S. B. M!u.AR., A workers" the company needs. Silent Success: Master's Education in the United States, Those involved in the professional master's move­ Johns Hopkins University Press, Baltimore, MD, 1993, ment believe that supplying the nonacademic pp. 17-20. workplace with mathematics (and science) profes­ [TCA95] S. TOBIAS, D. E CHUBIN, andK. AYLESWOR1H, Rethinking sionals will have four benefits: (1) an increase in Science As a Career: Perceptions and Realities in the the number of students willing to major in Physical Sciences, Research Corporation, Tucson, AZ, mathematics; (2) an increase in appreciation for 199S,p. 92. mathematics training among managers and [FrLa] A. FRIEDMAN and]. LAVERY, How to Start an Indus­ trial Mathematics Program in the University, SIAM, policy-makers; (3) an increase in funding for math­ 1993. See also A. Friedman and F. Santosa, Gradu­ ematics research; and, as a result of all of these, ate studies in industrial mathematics, Notices Amer. (4) an increase in the demand for faculty as the new Math. Soc. 43 (1996), 564-568. professional programs, now having total annual [MI95) The SIAM Report on Mathematics in Industry enrollments of only a few hundred, increase five- or (1995). See also The SIAM Report on Mathematics in ten-fold. But first, academic mathematicians will Industry (1998). have to become convinced that it is in their interest [DavisI P. DAVIS, First SIAM Regional Math in Industry Work­ to incorporate real-world problems in their teaching shop at WPI, published as "WPI hosts first in a series and to encourage workplace-oriented students to of SIAM's Regional Math in Industry workshops", SIAM News 31-7 (1998). Also online at [DAVIS]. pursue the master's degree in mathematics. [SIAMGuidelines01) Guidelines for a Professional Master's Degree, draft 3, SIAM, n.d. References [CRM] CHARLES MAcCLUER, Industrial Mathematics, Prentice­ [AMS] Undergraduate enrollments: Between 1985 and 1995, Hall, Englewood Cliffs, NJ, 2000. enrollments in calculus dropped 15%, from 63 7,000 to 539,000; in advanced mathematics, from 138,000 to 96,000, a decrease of 30% (at 30 students per sec­ tion, that's a loss of 1,400 sections per year; at 4 courses per year, 3 SO fewer mathematics faculty are needed). Mathematics majors: Between 1975 and 1995 a decrease from 18,883 to 12,456, a decrease of 33%. Graduate course enrollments: Between 1992 and 1997, 18.5% decrease (overall) and 28% in Group I Public

MAY 2001 NOTICES OF TIIE AMS 497 Book Review

The Universal Computer: The Road from Leibniz to Turing Reviewed by Brian E. Blank

The Universal Computer: The Road from Leibniz 1939 the German en­ to Turing gineer Konrad Zuse Martin Davis designed and con­ W. W. Norton and Company, 2000 structed two experi­ ISBN 0-393-04785-7 mental electro­ $26.95, 257pages mechanical digital computers, the Z1 If you teach a course on number theory nowa­ and Z2. In 1937 days, chances are it will generate more interest Howard Aiken sub­ among computer science majors than among mitted to IBM a for­ mathematics majors. Many will care little about mal proposal titled integers that can be expressed as the sum of two Proposed Automatic squares. They will prefer instead to learn how Calculating Machine. Alice can send a message to Bob without fear of The product of eavesdropper Eve deciphering it. No doubt they Aiken's initiative, the would be surprised to see the theory of numbers Harvard Mark I (also described as a "purely theoretical science without known as the ffiMAu­ practical applications" or, even more bluntly, as tomatic Sequence "useless". Yet, those are exactly the assessments Controlled Calculator) was placed in service in the of number theory that were given by Uspensky spring of 1944. It is considered the first electro­ and Heaslet in 1939 and by Hardy in 1940. It is with mechanical number-crunching computer. Mechan­ a sense of irony that we read these pronounce­ ical it certainly was. The 750,000 moving parts of ments now, knowing that the seeds of their Aiken's machine are said to have produced a roar like contradiction had already been sown. Work that that of a textile mill. Less than two years later, in would lead to the modern digital computer was February 1946, a computer known as the ENIAC was already under way. fully operational. This 30-ton behemoth, conceived The great theoretical advance that led to the and constructed by John Presper Eckert and John modern computer may be traced to 1936 when Alan William Mauchly, is considered to be the first elec­ Turing formulated a highly original concept that tronic computer. Electronic it certainly was. When the would eventually be called the Turing machine. At ENIAC went online, its 17,468 vacuum tubes are said the time, projects to build simpler computing to have dimmed lights throughout Philadelphia. devices were just aboutto begin. Between 1936 and The Mark I and the ENIAC were both funded Brian E. Blank is professor of mathematics at Washing­ by the military for the purpose of doing numeri­ ton University, St. Louis, Missouri. His e-mail address is cal calculations vital to the war effort. With the [email protected]. conclusion of the war, seminumerical commercial

498 NOTICES OF THE AMS VOLUME 48, NUMBER 5 applications such as accounting, scheduling, calculation. As he explains in his introduction, "A record-keeping, and billing were developed. As the computing machine is really a logic machine. computer rapidly evolved from its eponymous Its circuits embody the distilled insights of a function, the list of tasks assigned to it swelled. remarkable collection of logicians, developed over Even tasks that do not involve a single computa­ centuries. Nowadays, as computer technology tion have been taken over by the computer. Nowa­ advances with such breathtaking rapidity, as we days a book review, for example, is likely to be admire the truly remarkable accomplishments of solicited by computer communication, composed, the engineers, it is all too easy to overlook the researched, spell-checked, and typeset on a logicians whose ideas made it all possible. This computer, submitted by computer, and posted book tells their story." for access by a worldwide network of computers. One cannot imagine an author more qualified Even the "printer's proofs" might arrive in the than Martin Davis for such an endeavor. Many form of a computer file. Notices readers will be familiar with Davis from his The conversion of number theory from a "use­ contributions, both in research and exposition, to less" pursuit to an applied science has been due Hilbert's tenth problem. Others will know him in large part to an espe- from his excellent text­ cially ironic consequence books, which have become of the computer's evolu­ Davis's perspective standard references of the­ tion: in order that we may oretical computer science securely rely on the com­ is unique: he is [3), [6]. Those who keep puter for such noncompu­ track of awards will recog­ tational tasks as com­ concerned with the nize him as the recipient merce, communication, of the Chauvenet, the and archiving, we must development of the Lester R. Ford, and the first enlist the theory of Leroy P. Steele Prizes. In numbers to foil the com­ computer as an addition to his credentials putational power of the engine of logic as distinguished logician computer to decrypt. Like and honored expositor, the sea change in number rather than as an Davis is also a pioneer pro­ theory that it occasioned, grammer who wrote code the metamorphosis of the instrument of for both the Institute for computer from number Advanced Study computer, cruncher to all-purpose calculation. a historic machine that has logic machine has been a been in the collection of profound transformation the Smithsonian Institution that is now taken for since 1960, and for one of granted but was not origi- its clones, an Army Ord- nally transparent. Aiken, for example, did not rec­ nance "johnniac-class" computer known as the­ ognize the transition in progress. "If it should turn ORDVAC. His engaging autobiographical sketch out," he wrote in 1956, "that the basic logics of a [5] offers a rare glimpse of the programmer's craft machine designed for the numerical solution of dif­ as it existed in 1951, when the state of the art ferential equations coincide with the logics of a ma­ amounted to five kilobytes of random access mem­ chine intended to make bills for a department ory tenuously implemented as static charge on store, I would regard this as the most amazing co­ the surfaces of cathode-ray tubes. incidence I have ever encountered." In The Universal Computer Davis begins his tale Although the electronic digital computer is with Leibniz, whose proposal for an algebra of barely more than half a century old, its history has logic is the point of departure on the road to the attracted a devoted following. For more than twenty universal Turing machine. It is indicative of the years a scholarly journal, the Annals of the History enthusiasm with which Davis infuses his writing of Computing, has chronicled the development of that where others see "fragmentary anticipations computing in minute detail. A steady stream of of modern logic" [12], Davis perceives "a vision books-some erudite, some popular-has allowed of amazing scope and grandeur." As Davis tells engineers, historians, and journalists to delve into the story, Leibniz "dreamt of an encyclopedic nearly every facet of the computer revolution. compilation, of a universal artificial mathemati­ Martin Davis's new book, The Universal Computer: cal language in which each facet of knowledge The Road from Leibniz to Turing, is not like any of could be expressed, of calculational rules which them. would reveal all the logical interrelationships among Davis's perspective is unique: he is concerned these propositions. Finally, he dreamed of ma­ with the development of the computer as an chines capable of carrying out calculations, freeing engine of logic rather than as an instrument of the mind for creative thought." The chapter is

MAY 2001 NOTICES OF THE AMS 499 called "Leibniz's Dream", and that dream is a In keeping with the chronology, Davis interrupts sort of North Star toward which the axis of each Turing's biography to direct his attention to the subsequent chapter points. engineers who would take the next steps toward Following the style of "Leibniz's Dream", Davis the fulfillment of Leibniz's dream. He begins his devotes each of the next six chapters to the life eighth chapter, "Making the First Universal Com­ and contributions of a leading logician: the list puters", with thumbnail summaries of the contri­ comprises Boole, Frege, Cantor, Hilbert, GOdel, and butions of the hardware pioneers Aiken, Atanasoff, Turing. In making these choices, Davis has taken Eckert, and Mauchly. It may be argued that these great care not to stray from the road to Turing. Lo­ sketches are too brief, but in fact these hardware gicians such as Brouwer and Russell are discussed implementations fall outside the scope of Davis's in a fitting amount of detail, but De Morgan, Peano, book. That said, I do find it surprising that Davis and Skolem are mentioned only in passing, while accords only one paragraph to Claude Shannon, Peirce, Schroder, Lowenheim, and Zermelo are not whose 1938 master's thesis in electrical engineer­ mentioned at all. So coherent is the narrative, how­ ing showed how to apply Boole's algebra of logic ever, that one has the illusion that one is reading to electronic switching circuits. The complete omis­ the entire history of mathematical logic without any sion of Konrad Zuse is even more puzzling. In any discontinuity in its evolution. (The reader who is event, the early history of computing is well pro­ inspired to seek out a more conventional history vided for: readers who wish to learn more may con- oflogic may turn to [11] as sult [1], [2], [7], [8], [9], and well as to the references in [13]. [12, page 1].) The historian Tom Set­ Through the first seven Not only does Davis tle has used the death of chapters the principal log­ captivate us with a Galileo to illustrate how ical concepts of each pro­ elusive historical truth can tagonist are presented at a fascinating story, be. Despite an authentic level that is appropriate for death certificate that cites a general audience. It was a he caps it with a the evening of January 8, shrewd idea to embed these 1642, calendrical variation discussions inside capsule moral as well. renders uncertain which biographies of the logidans. one of four days is actu­ This stratagem serves both ally being specified. It is to lighten the load of the tempting to believe that reader who has no prior training in mathematical more recent events must prove less troublesome. logic and to maintain the interest of the more ex­ Indeed, the authors of anew book [10] about com­ perienced reader who is already familiar with the puter scientists assert that "in most sciences the logical theories. It is true that standard biogra­ seminal thinkers lived in the remote past. phies exist, and, with few exceptions, Davis does To uncover what they did... we must scavenge in not go beyond them. Nevertheless, most readers the historical record, picking among scraps of in­ will welcome his lively, informal synopses, replete formation, trying to separate facts from mythol­ as they are with amusing anecdotage. Perhaps the ogy. Computer science is different." Regrettably, best of these involves Davis himself. Driving in this plausible claim is not true. Above all, priority Princeton with his wife, Virginia, he happened to for one of the indispensable principles of modern pass the town's most famous denizen, dressed computing, the stored program concept, has like a tramp, walking with GOdel, nattily attired in proved to be hopelessly and bitterly controver­ suit and tie, briefcase in hand. "Einstein and his sial. lawyer," quipped Virginia. Naturally Godel and In a nutshell, John von Neumann, who worked Turing provide ample grist for the raconteur's with Eckert and Mauchly, has often been given full mill, but the fact is, every one of the featured lo­ credit for the stored program concept because he gicians, the dusty Victorian pedant George Boole advanced the idea in a widely circulated report that included, makes for a fascinating character study. he released under his name alone. Later both Eck­ By the end of the seventh chapter, Davis's ert and Mauchly disputed the importance of von readers will have learned about Boole's algebra Neumann's contribution. Their position is argued of logic, Frege's Begriffsschrift, the Continuum eloquently in a recent book about the ENIAC [7]. Al­ Hypothesis, Godel's theorem on undecidable though Davis admits that the question of von Neu­ propositions, Hilbert's Entscheidungsproblem, and mann's personal contribution "will probably never Turing machines. At this point the timeline of the fully be resolved," he seems to come down squarely narrative has reached the end of World War IT: all on von Neumann's side. His analysis is interesting, the developments in logic that are needed for the but in the big picture this acrimonious squabble universal computer are in place, and their physi­ lacks significance. For one thing, Zuse has a real cal realizations are literally on the drawing boards. claim to priority: he unmistakably proposed the

500 NOTICES OF THE AMS VOLUME 48, NUMBER 5 stored program concept as early as 1936 (but did [7] ScoTT McCARTNEY, ENIAC: The Triumphs and not pursue it, since it would have been of little use Tragedies of the World's First Computer, Walker and on his slow, mechanical memory machines). More Company, New York, 1999. importantly, the issue is something of a red herring. [8] BRIAN RANDELL, The Origins of Digital Computers, third ed., Springer-Verlag, New York, 1982. Davis himself first advanced this point of view in a [9] RAUL ROJAS and ULF HAsHAGEN (eds.), The First Com­ 198 7 article [4] that may be regarded as a skeleton puters: History and Architectures, The MIT Press, of the book under review. "What was really revolu­ Cambridge, MA, 2000. tionary about these machines," Davis points out, [10] DENNIS SHASHA and CATIIY LAZERE, Out of Their Minds: "was their universal all-purpose character, while The Lives and Discoveries ofFifteen Great Computer the stored program aspect was only a means to an Scientists, Springer-Verlag, New York, 1998. end." [11] N. I. STYAZHKIN, History of Mathematical Logic {rom Leibniz to Peano, The MIT Press, Cambridge, MA, That Turing had nailed the future of comput­ 1969. ing before all the others may be seen from several [12] HAo WANG, Popular Lectures on Mathematical Logic, of his statements, of which the following from Van Nostrand Reinhold Company, New York, 1981. 194 5 is typical: "There will positively be no inter­ Enlarged republication by Dover Publications, New nal alterations to be made even if we wish suddenly York, 1993. to switch from calculating the energy levels of the [13] KoNRAD ZusE, The Computer-My Life, Springer­ neon atom to the enumeration of groups of order Verlag, New York, 1993. 720." In 1948 he put it this way: "We do not need to have an infinity of different machines doing different jobs. A single one will suffice." Turing did not refer to this single machine by the misnomer that others with narrower visions were already using: he called it the universal machine, and, as Davis compellingly demonstrates, it was Turing's conception of the universal machine that influ­ enced von Neumann. When a distinguished expert offers a popular exposition of his subject, we greet the effort with keen anticipation. That is all the more true when the writer is as skilled as Martin Davis. It is a plea­ sure to report that in this case our anticipation is richly rewarded. Not only does Davis captivate us with a fascinating story, he caps it with a moral as well. I have echoed this moral at the beginning of this review, but it is worth repeating in the author's own words: "This book underscores the power of ideas and the futility of predicting where they will lead." Seldom has this point been made so well. Read this book and enjoy.

References [1] PAUL E. CERUZZI, A History ofModem Computing, The MIT Press, Cambridge, MA, 1999. [2] I. BERNARD CoHEN, Howard Aiken: Portrait of a Com­ puter Pioneer, The MIT Press, Cambridge, MA, 1999. [3] MARTIN DAVIS, Computability and Unsolvability, McGraw-Hill, 1958. Reprinted in an enlarged edi­ tion by Dover Publications, 1982. [4] __ , Mathematical logic and the origin of modern computers, Studies in the History of Mathematics (Esther R. Phillips, ed.), Math. Assoc. Amer., Wash­ ington, DC, 1987, pp. 137-165. [5] __ , From logic to computer science and back, People and Ideas in Theoretical Computer Science (C. S. Calude, ed.), Springer-Verlag, Singapore, 1999, pp. 53-85. [6] MARTIN DAVIS, RON SIGAL, and ELAINE WEYUKER, Com­ putability, Complexity, and Languages, second ed., Academic Press, New York, 1994.

MAY 2001 NOTICES OF THE AMS 501 Cannes Receives 2001 Crafoord Prize

The Royal Swedish Academy the College de France in Paris. He is a member of of Sciences will award the many scientific academies, including the Academie 2001 CrafoordPrizeinmath­ des Sciences de Paris and the National Academy ematics to ALAIN CONNES of of Sciences of the U.S. the Institut des Hautes The 2001 Crafoord Prize will be presented by Etudes Scientifiques and the the King of Sweden on September 26, 2001, at a College de France, Paris, for ceremony at the Royal Swedish Academy of Sciences his penetrating work on the in Stockholm. The prize consists of a gold medal theory of operator algebras and US$500,000. and for having been a The Anna-Greta and Holger Crafoord Foundation founder of noncommutative was established in 1980 for promoting basic geometry. research in mathematics, astronomy, the bio­ Alain Connes is counted sciences (particularly ecology), the geosciences, among the world's foremost and polyarthritis (joint rheumatism). Previous mathematicians. For his laureates in mathematics are Vladimir I. Arnold and work in operator algebras, Louis Nirenberg (1982), Pierre Deligne and Connes received the Fields Alexandre Grothendieck (1988) (Grothendieck Medal in 1983. Noncom­ declined the prize), and Simon Donaldson and mutative geometry is a new Shing-Tung Yau (1994). Alain Connes field of mathematics, and -From a Royal Swedish Academy news release Connes played a decisive role in its creation. His work has also provided pow­ erful new methods for treating renormalization the­ ory and the standard model of quantum and par­ ticle physics. He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis. Alain Connes was born in Draguignan, France, on April1, 1947. He attended the Ecole Normale Superieure in Paris during 1966-70. Since 1979 he has held the Leon Motchane Professorship at the lnstitut des Hautes Etudes Scientifiques in Bures­ sur-Yvette, outside Paris, and since 1984 he has also held a professorship in analysis and geometry at

502 NOTICES OF TilE AMS VOLUME 48, NUMBER 5 Arnold and Shelah Receive 2001 WolfPrize

The 2001 Wolf Prize in Mathematics has been awarded to VlADIMIR I. ARNoLD of the Steklov Mathematical Institute, Moscow, and the Universite de Paris-Dauphine, and to SAHARON SHEIAH of the Hebrew University of Jerusalem. Arnold is honored "for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singu­ larity theory." Shelah is honored "for his many fundamental contributions to math­ ematical logic and set theory and their ap­ plications within other parts of mathemat­ ics." The two share the $100,000 prize. Vladimir I. Arnold Saharon Shelah Vladimir I. Arnold Vladimir I. Arnold has made significant contribu­ Arnold was born in 1937 in Odessa, Russia. He tions to a large number of different mathematical received his B.Sc. (1954), his M.Sc. (1959), his Ph.D. disciplines. His many research papers, books, and (1961), and his D.Sc. (1963) all from Moscow State lectures, plus his enormous erudition and enthu­ University. He held positions at that institution siasm, have had a profound influence on an entire until 1986, when he became a professor at the generation of mathematicians. Arnold's Ph.D. the­ Steklov Mathematical Institute, a position he cur­ sis contained a solution to Hilbert's 13th problem. rently holds. In 1993 he also assumed his other His work on Hamiltonian dynamics, which includes current position as professor at the Universite de cocreation of KAM (Kolmogorov-Arnold-Moser) Paris-Dauphine. His previous honors include the theory and the discovery of "Arnold diffusion", Prize for Young Mathematicians of the Moscow made him world famous at an early age. Arnold's Mathematical Society (1958), the Lenin Prize (1965, contributions to the theory of singularities com­ shared with A. N. Kolmogorov), the Crafoord Prize plement Thorn's catastrophe theory and have trans­ of the Royal Swedish Academy of Sciences (1982, formed this field. Arnold has also made innumer­ shared with Louis Nirenberg), the Lobachevski able and fundamental contributions to the theory Prize of the Russian Academy of Sciences (1992), of differential equations, symplectic geometry, real and the Harvey Prize (1994). He is a member of the algebraic geometry, the calculus of variations, hy­ Russian Academy of Sciences, the Russian Acad­ drodynamics, and magneto-hydrodynamics. He emy of Natural Sciences, the U.S. National Acad­ has often discovered links between problems in di- emy of Sciences, the American Academy of Arts and verse areas. Sciences, the American Philosophical Society, the

MAY 2001 NOTICES OF TilE AMS 503 Academie des Sciences de Paris, the Royal Society Five annual Wolf Prizes have been awarded since of London, the Accademia dei Lincei, and the Acad­ 1978 to outstanding scientists and artists "for emia Europaea. He is also an honorary member of achievements in the interest of mankind and the London Mathematical Society. friendly relations among peoples, irrespective of nationality, race, color, religion, sex, or political Saharon Shelah view." The prizes of $100,000 apiece are given Saharon Shelah has for many years been the leading every year in four out of five scientific fields in ro­ mathematician in the foundations of mathematics tation: agriculture, chemistry, mathematics, med­ and mathematical logic. His staggering output of icine, and physics. The arts prize rotates among 700 papers and half a dozen monographs includes architecture, music, painting, and sculpture. The the creation of several entirely new theories that prize jury in each field is formed by three mem­ changed the course of model theory and modern bers: one from the United States, one from set theory and also provided the tools to settle old Europe, and one from Israel. New juries are problems from many other branches of mathemat­ appointed each year. The Wolf Foundation does not ics, including group theory, topology, measure disclose the names of the jury members in order theory, Banach spaces, and combinatorics. Shelah to allow them to make their decisions exclusively created a number of subfields of set theory, most on the basis of the candidates' achievements. notably the theory of proper forcing and the theory -Allyn Jackson of possible cofinalities, which is a remarkable refinement of the notion of cardinality and which led to proofs of definite statements in areas previously considered far beyond the limits of undecidability. His work on set theoretic algebra and its applica­ tions showed that many parts of algebra involve phenomena that are not controlled by universally recognized axioms of set theory. In model theory he carried through a monumental program of deep structural analysis known as "stability theory", which now dominates a large part of the field. Shelah was born in 1945 in Jerusalem, Israel. He received his B.Sc. (1964) from Tel Aviv University and his M.Sc. (1967) and Ph.D. (1969) from the Hebrew University of Jerusalem. He held positions at Princeton University (1969-70) and the University of California; Los Angeles (1970-71), before re­ turning to the Hebrew University of Jerusalem, where he is currently a professor. Since 1986 he has also been a Distinguished Visiting Professor at Rutgers University. His previous honors include the Erdos Prize (1977), the Rothschild Prize (1982), the C. Karp Prize of the Association for Symbolic Logic (1983), the George P6lya Prize of the Society for Industrial and Applied Mathematics (1992), the Israel Prize for Mathematical Research (1998), the Japanese Association of Mathematical Sciences Prize (1999), and the Janos Bolyai Prize of the Hungarian Academy of Sciences (2000). He is a member of the Israel Academy of Sciences and Humanities and is an honorary member of the American Academy of Arts and Sciences.

About the Wolf Prize The Wolf Foundation was established by the late German-born inventor, diplomat, and philan­ thropist Ricardo Wolf (1887-1981). A resident of Cuba for many years, he became Fidel Castro's ambassador to Israel and held this position until 1973, when Cuba severed diplomatic ties. Wolf decided then to stay on in Israel, where he lived until his death in 1981.

504 NOTICES OF THE AMS VOLUME 48, NUMBER 5 2001]PBM Communications Award

The Joint Policy Board for Mathematics (JPBM) icalAssociationof America. We Communications Award was established in 1988 recognize Keith for a prepon­ to reward and encourage journalists and other derance of highly public and communicators who, on a sustained basis, bring very popular work that covers accurate mathematical information to nonmathe­ a broad spectrum of topics and matical audiences. The 2001 award was presented has been delivered through a to KErrH J. DEVLIN at the Joint Mathematics Meetings variety of media to a worldwide in New Orleans in January 2001. What follows is audience. the citation for the award, a biographical sketch, and a response from Devlin upon receiving the Biographical Sketch award. Keith Devlin is dean of the School of Science at Saint Citation Mary's College in Moraga, The Joint Policy Board for Mathematics presents its California, and a senior re­ 2001 Communications Award to Dr. Keith Devlin searcher at the Center for the for his many contributions to public understand­ Study of Language and KeithJ. Devlin ing of mathematics through great numbers of radio Information at Stanford and television appearances; public talks; books; University. His current research work is centered and articles in magazines, newsletters, newspapers, on the application of mathematical techniques to journals, and online. For more than seventeen years, issues of language and information and the de­ Dr. Devlin's expository powers have furthered an sign of information systems. He is a member of the appreciation for the mathematical enterprise. Mathematical Sciences Education Board of the Dr. Devlin generates excitement for mathematical National Academy of Sciences and a fellow of the ideas without sacrificing accuracy. He is a regular American Association for the Advancement of correspondent on Scott Simon's Weekend Edition on Science. He is the author of twenty-two books, National Public Radio, and he regularly appears on ranging from research monographs to textbooks radio and television in the United Kingdom. It is to books aimed at a general audience. He has also most extraordinary for any newspaper, especially a written over sixty-five published research articles. major newspaper, to have a regular column on math­ Devlin writes a monthly column, "Devlin's ematics. Yet Dr. Devlin's column is a regular feature Angle", on the Web journal MAA Online and is a of the Manchester Guardian (England). Of twenty-two regular contributor to the National Public Radio books he has written, eleven are devoted to the program Weekend Edition. He also contributes to popular exposition of mathematics. He even wrote various other local and national radio programs in a mathematically inspired radio play. Keith can be the U.S. and Britain, commenting on advances in provocative; he is well known for writing such pieces mathematics and computing. In addition, he has as editor of Focus, the newsletter of the Mathemat- worked on and appeared in a number of

MAY 2001 NoTICES OF TIIE AMS 505 PICK UP A COPY TODAY! television programs, including Life by the Numbers, a six-part series broadcast on the Public Broadcasting System (PBS) in 1998, and GED Connections, a 13-part series aimed at the adult mathematics learner, to be broadcast by PBS in 2001. His most recent book for a general audience is The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip (Basic Books, 2000; reviewed in the Notices, February 2001).

Response How did I get into this situation? In late March 1983, on the spur of the moment, I dashed off a spoof mathematics article for publication as anApiil Fool's joke in the British newspaper The Guardian. (The spoof was that the mathematics described was correct, although hardly anyone would believe it and would assume it was a fake Apiil1 spoof!) The Guardian didn't publish it, but the editor called to say he liked my style and invited me to send in other pieces. I did, readers liked them, and by the end of the year I had a regular, 750-word math column that ran every two weeks. Unplanned, I found myself a "math popularizer". Being a sucker for flattery, when people said they liked my popular writing, I kept on doing it after I moved to the United States in 198 7. The following year my first "popular math book" was published by Penguin Books: Mathematics: The R. WILSON, The Open University, Milton Keynes, England New Golden Age. Despite its accidental beginnings, my side-career STAMPING THROUGH as a communicator of mathematics has developed into something I now take great pride in and like MATHEMATICS to think has value. Certainly, I devote a great deal of An Illustrated History of time and effort to it. Thus, being awarded the JPBM Mathematics Through Stamps Communications Award means a great deal to me. Postage stamps are an attractive vehicle for My sincere thanks to all concerned, not just to my presenting mathematics and its development. For colleagues at JPBM and in the mathematics profes­ many years the author has presented illustrated lectures sion in general, but to Tim Radford, my editor at entitled Stamping through Mathematics to school and The Guardian, who encouraged me in the early days college groups and to mathematical clubs and societies, and has become a good, lifelong friend; to the other and has written a regular "Stamps Corner" for a mathematical newspaper, magazine, and book editors who have journal. The book contains almost four hundred postage taught me-and continue to teach me-how to reach stamps relating to mathematics, ranging from the earliest a wider audience; and to Scott Simon, host ofNPR's Weekend Edition, senior producer, forms of counting to the modern computer age. The stamps and the program's to allow me onto appear enlarged and in full color with full historical Ken Hom, for having the courage Scott and I can commentary, and are listed at the end of the book. their show at regular intervals, where use mathematics to warm up the audience for Car 2001/128 PP., APPROX. 400 COLOR ILLUS./HARDCOVER/$29.95/ISBN 0-387-98949-8 Talk. ORDER TODAY! - From a JPBM announcement CALL: 1-800-SPRINGER • FAX: (201) 348-4505 • WRITE: Springer-Verlag New York, Inc., Dept. S2547, PO Box 2485, Secaucus, NJ 07096-2485 • VISIT: Your local technical bookstore • E-MAIL: [email protected] •INSTRUCTORS: call or write for info on textbook exam copies. YOUR 30-DAY RETURN PRIVILEGE IS ALWAYS GUARANTEED!

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506 NOTICES OF THE AMS VOLUME 48, NUMBER 5 MAAAwards Presented in New Orleans

At the Joint Mathematics Meetings in New Orleans school and university levels. The citation speaks of in January 2001, the Mathematical Association of her "remarkable dedication to teaching and great America (MAA) presented several prizes and success in all of its aspects." awards. "Leonard Klosinski is well known for his in­ valuable contribution to mathematics teaching Haimo Awards for Distinguished Teaching through his capable administration of the William The Deborah and Franklin Tepper Haimo Awards Lowell Putnam Mathematical Competition during for Distinguished College or University Teaching the last twenty-two years," the citation says. Under of Mathematics honor teachers who have been his leadership the number of contestants in the widely recognized as extraordinarily successful Putnam Competition has remained constant, or and whose teaching effectiveness has been shown even increased slightly, despite a drop in the to have had influence beyond their own institu­ number of mathematics majors. He is also an tions. Deborah Tepper Haimo was president of enormously popular teacher known for challeng­ the MAA during 1991-92. ing courses that have inspired a loyal following The 2001 Haimo Awards were presented to of students. EDwARD B. BURGER of Williams College, EVELYN Sn.VIA of the University of California, Davis, and LEoNARD F. Chauvenet Prize KLosiNSKI of Santa Clara University. The Chauvenet Prize for expository writing, first "Edward Burger possesses two great talents: he awarded in 192 5 to Gilbert Bliss of the University is a marvelous teacher and an exceptional lecturer of Chicago, is given for an outstanding expository and speaker," the citation says. "He inspires stu­ article on a mathematical topic by a member of the dents with his enthusiasm for mathematics and his MAA. The prize is named for William Chauvenet, ability to make it interesting, even fun." Known as an professor of mathematics at the U.S. Naval Acad­ engaging speaker, Burger has given conference talks, emy, and was established through a gift in 1925 presented popular lectures, and appeared on tele­ from J. L. Coolidge, then MAA president. vision and radio. The citation calls him "an inspiring The 2001 Chauvenet Prize was awarded to teacher" and "an ambassador of mathematics." CAROLYN S. GORDON and DAVID L. WEBB for their "Evelyn Sylvia is the consummate teacher, whose article "You can't hear the shape of a drum" (Amer­ hallmarks are complete dedication to the education ican Scientist, January-February 1996, pages 46-5 5). of her students; the ability to make difficult con­ The article describes work that answered a famous cepts comprehensible; great energy; and personal question raised by Mark Kac in a 1966 American qualities of integrity, helpfulness, and caring," the Mathematical Monthly article entitled "Can one citation says. Silvia has taught all levels of mathe­ hear the shape of a drum?", which won him the matics, from grade school to graduate school, and Chauvenet Prize in 1968. The citation says that the has served as a role model for teachers at the high Gordon-Webb article is "exciting, its mathematical

MAY 2001 NOTICES OF THE AMS 507 content understandable by anyone with a minimal knowledge of differential equations, group theory, and linear algebra; and it contains a great deal of historical information concerning what can be inferred about vibrating systems from their frequencies." Gung and Hu Award for Distinguished Service The Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics is the most prestigious award made by the MAA. The 2001 award was presented to MANuEL P. BERRiozABAL of the University of Texas at San Antonio. "Berriozabal is a mathematician, a college pro­ fessor, and a visionary with the unusual talent to turn his visions into reality," the citation says. "The best example-and the one for which he has become most widely known-is the incredibly successful Prefreshman Engineering Program (PREP)." Formed in 1979 in San Antonio, PREP aims to identify and encourage high-achieving students in grades six through eleven who are potential engineers and sci­ entists. The PREP program has expanded to other cities in Texas and has reached over 18,000 stu­ dents, 81 percent of them minorities and 54 per­ cent of them women. A 1999 study showed that 92 percent of these students entered college, and of Confidence in knowing. It's iaportant those, 90 percent completed college and 53 percent to feel secure about your iAsurance majored in science or engineering. In 1997 coverage. Bow you can. -Al!S carefully Berriozabal's achievements were recognized by the Presidential Award for Excellence in Science, selects experienced prov~ders with the Unancial stability to ensure Mathematics, and Engineering Mentoring. The ci­ coapetitive. in,urance options for tation for the Gung and Hu Award concludes by rec­ , aeaJ;)ers. ognizing "his extraordinary contributions to the mathematical community and .. .the vision that has Take advantage of one of your best benefited thousands of youth with potential to ex­ :' aeahership benefits. lffoz;ttable cov- . cel in mathematics, science, and engineering." erage. Reliable pro~iders. Portable Certificates of Meritorious Service he!lefits. Call. 600 lfi!lf•,663 to spe.ak to a cu.stoaer service representat:ive. Each year the MAA presents Certificates of Meri­ Because you-. want an insurance plan torious Service to recognize service to the MAA at count on. the national level or service to an MAA section. Hon­ ored with certificates for 2001 were: CARL LEINBACH of Gettysburg College, BERNARD SoHMER of City Col­ lege of the City University of New York, RALPH W. CARR of St. Cloud University, KENNETH A. Ross of the University of Oregon, and JoANNE PEEPLES of El Paso Community College. -From MAA announcements

program is administered by Seabury & Smith, an MMC Company. Some plans may not available in all states. The comprehensive healthcare insurance plan is underwritten by New York Ufe Insurance Company, 51 Madison Avenue, New York, NY 10010. The member assistance and disability income protection insurance plans are underwritten by Unum Life Insurance Company of America, 221 I Congress street, Portland, ME 04122. The catastrophe major medical and high limit accident insurance plans are underwritten by The United States Life Insurance Company, in the City of New York, 3600 Route 66, P.O. Box I 580, Neptune, NJ 07754-I 580. The term life insurance plan is underwritten by Connecticut General Life Insurance Company, a CJGNA Company, Hartford, CT 06152.

508 NOTICES OF THE AMS VOLUME 48, NUMBER 5 AWMAwards Presented in ·New Orleans

At the Joint Mathematics Meetings in New Orleans The award honors the memory of Louise Hay of the in January 2001, the Association for Women in University of Illinois at Chicago. Mathematics (AWM) awarded the Alice T. Schafer The AWM presented the 2001 Hay Award to Prize and the Louise Hay Award. PATRICIA D. SHURE of the University of Michigan in Ann Arbor. Each fall term at Michigan there are over Schafer Prize 120 sections of the mainstream precalculus and cal­ The annual Alice T. Schafer Prize recognizes excel­ culus courses, requiring about 115 instructors lent achievement in mathematics by an undergrad­ (graduate students, new assistant professors, and uate woman. The prize is named for former AWM visitors). Because around half of the instructors are president and one of its founding members, Alice T. unfamiliar with Michigan's mathematics program, Schafer, professor emerita of Wellesley College, who Shure developed a successful training program has contributed a great deal to women in mathe­ called the Professional Development Program. The matics throughout her career. material she developed for the program has been The AWM awarded the 2001 Schafer Prize to published and has been used to train instructors JACLYN (KOHLES) ANDERSON, a senior mathematics elsewhere in the U.S. and in Canada. In addition, major at the University of Nebraska at Lincoln. she ran a program to support underrepresented Anderson has written two research papers that minority students in mathematics, and she has grew out of her participation in summer programs worked throughout her career to attract young for undergraduates in mathematics. She has also women into mathematics. Since the early 1960s taken many graduate-level courses and served as Shure has been involved in curriculum reform. At a teaching assistant for honors calculus courses. Michigan she worked on a project to design and Last year she received an honorable mention evaluate ways to incorporate graphing calculators, for the Schafer Prize. The citation includes the writing, cooperative learning, and systematic test­ following quotations from her professors: her ing of symbolic skills into first-year undergradu­ work "far surpassed that of the rest of the stu­ ate mathematics courses. The citation for Shure dents," including graduate students; she is "the concludes by noting Shure's "tireless commitment most talented undergraduate I have encountered to improving mathematics education for countless in my 33 years of college teaching"; and she is "a students. Her professional contributions along bona fide star" with "impressive talent, drive, and with her personal commitment to improving math­ enthusiasm for mathematics." ematics education are noteworthy." -From A w:M announcements Hay Award The annual Louise Hay Award for Contributions to Mathematics Education recognizes outstanding achievements in any area of mathematics education, to be interpreted in the broadest possible sense.

MAY 2001 NOTICES OF THE AMS 509 AMERICAN MATHEMATICAL SOCIETY

Volume 7, 2001 (Most Recent Articles) MANAGING EDITOR Svetlana Katok Hans Ulrich Besche, Bettina Eick, and E. A. O'Brien, The groups of order at most 2000 EDITORIAL BOARD Stuart Antman John Fogarty, On Noether's bound for polynomial invariants of a finite David Benson group Dmitri Burago Simon Scott, Relative zeta determinants and the geometry of the Mark Freidlin determinant line bundle Ronald Graham Vadim Yu. Kaloshin and Brian R. Hunt, A stretched exponential bound Yitzhak Katznelson on the rate of growth of the number of periodic points for prevalent David Kazhdan diffeomorphisms I Alexander Kechris Alexandre Kirillov Vadim Yu. Kaloshin and Brian R. Hunt, A stretched exponential bound Frances Kirwan on the rate of growth of the number of periodic points for prevalent Krystyna Kuperberg diffeomorphisms II Robert Lazarsfeld Gregory Margulis Hugh Montgomery The American Mathematical Society's electronic-only journal, Electronic Walter Neumann Research Announcements of the AMS (ERA-AMS), is available on the World Klaus Schmidt Wide Web at www.ams.org/era. Richard Schoen Masamichi Takesaki ERA-AMS publishes high-quality research announcements of significant Michael Taylor advances in all branches of mathematics. Authors may submit manuscripts Guido Weiss to any editor. All papers are reviewed, and the entire Editorial Board must Zhihong Oeff) Xia approve the acceptance of any paper. Papers are posted as soon as they Don Zagier are accepted and processed by the AMS. Efim Zelmanov ERA-AMS offers you ... decreased turn-around time from submission to publication · fast access to your specific area of interest · up-to-the-minute research information To obtain submission information and the template, send email to: era­ [email protected] with the word "help" in the subject line. For more information, contact: [email protected] 1-800-321-4267,1-401-455-4000, fax 1-401 -455-4046 ~"t)UNi>ED \'t.~ AMS AMERICAN MATHEMATICAL SOCIETY

www.ams.org/era Mathematics People

prize for his work in complexity theory, probabilistic Nesterov Wins Dantzig Prize algorithms, and combinatorial optimization. The award YURU NESTEROV of the Catholic University of Louvain has been was presented on November 22, 2000, at the Mathemati­ awarded the 2000 George B. Dantzig Prize. According to the cal Research Institute in Oberwolfach, Germany, during a prize citation, Nesterov was selected "for his fundamental workshop on complexity theory. contributions to the theory of interior point methods for The Oberwolfach Prize is awarded by the Gesellschaft convex programming, including his introduction of the ftir Mathematische Forschung (Society for Mathematical concept of self-concordance." He has also made contribu­ Research) to European mathematicians not older than 3 5 tions to the study of convex optimization and automatic years. The prize recognizes excellent achievements in a differentiation. specific field of mathematics, which changes each time the The Dantzig Prize is awarded every three years by the prize is given. The prize carries a monetary award of Mathematical Programming Society and the Society for DM 10,000 (approximately US$5,000). Industrial and Applied Mathematics. Previous recipients of the Oberwolfach Prize are Peter Kronheimer (topology and geometry, 1991), Jorg Briidem -From a Dantzig Prize Committee announcement and Jens Franke (number theory and algebra, 1993), Gero Friesecke and Stefan Sauter (analysis and applied mathematics, 1996), and Alice Guionnet (stochastics, 1999).

Kleinberg Wins National -Elaine Kehoe Academy of Sciences Award The 2001 National Academy of Sciences Award for YangWms2001Frusill Initiatives in Research has been given to JoN M. KlEINBERG of Cornell University. Kleinberg, who received his Ph.D. from the International Prize Massachusetts Institute ofTechnologyin 1996, was chosen "for his development of deep and innovative algorithms CHEN NING YANG of the State University of New York at to solve fundamental problems in network, information Stony Brook has received the King Faisal International extraction, and discrete optimization." Prize for Science for 2001. Yang will receive a cash award The Award for Initiatives in Research carries a cash of $200,000. prize of $15,000 and is presented annually in various A 195 7 recipient of the Nobel Prize in Physics, Yang has fields to recognize innovative young scientists and to en­ made substantial contributions to mathematics and to courage new research. The 2001 award was designated for physics. He proposed a theoretical framework which later the fields of computational science and applied mathe­ became the basis of the present theory of the structure of matics. matter at the smallest scales and highest energies.

-From an NAS announcement -From a Faisal Foundation announcement Trevisan Awarded Ziegler Awarded Leibniz Prize Oberwolfach Prize The Deutsche Forschungsgemeinschaft (DFG) has selected a mathematical scientist as one of the eleven recipients LucA TREVISAN of the University of California, Berkeley, has of its Gottfried Wilhelm Leibniz Prize for the year 2001. received the 2000 Oberwolfach Prize for outstanding GtlNTHER ZIEGLER of the Technische Universitat Berlin will research in discrete mathematics (including logic and receive DM 1.5 million (approximately US$750,000) to theoretical computer science). Trevisan was awarded the support research over a period of five years.

MAY 2001 NOTICES OF THE AMS 511 Mathematics People

Gunther Ziegler, age 37, studied mathematics and physics at Ludwig-Maximillians-Universitat in Munich from AIM Five-Year Fellow 1981 to 1984. He received his Ph.D. in mathematics from Announced the Massachusetts Institute of Technology in 1987. He completed postdoctoral work at Augsburg University and The American Institute of Mathematics (AIM) has announced at the Institut Mittag-Leffler in Sweden. He received his ha­ that the recipient of the 2001 AIM Five-Year Fellowship is bilitation from the Technische Universitat Berlin, where he LENHARD L. NG of the Massachusetts Institute of Technol­ today holds the chair in discrete geometry. ogy. He was chosen out of more than 100 applicants. Ziegler's research includes work in discrete geometry, Ng received his A.B. in mathematics and physics from combinatorics, and polytopes. He has received numerous Harvard University in 1996 and was a Putnam Fellow for awards for his mathematical work, including the DFG's three years of his undergraduate career. He will receive his Gerhard Hess Award worth DM 1 million. Ph.D. from MIT in 2001. His interests are in the area of The aim of the Leibniz Prize program, which was differential geometry and, in particular, contact geometry instituted by the DFG in 1985, is to improve the working and symplectic geometry. His thesis, "Invariants of conditions of outstanding scientists and scholars, to Legendrian links," develops techniques to distinguish broaden their opportunities for research,. to ~lieve them between Legendrian knots and links in standard contact of administrative burdens, and to allow them to hire three-space. especially highly qualified young academics. The prizewin­ The AIM five-year fellowships are awarded each year to ners are permitted the greatest possible freedom in the way outstanding new Ph.D. recipients to support research in an they use the prize funds. The DFG is the main scientific area of pure mathematics. The fellowships cover sixty research funding agency of the German government. months of full-time research, as well as funds for travel and equipment. Each fellowship carries a stipend of $4,000 for - From a DFG announcement per month, with an additional $4,000 per year allocated travel and equipment. Humboldt Foundation -From an AIM announcement Research Awards ONR Young Investigators The Alexander von Humboldt Foundation annually grants up to 150 Humboldt Research Awards to scholars resident Awards Announced outside Germany whose academic qualifications enjoy in­ The Office of Naval Research (ONR) has announced the Among those receiving the awards ternational recognition. awarding of 26 grants totaling $8.5 million in the 2001 ONR in 1999 were nine mathematicians. What follows are their Young Investigator Program competition. Two individuals names, home affiliations, and the German institutes they in the mathematical sciences received awards. They are visited. CYNTHIA YOUNG HOPEN of the University of Central Florida DMITRYANosov: Steklov Institute of the Russian Academy and RONALD FEDKIW of Stanford University. of Sciences, Moscow; Universitat Ulm and Universitat Bonn; Hopen will do research in optimal signal processing DARYL JoHN DALEY: Australian National University; Technische methodologies for laser sensors, considering the effect Universitat Mlinchen; JoEL L. HoRoWITZ: University of Iowa; of atmospheric turbulence on laser beam propagation, Humboldt Universitat Berlin; YURI KIFER: Hebrew University divergence, coherence, and scintillation. This research can of Jerusalem; Universitat Bremen; KRlYszmF CZFSIAW: Systems improve long-range surveillance using laser radar. Fedkiw Research Institute of the Polish Academy of Sciences; will pursue algorithm design for computational fluid Humboldt Universitat Berlin; GusTAV I. LEHRER: University of dynamics, in particular to calculate pressures in flowing Sydney; Universitat Bielefeld; VlADIMIR G. MAz'YA: Linkoping fluids near an interface. These numerical techniques will University; Universitat Stuttgart; TEIMURZ Pnwmvru: Institute also be applied to three-dimensional visualization of data of Mathematics of the Georgian Academy of Sciences; and image processing. Universitat Bielefeld; and KARL RUBIN: Stanford University; The Young Investigator Program supports basic research Universitat Erlangen-Nfunberg. by exceptional faculty at U.S. universities who have The Humboldt Research Awards include the invitation received Ph.D.'s or equivalent degrees within the preced­ to undertake extended periods of research of the award­ ing five years. Grants to their institutions provide up winners' own choice at German research institutes (4-12 to $100,000 per year for three years. The funds may be months). The value of the awards ranges from DM 20,000 applied to a variety of research costs, including salary, to DM 150,000 (about US$10,000 to US$75,000). Nomina­ graduate student support, laboratory supplies, and tion s for the awards are made by leading German operating costs. Young Investigators are selected on the scholars or research institutions. Direct applications are basis of prior professional achievement, the submission of not accepted. a meritorious research proposal, and evidence of strong support by their respective universities. The program -Allyn Jackson supports outstanding research in a wide range of science

512 NOTICES OF 1HE AMS VOLUME 48, NUMBER 5 Mathematics People Financial Mathematics at and engineering fields that are critical to the evolution of King•s College London a first-rate Navy and Marine Corps. MSc and PhD programmes -From an ONR announcement King's College is a multi-faculty institution with over 16,000 students. The Department of Mathematics, located in the heart of London only a short walk from 'frafalgar Square and Covent Garden, has an international reputation for teaching and research. A graduate programme in Financial Correction Mathematics has now been initiated with the appointments of Professor Lane P. Hughston, Dr Giulia Iori and Dr Mihail Zervos to established posts in the Department. Applications arc invited for the MSc in Financial Mathematics, The November 2000 issue of the Notices, page 1284, which is being offered both on a full-time and a part-time basis. Some places carried an announcement about mathematicians elected are also available for suitably qualified research students who would like to to the Royal Society of Canada in 2000. One name pursue a PhD in this subject. was inadvertently omitted from the list: ]OHN McKAY of Since the pioneering days of Black, Scholes and Merton, the theory of Mathematical F"mance has developed into a substantial body of knowledge and Concordia University. its numerous applications have become vital to the day-to-day functioning of the world's financial institutions. As a consequence, a solid command of the principles and techniques of quantitative finance is essential for a responsible approach to trsding, asset management and risk control of complicated financial positions.

Deaths The F"mancial Mathematics MSc programme covers mainstream mathematical ]ON BARWISE, of Indiana University, Bloomington, died on finance and its applications. The curriculum includes, for example, derivatives pricing and hedging, asset price dynamics, risk analysis and extreme events, March 5, 2000. Born on June 29, 1942, he was a member interest rate and foreign exchange processes, credit and inflation-linked products, of the Society for 29 years. real options, energy derivatives, stochastic optimisation and control and ZYGMUND WIIllAM BIRNBAUM, professor emeritus, University investment decision making, as well as other mathematical subjects of relevance to practical financial modelling. The programme is run by the of Washington, Seattle, died on December 15, 2000. Born Financial Mathematics group in the Department of Mathematics at on October 18, 1903, he was a member of the Society for King's College London, and builds on the group's close links with financial institutions in the City of London and elsewhere throughout the world. 62 years. KAREL L. DE BoUVERE, professor emeritus, Santa Clara Uni­ The MSc is based on lecture courses and a project, and requires two years of pare-time study or one year of full-time study. Applications are currendy being versity, died on November 1, 2000. Born on November 15, considered for admission in September ZOO!. The part-time programme is 1918, he was a member of the Society for 40 years. compatible with the needs of those already employed in the fmancial sectOr. Candidates choose eight lecture courses in consultation with their course advisor. FRED T. DALY, ofXavier Jesuit Center, Denver, CO, died The present programme includes the following core courses and options: on December 5, 1998. Born on April 25, 1913, he was a Itttrtx.illaiotl to DrixJiioes Pridttg member of the Society for 49 years. AppiWJ ProiKJIJility IIIIIIStocM.rtit:s JosEPH LANDIN, professor emeritus, University of illinois Stodlastic haljsis AJrJattad Statistics at Chicago, died on February 20,2000. Born on January 25, FitlatJcial Mtrids 1913, he was a member of the Society for 59 years. E:x:oti& DrixJiir;es STANISLAW LEJA, professor emeritus, Western Michigan Nllltlerit:al MlfAotls for Partial Differmlia/ EtplatiqN University, died on September 27, 2000. Born on January 3, ltr~emt &te 111111 Forng, &t:Atmgr l>pramia Portfolio Risl M4llllpll#llt 1912, he was a member of the Society for 45 years. Ruther course options are available such as Neural Networks, Linear Systems JosEPHINE M. MITcHEll, professor emerita, State University and Control Theory, and the Spectral Theory of Markov Chains. Each candidate of New York, Buffalo, died on December 28, 2000. Born on also underta£s a project to study an area of finance in greater depth. June 30,1912, she was a member of the Society for 58 years. The fees are £9,ZOO for the full-time MSc programme beginning in September ZOOt and £4,800 per year for the part-time MSc programme beginning in PAUL OLUM, president emeritus, University of Oregon, September ZOO! . An entry requirement for the MSc is the equivalent of a first or Eugene, died on January 19, 2001. Born on August 16, upper second class degree in a mathematical discipline. 1918, he was a member of the Society for 54 years. A PhD degree in fmancial mathematics or a related area of applied probability is an asset that is often highly valued by employers in the financial sector. JosEPH W. SrRY, chief scientist, NASA, Goddard Space Applications from prospective PhD students are currently being solicited. Flight Center, Greenbelt, MD, died on January 4, 2001. A limited amount of funding may be available for highly qualified candidates. Born on August 7, 1920, he was a member of the Society Fbr further informacion and application forms, for both the MSe and PhD for 4 7 years. programmes, sec: www.mth.kcl.ac.uk YOSHIHITO TOMITA, of Kobe University, Japan, died in De­ Alternatively, please contact: cember 2000. Born on February 29, 1944, he was a mem­ The Postgraduate Secretary ber of the Society for 13 years. Department of Mathematics King's College London LAURENCE CHISHOLM YOUNG, of Madison, WI, died on The Strand, London WCZR ZLS, United Kingdom December 24, 2000. Born on July 14, 1905, he was a E-mail: [email protected] member of the Society for 55 years. Telephone: +44 (O)ZO 7848 Z107 VING'S Fax: +44 (O)ZO 7848 Z017 .&College LONDON EqiJa/ity of opportunily is College JM/icy RJundetii829

MAY 2001 NOTICES OF THE AMS 513 Mathematics Opportunities

apply. All non-U.S. citizens must have a current U.S. address. AWM Workshops for Women All applications should include a curriculum vitae, a Graduate Students and concise description (two to three pages) of research, and the title of the proposed talk or poster. Postdocs All applications should also include at least one letter of recommendation; in particular, a graduate student Over the past thirteen years the Association for Women in should include a letter of recommendation from her Mathematics (AWM) has held a series of workshops for thesis advisor. Nominations by other mathematicians (along women graduate .students and recent Ph.D.'s in conjunc­ with the information described above) are also welcome. tion with major mathematics meetings. The workshops Send five complete copies of the application materials are also supported by the Office of Naval Research and the (including a cover letter) to: Workshop Selection Commit­ National Science Foundation. The next AWM workshop will tee, Association for Women in Mathematics, 4114 Computer be held in conjunction with the annual Joint Mathematics & Space Sciences Building, University of Maryland, College Meetings in San Diego, California, January 6-9, 2002. The Park, MD 20742-2461; telephone 301-405-7892; e-mail: workshop will be held on Saturday, January 9, 2002, with [email protected]. The AWM Web site is at http:// an introductory dinner and discussion group on Friday www.awm-math.org/. evening, January 8. Please note that applications via e-mail or fax are not Twenty women will be selected in advance of the work­ acceptable. The deadline for receipt of applications at the shop to present their work; the selected graduate students AWM office is September 1, 2001. will present posters, and the postdocs will give 20-minute talks. AWM will offer funding for travel and two days' -AWM announcement subsistence for the selected participants. The workshop will also include a panel discussion on issues of career development, a luncheon, and a dinner with a discussion period. Participants will have the opportunity to meet with Nominations Sought for Peter other women mathematicians at all stages of their careers. Gruber Prize in Cosmology All mathematicians (female and male) are invited to attend the program. Departments are urged to help grad­ Nominations are invited for the 2001 Prize in Cosmology uate students and postdocs who do not receive funding to of the Peter Gruber Foundation. The prize is awarded obtain some institutional support to attend the workshop annually to an astronomer, physicist, or mathematician for presentations and the associated meetings. achievements that have advanced scientific understanding The AWM also seeks volunteers to lead discussion of the nature of the universe. The prize carries a cash groups and to act as mentors for workshop participants. award of $150,000. Anyone interested in volunteering should contact the AWM The deadline for nominations for the 2001 prize is office. April 30, 2001. Further information can be found on Applications are welcome from graduate students who the Gruber Foundation Web site, http: I /www. have made substantial progress toward their theses and GruberAwards. org/, or by c ontacting Larry E. Tise, from women who have received their Ph.D.'s within ap­ Peter Gruber Awards, P.O. Box 15792, Philadelphia, PA proximately the last five years. (The word "postdoc" refers 19103; telephone 215-765-4525; fax 215-765-2721; e-mail: to recent Ph.D.'s, whether or not they currently hold a [email protected]. postdoctoral or other academic position.) Women with grants or other sources of support are still welcome to - From a Gruber Foundation announcement

514 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Inside the AMS

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is to build the endowment to around $2 million through AMS Epsilon Fund Makes individual donations and grants from foundations. Once Awards the Epsilon Fund endowment has reached the targeted amount, the AMS intends to award a total of $100,000 in The AMS Epsilon Fund for Young Scholars was established Epsilon grants each year. in 1999 to provide financial assistance to summer pro­ For further information about the Epsilon Fund for grams for mathematically talented high school students in Young Scholars, visit the Web site http: I /www. ams. the United States. For many years these programs have org/giving-to-ams/;[email protected], provided these young people with their first serious telephone 800-321-4267, extension 4111, or 401-455-4111. mathematical experiences. The name for the fund was Information about how to apply for Epsilon grants is chosen in remembrance of the late Paul Erdos, who was available at http: I /www. ams. o rg/ emp 1 oyment/ fond of calling children "epsilons". epsi 1 on. htm1. A fairly comprehensive listing of summer The AMS has chosen eight summer mathematics pro­ programs for mathematically talented high school grams to receive Epsilon grants for activities in the summer students (including those with and without Epsilon grants) of 2001. The grants will support program expenses and is available at http: I jwww. ams. o rg/ emp 1 oyment/ student scholarships and, in some cases, scholarships only. mathcamps. htm1. The programs were chosen on the basis of mathematical excellence and enthusiasm by a selection committee chaired -Allyn jackson by Joel Spencer of the Courant Institute of Mathematical Sciences, New York University. Award amounts were governed by the varying financial needs of each program and totaled $80,000. The programs receiving grants are: All Girls/All Math (University of Nebraska); Mathcamp (Port Huron, Michigan); Michigan Math & Science Scholars (University of Michigan, Ann Arbor); Mathematics Scholars Academy (Oklahoma State University); Hampshire College Summer Studies in Mathematics (Hampshire College); PROMYS (Boston University); Young Scholars Program (University of Chicago); and Ross Mathematics Program (The Ohio State University). The grants for summer 2001 are paid for by the Society's Program Development Fund. The AMS has now begun an effort to raise an endownment for the Epsilon Fund and has pledged $500,000 to get the endowment started. The hope

MAY 2001 NOTICES OF THE AMS . 515 Reference and Book List

The Reference section of the Notices e-mail: ko l be@math. georgetown. edu telephone 202-334-2760; fax 202-334- is intended to provide the reader with or rosi er@math. georgetown. edu. 2759; e-mail: rap@nas . edu. frequently sought information in April 9-11, 2001: Full proposals April23-25, 2001: Full proposals an easily accessible manner. New for group projects for NSF Information for large projects for NSF Information information is printed as it becomes Technology Research Program. See Technology Research Program. See available and is referenced after the http:llwww.itr.nsf.govl. http:llwww.itr.nsf.govl. first printing. As soon as information April 13, 2001: Applications for April 30, 2001: Applications for is updated or otherwise changed, it Project NExT. See http: I larchi ves. the Peter Gruber Prize in Cosmology. will be noted in this section. math.utk.edulprojnextl. See http:llwww.GruberAwards. April 15, 2001: Applications for o rgl , or contact Larry E. Tise, Peter Contacting the Notices fall 2001 semester of "Math in Gruber Awards, P.O. Box 15792, Moscow" program for students. See The preferred method for contacting Philadelphia, PA 19103; telephone http: I lwww. mccme. rul the Notices is electronic mail. The 215-765-4525; fax 215-765-2721; editor is the person to whom to send mat hi nmoscowl or send e-mail to mi m@mccme. ru. e-mail: Lti se@attglobal. net. articles and letters for consideration. April 30, 2001: Nominations for Articles include feature articles, April 15, 2001: Applications for the second competition for NRC the Maria Mitchell Women in Science memorial articles, communications, Award. See http:llwww.mmo.orgl, opinion pieces, and book reviews. The Research Associateships. See http: I I www4.nationalacademies.orgl or contact the Maria Mitchell Women editor is also the person to whom to in Science Award Committee at the send news of unusual interest about oseplrapl, or contact the National Research Council, Associateship Pro­ Maria Mitchell Association, 2 Vestal other people's mathematics research. grams (TJ 2114), 2101 Constitution Street, Nantucket, MA 02554; tele­ The managing editor is the person Avenue, NW, Washington, DC 20418; phone 508-228-9198. to whom to send items for "Mathe­ matics People", "Mathematics Oppor­ tunities", "For Your Information", "Reference and Book list", and "Math­ ematics Calendar". Requests for permissions, as well as all other inquiries, go to the managing editor. The electronic-mail addresses are noti ces@math. tamu. edu in the case of the editor and noti ces@ams .org in the case of the managing editor. The fax numbers are 979-845-6028 for the editor and 401-331-3842 for the man­ aging editor. Postal addresses may be found in the masthead. Upcoming Deadlines April9, 2001: Proposals for 2002 NSF­ CBMS Regional Conferences. See http:llwww.maa.orglcbmslnsfl 2002_call . htm, or contact Confer­ ence Board of the Mathematical Sciences, 1529 18th Street, NW, Wash­ ington, DC 20036-1385; telephone: 202-293-1170; fax: 202-293-3412;

516 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Reference and Book List

May 1, October 1, 2001: Applica­ California mathematicians, students, and a sig­ tions for NSFIAWM Travel Grants for Karen Economopolous, TERC, nificant portion of the general public. Women. See http: I lwww. awm­ Cambridge, MA When a book has been reviewed in the math. orgl travel grants. html; Susan Eyestone, National Parent Notices, a reference is given to the telephone 301-405-7892; e-mail: Teacher Association (PTA), review. Generally the list will contain [email protected]. Minneapolis, MN only books published within the last May 1, 2001: Burroughs Wellcome Joan Ferrini-Mundy, Michigan State two years, though exceptions may be Fund Career Awards at the Scientific University made in cases where current events Interface. See http: I lwww. bwfund. Arthur Jaffe, Harvard University (e.g., the death of a prominent math­ orglinterfaces_in_science.htm, Dan Kennedy, The Baylor School, ematician, coverage of a certain piece or call Debi Linkous, program associ­ Chattanooga, TN of mathematics in the news) warrant ate, telephone 919-991-5116. joan Leitzel (chair), University of drawing readers' attention to older August 15, 2001: Applications for New Hampshire books. Suggestions for books to include the third competition for NRC Karen Longhart, Flathead High on the list may be sent to the manag­ Research Associateships. See http: I I School, MT ing editor, e-mail: noti ces@ams. org. www4.nationalacademies.orgl Miriam Masullo, ffiM Corporation osep/rap/, or contact the National Thomas Moore, Grinnell College The Advent of the Algorithm: The Research Council, Associateship Pro­ Deborah Paulson, Hornedo Middle Idea That Rules the World, by David grams (TJ 2114), 2101 Constitution School, El Paso, TX Berlinski. Harcourt, March 2000. ISBN Avenue, NW, Washington, DC 20418; Marge Petit, National Center for the 0-151-00338-6. telephone 202-334-2760; fax 202-334- Improvement of Educational Angles of Reflection: Logic and a 2759; e-mail: rap@nas. edu. Assessment Mother's Love, by Joan L. Richards. September 1, 2001: Deadline for Anthony Scott, Chicago Public W. H. Freeman, May 2000. ISBN AWM Workshops for Women Gradu­ Schools 0-7167-3831-7. ate Students and Postdocs. See William Steenken, GE Aircraft *Battle of Wits: The Complete Story http:l/www.awm-math.orgl, or Engines of Codebreaking in World War II, by contact Workshop Selection Commit­ Lee Stiff, North Carolina State Stephen Budiansky. Free Press, Octo­ tee, Association for Women in Math­ University ber 2000. ISBN 0-684-85932-7. ematics, 4114 Computer & Space Jim Stigler, University of California, The Bit and the Pendulum: How the Sciences Building, University of Los Angeles New Physics of Information Is Revolu­ Maryland, College Park, MD 20742- jerry Uhl, University of Illinois, tionizing Science, by Tom Siegfried. 2461; telephone 301-405-7892; e-mail: Urbana-Champaign John Wiley & Sons, February 2000. [email protected]. ISBN 0-47132-174-5. October 1, 2001: Nominations for MSEBStaff The Brain: Unraveling the Mystery the Emanuel and Carol Parzen Prize. of How It Works (The Neural Network Submit nominations to ]. H. Matis, Michael Feuer, Executive Director, Process), by Thomas L. Saaty. RWS Pub­ Department of Statistics, Texas A&M Center for Education lications, 2000. ISBN 1-888603-02-X. University, College Station, TX 77873- Gail Burrill, Director The Bride of Science, by Benjamin 3143. Bradford Findell, Program Officer Woolley. MacMillan, August 1999. ISBN Brian McQuillan, Senior Project 0-333-72436-4. Assistant Mathematical Sciences Chance Rules: An Informal Guide to Probability, Risk, and Statistics, by Education Board, National Mathematical Sciences Education Research Council Brian S. Everitt. Springer, August 1999. Board ISBN 0-387-98768-1. judy Ackerman, Montgomery National Research Council The Crest of the Peacock: The Non­ 2101 Constitution Avenue, NW College European Roots of Mathematics, by (HA 450) Richard A. Askey, University of George Gheverghese Joseph. Princeton Washington, DC 20418 Wisconsin-Madison University Press, October 2000 (new Tel: (202) 334-3294 Deborah Loewenberg Ball, University edition). ISBN 0-691-00659-8. Fax: (202) 334-1453 of Michigan Crypto: How the Code Rebels Beat e-mail: mseb@nas. edu Richelle Blair, Lakeland Community the Government-Saving Privacy in Web:http:l/www4. College the Digital Age, by Steven Levy. Viking nationalacademies.org/csmee/ ]ere Confrey (vice chair), University Press, January 2001. ISBN 0-6708- mseb.nsf/ of Texas, Austin 5950-8. Ingrid Daubechies, Princeton Divine Harmony: The Life and University Book List Teachings of Pythagoras, by John ]an de Lange, Freudenthal Institute, The Book List highlights books that Strohmeier and Peter Westbrook. The Netherlands have mathematical themes and hold Berkeley Hills Books, November 1999. Keith Devlin, St. Mary's College of appeal for a wide audience, including ISBN 0-965-37745-8.

MAY 2001 NOTICES OF THE AMS 517 Reference and Book List

The Dots and Boxes Game, by Elwyn Books, August 2000. ISBN 0-465-01618- Institute of Physics Publishing, July Berlekamp. A K Peters, July 2000. ISBN 9. (Reviewed February 2001.) 2000. ISBN 0-750-30648-3. 1-568-81129-2. Mathematics As Sign: Writing, Imag­ *Radical Equations: Math Literacy Duelling Idiots and Other Probabil­ ining, Counting, by Brian Rotman. and Civil Rights, by Robert P. Moses and ity Puzzlers, by Paul J. Nahin. Prince­ Stanford University Press, September Charles E. Cobb Jr. Beacon Press, ton University Press, October 2000. 2000. ISBN 0-804-73684-7. February 2001. ISBN 0-807-03126-7. ISBN 0-691-00979-1. Mathematics: Frontiers and Per­ Riemann, Topology, and Physics, by Education of a Mathematician, by spectives, V. Arnold, M. Atiyah, P. Lax, Michael Monastyrsky; translated by Philip J. Davis. A K Peters, August and B. Mazur, editors. AMS, December Roger Cooke, James King, and Victoria 2000. ISBN 1-568-81116-0. (Reviewed 1999. ISBN 0-8218-2697-2. King. Birkhauser, second edition, May January 2001.) Mathematics Success and Failure 1999. ISBN 3-764-33789-3. Einstein in Love: A Scientific among African American Youth: The Stephen Smale: The Mathematician Romance, by Dennis Overbye. Viking Roles of Sociohistorical Context, Com­ Who Broke the Dimension Barrier, Press, October 2000. ISBN 0-6708- munity Forces, School Influence, and by Steve Batterson. AMS, February 9430-3. Individual Agency, by Danny B. Martin. 2000. ISBN 0-8218-2045-1. (Reviewed Excursions into Mathematics: Mil­ Lawrence Erlbaum Associates, Decem­ December 2000.) lennium Edition, by Anatole Beck, ber 1999. ISBN 0-805-83042-1. Surfing through Hyperspace: Un­ Michael N. Cleicher, and Donald W. Mathematics Unlimited: 2001 and derstanding Higher Universes in Six Crowe. A K Peters, 2000. ISBN 1-568- Beyond, Bjorn Engquist and Wilfried Easy Lessons, by Clifford A. Pickover. 81115-2. Schmid,editors.Springer,September Oxford University Press, September 1999. ISBN 0-19-513006-5. *Exploring Randomness, by 2000. ISBN 3-540-66913-2. The Symbolic Universe: Geometry Gregory J. Chaitin. Springer, December My Numbers, My Friends: Popular and Physics 1890-1930, edited by 2000. ISBN 1-852-33417-7. Lectures on Number Theory, by Paulo Jeremy Gray. Oxford University Press, The Fermat Diary, by C. J. Mozzochi. Ribenboim. Springer, February 2000. September 1999. ISBN 0-19-850088-2. AMS, 2000. ISBN 0-8218-2670-0. ISBN 0-387-98911-0. Two Millennia ofMathematics: From Contributions to The Mystery of the Aleph: Mathe­ *Finite vs. Infinite, Archimedes to Gauss, by George M. and the Search an Eternal Dilemma, Cristian S. Calude matics, the Kabbalah, Phillips. Springer, July 2000. ISBN 0- and Gheorghe Paun, editors. Springer, for Infinity, by Amir D. Aczel. Four 387-95022-2. 2000. ISBN 1-852-33251-4. Walls Eight Windows, November 2000. Uncle Petros and Goldbach's Con­ The Game's Afoot! Game Theory in ISBN 1-56858-105-X. jecture, by Apostolos Doxiadis. Myth and Paradox, by Alexander Newton's Gift: How Sir Isaac New­ Bloomsbury USA, February 2000. ISBN Mehlmann. AMS, 2000. ISBN 0-8218- ton Unlocked the System of the World, 1-582-34067-6. (Reviewed November 2121-0. by David Berlinski. Free Press, October 2000.) Geometry from Africa: Mathemat­ 2000. ISBN 0-684-84392-7. The Universal Computer: The Road ical and Educational Explorations, Niels Hendrik Abel and His Times: from Leibniz to Turing, by Martin by Paulus Gerdes. Mathematical Called Too Soon by Flames Afar, Davis. W.W. Norton & Company, Octo­ Association of America, April 1999. by Arild Stubhaug, translated by ber 2000. ISBN 0-393-04785-7. ISBN 0-88385-715-4. R. Daly. Springer, May 2000. ISBN (Reviewed in this issue.) G6del: A Life of Logic, by John L. 3-540-66834-9. The Universal History of Computing: Casti and Werner DePauli. Perseus, Number: From Ahmes to Cantor, by From the Abacus to the Quantum Com­ August 2000. ISBN 0-738-20274-6. Midhat Gazale. Princeton University puter, by Georges Ifrah; translated G6del Meets Einstein: Time Travel in Press, March 2000. ISBN 0-691-00515-X. from the French and with notes by the G6del Universe, by Palle Yourgrau. The Parrot's Theorem, by Denis E. F. Harding, assisted by Sophie Wood, Open Court, November 1999. ISBN 0- Guedj. Weidenfeld & Nicolson, June Ian Monk, Elizabeth Clegg, and Guido 812-69408-2. 2000. ISBN 0-297-64578-1. (To be pub­ Waldman. John Wiley & Sons, Novem­ Hex Strategy: Making the Right Con­ lished in the U.S. by St. Martin's Press, ber 2000. ISBN 0-471-39671-0. nections, by Cameron Browne. A K September 2001, ISBN 0-312-28055-6.) The Universal History of Numbers: Peters, May 2000. ISBN 1-568-81117-9. (Reviewed March 2001.) From Prehistory to the Invention ofthe A History of Algorithms: From the Philosophy of Mathematics: A n Computer, by Georges Ifrah; translated Pebble to the Microchip, edited by Jean­ Introduction to a World of Proofs and from the French by David Bellas, E. F. Luc Chabert. Springer, September Pictures, by James Robert Brown. Harding, Sophie Wood, and Ian Monk. 1999. ISBN 3-540-63369-3. Routledge, August 1999. ISBN 0-415- John Wiley & Sons, December 1999. The Kingdom of Infinite Number: 12274-0. (Reviewed November 2000.) ISBN 0-4 71-3 7568-3. A Field Guide, by Bryan Bunch. W. H. *Ptolemy's Geography, translated The Unknowable, by Gregory Chaitin. Freeman, January 2000. ISBN 0-716- by J. Lennart Berggren and Alexander Springer, August 1999. ISBN 9-814- 73388-9. Jones. Princeton University Press, 02172-5. The Math Gene: How Mathematical November 2000. ISBN 0-691-01042-0. Where Mathematics Comes From: Thinking Evolved and Why Numbers The Pursuit of Perfect Packing, by How the Embodied Mind Brings Math­ Are Like Gossip, by Keith Devlin. Basic Tomaso Aste and Denis Weaire. ematics into Being, by George Lakoff

518 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Reference and Book List and Rafael Nu:iiez. Basic Books, Octo­ ber 2000. ISBN 0-465-03770-4. *White Light, by Rudy Rucker. Four Walls Eight Windows, April2001. ISBN 1-56858-198-X. High-Quality Paperbacks­ The Wild Numbers, by Philibert Schogt. Four Walls Eight Windows, As Little as $9.95! April 2000. ISBN 1-56858-166-1. (Reviewed November 2000.) Women Becoming Mathematicians: Creating a Professional Identity in Post­ World War IIAmerica, by Margaret A.M. Murray. MIT Press, September 2000. COMAP Membership ISBN 0-262-13369-5. Wonders ofNumbers: Adventures in Instant access to hundreds of math­ Math, Mind, and Meaning, by Clifford A. ematical materials for any course Pickover. Oxford University Press, September 2000. ISBN 0-19-513342-0. you teach at www.comap.com plus copies of The UMAP Journal. Developmental Mathematics and Its Applications Based on modeling real-world applications, this two semester course offers an alternative to elementary and intermediate algebra and provides an active discovery approach to learning mathematics. COMBINATORIAL OPTIMIZATION: Networks and Matrolds, Eugene Lawler. Perceptively Precalculus: written text examines shortest paths, network flows, bipartite matching, nonbipartite match­ Modeling Our World ing, matroids and the greedy algorithm, matroid intersections, and the matroid parity A new text that uses the model­ problems. Suitable lor courses in combinatorial computing and concrete computational com­ ing approach to teach traditional plexity. X+374pp. 5h 8Jf. 41453-1 Pa. $12.95 precalculus concepts. TOPOLOGICAL METHODS IN EUCUDEAN SPACES, Gregory L. Naber. Extensive devel­ Mathematical Contest opment of such topics as elementary combi­ natorial techniques, Sperner's Lemma, the in Modeling (MCMJ Brouwer Fixed Point Theorem, and the Stone­ Weierstrass Theorem. New section of solu­ tions to selected problems. 240pp. This contest offers students the 5Mx 8Jf. 41452-3 Pa. $9.95 opportunity to compete in a A ARST COURSE IN NUMERICAL ANALYSIS: Second Edition, Anthony Ralston and Philip team setting using applied Rabinowitz. Outstanding text, oriented toward computer solutions, stres,ses errors mathematics in the solving of in methods and computational efficiency. Problems at ends of chapters. 624pp. 5Mx 8Jf. real-world problems. 495 teams 41454-X Pa. $19.95 competed this year. N-PERSON GAME THEORY: Concepts and Applications, Anatol Rapoport. Sequel to Two-Person Game Theory introduces neces­ sary mathematical notation (mainly set theo­ ry), presents basic concepts and models, and provides applications to social situations. 332pp. 5Mx 8!>. 41455-8 Pa. $12.95 57 Bedford St. Suite 210 Request a FREE Dover Mathematics and Science Catalog (59065-8)-over 800 books, Lexington, MA 02420 most $10 to $20! Send to: DOVER PUBUCATIONS, Dept. 1-800-772-6627 AMS601, 31 E. 2nd St., Mi neola, NY 11501. Visit Dover online at www.comap.com www.doverpublications.com *Added to "Book List" since the list's last appearance.

MAY 2001 NOTICES OF THE AMS 519 From the AMS Secretary

Officers of the Members at Large Publications Committees All terms are for three years and Bulletin Editorial Committee 31 following the Society 2000 and expire on January Donald G. Saari 2/ 99-1/ 02 year given. Colloquium Editorial Committee 2001 2000 Susan]. Friedlander 2/ 96-1/ 02 Except for the members at large of Robert L. Bryant journal oftheAMS Editorial the Council, the month and year of the Jane M. Hawkins Committee first term and the end of the present Karen Parshall Carlos E. Kenig 2/ 00-1/ 04 term are given. For members at large M. Beth Ruskai Mathematical Reviews Editorial of the Council, the last year of the Michael Starbird Committee present term is listed. 2001 Hugh L. Montgomery 3/ 95-1/ 02 Haim Brezis Mathematical Surveys and Council Robert Fefferman Monographs Editorial Committee Presidents Donald G. Saari Michael P. Loss 2/ 01-1/ 03 Hyman Bass 2/ 01-1/ 03 Tatiana Toro Tudor Stefan Ratiu 2/ 96-1/ 01 Felix E. Browder 2/ 99-1/ 01 Nolan R. Wallach Mathematics ofComputation Immediate Past President 2002 Editorial Committee Felix E. Browder 2/ 01-1/ 02 Patricia E. Bauman Lars B. Wahlbin 1/ 90-1/ 01 President Elect William Fulton Proceedings Editorial Committee Golubitsky Hyman Bass 2/ 00-1/01 Martin Eric D. Bedford 2/ 01-1/ 05 Jonathan M. Rosenberg Vice Presidents Clifford J. Earle Jr. 1/ 89-1/ 01 Lisa M. Traynor James G. Arthur 2/ 99-1/ 02 Transactions and Memoirs Editorial 2003 Jennifer Tour Chayes 2/ 98-1/ 01 Committee Ingrid Daubechies 2/ 01-1/ 04 Walter L. Craig William Beckner 2/ 00-1/ 04 David Eisenbud 2/ 00-1/ 03 Keith J. Devlin Irene Fonseca Board of Trustees Secretary Alexander Nagel Roy L. Adler 2/ 93-1/ 03 Robert ]. Daverman 2/ 99-1/ 03 Louise A. Raphael Hyman Bass (ex officio) 2/ 01-1/ 03 Former Secretary Felix E. Browder (ex officio) 2/ 99-1/ 01 Robert M. Fossum 2/ 00-1/ 01 John B. Conway 2/ 01-1/ 06 Members of Executive Michael G. Crandall 2/ 96-1/ 01 Associate Secretaries Committee John M. Franks (ex officio) 2/ 99-1/ 03 John L. Bryant 2/ 99-1/ 03 Members of the Council, as provided Eric M. Friedlander 2/ 00-1/0 5 Susan]. Friedlander 2/ 96-1/ 02 Linda Keen 2/ 99-1/ 04 for in Article 7, Section 4 (last sen­ Bernard Russo 2/ 96-1/ 02 Andy R. Magid 2/ 97-1/ 02 tence), of the Bylaws of the Society. Lesley M. Sibner 2/ 93-1/ 03 B. A. Taylor (ex officio) 2/ 93-1/ 03 Treasurer RobertL.Bryant2/ 00-1/ 04 John M. Franks 2/ 99-1/ 03 John B. Conway 2/ 97-1/ 01 Associate Treasurer Joel Spencer 2/ 98-1/ 02 B. A. Taylor 2/ 93-1/ 03 Karen Vogtmann 2/ 99-1/ 03

520 NOTICES OF THE AMS VOLUME 48, NUMBER 5 he prize is awarded each year to an Tundergraduate student (or students having submitted joint work) for out­ standing research in mathematics. Any stu­ dent who is an undergraduate in a college or university in the United States or its posses­ sions, or Canada or Mexico, is eligible to be considered for this prize. he recipients of the prize are to be Tselected by a standing joint committee prize recipient's research ne~d not he of the AMS, MAA, and SIAM. The deci­ be confined to a single paper; 1t may T sions of this committee are final. The 2001 be contained in several papers. prize will be awarded for papers submitted However, the paper (or papers) to be consid­ for consideration no later than june 30, the prize must be submitted while ered for 2001, by (or on behalf of) students who were the student is an undergraduate; they can­ undergraduates in December 2000. not be submitted after the student's gradua­ tion. The research paper (or papers) may be submitted for consideration by the student or a nominator. All submissions for the prize must include at least one letter of sup­ Nominations and submissions should be sent to: port from a person, usually a faculty mem­ Morgan Prize Committee ber, familiar with the student's research. c/o Robert j. Daverman, Secretary Publication of research is not required. American Mathematical Society Department of Mathematics University of Tennessee • • • • Knoxville, TN 37996-1330 Questions may be directed to the chairperson of the Morgan Prize Committee: Dr. MarthaJ. Siegel, Chair Department of Mathematics Towson University Towson, MD 21252-0001 telephone: 410-704-4379 e-mail: siege1 @towson • edu Add this Cover Sheet to all of your Academic Job Applications

How to use this form The Joint Committee on Mathematics Depart­ JCEOReconunendations Employment Opportuni­ ments in Bachelor's, for Professional 1 . Using the facing page or a photocopy, ties has adopted the Master's and Doctorate Standards in Hiring (or a TEX version which cover sheet on the facing granting institutions Practices can be downloaded from page as an aid to job have been contacted and the e-math "Employment applicants and prospec­ are expecting to receive The JCEO believes that Information" menu, http://www.ams.org/ tive employers. The form the form from each every applicant is enti­ emp 1oyment/), is now available on applicant, along with tled to the courtesy of a fill in the answers which e-math in a TEX format any other application prompt and accurate apply to all of your which can be downloaded materials they require. response that provides academic applications. Make photocopies. and edited. The purpose Obviously, not all timely information about of the cover form is to departments will utilize his/her status. Specifi­ 2. As you mail each aid department staff in the cover form informa­ cally, the JCEO urges all application, fill in the man­ institutions to do the remaining questions tracking and responding tion in the same neatly on one cover to each application. ner. Please direct all following after receiving sheet and include it general questions and an application: on top of your applica­ comments about the tion materials. form to: (1) Acknowledge receipt [email protected] of the application­ or call the Professional immediately; and Programs and Services (2) Provide information Department, AMS, at as to the current status 800-321-4267 extension of the application, as 4105. soon as possible.

The JCEO recommends a triage-based response, informing the applicant that he/she (a) is not being consid­ ered further; (b) is not among the top candidates; or (c) is a strong match for the position. Academic Employment in Mathematics AMS STANDARD COVER SHEET Last Name First Name Middle Names Address through next June ------Home Phone

e-mail Address

Current Institutional Affiliation Work Phone

Highest Degree and Source

This form is provided Year of Ph.D. (optional) -::ourtesy of the American Ph.D. Advisor Mathematical Society. H the Ph.D. is not presently held, date on which you expect to receive ______This cover sheet is Indicate the mathematical subject area(s) in which you have done research using, if applicable, the Mathematics provided as an aid to Subject Classification printed on the back of this form or on e-MA TH. If listing more than one number, list first the departments in process­ one number which best describes your current primary interest. ing job applications. It should be included Primary Interest with your application Secondary Interests optional material.

Give a brief synopsis of your current research interests (e.g. finite group actions on four-manifolds). Please print or type. Avoid special mathematical symbols and please do not write outside of the boxed area. Do not send this form to theAMS.

Most recent, if any, position held post Ph.D. University or Company Position Title Indicate the position for which you are applying and position posting code, if applicable

If unsuccessful for this position, would you like to be considered for a temporary position? DYes D No If yes, please check the appropriate boxes. 0 Postdoctoral Position 0 2+ Year Position 0 1 Year Position List the names, affiliations, and e-mail addresses of up to four individuals who will provide letters of recom­ mendation if asked. Mark the box provided for each individual whom you have already asked to send a letter. 0 0 0 0 2000 Mathematics Subject Classification

00 General 51 Geometry 01 History and biography 52 Convex and discrete geometry 03 Mathematical logic and foundations 53 Differential geometry 05 Combinatorics 54 General topology 06 Order, lattices, ordered algebraic structures 55 Algebraic topology 08 General algebraic systems 57 Manifolds and cell complexes 11 Number theory 58 Global analysis, analysis on manifolds 12 Field theory and polynomials 60 Probability theory and stochastic processes 13 Commutative rings and algebras 62 Statistics 14 Algebraic geometry 65 Numerical analysis 15 Linear and multilinear algebra, matrix theory 68 Computer science 16 Associative rings and algebras 70 Mechanics of particles and systems 17 Nonassociative rings and algebras 74 Mechanics of deformable solids 18 Category theory, homological algebra 76 Fluid mechanics 19 K-theory 78 Optics, electromagnetic theory 20 Group theory and generalizations 80 Classical thermodynamics, heat transfer 22 Topological groups, Ue groups 81 Quantum theory 26 Real functions 82 Statistical mechanics, structure of matter 28 Measure and integration 83 Relativity and gravitational theory 30 Functions of a complex variable 85 Astronomy and astrophysics 31 Potential theory 86 Geophysics 32 Several complex variables and analytic spaces 90 Operations research, mathematical programming 33 Special functions 91 Game theory, economics, social and behavioral 34 Ordinary differential equations sciences 35 Partial differential equations 92 Biology and other natural sciences 37 Dynamical systems and ergodic theory 93 Systems theory, control 39 Difference and functional equations 94 Information and communication, circuits 40 Sequences, series, summability 97 Mathematics education 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus 45 Integral equations 46 Functional analysis 47 Operator theory 49 Calculus of variations and optimal control, optimization The selection committees for these prizes request nomina­ tions for consideration for the 2002 awards, which will be presented at the Joint Mathematics Meetings in San Diego, CA, in January 2002. Information about most of these prizes may be found in the November 1999 Notices, pp. 1258-1269. (Also available at http:/ /www.ams.org/ THE BacHER secretary /prizes.html). MEMORIAL PRIZE Beginning in 2002 the Bacher Memorial Prize will be pre­ sented at three-year intervals. It is awarded for a notable research memoir in analysis published during the preced­ ing five years.

The Award for Distinguished Public Service is presented DISTINGUISHED every two years to a research mathematician who has made PUBLIC a distinguished contribution to the mathematics profession during the preceding five years. SERVICE AWARD

Beginning in 2002 the Frank Nelson Cole Prizes will be pre­ sented at three-year intervals for outstanding contributions in number theory and algebra, respectively. FRANK NELSON COLE The Levi L. Conant Prize, first awarded in January 2001, is presented annually for an outstanding expository paper PRIZE IN published in either the Notices or the Bulletin of the American NUMBER THEORY Mathematical Society during the preceding five years.

Nominations should be submitted to the Secretary, Robert J. Daverman, American Mathematical Society, 312D Ayres Hall, University of Tennessee, Knoxville TN 37996-1330, LEVI L. CONANT and should include supporting materials. For the Public Service Award, include a short description of the pertinent PRIZE activities of the nominee; for the other prizes, include a short description of the work that is the basis of the nomi­ nation, including complete bibliographic citations. A (brief) curriculum vita should be included for all nominees. The nominations will be forwarded by the secretary to the appropriate prize selection committee which, as in the past, will make final decisions on awarding of the prizes.

Deadline for nominations is June 30, 2001. Geometriae Dedicata Edlton-ln=Cblef Benson s. Farb • Robert J. Zimmer University of Chicago, IL, USA

Editorial Board Tobias Holck COiding, New Yat UniYetsit;: l.54; Kenji Fukaya, A}oto UniYetsit;: .Japarr, William Mark Goldman, University of Maryland, College Park, l/54; Rotislav lvanovic:h Grigorchuk, SIE!kJov 1nsUtJJte of Mat11ernat:k5, ~ Russier, Gerard B.M. van der Geer, University ofAmsterdam, The Netherlands; Karsten Grove, University ofMary/and, Cdlege Park, l.54; Joel Hass, Univelsity of caJifomia, c.m.1s; l.54; Toshiyuki Kobayashi, University of ~ Japan, John W. loti:, University ofMichigan, Ann Arbol; l.54; Alex Lubotzky, The Hebrew Un~ Jerusalem, Israel; Lee D. Mosher, Rutgers Un~ Newark, to_ l.54; Jean-Pierre Otal, ENS de Lyon, France, Alan W. Reid, UniversityofTexas,Austin, l/54; RalfJ. Spatzier, UniversityofMichigan,AnnArtxx; l.54; Shmuel A. Weinberger, University of07iaJgo, IL, US4

its relationship to Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and topology, group theory and the theory of dynamical systems. Features of the journal include: 1. A fast turn-around time for articles. areas in geometry, 2. A "Surveys in Geometry" series: this will be a series of survey articles on active directed to other geometers. 3. Special issues centered on specific topics.

Forthcoming papers

. . J. Stallings "An extension theorem for Euler characteristics of groups" ...... and Frederic Paulin "Sure les immeubles hyperboliques" ...... Damian Gabotiau and Robert J. Zimmer "Actions of semisimple Lie groups on cirde bundles" ...... Dave Witte . G. N. Arzhantseva "On quasiconvex subgroups ofl.vord hyperbolic groups" ...... » "The lamplighter group as a group generated by a 2-state automaton, and its spectrum" ...... » ...... Rostlslav I. Grigorchuk and Andrzej Zuk . . . Caroline Series "On Kerckhoff minima and pleating loci for quasi-Fuchsian groups" ...... Evan Schwartz "Desargues theorem, dynamics, and hyperplane arrangements" ...... Richard . . . . . Herbert Abels "Properly discontinuous groups of affine transbrmations- A survey" ...... ·...... Igor Rlvln "Simple curves on surfaces" ...... Alain W. Reid "On subgroup separability in hyperbolic Coxeter groups" ...... D.D.Long, ...... » "The proalgebraic completion of rigid groups" ...... » ...... Hyman Bass, Alexander Lubotzky, Andy R Magid, Shahar Mozes . . . Pierre de Ia Harpe "Uniform growth in groups of exponential growth" ......

please contact: Kluwer academic publishers Oldar~P.O. Ib:322 • 3300AH DaRhrH. The ...... Fax: +31·7&8548474 • TeL: +31·7S«lll23&2 • E-mil: lldlii ......

SUBSCRIPTION INFORMATION ISSN 0046-5755, 2001, Volumes 84-88 (15 issues) Paper Version: NLG 1473.00 I USD 1400.00 Online Version: EUR01473.00 I USD 1400.00 Combined Paper & Online Version: EURO 1767.00 I USD 1680.00 ~ 0 w (!) 0 · ~ ..-- <(

lL w 0::: For up-to-date information, tables of contents & your FREE (online) sample copy Mathematics Calendar

The most comprehensive and up-to-date Mathematics Calendar information is available on e-MATII at http:llwww.ams . orglmathcall.

May 2001 City, NY 11530-6793. Information: R. Skurnick at 516-872-4682. * 5 East Coast Computer Algebra Day 2001, Florida State University, Tallahassee, Florida. * 22-27 Differential Equations and Related Topics, Dedicated to Program: The meeting is the eighth of the series, which has been the 1OOth Anniversary of I. G. Petrovskii, Moscow State University, held at a variety of U.S. East Coast locations since 1994. The meet­ Moscow, Russia. ing includes invited talks and contributed poster/software demo Organizing Committee: V. A. Sadovnichii (chair, president of MSU), sessions covering the breadth of computer algebra: algorithms, V. V. Alexandrov, N. S. Bakhvalov, A. A. Bolibrukh (vice chair), V. M. software, and applications. Buchstaber, G. A. Chechkin (executive secretary), A. T. Fomenko, A. Call for Posters: To submit a poster for one of the poster sessions, Yu. Goritski, Yu. S. ll'yashenko, G. M. Kobel'kov, V. A. Kondratyev, please send a title and a brief abstract to: accad01(Dmath.fsu. adu. V. V. Kozlov, 0. B. Lupanov, V. M. Millionshchikov, E. I. Moiseev, 0. Funding has been requested from the National Science Foundation A. Oleinik, A. S. Pechentsov, E. V. Radkevich, N. Kh. Rozov, T. A. to cover travel expenses for some of the U.S.-based participants. Shaposhnikova, A. A. Shkalikov (vice chair), and M. I. Vishik. As in previous years, preference will be given to recent Ph.D.'s who Program Committee: Yu. S. Osipov (chair, president of Russian present a poster at the conference. Academy of Sciences), H. Amann, A. Ambrosetti, D. V. Anosov, V. I. Information: http: I lwww .math. fsu. adulaccad01. Registration: Arnold, M. Atiyah, 0. V. Besov, J. Ball, A. Bensoussan, B. Bojarskii, See URL. Program: See URL. e-mail: accad01(Dmath.fsu.adu. A. A. Bolibrukh (vice chair), L. D. Faddeev, A. M. llyin, V. A. llyin, W. Jager, 0 . A. Ladyzhenskaya, G. I. Marchuk, V. P. Maslov, E. F. * 1 2 Graph Theory Day 41, Nassau Community College, Garden City, Mishchenko, S. M. Nikol'skii (vice chair), S. P. Novikov, 0. A. Oleinik New York. (vice chair), ]. Palls, S. I. Pohozaev, A. A. Shkalikov (vice chair), I. V. Sponsors: The New York Academy of Sciences; hosted by Depart­ Skrypnik, 0. Viro, V. S. Vladimirov. ment of Mathematics, Statistics, and Computer Processing, Nassau Contacts: Petrovskii Conference, Department of Mathematics and Community College, SUNY. Mechanics, Moscow M. V. Lomonosov State University, Vorobyevy Program: J. Malkevitch (Dept. of Math. and Comput. Sci., York Gory, Moscow 119899, Russia; e-mail: patr100(Dmach. math. msu. su; College, CUNY), "Gifts from Euler's Polyhedral Formula"; and R. fax: +7 (095) 939-2090. Kulkarni, (Dept. of Math., Queens College, CUNY), "Automorphisms of Finite Graphs"; and Graph Theory Notes of New York session. *25-27 Perspectives in Partial Differential Equations and Prob­ Registration: Fee $25 (includes lunch, refreshments, and a $5 ability: A Conference to Celebrate the 60th Birthday of Prof. contribution to NYAS Graph Theory Fund). NOTE: Lunch can be Nicolai Krylov, University of Minnesota, Minneapolis, Minnesota. guaranteed only for advance registrants. Make checks payable to Speakers: L. Caffarelli (Univ. of Texas, Austin), E. Dynkin (Cornell "Nassau Community College Foundation" and send to R. Skurnik, Univ.), W. Fleming (Brown Univ.), R. Khasminski (Wayne State Univ.), Dept. ofMAT/STNCMP, Nassau Community College, SUNY, Garden P. Malliavin (Univ. of Paris VI), A. Skorokhod (Michigan State Univ.),

This section contains announcements of meetings and conferences should be sent to the Editor of the Notices in care of the American of interest to some segment of the mathematical public, including ad Mathematical Society in Providence or electronically to noticas~Dams . org hoc, local, or regional meetings, and meetings and symposia devoted or mathcal~Cams . org. to specialized topics, as well as announcements of regularly scheduled In order to allow participants to arrange their travel plans, organizers of meetings of national or international mathematical organizations. A meetings are urged to submit information for these listings early enough complete list of meetings of the Society can be found on the last page of to allow them to appear in more than one issue of the Notices prior to each issue. the meeting in question. To achieve this, listings should be received in An announcement will be published in the Notices if it contains a call Providence six months prior to the scheduled date of the meeting. for papers and specifies the place, date, subject (when applicable), and The complete listing of the Mathematics Calendar will be published only the speakers; a second announcement will be published only if there in the September issue of the Notices. The March, June, and December are changes or necessary additional information. Once an announcement issues will include, along with new announcements, references to any has appeared, the event will be briefly noted in every third issue until previously announced meetings and conferences occurring within the it has been held and a reference will be given in parentheses to the twelve-month period following the month of those issues. New infor· month, year, and page of the issue in which the complete information mation about meetings and conferences that will occur later than the appeared. Asterisks (*) mark those announcements containing new or twelve-month period will be announced once in full and will not be revised information. repeated until the date of the conference or meeting falls within the In general, announcements of meetings and conferences held in North twelve-month period. America carry only the date, title of meeting, place of meeting, names of The Mathematics Calendar, as well as Meetings and Conferences of speakers (or sometimes a general statement on the program), deadlines the AMS, is now available electronically through e-MA1H on the World for abstracts or contributed papers, and source of further information. Wide Web. To access e-MA1H, use the URL: http: I I a-math. ams. orgl Meetings held outside the North American area may carry more detailed (or http: I lwww. ams. orgl). (For those with VT100-type terminals or for information. In any case, if there is any application deadline with respect those without WWW browsing software, connect to e-MA1H via Telnet to participation in the meeting, this fact should be noted. All commu­ (talnat a-math. ams. org; login and password a-math) and use the Lynx nications on meetings and conferences in the mathematical sciences option from the main menu.)

MAY 2001 NOTICES OF TilE AMS 527 Mathematics Calendar

of the genome, and evolutionary aspects N. Uraltseva (St. Petersburg State Univ.), S. R. S. Varadhan (Courant techniques of exploration Institute). · of cells. . Organizers: J. Baxter and M. Safanov (chair) (Univ. of Minnesota), Registration: ESMTB membership is open on http: I lwww. esmtb to and B. Rozovsky (Univ. of Southern California). Partially sponsored orgl join. html. Registration (or documented preregistration) fee. by the IMA, the NSF, and the Univ. of Minnesota. the ESMTB gives automatic entitlement to the reduced school 24 euros students) Information: Please contact safonov(Omath. umn. edu and visit the The ESMTB fee (40 euros full registration, with the school fee, Univ. of Minnesota School of Math. at http: I lwww. math. umn. edul. may be paid directly to ESMTB or together either by anticipation or at the school's venue. Please communicate to R. Bravo de la Parra (rafael. bravo

VOLUME 48, NUMBER 5 528 NOTICES OF TilE AMS Mathematics Calendar

July 2001 Information: Web page: http: I lwwr.r. math. ias. edu;-dgomesl sumsch. html, or e-mail: dgomes@ias. edu. * 2-13 ESSGT 2001 :The 11th European Summer School in Group Theory, CIRM, Marseille-Luminy, France. * 1 6-21 Workshop on the Skorokhod Problem, Mathematical Con­ Program: The European Summer Schools in Group Theory have been ference Center, Bedlewo, Poland. organized every year since 1991 in different European countries. Short Description: The purpose of the workshop is to bring together They are meant to train graduate students, postdocs, and young researchers working on various aspects of the Skorokhod problem researchers in various aspects of group theory and to introduce and its applications. them to recent developments in the field. Information: Further details can be found on the Web site http: I I Priority will Registration: The organizers expect 50 participants. wwr.r .mat.uni .torunlwosp2001l. be given to junior applicants (graduate students, postdocs, young fields. Deadline: researchers) working in group theory and related * 22-26 Experimental Chaos Conference 2001, University of Pots­ April15, 2001. dam, Germany. the Web site: http: Information: For more information visit Description: The conference is sponsored by the U.S. Office of u-nancy. fr ;-essgtlhomepage. html, or contact the I lwwr.r. iecn. Naval Research. Members of the worldwide scientific, medical, ofESSGT 2001: M. Douchez, Univ. Paris 7- Denis Diderot, secretariat and engineering communities interested in recent developments Jussieu, Theorie des Groupes, Case Institut de Mathematiques de and techniques of experimental nonlinear dynamics are invited to Jussieu, F-75251 Paris Cedex 05, France; fax: +33 (0)1 7012, 2 place attend the conference and to contribute to its technical sessions e-mail: douchez!Dmath.jussieu.fr. 44 27 78 18; and workshops. * 8-9 TA and Instructor Development Using Case Studies: A Sum­ Contributed Talks and Posters: Contributed posters will be mer Workshop for Faculty, Regis College, Weston, Massachusetts. presented at extended poster sessions throughout the meeting. talks of 15 minutes each will Organizers: BCCase, The Boston College Mathematics Case Studies Approximately 25-30 contributed those who Project. be selected by the program committee from among considered for a contributed talk. Contributed talks Description: This workshop will introduce participants to a newly wish to be by an 8-page paper which will be included developed tool, case studies, to prepare graduate students and will be accompanied the conference proceedings. Requests for special audio-visual junior faculty for their responsibilities as teaching assistants and in should be made as soon as possible after notification instructors. Learn how to use these materials and integrate them equipment acceptance of abstract. into a TA or faculty development program. The case studies we of will use are specific to university-level mathematics instruction and Conference Deadlines: The following deadlines should be noted: have been developed by a diverse group of mathematics researchers April 1: Abstracts for talks/posters due to program committee; and educators. April 31: Notification of acceptance of contributed talks/ posters; Housing: Dormitory accommodations and meals will be provided June 28: Final versions of papers due. by Boston College through a grant from FlPSE, The Fund for the Sessions and Invited Speakers: Applications: (Scholz-Reiter, Ger­ Improvement of Postsecondary Education. All transportation and many; Daw, U.S.; Eubank, U.S.; Helbing, Germany); Electronics: other costs are the responsibility of the participant. (Ogorzalek, Poland; Murali, India); Astrophysics: (J. Buchler, U.S.; (Arneodo, France; Sornette, France); Hy­ Application and Information: S. Friedberg (e-mail: friedber~bc. P. Boyd, U.S.); Geophysics: edu) or D. Foster, BCCase Project Administrator (e-mail: bccase@ drodynamics & Turbulence: (E. Villermaux, France; Pinton, France); bc.edu). Chemistry: Hudson, U.S.; Krischer, Germany); Neuroscience: (Tass, Germany; Feudel, Germany; Gluckman, U.S.; Selverston, U.S.); Con­ * 8-1 5 5th WSES/IEEE World Multiconference on Circuits, Systems, densed Matter: (Behn, Germany); Data Analysis: (Voss, Germany; Communications & Computers (CSCC 2001), Rethymnon, Crete, Dolan, U.S.); Optics: (Kim, U.S.; Tredice, France); Dynamo Machine: Greece. · (Mueller, Germany; Fauve, France); Synchronization: (Maza, Spain; Description: Joint conference with the 3rd WorldSES MCP and 3rd Tang, U.S.); Ecology: (Stone, Israel; Ben Jacob, Israel); Keynote WorldSES MCME, dedicated to Sir John Ambrose Fleming. Technical Speaker: I. Blekhman, Russia. cosponsorship by IEEE Signal Processing Society. Information: S. Boccaletti, Istituto Nazionale di Ottica Applicata, Deadline: For paper submission is March 20, 2001. Largo E. Fermi, 6, I 50125 Florence, Italy; tel: +39 055 23081; fax: Information: Send a message to [email protected] or to mbetini@ +39 055 2337755; e-mail: stefano@ino . it; or http: I lwwr.r. agnld. italymail.comortoCSCC~Worldses.org,orseehttp:llcscc2001 . uni-potsdam.del-shw1Workshopi04_ECC6Iindex.html. tripod.coml. * 30-August 3 6th International Conference on Difference Equa­ * 9-2 5 Summer School of Probability Theory, Saint-Flour, France. tions and Applications, University of Augsburg, Augsburg, Ger­ Speakers: 0. Catoni, "Statistical learning theory and stochastic many. optimization"; S. Tavare, "Ancestral inference from molecular data"; Description: Both in form and spirit this conference will resume the 0. Zeitouni, "Random walk in random environments: Asymptotic style established during the previous meetings in San Antonio, Texas results". (1994); Veszprem, Hungary (1995); Taipei, Taiwan (1997); Poznan, Deadline: Registration should be made before March 31, 2001. Poland (1998); and Temuco, Chile (2000). The conference will cover Information: http: I lwwr.rlma. uni v-bpclermont. fr I stflour I; all themes of the field of ordinary and partial difference equations, .! e-mail: stflour!Dmath. uni v-bpclermont. fr. classical and contemporary, theoretical and applied. There will be about 30 invited lectures in the morning and contributed talks in * 16-20 Summer School on Nonlinear Partial Differential Equa­ the afternoon sessions as well as an open problems seminar. tions, Instituto Superior Tecnico, Lisbon, Portugal. Contact:[email protected]. Program: Five 4-hour minicourses and contributed talks of 15-20 Information: http: I lwwr.r .math. uni-augsburg. de/icdea2001l. minutes each. Topics: Nonlinear wave and Schrtidinger equations, compressible August 2001 and incompressible Navier Stokes equations, and other equations of fluid mechanics. * 6-24 Mathematical Geophysics Summer School, Stanford Univer­ Confirmed Speakers: J. Colliander, N. Masmoudi, F. Planchon, I. sity, Stanford, California. Rodnianski, and M. Sammartino. Topics: Multiscale theory and computation with geophysical appli­ Organizers: P. Giraro (IST), D. Gomes (IST/IAS), J. Silva (1ST/Prince­ cations. ton Univ.), and J. Videman (1ST). Information: http: I I cartan . stanford. edulmgssl.

MAY 2001 NOTICES OF THE AMS 529 Mathematics Calendar

* 7-9 First Announcement-Nordic Conference on Topology and (Orsay, France); G. Frey (Essen, Germany); S. Goncharov (Brown, Applications, Sophus Lie Conference Centre, Nordfjordeid, Norway. USA), Y. Ihara (RIMS, Japan), U. Jannsen (Regensburg, Germany), Topics: The conference will be mostly devoted (but not limited) to P. Lochak (ENS, France), M. Matsumoto (Keio U., Tokyo, Japan), general and set-theoretic topology and applications. H. Nakamura (Tokyo Metropolitan U., Japan), M. Raynaud (Orsay, Organizing Committee: P. Collins, V. L. Hansen, B. Jahren, H. France), M. Saidi (Durham, UK), A. Schmidt (Heidelperg, Germany), L. Junnila, D. Repovs, and S. Watson. Schneps (Paris Vl, France), M. Spiess (Nottingham, UK), A. Tamagawa Program: A number of survey talks will be given by senior (RIMS, Japan), K. Wingberg (Heidelberg, Germany), Z. Wojtkowiak topologists, including A. V. Arhangel' skii. A full list of main (Nice, France). speakers will accompany the next announcement. There will be Deadline for Applications: May 11, 2001. parallel sessions in which participants can present shorter talks. Information: http: I lwww. esf . orgleurescol . Support: At present we cannot promise any financial support to participants. * 1 0-14 Workshop on Coding and Cryptography, Institute for Deadline: Registration at normal rate: April1, 2001. Abstracts and Mathematical Sciences, National University of Singapore, Singapore. late registration: July 1, 2001. Organizing Committee: S.-P. Chan (Singapore), R. Deng (Singapore), Information: For further information about the Sophus Lie Centre, S. Ling (Singapore), H. Niederreiter (Singapore, chair), E. Okamoto please contact T. Monsson, e-mail: ffhskule!Donline .no. All cor­ Uapan), I. E. Shparlinski (Australia), N.J. A. Sloane (USA), C. P. Xing respondence should be headed with the reference "Nordtop2001 ". (Singapore). http:llwww.math.uio.nolnordfjordeidlnordfjord.html. Description: The workshop is part of the inaugural program of the Institute for Mathematical Sciences on coding theory and data * 1 0-14 Svalbard Geometric Topology Conference, Radisson SAS integrity which will run from July to December 2001. There will be Polar Hotel Spitsbergen, Longyearbyen, Norway. invited talks and shorter contributed talks. Specific topics include Organizers: D. Repovs, S. Watson. (but are not limited to) constructions of codes, asymptotic theory Program: This will be a small and informal meeting, with only of codes, decoding algorithms, public-key cryptosystems, digital a few selected talks on topics, mostly from geometric topology. signature schemes, authentication schemes, applications of curves Propositions for talks (with abstracts) are welcome. Final program and codes to cryptography, and lattice-based cryptology. will be drawn on site. There might be a small registration fee to pay Call for Papers: Authors of contributed papers should e-mail an for the conference facilities, since no financial support is expected. abstract of 300-500 words to H. Niederreiter ( nied!Dmath. nus. edu. Deadline: July 1, 2001, for all participants' names and proposed sg) by June 22, 2001. Authors of accepted papers will be notified topics. by July 6, 2001. Information: e-mail: svalbard2001!Dat. yorku. ca. Information: http: I lwww. ims. nus. edu. sglprogramsl coding. html. * 1 8-2 3 Algebra and Discrete Mathematics, Hattingen (near Essen), Germany. * 24-28 Vertical Integration in Biology: From Molecules to Organ­ Description: The EuroConference on Classification, Non-classifica­ isms, Isaac Newton Institute, Cambridge, UK. tion and Independence Results for Modules, Groups and Model Themes: Biological systems exhibit prominent behavior at the levels Theory will highlight the interplay between model theory, infinite of both individuals and populations. Empirical and theoretical combinatorics, and various subfields of algebra. research at each of these levels has led to tremendous advances in Speakers (provisional): A. Baudisch (Berlin); E. Bouscaren (Paris); knowledge; however, much less is understood about the mechanisms P. Eklof (Irvine, USA); L. Fuchs (New Orleans); A. Glass (Cambridge, underlying the integration and coordination of behaviour at the UK); M. Goldstern (Wien, Austria); C. Gourion (Paris); W. C. Holland individual level to produce coherent population-level behaviour (Bowling Green, USA); M. Kojman (Beersheba, Israekl); V. Kopytov (vertical integration). This workshop will brin:g together leading (Novosibirsk, Russia); M. C. Laskowski (Maryland, USA); N. Mar­ experimental and theoretical researchers, together with those maridis (Ioannina, Greece); A. Oloshanskii (Moscow, Nashville); M. working at the interface between the two communities. The aim Prest (Manchester, UK); C. M. Ringel (Bielefeld); L. Salce (Padova, will be to identify and explore new interdisciplinary approaches to Italy); M. Sapir (Nashville); S. Shelah Uerusalem); D. Simson (Torun, the problem of vertical integration. PL); 0 . Spinas (Kiel, Germany); L. Striingmann (Essen, Germany; Confirmed Speakers: R. Adams (Bath), F. Ashcroft (Oxford), D. Bray Jerusalem); S. Thomas (New Brunswick, USA); J. Trlifaj (Prague); J, (Cambridge), G. Forgacs (Missouri), T. Hoefer (Berlin), P. Hunter Wilson (Birmingham, UK). (Auckland), J, Jack (Oxford), R. Keller (Virginia), P. Kulesa (Caltech), Deadline: Deadline for applications: May 1, 2001. M. Levine (Berkeley), J, Lewis (London), A. McCulloch (San Diego), Program and Information: Available at: http: I lwww . esf. orgl D. Noble (Oxford), J, Reinitz (New York), J, Sherratt (Edinburgh), K. eurescol01lpc01101a.htm. Weijer (Dundee). Location and cost: The workshop will take place at the Newton September 2001 Institute, and accommodations for participants will be provided in single study bedrooms with shared bathrooms at Wolfson Court, a * 1- 6 Number Theory and Arithmetical Geometry - .Arithmetic hall of residence adjacent to the Institute. The workshop package Aspects of Fundamental Groups, Acquafredda di Maratea (near costs £250, which includes registration fee, accommodations, Naples), Italy. breakfast, dinner, lunches, and refreshments on the days that Description: The conference will highlight recent progress in lectures take place. ' understanding arithmetic structures on algebraic fundamental Further Information and Application Forms: These are available groups of schemes and related topics. It will focus on number­ from the WWW at http: I lwww. newton. cam. ac. uklprogramsiiCBI theoretic aspects of the theory of algebraic fundamental groups icbw01.html, where further information about the workshop will of schemes, where much progress has been made in recent years. be posted and updated. Completed application forms should be The main themes of the meeting will be: Arithmetic aspects of sent to M. Clark at the above address or via e-mail to m. clark!D Galois groups, Algebraic fundamental groups of schemes, Galois newton. cam . ac .uk. action on fundamental groups, The Anabelian program, Higher Scientific Enquiries: May be addressed to N. Monk (n.monk!D class field theory and generalizations, Motivic Galois groups, sheffield. ac. uk) or P. Maini (maini!Dmaths. ox . ac. uk). Motivic structures on fundamental groups. The conference is open Deadline: For receipt of applications is April 30, 2001. to researchers worldwide, whether from industry or academia. Participation will be limited to 100. October 2001 Speakers (provisional): P. Debes (Lille, France), P. Deligne (lAS), I. Efrat (Beersheba, Israel), I. Fesenko (Nottingham, UK), J,-M. Fontaine * 1- 5 Aspects of Hyperbolic Geometry, University of Fribourg,

530 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Mathematics Calendar

Fribourg, Switzerland. Conference Committee: A. Alonso y Coria, F. Brambila Paz, P. Description: There will be about 22 invited talks covering a wide Exner, B. Grebert, R. Weder (chairman). range of hyperbolic geometry. Information: http: I /www. qmath-8. unam. mx/; e-mail: mmnsec2<0 Scientific Board: C. Bavard (Univ. of Bordeaux I), G. Besson (Univ. leibniz.iimas.unam.mx. QMath-8, Att. A.M. G. Ramirez, IIMAS­ of Grenoble I), R. Kellerhals (Univ. of Fribourg), V. Schroeder (Univ. UNAM, Apartado Postal20-726, Mexico, D.F. 01000, Mexico. of Zurich). Information: e-mail: hyp-geom(Dunifr. ch or http: I lwww. unifr. March 2002 chlmathlconferencel. * 21-22 8th Rhine Workshop on Computer Algebra, Mannheim, Germany. December 2001 Topics: The topics of the workshop include all aspects of computer algebra, from theory to * 1-3 First International Conference on Neutrosophy, Neutro­ applications and systems. Information: Updated sophic Logic, Set, Probability and Statistics, University of New information is available on the WWW at Mexico, Gallup, New Mexico. http://www.uni-mannheim.de/RWCA/. Organizer: F. Smarandache, Univ. of New Mexico, 200 College Road, Gallup, NM 87301; e-mail: smarand

* 1 0-14 Macroscopic Organisation from Microscopic Behaviour in Immunology, Ecology and Epidemiology, Isaac Newton Institute, Cambridge, UK. Themes: The purpose of this workshop is to bring together experimentalists and theoreticians working in immunology, ecology and epidemiology with the aim of fostering interaction and research. Speakers: R. Antia (Emory), C. Bangham (Imperial College), S. Bonheoffer (Friedrich Miescher Institute), C. Godfray (Imperial College), B. Grenfell (Cambridge), A. Hastings (UC Davis), S. Levin (Princeton), M. Lewis (Utah), A. lloyd (Princeton), A. Perelson (Los Alamos), D. Rand (Warwick), A. Sasaki (Kyushu), L. Segel (Weizmann Institute), D. Wodarz (Institute for Advanced Study). Location and Cost: The workshop will take place at the Newton Institute, and accommodations for participants will be provided in single study bedrooms with shared bathrooms at Wolfson Court, a hall of residence adjacent to the Institute. The workshop package costs £300, which includes registration fee, accommodations, breakfast, dinner, lunches, and refreshments on the days that lectures take place. Further Information and Application Forms: These are available from the WWW at http: I /www . newton. cam. ac. uklpr ogramsiiCBI icbw03. html, where further information aboutthe workshop will be posted and updated. Completed application forms should be sent to M. Clark at the above address or via e-mail to m. clark(Dnewton. cam. ac. uk. Scientific enquiries may be addressed to B. Sleeman (bds(Damsta .leeds. ac. uk). Closing date for receipt of applications and abstracts is June 30, 2001.

* 10- 14 QMath-8. Mathematical Results in Quantum Mechanics, Taxco, Mexico. Invited Speakers: J. M. Combes (Univ. de Toulon), M. Demuth (Technische Univ. Clausthal, V. Enss (RWTH-Aachen), A. Laptev (RIT, Stockholm), E. H. Lieb (Princeton Univ.), P.l. Naumkin (Univ. Michoacana), R. Schrader (Freie Univ. Berlin,) B. Simon (California Institute of Technology), G. Uhlmann (Univ. of Washington), A. Uribe (Univ. of Michigan), K. Yajima (Univ. of Tokyo). Topics: The following topics will be discussed: Bound state prob­ lems and scattering theory for Schrodinger operators; Inverse spectral and scattering theory of Schrodinger operators; Nonlin­ ear Schrodinger equations; Quantum chaos, quantum dots and wave guides; Parameter dependent Hamiltonians; Spectral and localization properties of Schrodinger operators.

MAY 2001 NOTICES OF 1HE AMS 531 New Publications Offered by the AMS

onal lie algebras; T. Nakajima and H.-F. Yamada, Schur's Q­ Algebra and Algebraic functions and twisted affine lie algebras; M. Nazarov, Capelli elements in the classical universal enveloping algebras; Geometry Y. Roichman, On permutation statistics and Heeke algebra characters; A. Schilling and M. Shimozono, Bosonic formula for level-restricted paths; T. Shoji, Length functions for Combinatorial G(r, p, n); T. Suzuki, Representations of degenerate affine Heeke algebra and g!n; H. Tagawa, A recursion formula of the Methods in weighted parabolic Kazhdan-Lusztig polynomials; Y. Yamada, Representation Special polynomials and generalized Painleve equations; M. Yamaguchi, A duality of a twisted group algebra of the Theory hyperoctahedral group and the queer lie superalgebra. Eiichi Bannai, Kyushu Advanced Studies in Pure Mathematics, Volume 28 University, Fukuoka, japan, January 2001, 422 pages, Hardcover, ISBN 4-314-10141-5, Kazuhiko Koike, Aoyama 2000 Mathematics Subject Classification: 05-XX, 17Bxx, 20Cxx; Gakuin University, Toyko, 05A15, 05A17, 05A19, 15A15, 16S30, 16S35, 16S99, 20F55, Japan, Masaki Kashiwara, 33£17, 82B23, Individual member $60, list $100, Institutional member $80, Order code ASPM/28N RIMS, Kyoto University, Japan, Soichi Okada, Nagoya University, japan, Itaru Terada, University of Tokyo, Japan, and ~ Frobenius Groups Hiro-Fumi Yamada, Okayama University, japan, MEMom.s \ 1 r 1 ~1 11 '' -. ct; and Classical Editors Maximal Orders A publication of the Mathematical Society of Japan. Frobenius Groups and Classical Ron Brown, University of This volume is a collection of papers written by the speakers Maximal Orders of two international conferences held at the Research Institute Hawaii, Manoa Ron Brown for Mathematical Sciences (RIMS) at Kyoto University Oapan). Contents: Introduction; Lemmas on Included are articles and surveys treating representations of @ truncated group rings; Groups of real (affine) Heeke algebras and affine lie algebras, combinatorial (1\.:.d~.h .. ~.. >

532 NOTICES OF THE AMS VOLUME 48, NUMBER 5 New Publications Offered by the AMS Integrales Orbitales Analysis Nilpotentes et Endoscopie Pour les Recommended Text Groupes Classiques A Modern Theory of non Ramifies Integration jean-Loup Waldspurger, Robert G. Bartle, Eastern CNRS, Universite de Paris, Michigan University, Ypsilanti, France and University of Illinois, Urbana A publication of the Societe Mathematique de France. The theory of integration is one of the twin pillars on which analysis is built. Let G be an unramified classical group over a p-adic local field The first version of integration that F, where p is sufficiently large. Let g be the lie algebra of G. students see is the Riemann integral. · Let 1J~ be the space of invariant distributions on g(F), with Later, graduate students learn that the Lebesgue integral is support included in the nilpotent set. Let 1JG,st be the space of "better" because it removes some restrictions on the inte­ stably invariant distributions on g(F). The author describes grands and the domains over which we integrate. However, explicitly 1J~ n 1JG,st. Let H be an elliptic unramified endo­ there are still drawbacks to Lebesgue integration, for instance, scopic group of G. For all D E D:fu n 1JH,st, an element of 1J~ dealing with the Fundamental Theorem of Calculus, or with that is a transfer of D is described. The paper also contains "improper" integrals. related results: a "Lusztig function" is equal to its Fourier transform up to a scalar and the scalar is described; the author This book is an introduction to a relatively new theory of the also describes the transfer factor for classical groups. integral (called the "generalized Riemann integral" or the "Henstock-Kurzweil integral") that corrects the defects in the This item will also be of interest to those working in number classical Riemann theory and both simplifies and extends the theory. Lebesgue theory of integration. Although this integral includes Distributed by the AMS in the United States, Canada, and Mexico. that of Lebesgue, its definition is very close to the Riemann Orders from other countries should be sent to the SMF, Maison de la integral that is familiar to students from calculus. One virtue SMF, B.P. 67, 13274 Marseille cedex 09, France, or to lnstitut Henri Poin­ care, 11 rue Pierre et Marie Curie, 75231 Paris cedex OS, France. of the new approach is that no measure theory and virtually no Members of the SMF receive a 30% discount from list. topology is required. Indeed, the book includes a study of measure theory as an application of the integral. Contents: Introduction; Definitions generales; Fonctions de Green; Une base de res.'J[(Dent); Stabilite; Transformees de Part 1 fully develops the theory of the integral of functions Fourier des fonctions de Lusztig; Demonstration de la proposi­ defined on a compact interval. This restriction on the domain tion IV.3 dans le cas symplectique; Demonstration de la is not necessary, but it is the case of most interest and does proposition IV.3 dans le cas orthogonal; Correspondance de not exhibit some of the technical problems that can impede Springer; Distributions stables a support nilpotent; Facteurs de the reader's understanding. Part 2 shows how this theory transfert; Induction endoscopique d'orbites nilpotentes; Trans­ extends to functions defined on the whole real line. The theory fert d'integrales orbitales nilpotentes; Index des notations; of Lebesgue measure from the integral is then developed, and Bibliographie. the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given Asterisque, Number 269 full exposure to the main classical results. January 2001, 455 pages, Softcover, ISBN 2-85629-096-5, The text is suitable for a first-year graduate course, although 2000 Mathematics Subject Classification: 22£35, 11F70, 20G40, much of it can be readily mastered by advanced undergraduate 20G25, Individual member $79, list $88, Order code students. Included are many examples and a very rich collec­ AST/269N tion of exercises. There are partial solutions to approximately one-third of the exercises. A complete solutions manual is available separately. Contents: Integration on compact intervals: Gauges and inte­ grals; Some examples; Basic properties of the integral; The fundamental theorems of calculus; The Saks-Henstock lemma; Measurable functions; Absolute integrability; Convergence theorems; Integrability and mean convergence; Measure, measurability, and multipliers; Modes of convergence; Applica­ tions to calculus; Substitution theorems; Absolute continuity; Integration on infinite intervals: Introduction to Part 2; Infinite intervals; Further re-examination; Measurable sets; Measurable functions; Sequences of functions; Limits superior and inferior; Unbounded sets and sequences; The arctangent lemma; Outer measure; Lebesgue's differentiation theorem; Vector spaces; Semimetric spaces; Riemann-Stieltjes integral; Normed linear

MAY 2001 NOTICES OF TilE AMS 533 New Publications Offered by the AMS

spaces; Some partial solutions; References; Index; Symbol index. '·*'hfiili Recommended text Graduate Studies in Mathematics, Volume 32 April2001, 458 pages, Hardcover, ISBN 0-8218-0845-1, Topics in Nonlinear LC 00-065063, 2000 Mathematics Subject Classification: 26-01; Functional Analysis 26A39, 26A42, 28-01, All AMS members $47, List $59, Order code GSM/32N Louis Nirenberg, New York Topics in University-Courant Institute of Nonlinear· · · Mathematical Sciences, NY Solutions Manual to Functional A Modern Theory of Analysis From reviews for the First Edition: These lecture notes are extremely stim­ Integration ulating. Robert G. Bartle, Eastern -Zentralblatt fiir Mathematik Michigan University, Ypsilanti, [The book] is short, concise, and to the and University of fllinois, point, and the proofs are unusually elegant, always with a geometric flavor and the best available. Urbana -Mathematical Reviews This solutions manual is geared Since its first appearance as a set of lecture notes published by toward instructors for use as a the Courant Institute in 1974, this book served as an introduc­ companion volume to the book, A tion to various subjects in nonlinear functional analysis. The Modem Theory of Integration, (AMS Graduate Studies in Mathe­ current edition is a reprint of these notes, with added biblio­ matics series, Volume 32). graphic references. Contents: Gauges and integrals; Some examples; Basic proper­ Topological and analytic methods are developed for treating ties of the integral; The fundamental theorems of calculus; The nonlinear ordinary and partial differential equations. The first Saks-Henstock lemma; Measurable functions; Absolute integra­ two chapters of the book introduce the notion of topological bility; Convergence theorems; Integrability and mean degree and develop its basic properties. These properties are convergence; Measure, measurability and multipliers; Modes of used in later chapters in the discussion of bifurcation theory convergence; Applications to calculus; Substitution theorems; (the possible branching of solutions as parameters vary), Absolute continuity; Infinite integrals; Measurable sets; Measur­ including the proof of Rabinowitz's global bifurcation theorem. able functions; Sequences of functions. Stability of the branches is also studied. The book concludes Graduate Studies in Mathematics with a presentation of some generalized implicit function theo­ rems of Nash-Moser type with applications to April2001, 72 pages, Softcover, ISBN 0-8218-2821-5, Kolmogorov-Arnold-Moser theory and to conjugacy problems. LC 2001022204, 2000 Mathematics Subject Classification: 26-01; 26A39, 26A42, 28-01, All AMS members $11, List $14, After more than 20 years, this book continues to be an excel­ Order code GSM/32.MN lent graduate level textbook and a useful supplementary course text. Titles in this series are copublished with the Courant Institute of Mathe­ ~ Multi-Interval Linear matical Sciences at New York University. MF-Moms Contents: Topological approach: Finite dimensions; Topolog­ I o 1 ot'" "' '' ~ Ordinary Boundary ical degree in Banach space; Bifurcation theory; Further topological methods; Monotone operators and the min-max Multi-Interval Unear Value Problems and Ordinary Boundary Value theorem; Generalized implicit function theorems; Bibliography. Problems and Complex Complex Symplectic Symplectic Algebra Courant Lecture Notes, Volume 6 W. N. Everitt Algebra L. Markus May 2001, approximately 160 pages, Softcover, ISBN 0-8218- W. N. Everitt, University of 2819-3, 2000 Mathematics Subject Classification: 46-:XX, All Birmingham, England, and AMS members $19, List $24, Order code CLN/6N L. Markus, University of Minnesota, Minneapolis Contents: Introduction: Goals, organization; Some definitions for multi-interval systems; Complex symplectic spaces; Single interval quasi-differential systems; Multi-interval quasi-differ­ ential systems; Boundary symplectic spaces for multi-intei,'Val systems; Finite multi-interval systems; Examples of complete Lagrangians; Bibliography. Memoirs of the American Mathematical Society, Volume 151, Number 715 March 2001, 64 pages, Softcover, ISBN 0-8218-2669-7, LC 2001018141, 2000 Mathematics Subject Classification: 34B10, 51A50; 34L05, Individual member $24, List $40, Institutional member $32, Order code MEM0/ 151/ 715N

534 NOTICES OF TilE AMS VOLUME 48, NUMBER 5 New Publications Offered by the AMS Wandering Solutions Selected Papers on MEiVIOll\S Classical Analysis of Delay Equations Brown with Sine-Like Katsumi Nomizu, Wandering Solutions of University, Providence, RI, Delay Equations with Feedback Editor Sine-Like Feedback Bernhard Lani-Wayda, This volume contains papers that orig­ University of Giessen, Germany inally appeared in Japanese in the journal Sugaku. Ordinarily the papers Contents: Introduction; Symbolic would appear in the AMS translation dynamics for maps; Composition of of that journal, but to expedite publi- 'local' and 'global' maps; Linking equa- cation, the Society has chosen to tions and maps; Explicit examples; publish them as a volume of selected papers. The papers here Appendix (Auxiliary results); References; Figures. are in the general area of mathematical analysis as it pertains Memoirs of the American Mathematical Society, Volume 151, to free probability theory. Number 718 Contents: F. Hiai, Free products of operator algebras and free March 2001, 121 pages, Softcover, ISBN 0-8218-2680-8, probability theory; K. Matsuzaki, Dynamics of Kleinian LC -2001018229, 2000 Mathematics Subject Classification: groups-The Hausdorff dimension of limit sets; S. Nii, Topolog­ 70K50, Individual member $28, List $47, Institutional member ical methods in stability analysis of travelling waves; $38, Order code MEM0/151/718N j.-i. Tanaka, Extension of almost periodic functions and analy­ ticity on flows; H. Umemura, Painleve equations in the past 100 years; M. Yamazaki, The Navier-Stokes equation in various function spaces. General and American Mathematical Society Translations-Series 2 Interdisciplinary June 2001, approximately 144 pages, Hardcover, ISBN 0-8218- 2780-4, LC 2001022373, 2000 Mathematics Subject Classification: 46J10, 34M55, 35K57, 35Q30, 57S30, 46Lxx, First International Individual member $45, List $75, Institutional member $60, Order code TRANS2-SUGAKU5N Congress of Chinese Mathematicians Equations Lo Yang, Chinese Academy of Differential Sciences, Beijing, and S.-T. Yau, Harvard University, The Dirichlet Cambridge, MA, Editors Problem for The International Congress of Mathe­ maticians was an historical event that Operators of The Dirichlet Problem Parabolic was held at the Morningside Center for Parabolic Operators with Singular Drift Mathematics of the Chinese Academy of Sciences (Beijing). It with Singular Drift Terms was the first occasion where Chinese mathematicians from all Steve Hofmann Terms over the world gathered to present their research. John L. Lew1s Steve Hofmann, University of The Morningside Mathematics lectures were given by R. Missouri, Columbia, and Borcherds, J. Coates, R. Graham, and D. Stroock. Other distin­ included J.-P. Bourguignon, J. Jost, M. Taylor, John L. Lewis, University of guished speakers and S. L. Lee. Topics covered in the volume include algebra and Kentucky, Lexington representation theory, algebraic geometry, number theory and automorphic forms, Riemannian geometry and geometric Contents: The Dirichlet problem and parabolic measure; analysis, mathematical physics, topology, complex analysis and Absolute continuity and the LP Dirichlet problem: Part 1; complex geometry, computational mathematics, and combina­ Absolute continuity and the LP Dirichlet problem: Part 2. torics. the American Mathematical Society, Volume 151, Memoirs of Titles in this series are copublished with International Press, Cambridge, Number 719 MA. March 2001, 113 pages, Softcover, ISBN 0-8218-2684-0, LC Contents: Morningside lectures: R. E. Borcherds, Problems in 2001018142, 2000 Mathematics Subject Classification: 42B20, moonshine; D. W. Stroock, The Ornstein-Uhlenbeck process on 35K05, Individual member $28, List $46, Institutional member a Riemannian manifold; Plenary addresses: W. E, Stochastic $37, Order code MEM0/151/719N PDEs in turbulence theory; H. Jiaxing, Recent developments of realization of surfaces in R 3; H. D. Ceniceros and T. Y. Hou, Numerical study of interfacial problems with small surface tension; J. Li, Selected topics in moduli spaces; B. H. Lian and S.-T. Yau, A tour of mirror symmetry; C.-S. Lin, The scalar curvature equation on sn; G. Tian, Symplectic isotopy in four dimension; Z. Xia, Stability in Hamiltonian dynamical systems;

MAY 2001 NOTICES OF THE AMS 535 ··::' New Publications Offered by the AMS

Z. Xin, On the behavior of solutions to the compressible Navier·Stokes equations; H.-T. Yau, Scaling limit of particle Council for African systems, incompressible Navier-Stokes equation and Boltz­ American mann equation; S. Zhang, Geometry of algebraic points; Invited COuncil for African American addresses: C.-H. Chu, Jordan structures in Banach manifolds; Researchering of Hopf alge­ !! bras; M.-C. Chang, On the Euler numbers of threefolds; W.-p. Li Alr.a.... _ fa. Noll Sciences: Volume III and Z. Qjn, Vector bundles, the S-duality conjecture of Vafa SOnyo A. F. st.pMns .e. Alfred G. Noel, University of and Witten, and vertex algebras; S.-L Tan, Cayley-Bacharach 'Q- property and k-very ampleness; K.-P. Lin and S. S.-T. Yau, Massachusetts, Boston, Sharp polynomial upper estimate of number of positive inte­ Earl Barnes, Georgia Institute gral points in tetrahedron and coordinate free characterization of Technology, Atlanta, and of homogeneous polynomials; L. Ji, Spectral theory and geom­ Sonya A. F. Stephens, Florida A & M University, etry of locally symmetric spaces; J.-s. Li, On the first eigenvalue Tallahassee, of Laplacian for locally symmetric manifolds; H.-D. Cao and Editors J. Zhou, DGBV algebras and mirror symmetry; L.-f. Tam, This volume presents research and expository papers Harmonic maps on noncompact manifolds; J.-Y. Wu, Curvature, presented at the third and fifth meetings of the Council for bounded cohomology and path integrals; K. Liu, Vanishing African American Researchers in the Mathematical Sciences theorems for loop spaces; D. H. Phong, Symplectic forms, 2+1 (CAARMS). The CAARMS is a group dedicated to organizing an soliton equations, and supersymmetric gauge theories; annual conference that showcases the current research X.-Y. Zhou, Quotients, invariant version of Cartan's lemma, and primarily, but not exclusively, of African Americans in the the minimum principle; T.-C. Kuo and L. Paunescu, an elemen­ mathematical sciences, including mathematics, operations tary expose on equisingularities; W.-H. Lin, A proof of the research, statistics, and computer science. Held annually since strong Kervaire invariant in dimension 62; X.-S. Lin, Melvin­ 1995, significant numbers of researchers have presented their Morton conjecture: A survey; G. Meng, The Seiberg-Witten current work in hour-long technical presentations, and grad­ equations and the Milnor torsion; S. Gong and X. Zheng, The uate students have presented their work in organized poster Bergman kernel function of some Reinhardt domains; sessions. The events create an ideal forum for mentoring and X. Huang, On some problems in several complex variables and networking where attendees can meet researchers and grad­ CR geometry; C.-Q. Cheng, A KAM theory for resonant tori and uate students interested in the same fields. For volumes based a generalization of Poincare-Birkhoff fixed point theorem; on previous CAARMS proceedings, see African Americans in P. w. Bates, K. Lu, and C. Zeng, Persistence of ck normally Mathematics ll (Volume 2 52 in the AMS series, Contemporary hyperbolic invariant manifolds for infinite dimensional dynam­ Mathematics), and African Americans in Mathematics (Volume ical systems; Y. Y. Li, Fine analysis of blow up and 34 in the AMS series, DIMACS). applications; F. Lin, Varifold type theory for Sobolev mappings; S.-S. Lin, Cellular neural networks: Pattern and waves; Contents: Research and expository papers: E. R. Barnes, A X.-P. Zhu, Long time behavior of curves evolving by curvature; lower bound for the chromatic number of a graph; G.-Q. Chen, Discontinuous solutions to nonlinear evolutionary M. R. Currie and E. H. Goins, The fractional parts of ~; partial differential equations; T.-P. Liu, Well-posedness theory D. E. Davenport, Ultrafilters and Ramsey theory; E. H. Goins, for hyperbolic conservation laws; J. Li and T. Zhang, Two­ Artins' conjecture and elliptic curves; C. Graham, Chaoticity dimensional Riemann problems for conservation laws in gas results for "join the shortest queue"; J. S. Ivy and dynamics; S. L. Lee and Q. Sun, Roundedness of cascade algo­ S. M. Pollock, Maintenance of deteriorating machines with rithms in Besov spaces; R. H. Chan, M.-h. Yip, C.-w. Leung, and probabilistic monitoring and silent failures; B. V. Saunders, M. K. Ng, Circulant preconditioners for ill-conditioned The application of numerical grid generation to problems in Hermitian Toeplitz matrices; R.-H. Wang, What is computa­ computational fluid dynamics; D. Stephens and G. Howell, The tional geometry; G. J. Chang, A survey on circular chromatic elementary residual method; M. Y. Stephens and Z. Liu, The numbers of graphs; L. Ruan, D.-z. Du, X. Hu, X. Jia, and D. Li, response of the upper ocean to surface buoyancy forcing: A Approximations for color-covering problems; Ust of partici­ characteristic solution to wave propagation; S. A. F. Stephens pants. and V. Lakshmikantham, An overview of integro-differential equations and variational Lyapunov method; Papers on philos­ AMS/IP Studies in Advanced Mathematics, Volume 20 ophy of mathematics: J. P. King, The art of mathematics; June 2001, 518 pages, Softcover, ISBN 0-8218-2652-2, W. A. Massey, Mathematics is four dimensional; Tutorials: LC 00-066350, 2000 Mathematics Subject Classification: 05-XX, S. W. Williams, Compact! A tutorial. 08-XX, 11-XX, 14-XX, 22-XX, 35-XX, 37-XX, 80-XX, All AMS Contemporary Mathematics, Volume 275 members $79, Ust $99, Order code AMSIP/20N June 2001, approximately 184 pages, Softcover, ISBN 0-8218- 2141-5, LC 2001022099, 2000 Mathematics Subject Classification: 33F05, 47N40, 68R05, 68R10, 65F25, 11R37, 60K20, 60K25, 68M20, OOA30, Individual member $29, Ust $49, Institutional member $39, Order code CONM/ 275N

536 NOTICES OF TIIE AMS VOLUME 48, NUMBER 5 New Publications Offered by the AMS

riences: The role of non-course experiences in mathematics One Field, Many education doctoral programs; Related issues: C. Thornton, Paths: U. S. Doctoral R. H. Hunting, J. M. Shaughnessy, J. T. Sowder, and K. C. Wolff, Organizing a new doctoral program in mathe­ Programs in matics education; D. B. Aichele, J. Boaler, C. A. Maher, Mathematics D. Rock, and M. Spikell, Reorganizing and revamping doctoral programs-Challenges and results; K. C. Wolff, Recruiting and Education funding doctoral students; C. E. Lamb, The use of distance­ Robert E. Reys, University of learning technology in mathematics education doctoral programs; R. Lesh, J. A. Crider, and E. Gummer, Emerging Missouri, Columbia, and possibilities for collaborating doctoral programs; Reactions and Jeremy Kilpatrick, University reflections: J. M. Bay, Appropriate preparation of doctoral of Georgia, Athens, Editors students: Dilemmas from a small program perspective; A. Flores, Perspectives from a newcomer on doctoral programs This book is the first to focus specifically on doctoral in mathematics education; T. Lingefjard, Why I became a programs in mathematics education. It reflects the proceedings doctoral student in mathematics education in the United of a National Conference on Doctoral Programs in Mathematics States; V. M. Long, Policy-A missing but important element in Education (Lake Ozark, MO) which was sponsored by the preparing doctoral students; G. A. Ragan, My doctoral program National Science Foundation. This conference was proceeded in mathematics education-A graduate student's perspective; by a comprehensive survey of programs conducted over the Ideas for action: J. Hiebert, J. Kilpatrick, and M. M. Lindquist, preceding year. The meeting was designed to generate dialog Improving U. S. doctoral programs in mathematics education; regarding the nature of current doctoral programs in mathe­ References: R. E. Reys and J. Kilpatrick, References; Appen­ matics education, to discuss ways to strengthen such dices: R. E. Reys and J. Kilpatrick, List of participants; programs, and to detail suggestions and guidelines for faculty R. E. Reys and J. Kilpatrick, Conference agenda. engaged in restructuring an existing program or in creating a new one. CBMS Issues in Mathematics Education, Volume 9 This volume outlines the results of the conference organized April2001, 192 pages, Softcover, ISBN 0-8218-2771-5, by the following sections: LC 2001018843, 2000 Mathematics Subject Classification: · Background, which includes papers providing different 97-06, All Individuals $17, List $29, Order code CBMATH/ 9N perspectives of doctoral programs in mathematics education liiiil§@h& in the U.S. and abroad. · Core Components, which highlights elements in common to most doctoral mathematic programs, including course work, Chaotic Elections! A A Mathematician Looks at Voting research, education, and teaching. -~ Mathematician Looks · Related Issues, which addresses the challenges of recruiting, organizing new programs, and restructuring existing at Voting programs. Donald G. Saari, University of · Reactions and Reflections, which contains the thoughts of California, Irvine recent graduates regarding their doctoral programs and observations on the importance of integrating policy issues What does the 2000 U.S. Presidential into doctoral programs. Election have in common with · Ideas for Action, which provides a brief synthesis of the selecting a textbook for a calculus conference and offers suggestions for future action to course in your department? Was Ralph improve future doctoral programs. Nader's influence on the election of This series is published in cooperation with the Mathematical Associa­ George W. Bush greater than the now-famous chads? In Chaotic tion of America. Elections!, Don Saari analyzes these questions, placing them in Contents: Background: E. F. Donoghue, Mathematics education the larger context of voting systems in general. His analysis in the United States: Origins of the field and the development shows that the fundamental problems with the 2000 presiden­ of early graduate programs; R. E. Reys, B. Glasgow, tial election are not with the courts, recounts, or defective G. A. Ragan, and K. W. Simms, Doctoral programs in mathe­ ballots, but are caused by the very way Americans vote for matics education in the U.S.: A status report; F. Fennell, president. D. Briars, T. Crites, S. Gay, and H. Tunis, Reflections on the This expository book shows how mathematics can help to match between jobs and doctoral programs in mathematics identify and characterize a disturbingly large number of para­ education; A. J. Bishop, International perspectives on doctoral doxical situations that result from the choice of a voting studies in mathematics education; Core components: J. T. Fey, procedure. Moreover, rather than being able to dismiss them Doctoral programs in mathematics education: Features, as anomalies, the likelihood of a dubious election result is options, and challenges; F. K. Lester, Jr. and T. P. Carpenter, surprisingly large. These consequences indicate that election The research preparation of doctoral students in mathematics outcomes-whether for president, the site of the next education; J. A. Dossey and G. Lappan, The mathematical Olympics, the chair of a university department, or a prize education of mathematics educators in doctoral programs in winner-can differ from what the voters really wanted. They mathematics education; N. C. Presmeg and S. Wagner, Prepara­ show that by using an inadequate voting procedure, we can, tion in mathematics education: Is there a basic core for inadvertently, choose badly. To add to the difficulties, it turns everyone?; D. V. Lambdin and J. W. Wilson, The teaching out that the mathematical structures of voting admit several preparation of mathematics educators in doctoral programs in strategic opportunities, which are described. mathematics education; L. V. Stiff, Discussions on different Finally, mathematics also helps identify positive results: By forms of doctoral dissertations; G. Blume, Beyond course expe- using mathematical symmetries, we can identify what the

MAY 2001 NOTICES OF THE AMS 537 New Publications Offered by the AMS phrase "what the voters really want" might mean and obtain a unique voting method that satisfies these conditions. Geometry and Topology Saari's book should be required reading for anyone who wants to understand not only what happened in the presidential elec­ tion of 2000, but also how we can avoid similar problems from Singularities­ appearing anytime any group is making a choice using a voting Sapporo 1998 procedure. Reading this book requires little more than high school mathematics and an interest in how the apparently Eiichi Bannai, Kyushu simple situation of voting can lead to surprising paradoxes. University, japan, Contents: A mess of an election; Voter preferences, or the jean-Paul Brasselet, Institut de procedure?; Chaotic election outcomes; How to be strategic; Mathematique de Luminy, What do the voters want?; Other procedures; other assump­ Marseille, France, and tions; Bibliography; Index. Tatsuo Suwa, Hokkaido May 2001, 159 pages, Softcover, ISBN 0-8218-2847-9, 2000 University, Sapporo, japan, Mathematics Subject Classification: 91B12, 91B14, OOA05, All AMS members $18, List $23, Order code ELECTN Editors A publication of the Mathematical Society of Japan. This volume presents the proceedings of a meeting held in Proceedings of the Sapporo Uapan) at Hokkaido University on singularities and St. Petersburg arrangements of hyperplanes. Participants from the interna­ tional community presented their research on singularities of Mathematical Society curves, characteristic classes of singular varieties, D-modules, Volume VII motives, resolution of singularities, Milnor fibration, Enriques diagrams, Hilbert schemes, and log-canonical singularities. N. N. Uraltseva, Saint Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, Petersburg State University, St. and distributed worldwide, except in Japan, by the AMS. Petersburg, Russia, Editor Contents: P. Aluffi, Weighted Chern-Mather classes and Milnor The articles in this collection present classes of hypersurfaces; V. Blanloeil, Cobordism of non-spher­ new results in combinatorics, algebra, ical knots; J.-P. Brasselet, From Chern classes to Milnor classes algebraic geometry, dynamical systems, analysis, and proba­ - A history of characteristic classes for singular varieties; J.· P. Brasselet and S. Yokura, Remarks on bivariant constructible bility. Of particular interest is the survey article by A. N. Kirillov devoted to combinatorics of Young diagrams and functions; J. Brianr;on, P. Maisonobe, and M. Merle, related problems of representation theory. Also included are Constructibilite de l'ideal de Bernstein; A. Campillo and articles devoted to the eightieth birthday of renowned Russian J. Olivares, Assigned base conditions and geometry of folia­ tions on the projective plane; G. Gonzalez·Sprinberg, mathematician, V. A. Rokhlin, "Remembrances of V. A. Rokhlin", by I. R. Shafarevich, and "An Unfinished Project of Generalized Enriques diagrams and characteristic cones; V.A. Rokhlin", by V. N. Sudakov. The results, ideas, and S. Ishii, The quotients of log-canonical singularities by finite methods given in the book will be of interest to a broad range groups; L D. Trang, Geometry of complex surface singulari­ of specialists. ties; D. Lehmann, A Chern-Weil theory for Milnor classes; F. Loeser, The Milnor fiber as a virtual motive; I. Nakai, Contents: A.M. Vershik and Yu. V. Yakubovich, Continuous Elementary topology of stratified mappings; M. Oka, Geometry lattices of partitions and lattices of continuous partitions; of cuspidal sextics and their dual curves; T. Suwa, Character­ A. N. Kirillov, Combinatorics of Young tableaux and configura­ istic classes of coherent sheaves on singular varieties; tions; B. Kunyavskii and B. Z. Moroz, On integral models of H. Tokunaga, Local types of singularities of plane curves and affine toric varieties; Ya. Yu. Nikitin and E. V. Ponikarov, the topology of their complements; Program; List of partici­ Rough asymptotics of probabilities of Chernoff type large devi­ pants. ations for von Mises functionals and U-statistics; A. P. Petukhov, Biorthogonal wavelet bases with rational Advanced Studies in Pure Mathematics, Volume 29 masks; A. L Pirozerski and M.A. Semenov-Tian-Shansky, January 2001, 322 pages, Hardcover, ISBN 4-314-10143-1, Q -pseudodifference universal Drinfeld-Sokolov reduction; 2000 Mathematics Subject Classification: 32Sxx; 14Cxx, 14Hxx, A. L. Chistov, Effective construction of local parameters of 14Jxx, 32Cxx, 57Rxx, 58Kxx, Individual member $52, List $86, irreducible components of an algebraic variety; I. R. Shafare­ Institutional member $69, Order code ASPM/29N vich, Remembrances of V. A. Rokhlin; V. N. Sudakov, An unfinished project of V. A. Rokhlin. American Mathematical Society Translations-Series 2, Volume 203 May 2001, 253 pages, Hardcover, ISBN 0-8218-2790-1, 2000 Mathematics Subject Classification: 01A70, 05£10, 06B35, 14Jxx, 14M25, 37Kxx, 42C40, 51-03, 60Fl0, Individual member $65, List $109, Institutional member $87, Order code TRANS2/203N

538 NOTICES OF TilE AMS VOLUME 48, NUMBER 5 New Publications Offered by the AMS

., "'I ~ topology, both by providing the algebraic tools that a A Course in Metric geometric topologist needs and by concentrating on those · A Cou1se m Geometry areas of algebraic topology that are geometrically motivated. Metnc Geometry Dmitri Burago, Pennsylvania Prerequisites for using this book are very modest and include basic set-theoretic topology, the definition of CW-complexes, State University, University some knowledge of the fundamental group/covering space Park, and Yuri Burago and theory, and the construction of singular homology. Most of Sergei Ivanov, Steklov Institute this material is briefly reviewed at the beginning of the book. of Mathematics, St. Petersburg, The topics discussed by the author include typical material for Russia first- and second-year graduate courses. The core of the expo­ sition consists of chapters on homotopy groups and on "Metric geometry" is an approach to spectral sequences. There is also material that would interest geometry based on the notion of geometric topologists (homology with local coefficients) and length on a topological space. This approach experienced a students interested more in algebraic topology (obstruction very fast development in the last few decades and penetrated theory, spectra, and generalized homology). Also included is into many other mathematical disciplines, such as group the material that prepares the student for more advanced theory, dynamical systems, and partial differential equations. topics of algebraic geometry (algebraic K-theory and the s­ The objective of this graduate textbook is twofold: to give a cobordism theorem). detailed exposition of basic notions and techniques used in the A unique feature of the book is the inclusion, at the end of theory of length spaces, and, more generally, to offer an each chapter, of several projects that require students to elementary introduction into a broad variety of geometrical present proofs of substantial theorems and to write notes topics related to the notion of distance, including Riemannian accompanying their explanations. Working on these projects and Carnot-Caratheodory metrics, the hyperbolic plane, allows students to grapple with the "big picture", teaches them distance-volume inequalities, asymptotic geometry (large scale, how to give mathematical lectures, and prepares them for coarse), Gromov hyperbolic spaces, convergence of metric participating in research seminars. spaces, and Alexandrov spaces (non-positively and non-nega­ tively curved spaces). The authors tend to work with The book is designed as a textbook for graduate students "easy-to-touch" mathematical objects using "easy-to-visualize" studying algebraic and geometric topology and homotopy methods. theory. Expositions are clear and special cases are presented over complex general statements. The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new Contents: Chain complexes, homology, and cohomology; concepts and methods are introduced and illustrated using Homological algebra; Products; Fiber bundles; Homology with simplest cases and avoiding technicalities. The book contains local coefficients; Fibrations, cofibrations and homotopy many exercises, which form a vital part of exposition. groups; Obstruction theory and Eilenberg-MacLane spaces; Bordism, spectra, and generalized homology; Spectral Contents: Metric Spaces; Length Spaces; Constructions; Spaces sequences; Further applications of spectral sequences; Simple­ of Bounded Curvature; Smooth Length Structures; Curvature of homotopy theory; Bibliography; Index. Riemannian Metrics; Space of Metric Spaces; Large-scale Geom­ etry; Spaces of Curvature Bounded Above; Spaces of Curvature Graduate Studies in Mathematics Bounded Below; Bibliography; Index. July 2001, 367 pages, Hardcover, ISBN 0-8218-2160-1, 2000 Graduate Studies in Mathematics Mathematics Subject Classification: 55-01, 57-01, All AMS members $44, List $55, Order code GSM-DAVISN July 2001,417 pages, Hardcover, ISBN 0-8218-2129-6, LC 2001022062, 2000 Mathematics Subject Classification: 51Kxx, All AMS members $35, List $44, Order code GSM-BURAGON *i''b'i,'* Recomme nded Text Recomme nde d Te xt Differential Lecture Notes in Geometry, Lie Algebraic Topology Groups, and James F. Davis and Paul Kirk, Symmetric Spaces -F. Davis Indiana University, ~.... ICirlc Bloomington Sigurdur Helgason, Massachusetts Institute of The amount of algebraic topology a Technology, Cambridge graduate student specializing in topology must learn can be intimi­ From reviews for the First Edition: dating. Moreover, by their second year A great book .. . a necessary item in of graduate studies, students must any mathematical library. make the transition from under­ - S. S. Chern, University of California standing simple proofs line-by-line to understanding the. overall structure of proofs of difficult theorems. Written with unmatched lucidity, systematically, carefully, beau­ tifully. To help students make this transition, the material in this -S. Bochner, Princeton University book is presented in an increasingly sophisticated manner. It is intended to bridge the gap between algebraic and geometric

MAY 2001 NOTICES OF THE AMS 539 New Publications Offered by the AMS

Helgason's monograph is a beautifully done piece of work and symmetric spaces; Solutions to exercises; Some details; Bibliog­ should be extremely useful for several years to come, both in raphy; list of notational conventions; Symbols frequently used; teaching and in research. Index; Reviews for the first edition. -D. Spencer, Princeton University Graduate Studies in Mathematics A brilliant book: rigorous, tightly organized, and covering a vast July 2001, 641 pages, Hardcover, ISBN 0-8218-2848-7, amount of good mathematics. LC 2001022205, 2000 Mathematics Subject Classification: -Barrett O'Neill, University of California 22E15, 22E46,22E60,22F30, 32M15,43A85,43A90, 53BOS, Renders a great service in permitting the non-specialist, with a 53B20, 53C35, All AMS members $55, list $69, Order code minimum knowledge of differential geometry and Lie groups, GSM-HELGASONN an initiation to the theory of symmetrical spaces.

-H. Cartan, Secretariat Mathematique, Paris ~ The mathematical community has long been in need of a book MEty\Olf\S Stable Homotopy on symmetric spaces. S. Helgason has admirably satisfied this \" \lrl ',, " over the Steenrod need with his book, Differential Geometry and Symmetric Spaces. It is a remarkably well-written book ... a masterpiece of Stable Homotopy Algebra concise, lucid mathematical exposition .. . it might be used as a over the H. textbook for "how to write mathematics". Steenrod Algebra John Palmieri, University of

-Louis Auslander John H. Palmieri Washington, Seattle [The author] will earn the gratitude of many generations of ® Contents: Preliminaries; Stable homo­ mathematicians for this skillful, tasteful, and highly efficient topy over a Hopf algebra; Basic presentation. It will surely become a classic. properties of the Steenrod algebra; -G. D. Mostow, Yale University Chromatic structure; Computing Ext The study of homogeneous spaces provides excellent insights with elements inverted; Quillen strati­ into both differential geometry and lie groups. In geometry, fication and nilpotence; Periodicity and other applications of for instance, general theorems and properties will also hold for the nilpotence theorems; Appendix A. An underlying model homogeneous spaces, and will usually be easier to understand category; Appendix B. Steenrod operations and nilpotence in and to prove in this setting. For lie groups, a significant Extt* (k, k); Bibliography; Index. amount of analysis either begins with or reduces to analysis on Memoirs of the American Mathematical Society, Volume 151, homogeneous spaces, frequently on symmetric spaces. For Number 716 many years and for many mathematicians, Sigurdur Helgason's March 2001, 172 pages, Softcover, ISBN 0-8218-2668-9, classic Differential Geometry, Lie Groups, and Symmetric LC 2001018228, 2000 Mathematics Subject Classification: Spaces has been-and continues to be-the standard source for SSS10, SSU15, 18G35, SSU35, SST15, SSP42, SSQ10, SSQ45, this material. · 18G15, 16W30, 18E30, 20]99, Individual member $32, list Helgason begins with a concise, self-contained introduction to $53, Institutional member $42, Order code MEM0/ 151/716N differential geometry. He then introduces lie groups and lie algebras, including important results on their structure. This Recommended Text sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text Geometry concludes with the classification of symmetric spaces by V. V. Prasolov, Independent means of the Killing-Cartan classification of simple lie alge­ bras over C and Cartan's classification of simple lie algebras University of Moscow, Russia, over R. and V. M. Tikhomirov, The excellent exposition is supplemented by extensive collec­ Moscow State University, tions of useful exercises at the end of each chapter. All the Russia problems have either solutions or substantial hints, found at This book provides a systematic intro­ the back of the book. duction to various geometries, For this latest edition, Helgason has made corrections and including Euclidean, affine, projective, added helpful notes and useful references. The sequels to the spherical, and hyperbolic geometries. present book are published in the AMS's Mathematical Surveys Also included is a chapter on infinite­ and Monographs Series: Groups and Geometric Analysis, dimensional generalizations of Euclidean and affine Volume 83, and Geometric Analysis on Symmetric Spaces, geometries. A uniform approach to different geometries, based Volume 39. on Klein's Erlangen Program is suggested, and similarities of Sigurdur Helgason was awarded the Steele Prize for Differential various phenomena in all geometries are traced. An important Geometry, Lie Groups, and Symmetric Spaces and Groups and notion of duality of geometric objects is highlighted Geometric Analysis. throughout the book. The authors also include a detailed presentation of the theory of conics and quadrics, including This item will also be of interest to those working in algebra the theory of conics for non-Euclidean geometries. The book and algebraic geometry. contains many beautiful geometric facts and has plenty of Contents: Elementary differential geometry; lie groups and lie problems, most of them with solutions, which nicely supple­ algebras; Structure of semisimple lie algebras; Symmetric ment the main text. spaces; Decomposition of symmetric spaces; Symmetric spaces With more than 150 figures illustrating the arguments, the of the noncompact type; Symmetric spaces of the compact book can be recommended as a textbook for undergraduate type; Hermitian symmetric spaces; Structure of semisimple lie and graduate-level courses in geometry. groups; The classification of simple lie algebras and of

540 NOTICES OF THE AMS VOLUME 48, NUMBER 5 New Publications Offered by the AMS

Contents: The Euclidean world; The affine world; The projec­ tive world; Conics and quadrics; The world of non-Euclidean Mathematical Physics geometries; The infinite-dimensional world; Addendum; Solu­ tions, hints, and answers; Bibliography; Author index; Subject index. Nonlinear Dynamics Translations of Mathematical Monographs and Renormalization July 2001, approximately 264 pages, Hardcover, ISBN 0-8218- 2038-9, LC 2001022063, 2000 Mathematics Subject Group Classification: 51-01, 51M04, 51M09; 46-01, Individual Nonlinear Dynamics I. M. Sigal and C. Sulem, member $59, List $99, Institutional member $79, Order code and Renonnallzatlon University of Toronto, ON, MMONO-TIKHOMIRN Group I.M.Sigal Canada, Editors C.Sulem. Edflcn This book contains the proceedings Logic and Foundations @ --- from the workshop, Nonlinear Dynamics and Renormalization Group, held at the Centre de recherches math­ ematiques (CRM) in Montreal (Canada), as part of the year-long Stable Groups program devoted to mathematical physics. In the book, active Bruno Poizat, Universite researchers in the fields of nonlinear partial differential equa­ Claude Bernard, Villeurbanne, tions and renormalization group contribute recent results on France topics such as Ginzburg-Landau equations and blow-up of solutions of the nonlinear Schroedinger equations, quantum From a review of the French edition: resonances, and renormalization group analysis in constructive quantum field theory. This volume offers the latest research in This is a beautiful book in which the rapidly developing fields of nonlinear equations and renor­ almost everything known about stable malization group. groups appears. -Zentralblatt fiir Mathematik Contents: S. Alama and L Bronsard, Analysis of some macro­ scopic models of high-Tc superconductivity; N. D. Alikakos This is the English translation of the and G. Fusco, The effect of distribution in space in Ostwald It book originally published in 1987. is a faithful reproduction ripening; D. Auckly and L Kapitanski, Mathematical problems of the original, supplemented by a new Foreword and brought in the control of underactuated systems; 0. I. Bogoyavlenskij, up to date by a short postscript. The book gives an introduc­ Axially and helically symmetric global plasma equilibria; tion by a specialist in contemporary mathematical logic to the 0. Costin, J. L Lebowitz, and A Rokhlenko, On the complete model-theoretic study of groups, i.e., into what can be said ionization of a periodically perturbed quantum system; about groups, and for that matter, about all the traditional J. Dimock, The sine-Gordon model at {3 = 4rr; G. M. Graf, algebraic objects. Ground states of supersymmetric matrix models; The author introduces the groups of finite Morley rank (those S. J. Gustafson, Some mathematical problems in the Ginzburg­ satisfying the most restrictive assumptions from the point of Landau theory of superconductivity; M. K.-H. Kiessling, view of logic), and highlights their resemblance to algebraic Renormalization in radiation reaction: New developments in groups, of which they are the prototypes. (All the necessary classical electron theory; C.-K. Lin, Singular limit of the modi­ prerequisites from algebraic geometry are included in the fied nonlinear Schrodinger equation; M. Merkli, Dynamics of book.) Then, whenever possible, generalizations of properties quantum resonances; H. Nawa, Nelson diffusions and blow-up of groups of finite Morley type to broader classes of supersta­ phenomena in solutions of the nonlinear Schrodinger equation bles and stable groups are described. with critical power; D. E. Pelinovsky and C. Sulem, Embedded The exposition in the first four chapters can be understood by solitons of the DSII equation; G. Perelman, On the blow up mathematicians who have some knowledge of logic (model phenomenon for the critical nonlinear Schrodinger equation in theory). The last three chapters are intended for specialists in 1D; S. Serfaty, Vorticity for the Ginzburg-Landau model of mathematical logic. superconductors in a magnetic field; A Soffer, Dissipation through dispersion; B. Vasilijevic, Quantum tunneling at posi­ Contents: A couple of words about groups; Introduction; tive temperature. Chain; Structure; Fields; Geometry; Generics; Rank; Weight; Bibliography; Index; Postscript: Thirteen years later. CRM Proceedings & Lecture Notes, Volume 27 Mathematical Surveys and Monographs, Volume 87 April2001, 192 pages, Softcover, ISBN 0-8218-2802-9, LC 2001022066, 2000 Mathematics Subject Classification: 35105; June 2001, approximately 132 pages, Hardcover, ISBN 0-8218- 35J05, 35Q55, 35Q99, 70K99, 82C28, 81Tl5, 37K40, Individual 2685-9, LC 2001022098, 2000 Mathematics Subject member $35, List $59, Institutional member $47, Order code Classification: 03C45, 03C60, 14Lxx, 20Gxx, Individual CRMP/27N member $29, List $49, Institutional member $39, Order code SURV/87N

MAY 2001 NOTICES OF THE AMS 541 Previously Announced Publications

Hilbert spaces of Sobolev type and diffeomorphism groups. Previously Announced Other useful tools are the algebraic geometry of Riemann surfaces and infinite-dimensional determinants. Also discussed Publications are the boundary regularity questions. The main result is a presentation of the string partition function as an integral over a moduli space of Riemann surfaces. Some new physical Study Inde pe nde nt concepts, such as D-branes, are also discussed. Laguerre Calculus and Its Applications This volume offers a mathematically rigorous treatment of on the Heisenberg Group some aspects of string theory, employs a global geometry approach, systematically treats strings with boundary, and of Maryland, College Park, Carlos Berenstein, University carefully explains all mathematical concepts and tools. Der-Chen Chang, Georgetown University, Washington, Titles in this series are copublished with International Press, Cambridge, DC, and Jingzhi Tie, University of Georgia, Athens MA. For nearly two centuries, the relation between analytic func­ AMS/IP Studies in Advanced Mathematics tions of one complex variable, their boundary values, harmonic March 2001, approximately 112 pages, Hardcover, ISBN 0-8218- functions, and the theory of Fourier series has been one of the 2644-1, LC 00-067543, 2000 Mathematics Subject Classification: central topics of study in mathematics. The topic stands on its 81T30; 83E30, 81T50, 58D30, 32G15, 53A10, All AMS own, yet also provides very useful mathematical applications. members $31, list $39, Order code AMSIP-JOST1RT105 This text provides a self-contained introduction to the corre­ sponding questions in several complex variables: namely, ed Text analysis on the Heisenberg group and the study of the solu­ Recommend tions of the boundary Cauchy-Riemann equations. In studying Analysis this material, readers are exposed to analysis in non-commuta­ tive compact and lie groups, specifically the rotation group Second Edition and the Heisenberg groups-both fundamental in the theory of Elliott H. lieb, Princeton University, N], and group representations and physics. Michael Loss, Georgia Institute of Technology, Atlanta Introduced in a concrete setting are the main ideas of the Praise for the previous edition ... Calder6n-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of I find the selection of the material covered in the book very harmonic and complex analysis, in particular, the Morera and attractive and I recommend the book to anybody who wants to Pompeiu problems for the Heisenberg group, which relates to learn about classical as well as modern mathematical analysis. questions in optics, tomography, and engineering. -European Mathematical Sodety Newsletter The book was borne of graduate courses and seminars held at The essentials of modern analysis ... are presented in a rigorous the University of Maryland (College Park), the University of and pedagogical way . . . readers ... are guided to a level where Toronto (ON), Georgetown University (Washington, DC), and they can read the current literature with understanding ... the University of Georgia (Athens). Readers should have an treatment of the subject is as direct as possible. advanced undergraduate understanding of Fourier analysis and -Zentralblatt (iir Mathematik complex analysis in one variable. Lieb and Loss offer a practical presentation of real and func­ Titles in this series are copublished with International Press, Cambridge, tional analysis at the beginning graduate level ... could be used MA. as a two-semester introduction to graduate analysis ... not all of The authors introduce the subject AMS/IP Studies in Advanced Mathematics, Volume 22 the topics covered are typical. with a thorough presentation ... [an] informative exposition. May 2001, approximately 328 pages, Hardcover, ISBN 0-8218- -CHOICE 2761-8, LC 00-054826, 2000 Mathematics Subject Classification: and expanded, this new Second Edition 22E30,33C20,42C10,43A80,47G30,32VVOS, 32A10,30E99, Significantly revised from beginning students to All AMS members $51, list $64, Order code AMSIP/22RT105 provides readers at all levels- practicing analysts-with the basic concepts and standard tools necessary to solve problems of analysis, and how to Bosonic Strings: A Mathematical apply these concepts to research in a variety of areas. Treatment Authors Elliott lieb and Michael Loss take you quickly from to methods that work successfully in mathematics Max Planck Institute for Mathematics in the basic topics Jiirgen Jost, and its applications. VVhile omitting many usual typical text­ Sciences, Leipzig, Germany book topics, Analysis includes all necessary definitions, proofs, Presented in this book is a mathematical treatment of Bosonic explanations, examples, and exercises to bring the reader to an string theory from the point of view of global geometry. As advanced level of understanding with a minimum of fuss, and, motivation, the author presents the theory of point particles at the same time, doing so in a rigorous and pedagogical way. and Feynman path integrals. He considers the theory of strings Many topics that are useful and important, but usually left to as a quantization of the classical Plateau problem for minimal advanced monographs, are presented in Analysis, and these surfaces. The conformal variance of the relevant functional, give the beginner a sense that the subject is alive and growing. the Polyakov action or (in mathematical terminology) the This new Second Edition incorporates numerous changes since Dirichlet integral, leads to an anomaly in the process of quanti­ the publication of the original1997 edition, and includes: zation. The mathematical concepts needed to resolve this anomaly via the Faddeev-Popov method are introduced, specifi­ cally the geometry of the Teichmuilller and moduli spaces of Riemann surfaces and the corresponding function spaces, i.e.,

542 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Previously Announced Publications Features: Simplicial and Operad Methods in . a new chapter on eigenvalues that covers the min-max prin­ ciple, semi-classical approximation, coherent states, Algebraic Topology lieb-Thirring inequalities, and more V. A. Smirnov, Moscow State Pedagogical Institute, . extensive additions to chapters covering Sobolev Inequalities, Russia including the Nash and Log Sobolev inequalities . new material on Measure and Integration In recent years, for solving problems of algebraic topology and, . many new exercises in particular, difficult problems of homotopy theory, algebraic · and much more ... structures more complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and more The Second Edition continues its no-nonsense approach to the often. A convenient language for describing various structures topic that has made it one of the best selling books on the arising naturally on topological spaces and on their coho­ subject. It is an authoritative, straight-forward volume that mology and homotopy groups is the language of operads and readers-from the graduate student, to the professional math­ algebras over an operad. This language was proposed by J. P. ematician, to the physicist or engineer using analytical May in the 1970s to describe the structures on various loop methods-will find useful both as a reference and as a guide to spaces. real problem solving. This book presents a detailed study of the concept of an About the authors: Elliott lieb is Professor of Mathematics operad in the categories of topological spaces and of chain and Physics at Princeton University and is a member of the US, complexes. The notions of an algebra and a coalgebra over an Austrian, and Danish Academies of Science. He is also the operad are introduced, and their properties are investigated. recipient of several prizes including the 1988 AMS/SIAM Birk­ The algebraic structure of the singular chain complex of a hoff prize. Michael Loss is Professor of Mathematics at the topological space is explained, and it is shown how the Georgia Institute of Technology. problem of homotopy classification of topological spaces can Graduate Studies in Mathematics, Volume 14 be solved using this structure. For algebras and coalgebras over operads, standard constructions are defined, particularly ISBN 0-8218- May 2001, approximately 326 pages, Hardcover, the bar and cobar constructions. Operad methods are applied Subject Classification: 42-01, 2783-9, 2000 Mathematics 28-01, to computing the homology of iterated loop spaces, investi­ 46-01,49-01;26010, 26D15,31B05,31B15,46E35,46F05, gating the algebraic structure of generalized cohomology $39, Order 46F10, 49-XX, 81Q05, All AMS members $31, list theories, describing cohomology of groups and algebras, code GSM/14.RRT105 computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems. Supple me nta ry Read1ng Translations of Mathematical Monographs, Volume 198 Geometry of Characteristic Classes March 2001, 235 pages, Hardcover, ISBN 0-8218-2170-9, LC 00- Shigeyuki Morita, Tokyo Institute of Technology, Japan 068957, 2000 Mathematics Subject Classification: 55P15, 55S20, Individual member $53, list $89, Institutional member $71, Characteristic classes are central to the modern study of the Order code MMON0/198RT105 topology and geometry of manifolds. They were first intro­ duced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined (via the Chern-Weil theory) using connections on vector bundles, thus revealing their geometric side. In the late 1960s new theories arose that described still finer structures. Examples of the so-called secondary characteristic classes came from Chern-Simons invariants, Gelfand-Fuks cohomology, and the characteristic classes of flat bundles. The new techniques are particularly useful for the study of fiber bundles whose structure groups are not finite dimensional. The theory of characteristic classes of surface bundles is perhaps the most developed. Here the special geometry of surfaces allows one to connect this theory to the theory of moduli space of Riemann surfaces, i.e., Teichmilller theory. In this book Morita presents an introduction to the modern theo­ ries of characteristic classes. This item will also be of interest to those working in algebra and algebraic geometry. Translations of Mathematical Monographs (Iwanami Series in Modem Mathematics), Volume 199 May 2001, 180 pages, Softcover, ISBN 0-8218-2 139-3, LC 00- 054312, 2000 Mathematics Subject Classification: 54C40, 14E20; 46E25, 20C20, All AMS members $24, list $30, Order code MMON0/199RT105

MAY 2001 NOTICES OF TilE AMS 543 Graduate Textbooks Recommended by the AMS Included here are affordable and comprehensive books for graduate-level courses. These volumes are also finely suited for independent study or supplementary reading. For more AMS textbooks, visit the AMS Bookstore at www.ams.org/bookstore.

A Course in Operator Theory Partial Differential Equations John B. Conway, University of Tennessee, Knoxville, TN P. R. Garabedian, New York University-Courant Institute of Graduate Studies In Mathematics, Volume 21; 2000; ISBN 0-8218-2065-6; Mathematical Sciences, NY 372 pages; Hardcover; All AMS members $39, List $49, Order Code From a review for the original edition: GSM/21CT105 [T]he author has made use of an interesting combination of classical C*-Aigebras by Example and modern analysis in his proofs ... Because of the author's Kenneth R. Davidson, University of Waterloo, ON, Canada emphasis on constructive methods for solving problems which are of physical interest, his book will likely be as welcome to the engineer One can assign parts of Davidson's book to good students learning and the physicist as to the mathematician ... The author and the subject and expect good results ... the details persist even when publisher are to be complimented on the general appearance of the the going gets tough ... this is the only book I know in which one can book. go through the BDF classification of essentially normal operators and -Mathematical Reviews follow, point by point, to the end. AMS Chelsea Publishing; 1964; ISBN 0·8218-1377-3; 672 pages; Hardcover; -Bulletin of the American Mathematical Society All AMS members $41, List $45, Order Code CHEU325.HCT105 The writing is exceptionally clear and easy to follow ... an Algebraic Topology: An Intuitive Approach outstanding book that should be on every operator algebraist's book­ Hajirne Sato, Nagoya University, Japan shelf. -Mathematical Reviews This is an uncommon book with an interesting idea behind it, which Customers in India, Sri Lanka, Bangladesh, and Pakistan, please contact Hindustan is given in its title: to give an intuitive approach to algebraic topology. Book Agency (India), 17 U BJawahar Nagar, Delhi 1.10 007, India Instead of stating theorems in full generality or proving them rigor­ Fields Institute Monographs, Volume 6; 1996; ISBN 0-8218-0599-1 ; 309 ously with all technical details (or proving them at all), the author pages; Hardcover; All AMS members $47, List $59, Order Code FIM/6CT1 05 rather tries to make the reader familiar with "the idea• of the central Enveloping Algebras notions of algebraic topology. -Zentralblatt fiir Mathematik Jacques Dixmier, Paris, France The present book is an interesting, perhaps radical, survey of current Self-contained, written with precision and elegance .. . an excellent algebraic topology. For a student wh.o finds topology to be a forest of textbook for the graduate student, a very good background for the details, this text offers a chance to get an overview of the whole field. professional algebraist not very familiar with the subject and a very -Mathematical Reviews useful source of references for the expert. Translations of Mathematical Monographs (lwanami Series in Modem -Zentralblatt fiir Mathematik Mathematics), Volume 183; 1999; ISBN Q-8218·1 046-4; 118 pages; Softcover; Graduate Studies In Mathematics, Volume 11 ; 1996; ISBN 0·8218·0560·6; All AMS members $16, List $20, Order Code MMON0/183CT1 05 379 pages; Hardcover; All AMS members $47, List $59, Order Code GSM/11 CT1 05 Introduction to Riemann Surfaces Second Edition An Introduction to Superprocesses George Springer Alison M. Etheridge, University of Oxford, England Written with unusual clearness. As in the Introduction, which outlines University Lecture Series, Volume 20; 2000; ISBN 0·8218·2706·5; 187 the whole book, similar [outlines] appear in each chapter ... a pages; Softcover; All AMS members $26, List $33, Order Code modern treatment in a self-contained manner with a minimum ULECT/20CT1 05 assumption of knowledge. He is most successful in this magnificent Dirac Operators in Riemannian Geometry project .. . It is highly recommended. Thomas Friedrich, Humboldt-Universitat, Berlin, Germany -American Mathematical Monthly From a review of the German edition: The book is written specifically with graduate (and advanced under­ graduate) students in mind .. . Concepts and theorems are The text contains full, detailed and elegant proofs throughout, all illuminated by examples and excellent figures, proofs are clarified by calculations are carefully performed, and considerations are well heuristic remarks, and the inventiveness of even the good student is formulated and well motivated. This style is typical of the author. It is challenged by a well chosen problem collection. The style, while very a pleasure to read the book; any beginning graduate student should readable, never becomes "insultingly simple" and even the specialist have access to it. can derive pleasure from reviewing basic material in a well-organized -Mathematical Reviews form. -Mathematical Reviews Graduate Studies In Mathematics, Volume 25; 2000; ISBN 0-8218-2055-9; 195 pages; Hardcover; All AMS members $27, List $34, Order Code AMS Chelsea Publishing; 1981 ; ISBN 0-8284-0313-9; 309 pages; Hardcover; GSM/25CT1 05 All AMS members $23, List $25, Order Code CHEU313CT105

To order, call: 1-800-321-4AMS (4267), in the U.S. and Canada, or 1·401-455-4000; fax: 1401-455- 4046; email: cust·[email protected]. Visit the AMS Bookstore and order online at www.ams.org/bookstore. Or write to: American Mathematical Society, P. 0. Box 6248, Providence, Rl 02940·6248. Prices subject ci)AMS to change without notice. ~ AMERICAN MATHEMATICAL SOCIETY

For more publications in your subject area visit the AMS Bookstore: www.ams.org/ bookstore.

544 NOTICES OF THE AMS VOLUME 48, NUMBER 5 QUANTITATIVE FINANCE

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Department Chair KENTUCKY 312 Wallace Building NEW YORK Department of Mathematics EASTERN KENTUCKY UNIVERSITY and Statistics ST. JOHN'S UNIVERSITY Department of Mathematics and 521 Lancaster Avenue Department of Mathematics & Statistics Eastern Kentucky University Computer Science Tenure-Track Faculty Position Richmond, KY 40475-3102 Three Assistant/Associate Professors The Department of Mathematics and Sta­ Review of applications will begin immedi­ St. John's University, with campuses in the ately and will continue until the position tistics at Eastern Kentucky University in­ Hillcrest, Jamaica, and Staten Island sec­ is filled. Eastern Kentucky University is vites applications for a tenure-track posi­ tions of New York City, is an independent an Equal Opportunity/Affirmative Action tion. The primary teaching responsibilities Catholic coeducational institution in the Employer. are teaching courses in developmental Vincentian tradition. mathematics, mathematics education, or Applications are invited for a tenure­ general education mathematics. Experi­ track position at the assistant or asso­ ence with large lecture classes preferred. NEW JERSEY ciate professor rank in mathematics on ·" Although the field is open, additional the Queens campus for September 2001. ability to contribute to the teaching and INSTITUTE FOR ADVANCED STUDY Applicants should possess a Ph.D. in math­ student research in the department's grad­ The School of Mathematics has a limited ematics. There are also two tenure-track positions at the assistant or associate uate and undergraduate programs is a number of memberships, some with finan­ professor rank in computer science on plus. Duties include a 12-hour teaching cial support for research in mathematics at load per semester, service activities, and the Institute during the 2002-03 academic the Staten Island campus for September continued scholarly activity. The applicant year. Candidates must have given evidence 2001. Applicants should possess a Ph.D. in must provide a record of excellence in of ability in research comparable at least computer science or mathematics with a teaching and on-going research. Educa­ with that expected for the Ph.D. degree. computer science background. A commit­ tional qualifications: a doctoral degree in The special program for the year will focus ment to teaching and research is essential. mathematics, statistics, or related field. A on stochastic PDE and models of turbu­ Candidates should submit a letter of ap­ complete application package includes: a lence, and Weinan E will be in residence. plication, a resume, three letters of recom­ letter of application addressing qualifica­ For a brief description of the program and mendation, undergraduate and graduate tions and developmental or service course information about application materials transcripts to: experience; a detailed curriculum vitae, a and deadline, please consult "Activities" Search Committee statement of teaching philosophy and re­ and "How To Apply" on our home page at Department of Mathematics search interest; unofficial copies of all aca­ http://www.math.ias .edu/. & Computer Science demic transcripts; and the names, phone St. John's University numbers, and e-mail addresses of four pro­ 8000 Utopia Parkway fessional references. For further informa­ Jamaica, NY 11439 tion about the position, department, uni­ e-mail: trainacCQstjohns. edu versity, or community, please visit http: I I Review of the applications will begin im­ www .math. eku. edu/. Send application ma­ mediately, and will continue until the terials to: positions are filled.

Suggested uses for classified advertising are positions available, books issue-June 26, 2001; October 2001 issue-July 26, 2001; November 2001 or lecture notes for sale, books being sought, exchange or rental of issue-August 27, 2001; December 2001 issue-September 27, 2001. houses, and typing services. U.S. laws prohibit discrimination in employment on the basis of color, The 2001 rate is $100 per inch or fraction thereof on a single column age, sex, race, religion, or national origin. "Positions Available" advertise­ (one-inch minimum), calculated from top of headline. Any fractional text ments from institutions outside the U.S. cannot be published unless they of lf2 inch or more will be charged at the next inch rate. No discounts for are accompanied by a statement that the institution does not discriminate multiple ads or the same ad in consecutive issues. For an additional $10 on these grounds whether or not it is subject to U.S. laws. Details and charge, announcements can be placed anonymously. Correspondence will specific wording may be found on page 13 73 (vol. 44). be forwarded. Situations wanted advertisements from involuntarily unemployed Advertisements in the "Positions Available" classified section will be set mathematicians are accepted under certain conditions for free publi­ with a minimum one-line headline, consisting of the institution name cation. Call toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or above body copy, unless additional headline copy is specified by the 401-455-4084 worldwide for further information. advertiser. Headlines will be centered in boldface at no extra charge. Ads Submission: Promotions Department, AMS, P.O. Box 6248, Providence, will appear in the language in which they are submitted. Rhode Island 02940; or via fax: 401-331-3842; or send e-mail to classadsl!l There are no member discounts for classified ads. Dictation over the ams. org. AMS location for express delivery packages is 201 Charles telephone will not be accepted for classified advertising. Street, Providence, Rhode Island 02904. Advertisers will be billed upon Upcoming deadlines for classified advertising are as follows: June/ July publication. issue-April 25, 2001; August 2001 issue-May 24, 2001; September 2001

546 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Classified Advertisements

St. John's University is an Equal Oppor­ to whole libraries sought. Contact: Collier tunity Employer; women and minorities PUERTO RICO Brown or Ryan Thomas @ Powell's Techni­ are encouraged to apply. UNIVERSITY OF PUERTO cal Bks., Portland, OR. Call 800-225-6911, fax 503-228-0505, or e-mail ryan. thomas® RICO AT MAYAG0EZ powells. com. Department of Mathematics The mathematics department is seeking a VIRGINIA mathematician for a tenure-track faculty position at the assistant professor level, to begin in August 2001. Applicants are expected to have a Ph.D. in mathematics, NATIONAL SCIENCE FOUNDATION a demonstrated potential for outstanding Division of Mathematical Sciences research, and strong teaching skills. Can­ Employment Opportunities didates should be able to teach at both the undergraduate and graduate levels. POSillONS. Several of the technical staff Specialities considered include algebra or of the Division of Mathematical Sciences fields closely related to algebra. Prefer­ of the National Science Foundation serve ence for Spanish speakers. To apply, send 1-2-year "Visiting Scientist" or "Intergov­ a current curriculum vitae, three letters ernmental Personnel Act" appointments of recommendation, and graduate school as program directors while on leave from transcripts to: universities, colleges, industry, or national Personnel Committee laboratories. Since the timing of these posi­ Department of Mathematics tions is staggered, the division continually University of Puerto Rico at Mayagiiez seeks talented applicants. In 2001 the divi­ P. 0. Box 9018 sion will be seeking to make appointments Mayagiiez, PR 00681-9018 in all areas. "Permanent" program director Tel: 787-265-3848 appointments will also be considered. Fax: 265-5454 The positions involve responsibility for the planning, coordination, and manage­ ment of support programs for research SWITZERLAND (including multidisciplinary projects), in­ frastructure, and human resource develop­ ETH Zurich ment for the mathematical sciences. Nor­ Assistant Professor in mally, this support is provided through Applied Mathematics merit-reviewed grants and cooperative agreements that are awarded to academic Duties of this open position include, in ad­ institutions and nonprofit nonacademic dition to research, an active participation research institutions. in the teaching of courses for students of mathematics, natural sciences, and engi­ QUAilFICATIONS. Applicants should neering. have a Ph.D. or equivalent training in a field of the mathematical sciences, a Candidates should have the doctorate broad knowledge of one of the rele­ or equivalent and have demonstrated the ability to carry out independent research. vant disciplinary areas of the Division of Willingness to teach at all university lev­ Mathematical Sciences, some administra­ els and to collaborate with colleagues is tive experience, a knowledge of the general expected. scientific community, skill in written com­ munications and preparation of technical This assistant professorship has been reports, an ability to communicate orally, established to promote the careers of and several years of successful indepen­ younger scientists. Initial appointment is for three years, with the possibility of dent research normally expected of the renewal for an additional three years. academic rank of associate professor or Please submit your application together higher. Skills in multidisciplinary research are highly desirable. with a curriculum vitae and a list of pub­ lications to the president of ETH Zurich, NSF is an Equal Opportunity Employer Prof. Dr. 0. Kuebler, ETH Zentrum, CH- committed to employing a highly qualified 8092 Zurich, no later than May 31, 2001 (or staff that reflects the diversity of our by e-mail also to the assistant to the presi­ nation. dent, Dr. Th. Eichenberger, eichenberger!D Applicants should send a letter of inter- sl. ethz . ch). The ETHZ specifically en­ est and vita to: courages female candidates to apply, with a view towards increasing the proportion Dr. Bernard R. McDonald, of female professors. Executive Officer Division of Mathematical Sciences National Science Foundation 4201 Wilson Boulevard, Suite 1025 PUBLICATIONS WANTED Arlington, VA 22230 MATHEMATICS BOOKS PURCHASED Tel: 703-292-4851 Fax: 703-292-9032 Pure & appl. adv. & research level, any e-mail: bmcdonal\Dnsf. gov age, usable cond. Reprints OK. One box

MAY 2001 NOTICES OF THE AMS 547 OUTSTANDING MATHEMATICS TITLES FROM CAMBRIDGE

Measure~Preserving An Introduction to Homeomorphisms Fluid Dynamics congratulates Steve Alpern and V. S. Prasad G.K Batchelor · A self-contained introduction to typical First published in 1967, this classic work Richard Stanley, properties of volume-preserving homeo­ is still one of the foremost texts on fluid Winner of Steele Prize for morphisms, examples of which include dynamics. Its careful presentation of the. , . transitivity, chaos and ergodicity. the underlying theories of fluids is still Mathematicil Expositio!l The authors focus first on volume, timely and applicable, even in these days preserving homeomorphisms of the of almost limitless computer power. Enumerative unit n -dimensional cube. They also Cambridge Mathematical Library prove fixed point theorems (Conley­ 2000 635 pp. Combinatorics Zehnder-Franks). 0-521-66396-2 Paperback $29.95 Volume I Cambridge Tracts in Mathematics 139 2000 224 pp. Mathematical Tools for Richard P. Stanley Q-521-58287-3 Hardback $49.95 Probabilistic Risk Analysis ''A standard as an introductory Tim Bedford and Roger M. Cooke graduate text in combinatorics." The Beginner's Discusses the fundamental notion -Bulletin ofthe AMS Guide to Mathematlca®, of uncertainty, its relationship with Version 4 probability, and the limits to quanti­ "Very readable and a mine of jerry Glynn and Theodore Gray fication. The authors focus on the con­ information." ''A user-friendly tutorial that will allow ceptual and mathematical foundations - journal ofthe LMS first-time users to quickly acquire the underlying the quantification, inter­ "Will engage from start to finish the skills to productively use this extensive pretation and management of risk. attention of any mathematician who computer algebra system." 2001 416 pp. Q-521-77320.2 Hardback $54.95 will open it at page one." -Choice -Gian-Carlo Rota The book teaches new Mathematica Galois Theories ''An excellent and valuable book." users some of the important basics: Fronds Borceux and George janelidze -Mathematical Reviews using the typesetting features, pro­ The authors first formalize the cate­ gramming palettes, defining functions, Cambridge Studies in Advanced Mathematics 49 gorical context in which a general and notebooks, 1997 337 pp. creating graphs Galois theorem holds, and then apply 0-521-55309-1 Hardback $85.00 and applying useful problem-solving Galois theory to commutative rings, 0-521-66351-2 Paperback $29.95 techniques. central extensions of groups, the 2000 442 pp. Volume 2 0-521-77769-0 Paperback $29.95 topological theory of covering maps, and a Galois theorem for toposes. Richard P. Stanley Quantum Computation and Cambridge Studies in Advanced Mathematics 72 2001 360 pp. "Volume 2 not only lives up to the Quantum Information Q-521-80309-8 Hardback $74.95 high standards set by Volume 1, but Michael A. Nielsen and Isaac L Chuang surpasses them... Stanley's book is a The authors ask the question: What are Derivation and Integration valuable contribution to enumerative the ultimate physical limits to computa­ Washek F. Pfeffer combinatorics. Beginners will find it tion and communication? They detail to an invariant an accessible introduction to the This book, devoted such remarkable effects as fast quantum process of recovering subject, and experts will still find multidimensional algorithms, quantum teleportation, its derivative, considers much to learn from it." a function from quantum cryptography and quantum additive functions defined on the family -Mathematical Reviews error correction. of all bounded BV sets that are contin­ Cambridge Studies in Advanced Mathematics 62 Cambridge Series on Information and the uous with respect to a suitable topology. 1999 594 pp. Natural Sciences 2 The main applications are related to the 0-521-56069-1 Hardback $74.95 2000 700 pp. 0-521-78987-7 Paperback $34.95 0-521-63503-9 Paperback $47.95 Gauss-Green and Stokes theorems. Cambridge Tracts in Mathematics 140 2001 288 pp. For more ltiformatiory, us.cambridge.org/mathemat.ics Q-521-79268-1 Hardback $64.95

~/~ ~"1 c A M B I) I J) ( " E 4 0 w e,t 20th Stt-P<'l, N f' w Yorl<, N y I 00 I I -4 2 1 I ''- L '- T C II II f., , 800 8 7 2 7423 Avadable "'bookstores · ·- "'1 a to - "c - - or fro"' ?_;J ~ l ' :'\I V I _!{ S I I Y P !{ L S S A mEx/ M,hle>t Carrl!VIS A acce pte d. P rtces ' u bject to c h ange Abstract and Applied Analysis Editor-in-Chief: A. C. Kartsatos (University of South Florida)

Journal of • Aims & Scope: AAA is devoted exclusively to the publication of original research APPlied Matbemati~ papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, - ··-··_, ordinary and partial differential equations, optimization theory, and control theory. Subscription Information (ISSN 1085-3375, 2001, volume 6, 8 issues): $295 (print and electronic), $195 (electronic only), $195 (personal print and electronic), $95 (personal electronic only). Journal's web site: http://aaa.hindawi.com. International Journal of Mathematics and Mathematical Sciences Founding Managing Editor: L. Debnath (University of Central Florida)

Aims & Scope: IJMMS is devoted to the publication of original research papers, research notes, and research expository and survey articles with emphasis on unsolved problems ~~=- and open questions in mathematics and mathematical sciences. All areas listed on the "'"i$A.__,.\IIIIIIOM .. Wo.t...,_,"E._ cover of Mathematical Reviews are included within the scope of the journal. -- Subscription Information (ISSN 0161-1712, 2001, volumes 25-28, 48 issues): $495 (print and electronic), $395 (electronic only), $395 (personal print and electronic), $95 (personal electronic only). Journal's web site: http://ijmms.hindawi.com. Journal of Applied Mathematics Editors-in-Chief: C. Brezinski (Universite des Sciences et Technologies de Lil/e), L. Debnath (University of Central Florida), H . Nijmeijer (Eindhoven University of Technology) Aims & Scope: JAM is devoted to the publication of original research papers and review articles in all areas of applied, computational, and industrial mathematics. Subscription Information (ISSN 1110-757X, 2001, volume 1, irregular): $295 (print and electronic), $195 (electronic only), $195 (personal print and electronic), $95 (personal electronic only). Journal's web site: http://jam.hindawi.com.

Bundle Subscription: All three journals for $995 (print and electronic), $695 (electronic only), $695 (personal print and electronic), $195 (personal electronic only). Online availability: All journals' back volumes are available online. Subscription to cur­ rent year enables online access to all back volumes during the subscription period. Hindawi Alert: Register with " Hindawi Alert" at http://www.hindawi.com/alert.html to receive email notifications with tables of contents as new issues appear online. Backlog Information: Current detailed backlog information of all journals is available at http://www.h i ndawi .com/backlog.html. Electronic Submission: Submit manuscripts to IJMMS at [email protected] and to JAM at submit®jam.hindawi.com.

Books online: Symmetry in Nonlinear Mathematical Physics. Proceedings of EWM 8th and 9th general meetings. EWM Workshop on Moduli Spaces in Mathematics and Physics. Visit http://www.hindawi.com for more mathematics publications.

Hindawi Pu blishing Corporation, P.O. Box 5873, Akron, OH 44334..-QS Fax: 1 888 5224561 (U SA toll-free), 31 20 5248282 (Amsterdam, Hoi t=IIN DAWI URL: http://www.hindawi.com, Email: hindawi®hindaw.i.com NEW from de Gruyter

Groups and Computation III Proceedings of the International Conference at The Ohio State University, June 15-19, 1999 Edited by William M. Kantor and Akos Seress 2001. viii + 368 pages. ISBN 3-11-016721-2. Cloth $128.95 (Ohio State University Mathematical Research Institute Publications 8) This volume contains contributions by the participants of the conference "Groups and Computation'', which took place at The Ohio State University in Columbus, Ohio, in June 1999. This conference was the successor of two workshops on "Groups and Computation'' held at DIMACS in 1991 and 1995. There are papers on permutation group algorithms, finitely presented groups, polycyclic groups, and parallel computation, providing a representative sample of the breadth of Computational Group Theory. On the other hand, more than one third of the papers deal with computations in matrix groups, giving an in-depth treatment of the currently most active area of the field. The points of view of the papers range from explicit computations to group-theoretic algorithms to group-theoretic theorems needed for algoritlun development. Number Theory Proceedings of the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, May 31 -June 4, 1999 Edited by Matti Jutila and Tauno Metsiinkylii 2001. ix + 327 pages. ISBN 3-11-016481-7. Cloth $128.95 These Proceedings contain 22 refereed research and survey articles based on lectures given at the Turku Symposium on Number Theory in Memory ofKustaa Inkeri, held in Turku, Finland, from May 31 to June 4, 1999. The subject of the symposium was number theory in a broad sense with an emphasis on recent advances and modern methods. The topics covered in this volume include various questions in elementary number theory, new developments in classical Diophantine problems - in particular of the Fermat and Catalan type, the ABC­ conjecture, aritlunetic algebraic geometry, elliptic curves, Diophantine approximations, Abelian fields, exponential sums, sieve methods, box splines, the Riemann zeta-function and other Dirichlet series, and the spectral theory of automorphic functions with its aritlunetical applications. DE GRUYTER EXPOSITIONS IN MATHEMATICS Volume 32 Boris N. Apanosov . Conformal Geometry of Discrete Groups and Manifolds 2000. xiii + 523 pages. ISBN 3-11-014404-2. Cloth $128.95 This book presents the first systematic account of conformal geometry of n-manifolds, as well as its Riemannian counterparts. A unifying theme is their discrete holonomy groups. In particular, hyperbolic manifolds, in dimension 3 and higher, are addressed. The treatment covers also relevant topology, algebra (including combinatorial group theory and varieties of group representations), aritlunetic issues, and dynamics. Progress in these areas has been very fast over the last two decades, especially due to the Thurston geometrization program, leading to the solution of many difficult problems. A strong effort has been made to point out new connections and perspectives in the field and to illustrate various aspects of the theory. An intuitive approach which emphasizes the ideas behind the constructions is complemented by a large number of examples and figures which both use and support the reader's geometric imagination. The text will be of value to graduate students and researchers in topology, geometry, group representations and theoretical physics. Volume 34: Marek J arnicki and Peter Pflug Extension of Holomorphic Functions 2000. x + 490 pages. ISBN 3-11-015363-7. Cloth $98.95 This monograph is devoted to a systematic exposition of the theory of extension of holomorphic functions, e. g. characterizations of envelopes of holomorphy with respect to various families of holomorphic functions. Therefore, there is emphasis on a detailed presentation of holomorphic convexity and pseudoconvexity of Riemann domains over c". Our interest in this area of complex analysis started directly after our studies when both of us were interested in continuation of holomorphic functions. During the years, we got the impression that there is a need to have a source where the main results could be found. We hope this book can serve_as such a source. The choice of topics obviously reflects our personal preferences. For example, we will solve the Levi problem via the a -problem and functions of restricted growth. Most of the results have not yet been published in book form. The text will be of interest both to students and experts. Meetings & Conferences oftheAMS

' '• IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information "' withlinkstotheabstractforeachtalkcanbefoundontheAMSWebpages.Seehttp://www.ams.org/meetings/.Programs and abstracts will continue to be displayed on the AMS Web pages in the Meetings and Conferences section until about three weeks after the meeting is over. Final progranis for Sectional Meetings will be archived on the AMS Web pages in an electronic issue of the Notices as noted below for each meeting.

Geometric and Computational Group Theory, Eric M. Las Vegas, Nevada Freden, Southern Utah University, and Eric L. Swenson, Brigham Young University. · University of Nevada Graphs and Digraphs, Michael Jacobson, University of April21-22, 2001 Louisville, and K. Brooks Reid, California State University, San Marcos. Meeting #965 History ofMathematics, Shawnee L. McMorran, California Western Section State University, San Bernadino, Adrian Rice, Randolph­ College. Associate secretary: Bernard Russo Macon College, and James Tattersall, Providence Announcement issue of Notices: February 2001 Number Theory with a Geometric Flavor, Arthur Baragar, Program first available one-MATH: March 8, 2001 University of Nevada, Las Vegas. Program issue of electronic Notices: May 2001 PDEs (rom Fluid Mechanics: Applied Analysis and Numerical Issue of Abstracts: Volume 22, Issue 2 Methods, L. Steven Hou, York University and Iowa State University, and Xiaoming Wang, Iowa State University. Deadlines Physical Knotting and Unknotting, Jorge Alberto Calvo, For organizers: Expired North Dakota State University, Kenneth C. Millen, University For consideration of contributed papers in Special Ses­ of California Santa Barbara, and Eric J. Rawdon, Chatham sions: Expired College. For abstracts: Expired Set Theory, Douglas Burke and Derrick Dubose, University of Nevada, Las Vegas. Invited Addresses Topology ofLinks, Jeff Johannes, University of Nevada, Las Panagiota Daskalopoulos, University of California Irvine, Vegas, and Swatee Naik, University of Nevada, Reno. Regularity of interfaces in degenerate diffusion. Waves in Heterogeneous Media, Randall J. LeVeque, Randall J. LeVeque, University of Washington, Numerical University of Washington, and Knut Solna, University of methods for wave propagation in heterogeneous media: California Irvine. Solving hyperbolic PDEs with discontinuous coefficients. Vera Serganova, University of California Berkeley, Lie su­ peralgebras: Representation theory and geometry. Hoboken, New Jersey Lynne H. Walling, University of Colorado, How much do Stevens Institute of Technology the Heeke operators actually tell us about modular forms? April28-29, 2001 Special Sessions Meeting #966 and Applications of Nonlinear PDEs, David G. Analysis Eastern Section Hossein Tehrani, University of Costa, Zhonghai Ding, and Associate secretary: Lesley M. Sibner Nevada, Las Vegas. Announcement issue of Notices: February 2001 Finite Element Analysis and Applications, Jichun li, Michael Program first available one-MATH: March 15, 2001 Marcozzi, George Miel, and Darrell W. Pepper, University Program issue of electronic Notices: May 2001 of Nevada. Issue of Abstracts: Volume 22, Issue 2

MAY 2001 NOTICES OF THE AMS 551 Meetings & Conferences

Deadlines Wavelets, Multiscale Analysis, and Applications, Ivan For organizers: Expired Selesnick, Polytechnic University, and Gerald Schuller, For consideration of contributed papers in Special Ses­ Bell Laboratories. sions: Expired For abstracts: Expired Morelia, Mexico Invited Addresses Hotel Fiesta Inn Alexander Barvinok, University of Michigan, Ann Arbor, Complexity and geometry of counting. May 23-26,2001 Robert Calderbank, AT&T Laboratories Research, Combi­ Meeting #967 natorics, quantum computers, and cellularphones. Fifth International ]oint Meeting of the AMS and the Sociedad Alexei Miasnikov, City College, New Title to be an­ York, Matematica Mexicana (SMM). nounced. Associate secretary: John L. Bryant Frank Sottile, University of Massachusetts at Amherst, Announcement issue of Notices: April2001 A Gromov-Witten invariant in the real world. Program first available one-MATH: Not applicable Program issue of electronic Notices: Not applicable Special Sessions Issue of Abstracts: Not applicable Analytic Number Theory, Milos A. Dostal, Stevens Institute of Technology, and Werner G. Nowak, Vienna, Austria. Deadlines For organizers: Expired Computational Algebraic Geometry and Its Applications, For consideration of contributed papers in Special Ses­ Serkan Hasten, San Francisco State University, and Frank sions: To be announced Sottile, University of Massachusetts at Amherst. For abstracts: Expired Computational Group Theory, Robert Gilman, Stevens Institute of Technology, and Alexei Myasnikov, Vladimir Invited Addresses Shpilrain, and Sean Cleary, City College, New York. Victor Perez Abreu, CIMAT, Gaussian measures and some Deformation Quantization and Its Applications, Siddhartha extensions. Sahi, Rutgers University, and Martin J. Andler, University Eric M. Friedlander, Northwestern University, Between of Versailles. algebraic and topological K -theory. Graph Theory (Dedicated to Frank Harary on His BOth Helmut H. W. Hofer, Courant Institute, New York University, Birthday), Michael L. Gargano and Louis V. Quintas, Periodic orbits, holomorphic curves, and algebraic invariants. Pace University, and Charles Suffel, Stevens Institute of Ernesto A. Lacomba, Singularities Technology. UAM-1, and chaos in classical and celestial mechanics. History ofMathematics, Patricia R. Allaire, Queensborough Claude R. LeBrun, SUNY at Stony Brook, On the curvature Community College, CUNY, and Robert E. Bradley, Adelphi of 4-manifolds. University. Antonmaria Minzoni, IIMAS-UNAM, Interaction of two Matchings in Graphs and Hypergraphs, Alexander Barvinok, dimensional coherent structures and radiation in integrable University of Michigan, and Alex Samorodnitsky, Institute and nonintegrable problems. for Advanced Study. Quantum Error Correction and Related Aspects of Coding Special Sessions Theory, Harriet S. Pollatsek, Mount Holyoke College, and Algebraic Geometry, Pedro Luis del Angel, CIMAT, Javier M. Beth Ruskai, University of Massachusetts at Lowell. Elizondo, IMATE-UNAM, Charles A. Weibel, Rutgers Ricci Curvature and Related Topics, George I. Kamberov, University, James D. Lewis, University of Alberta, and Marc Stevens Institute of Technology, Christina Sormani, Lehman N. Levine, Northeastern University. College, CUNY, and Megan M. Kerr, Wellesley College. Algebraic Topology and K-Theory, Miguel Xicotencatl, Singular and Degenerate Nonlinear Elliptic Boundary Value CINVESTAV, and Ernesto Lupercio, University of Wiscon­ Problems, Joe McKenna, Changfeng Gui, and Yung Sze sin-Madison. Choi, University of Connecticut. Biomathematics, Jorge Velasco, UAM-1, and Zhilan Feng, Stability of Nonlinear Dispersive Waves, Yi Li, Stevens Purdue University. Institute of Technology, and Keith S. Promislow, Simon Combinatorics and Graph Theory, Ernesto Vallejo, IMATE­ Fraser University. UNAM, and Igor Pak, Massachusetts Institute of Technology. Surface Geometry and Shape Perception, Gary R. Jensen, Complex Analysis, Enrique Ramirez de Arellano, CINVES­ Washington University, and George I. Kamberov, Stevens TAV, and John E. Fornaess, University of Michigan, Ann Institute of Technology. Arbor.

552 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Meetings & Conferences

Differential Geometry, Adolfo Scinchez Valenzuela, CIMAT, Invited Addresses Raw Quiroga, CINVESTAV, and Charles P. Boyer, Univer­ Sun-Yung Alice Chang, Princeton University, Title to be an­ sity of New Mexico. nounced. Dynamical Systems Emphasizing Geometrical Aspects and Jean-Pierre Demailly, Universite de Grenoble, Title to be Dynamics, Gelasio Salazar and Jestls Urias, UASLP, Symbolic announced. and Nicolai T. A. Haydn, University of Southern California. Persi Diaconis, Stanford University, Title to be announced. Dynamical Systems with Emphasis on Holomorphic Dynamics, Hamiltonian Systems and Variational Systems, Robert Gardner, University of Massachusetts at Amherst, Joaquin Delgado-Fernandez, UAM-1, Hector Sanchez­ Title to be announced. Morgado, IMATE-UNAM, and David Fried, Boston University. Claude Le Bris, Universite de Paris IX-Dauphine, Title to be Functional and Harmonic Analysis, Lourdes Palacios, UAM- announced. 1, Salvador Perez-Esteva, IMATE-UNAM, Josefina Alvarez, Yves Meyer, Ecole Normale Superieure de Cachan, Title to University of New Mexico, and Thomas V. Tonev, University be announced. of Montana-Missoula. Michele Vergne, Ecole Polytechnique, Title to be announced. Mathematical Physics, Peter Zhevandrov, University Michoacan, and Eric A. Carlen, Georgia Institute of Special Sessions Technology. Additive Number Theory, Melvyn B. Nathanson, Herbert H. Nonlinear Analysis, Jorge Ize, IIMAS-UNAM, Monica Oapp, Lehman College (CUNY), and Jean-Marc Deshouillers, IMATE-UNAM, and Paul H. Rabinowitz, University of Universite de Bordeaux II. Wisconsin-Madison. Commutative Algebra and Its Interactions with Algebraic Number Theory, Eugenio Balanzario and Florian Luca, Geometry, Marc F. Chardin, Universite Pierre et Marie IMATE-Morelia, and Harold G. Diamond, University of Curie-Paris VI, and Claudia Polini, University of Oregon. Illinois, Urbana-Champaign. Differential Geometric Methods in Mathematical Physics, Numerical Methods in Differential Equations, Pablo Barrera, Johannes Huebschmann, Universite Lille I, Yvette Kosmann­ UNAM, Francisco Solis, CIMAT, and Benito Chen, Univer­ Schwarzbach, Ecole Polytechnique, and Richard W. sity of Wyoming. Montgomery, University of California Santa Cruz. Quasigroups, Loops, Nonassociative Algebras and Their Dynamics ofNonlinear Waves, Christopher K. R. T. Jones, Applications, J. D. Phyllips, and Lev V. Sabinin, University Brown University, and Jean-Michel Roquejoffre, Universite of Quintana Roo. Toulouse III. Ring Theory, Francisco Raggi, IMATE-UNAM, and Sergio R. Fractal Geometry, Number Theory, and Dynamical Systems, Lopez-Permouth, Ohio University. Michel Lapidus, University of California Riverside, Michel Stochastical A nalysis and Probability, Mogens Bladt, liMAS· Mendes-France, Universite de Bordeaux, and M~chiel van UNAM, Maria E. Caballero, IMATE-UNAM, and Thomas G. Frankenhuysen, University of California Riverside. Kurtz, University of Wisconsin-Madison. Gauge Theory, jean-Claude Sikorav, Ecole Normale Theory of Representation of Algebras and Its Applications, Superieure de Lyon, and Ronald Fintushel, Michigan State Raymundo Bautista, IMATE-UNAM, and AlexMartsinkovsky, University. Northeastern University. Geometric Group Theory, Gilbert Levitt, Universite Toulouse III, and Karen Vogtmann, Cornell University. Geometric Methods in Low Dimensional Topology, Hamish Lyon, France Short, and Daryl Cooper, University of California Santa July 17-20, 2001 Barbara. Geometric Structures in Dynamics, M. Lyubich, SUNY at Meeting #968 Stony Brook, Etienne Ghys, Ecole Normale Superieure de First ]oint International Meeting between the AMS and the Lyon, and Xavier Buff, Universite Toulouse III. Socilfte Mathematique de France. Geometry and Representation Theory ofA lgebraic Groups, Associate secretary: Lesley M. Sibner Michel Brion, Universite de Grenoble I, and Andrei Announcement issue of Notices: April2001 Zelevinsky, Northeastern University. Program first available one-MATH: Not applicable History of Mathematics, Thomas W. Archibald, Acadia Program issue of electronic Notices: Not applicable University, Christian Gilain, Universite Pierre et Marie Issue of Abstracts: Not applicable Curie-Paris VI, and James J. Tattersall, Providence College. Deadlines Logic and Interaction: From the Rules ofLo gic and the Logic For organizers: Expired of Rules, Jean-Yves Girard, Universite de Marseille, and For consideration of contributed papers in Special Ses­ Philip Scott, University of Ottawa. sions: To be announced Mathematical Fluid Dynamics, Yann Brenier, Universite For abstracts: To be announced Pierre et M arie Curie-Paris VI, Susan J. Friedlander,

MAY 2001 NOTICES OF THE AMS 553 Meetings & Conferences

University of illinois at Chicago, and Emmanuel Grenier, Cryptography and Computational and Algorithmic Number Ecole Normale Superieure de Lyon. Theory(Code: AMS SS E1), Eric Bach, University of Wisconsin­ Mathematical Methods in Financial Modelling, Marco Madison, and Jonathan Sorenson, Butler University. Avellaneda, Courant Institute, New York University, and Fractals (Code: AMS SS P1), Gerald Edgar, Ohio State Rama Cont, Ecole Polytechnique. University. Model Theory, Gregory L. Cherlin, Rutgers University, and Group Theory (Code: AMS SS Fl), Koichii-o Harada, Surinder Frank Wagner, Universite Claude Bernard Lyon I. Seghal, and Ronald Solomon, Ohio State University. Partial Differential Equations and Geometry, Fabrice Bethuel, Multivariate Generating Functions and Automatic Universite Pierre et Marie Curie-Paris VI, and Paul C. Yang. Computation (Code: AMS SS H1), Robin Pemantle, Ohio Probability, Gerard Benarous, Ecole Normale Superieure, State University. and George C. Papanicolaou, Stanford University. Number Theory(Code: AMS SS Jl}, David Goss, Ohio State University. Proof Theory and the Foundations of Mathematics (Code: Columbus, Ohio AMS SS K1), Timothy Carlson, Ohio State University. Ohio State University Quantum Topology(Code: AMS SS Ll), Thomas Kerler, Ohio State University. September 21-23,2001 Rings and Modules (Code: AMS SS M1), S. K. Jain, Ohio Meeting #969 University, and Tariq Rizvi, Ohio State University. Central Section Spectral Theory of Schrodinger Operators (Code: AMS SS Associate secretary: Susan J. Friedlander N1), Boris Mityagin, Ohio State University, and Sergei Announcement issue of Notices:. June 2001 Novikov, University of Maryland. Program first available one-MATH: August 9, 2001 Stochastic Modeling in Financial Mathematics (Code: AMS Program issue of electronic Notices:. October 2001 SS D1), Ronnie Sircar, Princeton University. Issue of Abstracts:. Volume 22, Issue 3 Deadlines For organizers: Expired Chattanooga, For consideration of contributed papers in Special Ses- sions: June 5, 2001 Tennessee For abstracts: July 13, 2001 University of Tennessee, Chattanooga Invited Addresses October 5-6,2001 Alex Eskin, University of Chicago, Title to be announced. Dennis Gaitsgory, University of Chicago, Title to be an­ Meeting #970 nounced. Southeastern Section Associate secretary: John L. Bryant Yakov B. Pesin, Pennsylvania State University, Title to be Notices:. August 2001 announced. Announcement issue of Program first available one-MATH: August 23, 2001 Title Thaleia Zariphopoulou, University of Texas at Austin, Program issue of electronic Notices:. November 2001 to be announced. Issue of Abstracts:. Volume 22, Issue 3 Special Sessions Deadlines L 2 Methods in Algebraic and Geometric Topology (Code: AMS For organizers: Expired SS G1), Dan Burghelea and Michael Davis, Ohio State For consideration of contributed papers in Special Ses- University. sions: June 19, 2001 Algebraic Cycles, Algebraic Geometry (Code: AMS SS A1), For abstracts: August 14, 2001 Roy Joshua, Ohio State University. Sessions Coding Theory and Designs (Code: AMS SS B1), Tom Special Dowling, Ohio State University, and Dijen Ray-Chaudhuri. Asymptotic Behavior of Solutions of Differential and Commutative Algebra (Code: AMS SS Cl), Evan Houston, Difference Equations (Code: AMS SS B1), John R. Graef, University of North Carolina, Charlotte, and Alan Loper, University of Tennessee at Chattanooga, and Chuanxi Qjan, Ohio State University. Mississippi State University. Complex Approximation Theory via Potential Theory (Code: Commutative Ring Theory (Code: AMS SS A1), David F. AMS SS R1), V. V. Andrievskii and Richard S. Varga, Kent Anderson and David E. Dobbs, University of Tennessee at State University. Knoxville.

554 NOTICES OF THE AMS VOLUME 48, NUMBER 5 Meetings & Conferences

Mathematical and Numerical Aspects of Wave Propagation Special Sessions and Yongzhi Xu, (Code: AMS SS Fl), Boris P. Belinskiy Abelian Varieties (Code: AMS SS K1), Alexander Polishchuk University of Tennessee at Chattanooga. and Emma Previato, Boston University. New Directions in Combinatorics and Graph Theory(Code: Algebraic and Topological Combinatorics (Code: AMS SS AMS SS Cl), Teresa Haynes and Debra J. Knisley, East D1), Eva Maria Feichtner, ETH, Ziirich, Switzerland, and Tennessee State University. Dmitry N. Kozlov, KTH, Stockholm, Sweden. Numerical Analysis and Approximation Theory (Code: AMS Commutative Algebra (Code: AMS SS Cl), Susan R. Loepp, SSG 1), Tian-Xiao He, illinois Wesleyan University, and Don Williams College, and Graham J. Leuschke, University of Hong, Eastern Tennessee State University. Kansas. Real Analysis (Code: AMS SS D1), Paul D. Humke, Saint Olaf Diophantine Problems (Code: AMS SS Fl), Edward B. Burger, of Texas College, and Harry I. Miller, University of Tennessee at Williams College, and Jeffrey D. Vaaler, University at Austin. Chattanooga. Ergodic Theory (Code: AMS SS Hl), Cesar Silva, Williams in Geometric Function Theory (Code: AMS SS E1), Lelia Topics College. Miller-Van Wieren, Penn State Berks Campus, and Bruce P. Geometry and Topology of the Universe (Code: AMS SS El), Palka, University of Texas at Austin. Colin C. Adams, Williams College, Glenn Starkmann, Case Accommodations Western Reserve University, and Jeffrey R. Weeks, Canton, New York. Information on accommodations and airfare is included on Conference of the AMS Web site for those who prefer to make arrange­ Harmonic Analysis Since the Williamstown 1978 (Code: AMS SS G1), Janine E. Wittwer, Williams ments well in advance, given that hotels may be sold out College, and David Cruz-Uribe, Trinity College. early. Follow the links through http: I lwww. ams . orgl History of Mathematics (Code: AMS SS A1), Glen R. Van amsmtgs/secti onal . html for this meeting to the section Brummelen, Bennington College, Della D. Fenster, on "Registration/Housing, etc.". Richmond University, and James J. Tattersall, Providence College. Integrable Systems and Quantum Groups (Code: AMS SS Ll), Williamstown, Pavel I. Etingof, Massachusetts Institute of Technology, and Emma Previato, Boston University. Massachusetts Nonlinear PDEs and Calculus of Variations (Code: AMS SS Williams College ]1), Yisong Yang, Polytechnic University, and Fanghua Lin and Nader Masmoudi, Courant Institute, New York October 13- 14, 2001 University. Number Theory, Holomorphic Dynamics, and Algebraic Meeting #971 Dynamics (Code: AMS SS Bl), Robert L. Benedetto, Eastern Section University of Rochester, John W. Milnor, IMS and SUNY Associate secretary: Lesley M. Sibner Stony Brook, and Kevin M. Pilgrim, University of Missouri Announcement issue of Notices: August 2001 at Rolla. Program first available one-MATH: August 30, 2001 Accommodations Program issue of electronic Notices: November 2001 Information on accommodations and airfare is included on Issue of Abstracts: Volume 22, Issue 4 the AMS Web site for those who prefer to make arrange­ ments well in advance, given that hotels may be sold out Deadlines early. Follow the links through http: I lwww. ams. orgl For organizers: Expired amsmtgs/secti onal. html for this meeting to the section For consideration of contributed papers in Special Ses­ on "Registration/Housing, etc.". sions: June 26, 2001 For abstracts: August 21, 2001 Irvine, California Invited Addresses University of California Irvine Hubert Bray, Massachusetts Institute of Technology, Title to be announced. November 10-11,2001 Robin Forman, Rice University, Title to be announced. Meeting #972 Emma Previato, Boston University, Theta Functions, Old and Western Section New. Associate secretary: Bernard Russo Yisong Yang, Polytechnic University, Title to be announced. Announcement issue of Notices: September 2001

MAY 2001 NOTICES OF THE AMS 555 Meetings & Conferences

Program first available one-MATH: September 27, 2001 Program issue of electronic Notices: December 2001 Ann Arbor, Michigan Issue of Abstracts: Volume 22, Issue 4 University of Michigan Deadlines March 1-3,2002 For organizers: April10, 2001 Central Section For consideration of contributed papers in Special Ses­ Associate secretary: Susan J. Friedlander sions: July 24, 2001 Announcement issue of Notices: To be announced For abstracts: September 18, 2001 Program first available one-MATH: To be announced Program issue of electronic Notices: To be announced Invited Addresses Issue of Abstracts: To be announced William Duke, University of California Los Angeles, Title Deadlines to be announced. For organizers: August 3, 2001 Grigory Mikhalkin, University of Utah, Title to be an­ For consideration of contributed papers in Special Ses­ nounced. sions: To be announced Gigliola Staffilani, Stanford University, Title to be an­ For abstracts: To be announced nounced. Special Sessions Jonathan Weitsman, University of California Santa Cruz, Title to be announced. Quantum Topology in Dimension Three (Code: AMS SS A1), Charles Frohman, University of Iowa, and joanna Kania­ Special Sessions Bartoszynska, Boise State University. Quantum Topology (Code: AMS SS A1), Louis Kauffman, University of Illinois at Chicago, and Fernando Souza, Uni­ versity of Waterloo. Atlanta, Georgia Georgia Institute of Technology San Diego, California March 8-1 0, 2002 Southeastern Section San Diego Convention Center Associate secretary: John L. Bryant Announcement issue of Notices: To be announced january 6-9, 2002 Program first available one-MATH: To be announced Program issue of electronic Notices: To be announced Meeting #973 Issue of Abstracts: To be announced ]oint Mathematics Meetings, including the 1 OBth Annual Meeting of the AMS, 85th Meeting of the Mathematical Deadlines Association of America (MAA), annual meetings of the For organizers: October 8, 2001 Association for Women in Mathematics (A lt'M) and the For consideration of contributed papers in Special Ses­ National Association of Mathematicians (NAM), and the sions: To be announced winter meeting of the Association for Symbolic Logic (ASL). For abstracts: To be announced Associate secretary: John L. Bryant For summaries of papers to MAA organizers: To be an­ nounced Announcement issue of Notices: October 2001 Program first available one-MATH: November 1, 2001 Program issue of electronic Notices: January 2002 Issue of Abstracts: Volume 23, Issue 1 Montreal, Quebec, Deadlines Canada For organizers: Expired Centre de Recherches Mathematiques, For consideration of contributed papers in Special Ses­ sions: August 7, 2001 Universite de Montreal For abstracts: October 2, 2001 May 3-5, 2002 For summaries of papers to MAA organizers: To be an­ Eastern Section nounced Associate secretary: Lesley M. Sibner AMS Invited Addresses Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Michael V. Berry, Bristol University, Title to be announced Program issue of electronic Notices: To be announced (AMS Josiah Willard Gibbs Lecture). Issue of Abstracts: To be announced

556 NoTICES OF THE AMS VOLUME 48, NUMBER 5 Meetings & Conferences

Deadlines For organizers: October 3, 2001 Boston, For consideration of contributed papers in Special Ses­ sions: To be announced Massachusetts For abstracts: To be announced Northeastern University October 5-6, 2002 Pisa, Italy Eastern Section Associate secretary: Lesley M. Sibner June 12-16, 2002 Announcement issue of Notices: To be announced First ]oint International Meeting between the AMS and the Program first available one-MATH: To be announced Unione Matematica Italiana. Program issue of electronic Notices: To be announced Associate secretary: Lesley M. Sibner Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: March 6, 2002 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses­ sions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Ses­ sions: To be announced Madison, Wisconsin For abstracts: To be announced University of Wisconsin-Madison Invited Addresses October 12-1 3, 2002 Luigi Ambrosio, Scuola Normale Superiore, Title to be an­ Central Section nounced. Associate secretary: Susan J. Friedlander Luis A. Caffarelli, University of Texas at Austin, Title to be Announcement issue of Notices: To be announced announced. Program first available one-MATH: To be announced Notices: To be announced Claudio Canuto, University of Torino, Title to be announced. Program issue of electronic Issue of Abstracts: To be announced L. Craig Evans, University of California Berkeley, Title to be announced. Deadlines Giovanni Gallavotti, University of Rome I, Title to be an­ For organizers: March 12, 2002 nounced. For consideration of contributed papers in Special Ses­ Sergiu Klainerman, Princeton University, Title to be an­ sions: To be announced nounced. For abstracts: To be announced Claudio Procesi, University of Rome, Title to be announced. Baltimore, Maryland Portland, Oregon Baltimore Convention Center Portland State University january 1 5-18, 2003 ]oint Mathematics Meetings, including the 1 09th Annual 20-22, 2002 june Meeting of the AMS, 86th Annual Meeting of the Mathematical Western Section Association of America (MAA), annual meetings of the Associate secretary: Bernard Russo Association for Women in Mathematics (A WM) and the Announcement issue of Notices: To be announced National Association of Mathematicians (NAM), and the Program first available one-MATH: To be announced winter meeting of the Association for Symbolic Logic (ASL). Program issue of electronic Notices: To be announced Associate secretary: Susan J. Friedlander Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Deadlines .. : Program issue of electronic Notices: To be announced For organizers: November 20, 2001 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses­ sions: To be announced Deadlines For abstracts: To be announced For organizers: April15, 2002

MAY 2001 NOTICES OF THE AMS 557

· ·,: Meetings & Conferences GRANT FUNDING AVAILABLE

The Calculus Consortium for Higher Education, For consideration of contributed papers in Special Ses­ a small non-profit public charity, seeks to sions: To be announced For abstracts: To be announced improve the teaching of mathematics in For summaries of papers to MAA organizers: To be an­ secondary schools, two-year colleges, colleges nounced and universities. It supports workshops, meetings, conferences or research projects that deal with that mission. Grant requests are Seville, Spain hereby solicited in those four areas. Grants are june 25-28,2003 usually for 1 year and for less than $25,000. First ]oint International Meeting between the AMS and the Proposals should be less than 5 pages, Real Sociedad Matematica Espanola (RSME) accompanied by a budget using NSF Form Associate secretary: Susan]. Friedlander 1030. Send proposals by November 1st to: Announcement issue of Notices: To be announced CCHE, P.O. Box 22333, Carmel, CA 93922 or Program first available on e-MATH: To be announced Email: cche@redshi ft. com; Fax: (831) 624- Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced 7 5 71 for consideration by the Board of Directors in early January. Requests for an Deadlines earlier review date will be considered on an For organizers: To be announced individual basis. If you have any questions, For consideration of contributed papers in Special Ses­ please contact Thomas Tucker, Mathematics sions: To be announced Department, Colgate University, Hamilton, NY For abstracts: To be announced 13346, Email (preferred): ttucker@mai 1. colgate. edu. Phoenix, Arizona Phoenix Civic Plaza january 7-10, 2004 Associate secretary: Bernard Russo Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines For organizers: April2, 2003 For consideration of contributed papers in Special Ses­ sions: To be announced For abstracts: To be announced For summaries of papers to MAA organizers: To be an­ nounced

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558 NOTICES OF THE AMS VOLUME 48, NUMBER 5 A ,\I E R I C A i'.: Nl A T H E ,\I A T I C A L S 0 C I E T Y

NE.W JOURNAL! 2002 subscriptions available now from the American Mathematical Society

MOSCOW MATHEMATICAL JOURNAL Published by the Independent University of Moscow Editorial Board: Editors: Yulij llyashenko and Michael Tsfasman Dmitri Anosov Vladimir Arnold This international quarterly publishes highest quality research and research-expository papers Sergei Artemov from all over the world. Its purpose is to bring together different branches of our science and to Alexander Belavin achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular. Victor Buchstaber An important specific trait of the journal is that it especially encourages research-expository Pierre Cartier papers, which must contain new important results and include detailed introductions, placing the Boris Feigin achievements in the context of other studies and explaining the motivation behind the research. Victor Ginzburg The aim is to make the articles-at least the formulation of main results and their significance-­ John Guckenheimer readable not only to a narrow class of specialists. Sabir Gusein-Zade 2002 Subscription: Price (four issues) US$150.00* Anatol Katok Askold Khovanski FREE SAMPLE AVAILABLE TO INSTITUTIONS UPON REQUEST* Alexander Kirillov The journal is published in English beginning in 200 I.An electronic version will be available for Igor Krichever viewing by both subscribers and nonsubscribers through December 200 I. Check Robert MacPherson www.ams.org/distribution/mmj for updated details. Gregory Margulis The first issue appears in spring 200 I. Subscriptions are available from AMS for the 2002 Nikolai Nadirashvili subscription year. All 2002 subscribers to Moscow Mathematical journal will receive the inaugural Yuri Neretin 200 I issues free of charge as soon as they are published. Sergei Novikov Here is a sample of the papers that will be appearing in this prestigious journal: jacob Palis • J. Gucken~eimer and Yu. llyashenko: The duck and the devil: canards on the staircase. Senya Shlosman • A. Kirillov: Some results on the structure of quantum family algebras. Stephen Smale • Yu. Neretin: Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants. Alexei Sossinski • S. Shlosman and M.Tsfasman: Random lattices and random sphere packing: typical properties. Victor Vassiliev • V. Vassiliev: On cpmbinatorical formulas for cohomology of spaces of knots. Serge Vladut • S. Vladut: lsogeny classes statistics for abelian varieties over finite fields.

Papers are welcome by email sent to [email protected]. Subscriptions for 2002 can be placed through your usual subscription agents, or can be ordered directly from theAMS*: Moscow Mathematical Journal 2002 American Mathematical Society, P. 0. Box 6248, Providence, Rl 02940-6248, USA Tel: 401-455-4000, Fax: 40 1-331-3842, Email: [email protected]

*Institutional sample copies and subscriptions are available from the AMS in the following world regions: North America, United Kingdom, European Union countries, Scandinavia, Switzerland, lsraei,Japan,Australasia. For individuals worldwide and institutions in other regions, special prices are applicable (still lower prices for Russian residents and libraries); please direct these subscription inquiries and sample requests to: Moscow Mathematical Journal; Independent University of Moscow, II , B.VIasievsky per., Moscow 121002, Russia; Tel: 7-095-241-0500, Fax: 7-095-291-650 I, Email: [email protected]. Meetings and Conferences of the AMS

Associate Secretaries of the AMS Western Section: Bernard Russo, Department of Mathe­ Eastern Section: Lesley M. Sibner, Department of Mathe- . matics, University of California Irvine, CA 92697; e-mail: matics, Polytechnic University, Brooklyn, NY 11201-2990; [email protected]; telephone: 949-824-5505. e-mail: [email protected]; telephone: 718-260-3505. Central Section: Susan J. Friedlander, Department of Math­ Southeastern Section: John L Bryant, Department of Math­ ematics, University of Illinois at Chicago, 851 S. Morgan (M/C 249), ematics, Florida State University, Tallahassee, FL 32306-4510; e­ Chicago, IL 60607 -7045; e-mail: susan@math. nwu. edu; telephone: mail: bryant@math. fsu. edu; telephone: 850-644-5805. 312-996-3041.

The Meetings and Conferences section of the Notices gives information on all AMS meetings and conferences approved 2003 by press time for this issue. Please refer to the page numbers January 15-18 Baltimore, Maryland p. 557 cited in the table of contents on this page for more detailed Annual Meeting information on each event. Invited Speakers and Special Ses­ June 25-28 Seville, Spain p. 558 sions are listed as soon as they are approved by the cognizant program committee; the codes listed are needed for electronic 2004 abstract submission. For some meetings the list may be in­ January 7-10 Phoenix, Arizona p. 558 complete. Information in this issue may be dated. Up-to­ Annual Meeting date meeting and conference information is available on the World Wide Web at www. ams. orglmeeti ngsl. Important Information regarding AMS Meetings Potential organizers, speakers, and hosts should refer to · Meetings: page 87 in the January 2001 issue of the Notices for general 2001 information regarding participation in AMS meetings and · April21-22 Las Vegas, Nevada p. 551 conferences. April28-29 Hoboken, New Jersey p. 551 Abstracts 552 May 23-26 Morella, Mexico p. Several options are available for speakers submitting abstracts, July 17-20 Lyon, France p. 553 including an easy-to-use interactive Web form. No knowledge September 21-23 Columbus, Ohio p. 554 of LaTeX is necessary to submit an electronic form, although October 5-6 Chattanooga, Tennessee p. 554 those who use LaTeX or AMS-LaTeX may submit abstracts October 13-14 Williamstown, MA p. 555 with such coding. To see descriptions of the forms available, November 10-11 Irvine, California p. 555 visit http: I lwww. ams. orglabstractsli nstructi ons. html, or send mail to abs-submit@ams. org, typinghel pas the sub­ ject line; descriptions and instructions on how to get the tem­ 2002 plate of your choice will bee-mailed to you. January 6-9 San Diego, California p. 556 Completed abstracts should be sent to abs-submi t@ Annual Meeting ams. org, typing submission as the subject line. Questions March 1-3 Ann Arbor, Michigan p. 556 about abstracts may be sent to abs-i nfo@ams. org. March 8-10 Atlanta, Georgia p.556 Paper abstract forms may be sent to Meetings & Conferences May 3-5 Montreal, Quebec, Canada p.556 Department, AMS, P.O. Box 6887, Providence, RI 02940. There There is no June 12-16 Pisa, Italy p. 557 is a $20 processing fee for each paper abstract. for electronic abstracts. Note that all abstract deadlines June 20-22 Portland, Oregon p. 557 charge are strictly enforced. Close attention should be paid to spec­ Boston, Massachusetts p. 557 October 5-6 ified deadlines in this issue. Unfortunately, late abstracts can­ October 12-13 Madison, Wisconsin p. 557 not be accommodated. .

Conferences: (See http: I lwww. ams. org/meeti ngsl for the most up-to-date information on these conferences.) June 10-August 9, 2001: Joint Summer Research Conferences in the Mathematical Sciences, Mount Holyoke College, South Hadley, MA. See pages 1327-1331, November 2000 issue, for details. Cosponsored Conferences: May 28-30, 2001: 20011nternational Conference on Computational Science, Hilton San Francisco and Towers Hotel, San Fran­ cisco, CA. june 3-8, 2002: Abel Bicentennial Conference 2002, University of Oslo, Norway.

560 NOTICES OF THE AMS VOLUME 48, NUMBER 5