comm-ucla.qxp 9/11/00 3:11 PM Page 1271 Mathematical Challenges of the 21st Century: A Panorama of Mathematics On August 7–12, 2000, the AMS held the meeting lenges was a diversity Mathematical Challenges of the 21st Century, the of views of mathemat- Society’s major event in celebration of World ics and of its connec- Mathematical Year 2000. The meeting took place tions with other areas. on the campus of the University of California, Los The Mathematical Angeles, and drew nearly 1,000 participants, who Association of America enjoyed the balmy coastal weather as well as held its annual summer the panorama of contemporary mathematics Mathfest on the UCLA Reception outside Royce Hall. provided in lectures by thirty internationally campus just prior to renowned mathematicians. the Mathematical Chal- This meeting was very different from other na- lenges meeting. On Sunday, August 6, a lecture by tional meetings organized by the AMS, with their master expositor Ronald L. Graham of the Uni- complicated schedules of lectures and sessions versity of California, San Diego, provided a bridge running in parallel. Mathematical Challenges was between the two meetings. Graham discussed a by comparison a streamlined affair, the main part number of unsolved problems that, like those of which comprised thirty plenary lectures, five per presented by Hilbert, have the intriguing combi- day over six days (there were also daily contributed nation of being simple to state while at the same paper sessions). Another difference was that the time being difficult to solve. Among the problems Mathematical Challenges speakers were encour- Graham talked about were Goldbach’s conjecture, aged to discuss the broad themes and major the “twin prime” conjecture, factoring of large outstanding problems in their areas rather than numbers, Hadwiger’s conjecture about chromatic their own research. Many of the speakers made numbers of graphs, and the Riemann hypothesis. serious efforts to communicate to a wide mathe- The comfortable and elegant setting for the matical audience rather than to specialists. Taken Mathematical Challenges lectures was UCLA’s together, the lectures provided a captivating stately Royce Hall, which was built in 1929 and portrait of a field with a seemingly inexhaustible modeled on a basilica in Milan, Italy. Graham’s lec- appetite for intellectual challenges. ture was followed by the Opening Ceremonies and It was on the morning of August 8, 1900, that a reception held on the terrace outside of Royce David Hilbert delivered his historic lecture at the Hall. The meeting began in earnest the next day, International Congress of Mathematicians in Paris. Monday, August 7, with a lecture by Charles In connection with that lecture, Hilbert posed Fefferman of Princeton University. Speaking about twenty-three outstanding problems that subse- the Navier-Stokes and Euler equations of fluid quently had an important impact on mathemati- mechanics, Fefferman set a high standard for cal research. While Mathematical Challenges was clarity of exposition at the meeting. A central in part a celebration of Hilbert’s lecture, no single question about these equations is whether or not individual’s view of mathematics dominated the solutions develop singularities in finite time. meeting. In fact, a hallmark of Mathematical Chal- Fefferman drew on mathematical analysis of the NOVEMBER 2000 NOTICES OF THE AMS 1271 comm-ucla.qxp 9/11/00 3:11 PM Page 1272 Michael Freedman James Arthur Left to right: Sergiu Klainerman, Helmut Hofer, and Peter Sarnak equations as well as on information about simu- Geometry and geometrical ideas arose in many lations and physical experiments to describe the of the talks. William P. Thurston of the University current state of knowledge. of California, Davis, discussed his “geometriza- Fefferman’s Princeton colleague Sergiu tion conjecture”, a generalization of the Poincaré Klainerman also presented a lecture about conjecture that provides a way of classifying nonlinear partial differential equations (PDEs), 3-manifolds. Although the conjecture is not proved but from a completely different viewpoint. Rather yet, it is widely believed to be correct and to pro- than focusing on specific equations, Klainerman vide deep insights. Using two laptop computers, presented an overview of a wide swath of the Thurston dazzled the audience with an array of field of PDEs, taking a conceptual approach to a computer software tools that are designed to help subject that is often shrouded in technicalities. mathematicians develop new intuition about Klainerman categorized types of equations ac- 3-manifolds. Clifford Taubes of Harvard University cording to their origins, such as, for example, equa- talked about the very different world of tions arising from fundamental notions in 4-manifolds. The topological classification of geometry or mathematical physics, or equations that emerge from taking limits or symmetry 4-manifolds was done by Michael Freedman in reductions of more complicated equations. Two the 1980s, but the plethora of smooth structures essential goals are understanding the range of va- possible on 4-manifolds makes a smooth classifi- lidity of approximations and developing useful cation extremely difficult. Taubes pointed to notions of generalized solutions. certain research directions that look promising. Plenary Speakers at Mathematical Challenges of the 21st Century James G. Arthur, University of Toronto Simon A. Levin, Princeton University Alexander A. Beilinson, University of Chicago László Lovász, Microsoft Research Michael V. Berry, University of Bristol David Mumford, Brown University Haim Brezis, Université de Paris VI and Rutgers Peter Sarnak, Princeton University University Saharon Shelah, Hebrew University and Rutgers Alain Connes, Collège de France and Institut University des Hautes Études Scientifiques Peter W. Shor, AT&T Laboratories David L. Donoho, Stanford University Yakov G. Sinai, Princeton University Charles L. Fefferman, Princeton University Richard P. Stanley, Massachusetts Institute of Michael H. Freedman, Microsoft Research Technology Ronald L. Graham, University of California, San Dennis Sullivan, City University of New York Diego Graduate School Helmut H. W. Hofer, Courant Institute, New Clifford H. Taubes, Harvard University York University Jean E. Taylor, Rutgers University Richard M. Karp, International Computer William P. Thurston, University of California, Science Institute Davis Sergiu Klainerman, Princeton University Karen Uhlenbeck, University of Texas, Austin Maxim Kontsevich, Institut des Hautes Études S. R. S. Varadhan, Courant Institute, New York Scientifiques University Peter Lax, Courant Institute, New York Edward Witten, Institute for Advanced Study University S. T. Yau, Harvard University 1272 NOTICES OF THE AMS VOLUME 47, NUMBER 10 comm-ucla.qxp 9/11/00 3:11 PM Page 1273 All photographs by Allyn Jackson. Edward Witten Program Committee Chair Felix Browder (left) and Local Arrangements Committee Chair Tony Chan. Another theme was the use of mathematics in of modular forms in mathematics” and the mys- science and technology. David Donoho of Stan- teries of the Riemann hypothesis. Another highlight ford University discussed the field of data analy- was the lecture by Edward Witten of the Institute sis, which is becoming increasingly important as for Advanced Study, who discussed the impor- humankind amasses ever more, and ever more tance of quantum field theory in physics and math- complicated, data. He pointed out that most of the ematics. The last lecture of the meeting, presented techniques now used in data analysis were devel- by Alain Connes of the Collège de France and the oped sixty years ago. Ever more powerful com- Institut des Hautes Études Scientifiques, provided puters have pressed these techniques to their a marvelous ending. Connes presented a lucid de- limits, and now all the barriers to improvements scription of noncommutative geometry, from its are mathematical. The keen need for ideas from roots through its current directions. He also briefly mathematics was also illustrated in the lecture by outlined a program for proving the Riemann David Mumford of Brown University, who talked hypothesis using the ideas of noncommutative about the use of statistical methods in modeling geometry. visual perception, and in the lecture by Richard Mathematical Challenges was first proposed by Karp of the International Computer Science Insti- AMS president Felix Browder, who chaired the pro- tute, who spoke on the use of mathematics in gram committee. “The meeting provided a way of molecular biology, particularly genomics. crystallizing and focusing attention on classical Many of the lectures provided vivid illustration problems in mathematics and on the diversity of of the impact of computers on mathematical re- mathematical development,” he said. One of the search, not only in the way that they provide means purposes of the meeting was to demonstrate the for experimentation, but also in the deep mathe- dynamism of mathematics and its connections to matical problems they raise. One such problem is other areas of science, “and this came through that of quantum computing, the subject of two lec- sharply.” Browder is now busy collecting papers tures at the meeting. Peter Shor of AT&T Labora- based on the talks in preparation for publication tories talked about his groundbreaking work on a of a proceedings volume. “It was a very unusual quantum algorithm for factoring integers and on meeting,” Browder