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Mathematical Challenges of the 21st Century: A Panorama of Mathematics


`The meeting provided a way of crystallizing and focusing attention on classical problems in mathematics and on the diversity of mathematical development.'
On August 7-12, 2000, the AMS held the meeting Mathematical Challenges of the 21st Century, the Society's major event in celebration of World Mathematical Year 2000. The meeting took place on the campus of the University of California, Los Angeles, and drew nearly 1,000 participants, who enjoyed the balmy coastal weather as well as the panorama of contemporary mathematics provided in lectures by thirty internationally renowned mathematicians.

The Mathematical Challenges speakers were encouraged to discuss the broad themes and major outstanding problems in their areas rather than their own research. Many of them made serious efforts to communicate to a wide mathematical audience rather than to specialists. Taken together, the lectures provided a captivating portrait of a field with a seemingly inexhaustible appetite for intellectual challenges.

It was on the morning of August 8, 1900, that David Hilbert delivered his historic lecture at the International Congress of Mathematicians in Paris in 1900. In connection with that lecture, Hilbert posed 23 outstanding problems that subsequently influenced much of 20th century mathematical research. While Mathematical Challenges was in part a celebration of Hilbert's lecture, no single individual's view of mathematics dominated the meeting. In fact, a hallmark of Mathematical Challenges was a diversity of views of mathematics and of its connections with other areas.

The Mathematical Association of America held its annual summer Mathfest on the UCLA campus just prior to the Mathematical Challenges meeting. On Sunday, August 6, a lecture by master expositor Ronald L. Graham of the University of California, San Diego, provided a bridge between the two meetings. Graham discussed a number of unsolved problems that, like those presented by Hilbert, have the intriguing combination of being simple to state, while at the same time being difficult to solve. The comfortable and elegant setting for the Mathematical Challenges lectures was UCLA's stately Royce Hall, which was built in 1929 and modeled on a basilica in Milan, Italy. Graham's lecture was followed by the Opening Ceremonies and a reception held on the terrace outside of Royce Hall.

The meeting began in earnest the next day, Monday, August 7, with a lecture by Charles Fefferman of Princeton University, who discused the Navier-Stokes and Euler equations of fluid mechanics. Fefferman's Princeton colleague, Sergiu Klainerman, also presented a lecture about nonlinear partial differential equations (PDEs), but from a completely different viewpoint. Rather than focusing on specific equations, Klainerman presented an overview of a wide swath of the field of PDEs.

Geometry and geometrical ideas arose in many of the talks. William P. Thurston of the University of California, Davis, discussed his "geometrization conjecture." Using two laptop computers, he dazzled the audience with an array of computer software tools designed to help mathematicians develop new intuition about 3-manifolds. Clifford Taubes of Harvard University talked about the very different world of 4-manifolds.

Another theme was the use of mathematics in science and technology. David Donoho of Stanford University discussed the field of data analysis, which is becoming increasingly important as humankind amasses ever more, and ever more complicated, data. The keen need for ideas from mathematics was also illustrated in the lecture by David Mumford of Brown University, who talked about the use of statistical methods in modeling visual perception, and in the lecture by Richard Karp of the International Computer Science Institute, who spoke on the use of mathematics in molecular biology, particularly genomics.

Quantum computing was the subject of two lectures at the meeting. Peter Shor of AT&T Laboratories talked about his ground-breaking work on a quantum algorithm for factoring and on quantum error-correcting codes. Providing a different take on quantum computing, Michael Freedman of Microsoft Research described his ideas for exploiting topology, in particular braid groups and the Jones polynomial, to model quantum computation.

Many at the meeting appreciated the lecture by James Arthur of the University of Toronto, who gave an especially accessible and clear overview of the Langlands program. Touching on some of the same themes was the lecture by Peter Sarnak of Princeton University, in which he spoke of "the unreasonable effectiveness of modular forms in mathematics" and the mysteries of the Riemann Hypothesis.

Another highlight was the lecture by Edward Witten of the Institute for Advanced Study, who discussed the importance of quantum field theory in physics and mathematics. The last lecture of the meeting, presented by Alain Connes of the College de France and the Institut des Hautes Etudes Scientifiques, provided a marvelous ending. Connes presented a lucid description of noncommutative geometry, from its roots through its current directions.

Mathematical Challenges was first proposed by AMS president Felix Browder, who chaired the program committee. "The meeting provided a way of crystallizing and focusing attention on classical problems in mathematics and on the diversity of mathematical development," said Browder. "It was a very unusual meeting and one that may have a significant influence on the future of mathematics."

--- Allyn Jackson

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