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`The meeting provided a way of crystallizing
and focusing attention on classical problems in mathematics and on the
diversity of mathematical development.'
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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."
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