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The video to watch first would definitely be "The CMI Millennium Meeting."
This video gives an overview of the meeting and lasts 30 minutes. There
are interviews with leading CMI-affiliated mathematicians, an awards
ceremony, an audio tape of the end of Hilbert's 1930 retirement radio
address, snippets of the plenary lectures by Gowers, Tate, and Atiyah, an
understated announcement of the million dollar prize for each problem, and
shots of the evening banquet.
Hilbert's 1900 lecture began with general themes and concluded with
specific problems. Gowers as the first lecturer is concerned with general
themes, Tate follows by presenting three prize problems, and Atiyah
concludes by presenting the remaining four. Gowers' lecture is aimed at a
wide audience. He mentions "journalists and other influential
non-mathematicians." Certainly undergraduate math majors could be added to
the list. Tate's and Atiyah's lectures are pitched for mathematicians who
are not specialists in the areas discussed.
Gowers' title is "The Importance of Mathematics" and he interweaves many
themes. One particularly novel theme — tentatively advanced —
uses society at large as a metaphor for the mathematical research
community. In society at large, people are ideally left to pursue their
self-interest, and the net result economically advances society as a whole.
In mathematics, things work best if individual researchers are left free to
pursue what interests them most, even if it their interests seem completely
non-applicable. The result is a vast world-wide "encyclopedia" of
mathematics, of wide applicability. Another theme is that mathematics is
extremely interconnected and as a consequence mathematical knowledge
expands in unpredictable ways. A third theme is how mathematicians are
guided by their sense of beauty. The themes are appealingly presented,
with recurring reference to a hypothetical "miserly finance minister."
Throughout, the themes are illustrated by extremely well chosen nuggets of
mathematics. For example, unexpected applicability is illustrated by graph
coloring as applied to scheduling. The most advanced of these nuggets is
the Erdös-Kac theorem, used to illustrate the role of beauty in
mathematics. Gowers presents experimental evidence and then the statement
in the informal version that the number of prime factors of a randomly
chosen number near n is normally distributed with mean
ln(ln(n)) and variance also ln(ln(n)). The accessibility of
the mathematics in Gowers' lecture provides a nice balance to the more
advanced mathematics in the remaining two lectures.
Atiyah was assigned to lecture on the
Poincaré conjecture, the Hodge conjecture, the Yang-Mills problem,
and the Navier-Stokes problem. He introduces a number of general themes
which quite remarkably each apply to every problem. For example, he
discusses how increasing dimension by one can vastly change a problem. He
somewhat jokingly says that his job is to give "hints" to younger
researchers on these problems, and viewers are sure to appreciate his
expertise. In discussing the Poincaré conjecture, for example, he
emphasizes how geometrical analysis might be essential in solving this
apparently purely topological problem, and how the three-dimensional
Poincaré conjecture finds its natural home as part of the Thurston
geometrization program for three manifolds. If Perelman's currently
circulating manuscripts establishing this program stand up to scrutiny, I
would say that Atiyah had pointed his audience in the right general
direction. Conversely, in discussing the highly analytical Navier-Stokes
problem, Atiyah emphasizes that topology might play a critical role. Very
interestingly, Atiyah cautions listeners that topology might force the
Navier-Stokes problem to have a negative solution: the Navier-Stokes
equations might possibly work only within a certain regime, so that their
solutions might ultimately develop singularities which are not physically
meaningful.
The collection of seven problems the CMI chose has a Hilbertian balance.
In his retirement address mentioned earlier, Hilbert characterized number
theory as having a richness "which far surpasses that of any other field of
mathematics." Number theory is well-represented by the Riemann hypothesis
and the Birch-Swinnerton-Dyer conjecture. Hilbert's lifelong interest in
mathematical logic and solvability or unsolvability of problems carries on
in P vs. NP. Hilbert would have appreciated both the geometrical aspect
and the centrality to mathematics of the Poincaré and Hodge
conjectures. Finally, Hilbert's sixth problem was to mathematicize "those
physical sciences in which mathematics plays an important part." Much of
Hilbert's later work was in this general direction. This interest is
represented in the Yang-Mills problem and the Navier-Stokes problem. These
problems, like the other five, have a long history. However, unlike the
other five, they were not previously well-known as precise conjectures. I
think the inclusion of these two problems, a forceful affirmation of the
importance of physics-inspired mathematics, may be the most influential
aspect of the conference recorded in these videos. Their inclusion may
have surprised many mathematicians, but certainly would have pleased David
Hilbert.
Publication Data:
The
Millennium Meeting Collection, available either as a boxed set or
individually. The Clay Mathematical Institute has both general interest
and detailed mathematical descriptions of its
seven
millennial prize problems on its website.
The written
version of Hilbert's 1900 address and his
retirement
radio address are both available in English translation.
Victor Vinnikov's article "We shall know: Hilbert's apology" in
Mathematical Intelligencer 21 (1999) no. 1, 42-46 offers
insightful commentary on the latter.
The result on primality testing, due to Manindra Agrawal, Neeraj Kayal,
and Nitin Saxena, is nicely described by Folkmar Bornemann in
Primes
Is in P: A Breakthrough for "Everyman", Notices of the AMS,
May 2003, 545-552. It was also discussed by Carl Pomerance in the
November 2002 issue of FOCUS. The three books on the Riemann
hypothesis are by John Derbyshire,
Karl Sabbagh, and
Marcus du Sautoy.
David Roberts is an associate
professor of mathematics at the University of Minnesota, Morris.
Posted November 3, 2003
Read This! is the MAA
Online book review column. Contributions are welcome; contact the
editor if you'd like to be one of our reviewers. Books for review should be
sent to the editor: Fernando Gouvêa, Dept. of Mathematics, Colby
College, Waterville, ME 04901. Publishers, please check our
reviews information page.
The three other videos each record a 60-minute
plenary lecture. A single lecturer working an overhead is something that
doesn't naturally transfer to exciting video, but Tisseyre and his team do
a very professional job here. The standard view has the speaker in the
bottom right corner and the slide dominating. Regularly one gets other
views as well, close-ups, different angles, audience shots, computer
images. Particularly effective for mathematical understanding is that the
currently relevant portion of a slide is often overlaid on the main image.
Tate was assigned to lecture on the Riemann
Hypothesis, the Birch-Swinnerton-Dyer conjecture, and the P vs. NP problem.
He takes a "historical and elementary" point of view. Like the two other
speakers, he heavily emphasizes the unity of mathematics as a theme.
When discussing the Riemann Hypothesis, he introduces generalized zeta
functions. Some of these zeta functions reappear in the statement of the
Birch-Swinnerton-Dyer conjecture. Also he explains that the best evidence
for both is that characteristic p analogs are proved in the case of
the Riemann Hypothesis and well-advanced in the case of the
Birch-Swinnerton-Dyer conjecture. In his explanation of the P vs. NP
problem, he points out that establishing the Riemann hypothesis would place
the problem of distinguishing prime from composite numbers in the class P.
That this problem is indeed in the class P has since been proven
unconditionally, an illustration of related recent progress in mathematics,
although not progress on the problems Tate presented themselves. 
The main way the 2000 CMI conference will be
influential is through the attention it will focus on the problems
announced there. Since there are only seven problems, rather than
Hilbert's twenty-three, the focus is somewhat sharper. The million-dollar
prize for each will naturally focus attention too. Evidence of this
attention-focusing is already clear. For example, one of the seven
problems, the Riemann hypothesis, has been with us for more than 140 years.
Its profile was substantially raised by being part of Hilbert's eighth
problem. In all this time, one might have expected a book-length popular
account of the long and ongoing struggle towards a proof. But it is only
in the last two years that such a book has appeared, in fact three such
books, all prominently referring to the million-dollar prize.
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