David Hilbert's 1900 lecture at the Paris International Congress of Mathematicians is described on the Clay Mathematics Institute's website as "more influential in setting the direction for 20th century mathematics than any other single event." The four videos under review record a May 2000 CMI conference in Paris which I think will be comparable in influence, mainly for the announcement there of the seven Clay Millennium Prize Problems. The videos are sold separately and certainly can be watched and appreciated separately. However they are also sold together — at a substantial discount — and I would say are best appreciated together.
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.
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.
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.
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.
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 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.
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.
The Millennium Meeting Collection is 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.