The CMI millennium meeting collection, Clay Mathematics Institute millennium meeting, Collège de France, Paris, France, May 24--25, 2000 (Q5959350)

The first of these video tapes, as a good introduction for a broad audience, brings a nice film of 30 minutes by \textit{Francois Tisseyre} on the Clay Mathematics Institute (CMI) Millenium Meeting at the Collège de France, Paris, May 24-25, 2000. After a short view on Paris, \textit{Andrew Wiles} and \textit{Arthur Jaffe}, the President of the CMI, appear and talk about the CMI. At a press conference, \textit{Landon T. Clay} explains that the meeting takes place in Paris to honor the 100th anniversary of the talk of David Hilbert at the International Congress of Mathematicians in Paris in which he gave his famous collection of problems. The CMI will now present the seven ``Millenium Prize Problems'' of 2000 for the solution of each one of which mathematicians can earn one million dollars. Then the Opening Ceremony is documented, starting with an amphitheatre which is slowly filling up. Andrew Wiles talks about correct proofs, \textit{Alain Connes} about intuition: ``mathematics is a science and an art''. Next, the Clay Mathematics Awards are presented by Lavinia Clay to L. Lafforge and to Alain Connes. Laurent Schwartz can be seen in the audience. The film proceeds to highlights of the Keynote Address of Timothy Gowers on the importance of mathematics. The recorded voice of David Hilbert expresses his belief that in mathematics, we will not know an ``ignorabimus''. After this, excerpts of the talks of John Tate (on the Riemann hypothesis) and Michael Atiyah (on the Navier-Stokes equations) are given. Alain Connes tells a nice joke: It is about a mathematician who has worked many years on the Riemann hypothesis and in the end desperately wants to know if it is true or not. He promises his soul to the devil if the devil can bring him this information in three days' time. After three days the devil reappears, completely disheveled and exhausted, saying that he could not solve the problem, but found this nice little formula\dots [The reviewer has to admit that he had to listen to Connes' voice twice to understand what he was saying with his charming French accent.] The meeting ends with an invitation to champagne, and the film finishes with an epilogue ``between friends'' in which, among others, J. P. Bourguignon and P. Malliavin can be seen. This first video tape is highly recommended, as are the second and fourth tapes. The second video tape documents the impressive Keynote Address of \textit{Timothy Gowers} ``The Importance of Mathematics''. After a short introduction of the speaker by A. Jaffe, Gowers starts with a real lecture of several minutes before his talk is enhanced by several well-prepared transparencies. Concerning the attitude of mathematicians towards applications, G. H. Hardy's opinion that his mathematics (number theory) needed no applications, but just beauty is contrasted with the fact that nowadays number theory definitely has important applications. At the other end of the spectrum, the Black-Scholes formula is given as an example of directly applicable mathematics. The first slides are dedicated to the problem of two-coloring and three-coloring a graph. A large part of Gowers' talk is addressed to an imaginary Finance Minister, explaining to him why mathematics should be supported. First of all, it is cheap, but occasionally produces big breakthroughs, hence gives a big return on a small investment. Second, the mathematicians also teach, not only students of mathematics, but also students of many other fields. Well, the Finance Minister might have a naive view of mathematics and could come up with the idea to support only those parts of mathematics which have applications or might have applications in the (near) future. Gowers points out the fallacies behind such a proposal: To start with, it is notoriously hard to find these areas; examples: non-Euclidean and Riemannian geometry and Einstein's relativity theory, knot theory and its unexpected links to physics by the work of Vaughan Jones and Witten. Next, mathematics is very interconnected; if one cuts the links between certain fields, then one loses too much also for the applications. Mathematics is just a huge body of knowledge with an enormous number of cross-references. [The reviewer has to applaud the speaker for the excellent illustration of this point.] Gowers gives an example in which he took part and mentions arithmetic progressions, the Balog-Szemerédi theorem, the Kakeya problem, and partial differential equations. The problems are not as different as one might think, and the interconnections are explained as a cautionary tale for the Finance Minister. Also, Gowers now gives one direct, practical application of the problem of coloring graphs which had looked like a problem from pure mathematics in which no non-mathematician would take interest. In the last part of his talk, Gowers turns from an economic case for mathematics to a cultural case. He tries to explain the beauty in mathematics. The Bell curve is shown. Gowers tells about a question of Kac during a talk at Princeton, a question which by the end of the talk was answered by Erdős, leading to the Erdős-Kac theorem. Beauty and symmetry, Fibonacci numbers are the next keywords. ``Beauty contributes to the importance of mathematics.'' -- It is surprising to see how long the audience had remained quite reserved during Gowers' excellent talk! In the other two videos, \textit{J. Tate} and \textit{Sir Michael Atiyah} present the seven Millenium Prize Problems. Tate starts with Riemann hypothesis and its history. He also mentions the Extended Riemann Hypothesis, but does not report on the consequences which a positive solution of the Riemann hypothesis would have. Next, he turns to the conjecture of Birch and Swinnerton-Dyer, closing his talk of 60 minutes with the \(P=NP\) problem. [On an intervention from the audience Tate answers something like ``I don't know very much about this problem''. In the reviewer's opinion, Atiyah does a much better job in his talk than Tate.] The fourth video documents Atiyah's talk of also 60 minutes, and it is absolutely fascinating to listen to him. The four problems which he addresses are presented in a very regular way: first the field from which they come, some background information. Then the problem is formulated, comments are given, and the methods which might help with the solution are outlined. He starts with the Poincaré conjecture. Next comes the Hodge conjecture. The problem about Yang-Mills theory is formulated as follows: ``Establish Yang-Mills theory as a rigorous quantum field theory.'' But the way this problem is commented, it looks as if the CMI is asking for a rigorous mathematical foundation of all of modern physics, including quantum field theory, gravity, and general relativity, certainly ``a big challenge for the 21st century'' -- for which in this reviewer's opinion a million dollars do not seem to be enough! Finally, Atiyah asks: ``Prove or disprove the existence of smooth solutions of the Navier-Stokes equations for all times, for all times.'' Numerical indications that a blow up might occur in finite time, that the model may no longer be valid are mentioned. Each mathematical institute should own the four videos. Parts of them could be presented to the students from time to time. Also, on open days at universities (parts of) the first two videos might give a good introduction to an active discussion on the value of modern mathematics.