After a lunch-time “reception” (the less said, the better), the first afternoon session is dedicated to the laudationes. An expert in the area speaks on the work of each Field Medalist and of Jon Kleinberg.
First, a mathematician from ETH Zurich (who is, however, Italian) speaks on the work of Andrei Okounkov. He studies “partitions,” which, in the simplest case, are ways of writing an integer as the sum of a non-decreasing sequence of integers. Partitions are related to the representation theory of the symmetric group, and apply in a variety of problems that were too technical for me to follow.
John Lott spoke on Perelman’s work. Lott is part of one of the three teams that have been working out complete expositions of Perelman’s work. His talk is extremely clear. The Poincare conjecture states that
every 3-dimensional manifold that is simply connected and compact is diffeomorphic to the boundary of a
sphere ball in R4.
Where “simply connected” means that a simple loop can always be shrunk to a point via a continous transformation. “Diffeomorphic” means, probably, “equivalent under the appropriate type of nice continuous transformations.” The lower dimensional analog is that any surface with “no holes” can be “stretched” into the surface of a
sphere ball. Even though this is a purely topological problem (the assumption is a topological property and the conclusion is a topological property), the big idea was to use methods from analysis. Specifically, Hamilton suggested in the 80s to consider a physics-inspired continuous transformation (the “Ricci flow”) on the manifold until it “rounds up” and becomes a sphere. (The continous transformation maps the starting manifold into a “diffeomorphic” one.) Unfortunately, the process produces singularities, and Hamilton had found ways to remove them in certain nicely-behaved special case. Perelman’s work shows how to always remove such singularities. This was a major tour de force. Before Perelman’s work, apparently, the experts in the area considered the known obstacles to eliminating singularities in all cases to be almost insurmountable, and did not consider Hamilton’s program to be too promising. Hamilton said that he was “as surprised as anybody else” by the fact that Perelman made the Ricci flow approach work.
Next, Fefferman delivers the laudatio for Tao. It is a daunting task given the variety of areas that Tao has worked on. Fefferman choses to talk about the Kakeya problem, about non-linear Schrodinger equations and about arithmetic progressions in the primes. By the way, both his work on the Kakeya problem and the work about arithmetic progressions (though not the work on the primes in particular) have applications in theoretical computer science.
The laudatio for Werner is a complete disaster. The speaker reads from a prepared speech, and the big screen (that, for the previous speakers, had shown slides of the presentation) shows the text of the speech. This doesn’t do justice to Werner’s work on proving rigorous results in statistical physics which were previously derived via non-rigorous methods. His work includes problems of the type that computer scientists work on. In his interview, Werner says that he feels he shares the medal with Oded Schramm and Greg Lawler. Oded Schramm, in particular, was a leading figure in this line of work, but he was already over the age limit in 2002, and this work was done after 1998, so he could never be considered for the award.
John Hopcroft delivers the laudatio for Jon Kleinberg. He emphasizes five results: the hubs-authority idea for web search ranking, his work on small-world models and algorithms, the work on “bursts” in data, the work on nearest neighbor data structures and the work on collaborative filtering. Like Fefferman, Hopcroft runs out of time.
In the late afternoon (this is a LONG day), Hamilton gives a plenary talk on his work on the Poincare conjecture. Despit Lott’s nice earlier introduction, I get rapidly lost in what seems a series of unrelated technical statements. Avi seems to follow. “No, no, they are not unrelated, he is building up an exposition of the proof by explaining all the problems and how they are overcome.” He starts late because of projector problems, and he runs enormously out of time. At one point the session chair timidly stands up, not quite sure what to do, then walks towards the podium, then walks back, apparently with Hamilton not noticing. Where is Johan Hastad when you need him?