The first thing that stood out when H-T Yau got up on to the stage was his relative youth. (I’ve just looked him up and he was born in 1959.) He began with an amusing quote from von Neumann, who advised Shannon to use the word “entropy” on the grounds that “Nobody knows what entropy really is, so in a debate you will always have the advantage.” Part of the reason von Neumann said that was that there has always been a tension between the irreversible nature of entropy and the reversibility of the Newtonian mechanics that is supposed to underpin it. How can the two be reconciled? My impression is that this quasi-philosophical problem has largely been sorted out (so in particular, I’m not about to say that Villani has “solved the mystery of entropy” or something like that). In fact, let me reproduce Yau’s list of Villani’s three major achievements.
1. He established a rigorous connection between entropy and entropy production. (I don’t actually quite know what he meant by this.)
2. He established entropy as a fundamental tool in optimal transport, and curvature in metric spaces.
3. He rigorously proved a phenomenon known as Landau damping, a very surprising decay of the electric field in a plasma without particle collisions (and therefore without entropy increase).
I’ve just looked at Tao’s post on the Fields medallists and my understanding is such that I’m not even quite certain which of the above three achievements he is describing in detail. (That’s a comment about the headings — Tao writes with his usual clarity.)
I’m not going to try to explain Villani’s work beyond this. Let me just mention a few random things from what Yau said, and some even more random thoughts that I had during the talk. One of the latter was that amongst the other mathematicians Yau mentioned were Cergignani, who conjectured that the decay to global equilibrium of, I think, solutions to the Boltzmann equation is exponentially fast, Toscani, who proved with Villani that this conjecture is almost always (in a certain precise sense)correct (which was interesting as there are counterexamples due to Bobylev and Cergignani himself), and Gualdani, whose role in the story I did not write down and have forgotten. Could there be a pattern here?
Here, just to give you an idea of what being at the talk was like, is another note I wrote: existence of renormalized solution … other stuff … slide too quick.
I think I’ll finish this short post with a summary that gives some idea of why Villani’s results (some proved with Clement Mouhot) about Landau damping are considered so spectacular. They are the first rigorous results to establish fast decay to equilibrium in confined collisionless time-reversible dynamics.
There was also an interesting discussion about a function — which, looking at an earlier part of my notes, probably should in fact be the Boltzmann H-functional (whatever that is). Apparently, it carries all the information of the initial data for all time, which might sound hard to reconcile with fast decay to equilibrium, but the point is that it pushes this information to higher and higher frequencies. (I’m mentioning this as another example of how it is possible for a sentence that one doesn’t understand at all well nevertheless to create a mental picture that provides some help when thinking about an unfamiliar subject.)
One other point that was strongly emphasized was that Villani has been an inspiration to a generation of younger mathematicians. You can get some idea of why this might be by visiting his home page, which includes a link to informal presentations of his research. It also includes a statement that becoming, at a remarkably young age, director of the Institut Henri Poincaré will not stop him being an active researcher.
Based on what I’ve seen from his home page and what I hear about his interactions with younger mathematicians, I can’t help thinking that Villani is a blogger who just doesn’t realize it yet. Cédric, if you’re reading this, please start a blog. Post as frequently or infrequently as you like (as long as it’s a non-zero amount). It would be great to have somebody in your area to tell the rest of us about it, not to mention your general perspective on mathematics: you would be guaranteed a large and enthusiastic readership.