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.

August 22, 2010 at 8:15 am |

From your description I imagine the paper of Desvilletes and Villani that I mentioned in my own blog was what HT Yau had in mind with his point 1. In that paper Desvillettes and Villani introduce a number of physically interesting quantities (including entropy and entropy production, but also some statistics that measure how non-homogeneous in space the distribution is, and how close one is to a Gaussian) and manage to establish enough inequalities between them that one can prove near-exponential convergence to the Maxwellian distribution even if one starts very far from that distribution. (Actual exponential decay is, however, known to be false; one gets something like decay instead, if I understand correctly.)

August 22, 2010 at 11:18 am |

It is a joy to read your lively and witty accounts of the ICM events, but one cannot help but wondering about a few things.

It seems that for quite a few huge chunks of the talks you were puzzled and also taking notes like the rest of us humans listening on the subject for the first time. So how did you and other members of the Fields medal committee come up with the decision on the selection of these medal recipients on our behalf? Any probabilistic tools used?

In recent years most of the gossips and speculations were on names such as Avila, Bhargava, Green, Calegari, Hacon, etc. How could the mathematical community be so wrong in their predictions? Or to put it another way, how could the viewpoint of the nine members of the committee be so different from the rest of the community? Anything to do with the recipients approaching the critical point of needing spectacles and will not attend ICM 2014?

August 22, 2010 at 11:34 am

The most informative answer I can give you is … no comment.

August 22, 2010 at 5:33 pm

I’ve not served on the Fields committee, but I know that rumours among the community have been only partially accurate at best in predicting past winners of the Fields medal (or indeed for other major prizes in maths or science, e.g. the Nobels). I think the dynamics of rumour propagation tend to emphasise some names over others for reasons other than the criteria for the award.

August 22, 2010 at 6:46 pm

Is there anywhere a careful discussion of these issues? I ask sincerely.

I guess there is a certain respectful shyness of the blogosphere in avoiding the many questions about prizes at this level… Can I say this?

I guess I hope mathematicians’ highest prizes’ image does not stray closer to that I have of the Nobel prize -I am thinking of the recent Nobel triplet vs. Higgs inventors problem, among other examples.

And I am sure the discussion could lead to plenty of social science modeling many here could enjoy.

August 22, 2010 at 7:34 pm

While I can certainly understand the fascination with such topics, I find it unlikely that there will be a good public forum for an informed discussion. Committee work, by its nature, is bound by confidentiality – it is nearly impossible to solicit or discuss candid evaluations of nominees without such a guarantee, as the possibility of future public exposure would inevitably lead to political or personal pressures or other conflicts of interest, both for committee members and for reference letter writers which are so crucial to the process. Indeed, this may be the one area in mathematics in which increased transparency may actually be detrimental to the quality of work.

The usual compromise between protecting the confidential and candid nature of process, and ensuring the integrity of the system, is to have oversight of the committee by a publicly appointed independent panel. However, the work of the panel must also be confidential, for the same reasons as for the committee.

In some cases, it is possible to have some public discussion of a committee process by anonymising all the information, but with an award as high-profile as the Fields Medal, I doubt that any cloak of anonymity would last for long.

Ultimately, one will probably have to wait a decade or so before one can properly judge whether the committees chose the right recipients. Looking at the medals awarded a decade a more ago, though, I would have to say that the past Fields committees tended to do a pretty good job.

July 25, 2013 at 6:31 pm

I don’t know if you can give an informative answer to a different question but let me ask anyway. Do candidates who were not selected in the top four find out later (or know at any time) that they were being considered? Bragging rights could be a consolation prize!

July 26, 2013 at 4:11 pm

@anonymo2+: There is a nice interview with Tim Gowers here: http://www.ems-ph.org/journals/newsletter/pdf/1999-09-33.pdf

In particular, read his answer to the following question.

Q: When did you first think that you might win a Fields medal?

A: Initially, I assumed that it was impossible to get one for work in Banach spaces, which at least saved me the bother of thinking about it. However, about two years before the last International Congress I started getting mysterious e-mail messages asking for lists of publications, copies of papers, and so on. The messages were usually labelled urgent, but did not explain why they were urgent. Even then, it was a long time before I dared to wonder whether I was being considered.

I read similar comments, though less explicit, in interviews of other Fields medalists. So, yes, they do know they are being considered, although my guess is that they don’t receive a formal rejection letter to hang on the wall.

@gowers: Nice solution to combat spam, I had to type the quotes above including the link to the interview, could not just copy and paste.

August 22, 2010 at 12:32 pm |

This is interesting … said, “The principle of functoriality awaits the efforts of future Fields medallists.”

August 24, 2010 at 3:33 pm |

I have noticed that pure mathematicians consider mainstream nonlinear problems such as Navier-Stokes ugly. Awards in this area are rare and appear “out of the hat”. However, nontrivial nonlinear problems are getting increasingly popular in mainstream physics. Let me mention quite a neat one here. It is the construction of a rolling tachyon in p-adic string theory. It boils down to a study of a semilinear Fredholm equation on the real line. The problem was mentioned by Witten et. al. more than 20 years ago. A complete solution requires calculations beyond the variational methods of Lions. Here is a more recent partial solution by a guy from Steklov: http://arxiv.org/abs/math-ph/0611068

November 18, 2011 at 11:31 pm

A discrete system modeling a biological circuit may be of more interest to the maths community, see the latest discussion in PAMS Vol. 139, No. 10, Oct. 2011, 3537-3551. Unfortunately, they have made no progress on the cubic case. Maybe they decided that it would not be appropriate to send a solution to the managing editor Prof. Bill Chen at Annals of Combinatorics.

September 23, 2010 at 9:41 pm |

Apparently Villani has read this post, since on his webpage one can read:

“Answer to Timothy Gowers’s Weblog: Thanks for the advice Tim, I’ll probably set up a blog one of these days. ”

http://math.univ-lyon1.fr/homes-www/villani/

That’s nice, and hopefully several others might want to do the same. For example Bill Thurston has started to contribute to MathOverflow with great effect, and would be an outstanding blogger, too.

December 27, 2011 at 8:12 am

There is also a contribution to open problems: Model Problem 43 page 67 http://math.univ-lyon1.fr/homes-www/villani/Cedrif/B09.Hypoco.pdf