I haven’t kept up anything like the frequency of posts at this ICM that I managed at the last one. There are at least three reasons for this. One is that I was in the middle of writing up a result, so I devoted some of my rare free moments to that. Another is that the schedule was so full of good talks that I hardly skipped any sessions. And the third is that on the last day I was taken ill: I won’t go into too much detail, but let’s say that what I had sort of rhymed with “Korea”, but also left me feeling fairly terrible. So I didn’t much enjoy the conference banquet — at least from the food point of view — and then the next day, which I can’t quite believe was actually yesterday, when I got up at 5am in order to catch the bus from the hotel to the airport in time for my 9:30 flight back to Paris, I felt sufficiently terrible that I wasn’t quite sure how I would get through the 11-hour flight, four-hour stopover in Paris and four-and-a-half-hour train journey from Paris to Béziers.
I was rescued by an extraordinary piece of luck. When I got to the gate with my boarding card, the woman who took it from me tore it up and gave me another one, curtly informing me that I had been upgraded. I have no idea why. I wonder whether it had anything to do with the fact that in order to avoid standing any longer than necessary I waited until almost the end before boarding. But perhaps the decision had been made well before that: I have no idea how these things work. Anyhow, it meant that I could make my seat pretty well horizontal and I slept for quite a lot of the journey. Unfortunately, I wasn’t feeling well enough to make full use of all the perks, one of which was a bar where one could ask for single malt whisky. I didn’t have any alcohol or coffee and only picked at my food. I also didn’t watch a single film or do any work. If I’d been feeling OK, the day would have been very different. However, perhaps the fact that I wasn’t feeling OK meant that the difference it made to me to be in business class was actually greater than it would have been otherwise. I rather like that way of looking at it.
An amusing thing happened when we landed in Paris. We landed out on the tarmac and were met by buses. They let the classy people off first (even we business-class people had to wait for the first-class people, just in case we got above ourselves), so that they wouldn’t have to share a bus with the riff raff. One reason I had been pleased to be travelling business class was that it meant that I had after all got to experience the top floor of an Airbus 380. But when I turned round to look, there was only one row of windows, and then I saw that it had been a Boeing 777. Oh well. It was operated by Air France. I’ve forgotten the right phrase: something like “shared code”. A number of little anomalies resolved themselves, such as that that take-off didn’t feel like the one in Paris, that the slope of the walls didn’t seem quite correct if we were on the top floor, etc.
I thought that as an experiment I would see what I could remember about the laudatio for Martin Hairer without the notes I took, and then after that I would see how much more there was to say with the notes. So here goes. The laudatio was given by Ofer Zeitouni, one of the people on the Fields Medal committee. Early on, he made a link with what Ghys had said about Avila, by saying that Hairer too studied situations where physicists don’t know what the equation is. However, these situations were somewhat different: instead of studying typical dynamical systems, Hairer studied stochastic PDEs. As I understand it, an important class of stochastic PDEs is conventional PDEs with a noise term added, which is often some kind of Brownian motion term.
Unfortunately, Brownian motion can’t be differentiated, but that isn’t by itself a huge problem because it can be differentiated if you allow yourself to work with distributions. However, while distributions are great for many purposes, there are certain things you can’t do with them — notably multiply them together.
Hairer looked at a stochastic PDE that modelled a physical situation that gives rise to a complicated fractal boundary between two regions. I think the phrase “interface dynamics” may have been one of the buzz phrases here. The naive approach to this stochastic PDE led quickly to the need to multiply two distributions together, so it didn’t work. So Hairer added a “mollifier” — that is, he smoothed the noise slightly. Associated with this mollifier was a parameter : the smaller was, the less smoothing took place. So he then solved the smoothed system, let tend to zero, showed that the smoothed solutions tended to a limit, and defined that limit to be the solution of the original equation.
The way I’ve described it, that sounds like a fairly obvious thing to do, so what was so good about it?
A first answer is that in this particular case it was far from obvious that the smoothed solutions really did tend to a limit. In order to show this, it was necessary to do a renormalization (another thematic link with Avila), which involved subtracting a constant . The only other thing I remember was that the proof also involved something a bit like a Taylor expansion, but that a key insight of Hairer was that instead of expanding with respect to a fixed basis of functions, one should instead let the basis of functions depend on the function was expanding — or something like that anyway.
I was left with the feeling that a lot of people are very excited about what Hairer has done, because with his new theoretical framework he has managed to go a long way beyond what people thought was possible.
OK, now let me look at the notes and see whether I want to add anything.
My memory seems to have served me quite well. Here are a couple of extra details. An important one is that Zeitouni opened with a brief summary of Hairer’s major contributions, which makes them sound like much more than a clever trick to deal with one particular troublesome stochastic PDE. These were
1. a theory of regularity structures, and
2. a theory of ergodicity for infinite-dimensional systems.
I don’t know how those two relate to the solution of the differential equation, which, by the way, is called the KPZ equation, and is the following.
It models the evolution of interfaces. (So maybe “interface dynamics” was not after all the buzz phrase.)
When I said that the noise was Brownian, I should have said that the noise was completely uncorrelated in time, and therefore makes no sense pointwise, but it integrates to Brownian motion.
The mollifiers are functions that replace the noise term . The constants I mentioned earlier depend on your choice of mollifier, but the limit doesn’t (which is obviously very important).
What Zeitouni actually said about Taylor expansion was that one should measure smoothness by expansions that are tailored (his word not mine) to the equation, rather than with respect to a universal basis. This was a key insight of Hairer.
One of the major tools introduced by Hairer is a generalization of something called rough-path theory, due to Terry Lyons. Another is his renormalization procedure.
Zeitouni summarized by saying that Hairer had invented new methods for defining solutions to PDEs driven by rough noise, and that these methods were robust with respect to mollification. He also said something about quantitative behaviour of solutions.
If you find that account a little vague and unsatisfactory, bear in mind that my aim here is not to give the clearest possible presentation of Hairer’s work, but rather to discuss what it was like to be at the ICM, and in particular to attend this laudatio. One doesn’t usually expect to come out of a maths talk understanding it so well that one could give the same talk oneself. As I’ve mentioned in another post, there are some very good accounts of the work of all the prizewinners here. (To see them, follow the link and then follow further links to press releases.)
Update: if you want to appreciate some of these ideas more fully, then here is a very nice blog post: it doesn’t say much more about Hairer’s work, but it does a much better job than this post of setting his work in context.