As I said in my previous post, I don’t think I’m going to try all that hard to explain the work of the prizewinners, since it has been very well explained in other places (except that much more attention has gone to the Fields medallists than to the Nevanlinna prize winner — maybe I’ll try to redress the balance a little bit there). Instead, I’d just like to mention a few things that I found interesting or amusing during the laudationes.
The first one was an excellent talk by Etienne Ghys on the work of Artur Avila. (The only other talk I’ve heard by Ghys was his plenary lecture at the ICM in Madrid in 2006, which was also excellent.) It began particularly well, with a brief sketch of the important stages in the history of dynamics. These were as follows.
1. Associated with Newton is the idea that you are given a differential equation, and you try to find solutions. This has of course had a number of amazing successes.
2. However, after a while it became clear that the differential equations for which one could hope to find a solution were not typical. The next stage, initiated by Poincaré, was to aim for something less. One could summarize it by saying that now, given a differential equation, one tries merely to say something interesting about its solutions.
3. In the 1960s, Smale and Thom went a stage further, trying to take on board the realization that often physicists don’t actually know the equation that models the phenomenon they are looking at. As Ghys put it, the endeavour now can be summed up as follows: you are not given a differential equation and you want to say something interesting about its solutions.
Of course, once the well-deserved laugh had died down, he explained a bit further what he meant. One way he put it was to ask what a typical dynamical system looks like.
He then talked about four important results of Avila that fit into this broad framework. One concerns iterates of unimodal maps, which are maps that look like upside-down parabolas (they are zero at 0 and 1 and have a single local maximum in between, which lies above the line ). Avila showed that given an analytic family of such maps, almost every function in the family gives rise either to a very structured dynamical system or a rather random-like one. More precisely, for almost every in the family, either almost every orbit converges to an attracting cycle (such systems are called regular) or there is an absolutely continuous measure such that almost every orbit in is distributed according to .
The main tool in the proof is something called the renormalization operator. I didn’t fully understand what this was, but I got a partial understanding. A discrete dynamical system is a set together with a map (usually assumed to have extra properties such as continuity or preservation of measure, which of course requires to have some structure so that those properties make sense) that one iterates. We are interested in orbits, which are simply sequences of the form .
Now suppose you have a subset of . Often you can define a dynamical system on by simply setting to be for the smallest positive integer for which . And often this dynamical system is closely related to the big dynamical system on . In a way I didn’t pick up from the lecture, the renormalization operator exploits this close relationship to turn maps from to into maps from to . We can use this basic idea to define a renormalization operator on the space of all unimodal maps.
It is not obvious to me why this is a good thing to do, except that it fits into the general philosophy, that applies in many many contexts, that considering a lot of objects of a certain type at once is often a great way to learn about individual objects of that type. (This theme was to reappear in a big way in the talk about Mirzakhani’s work.) Avila did not invent the renormalization map, but according to Ghys he is an absolute master at using it, and has in that way made it his own.
The second result was about interval exchange maps. These are maps that take a unit interval, chop it up into finitely many pieces (of varying lengths if you want the map to be interesting) and reassemble them in a different order. In 2007, Avila and Giovanni Forni proved that almost all interval exchange maps are weak mixing. This means that if you take any two sets and , then for almost every the measure of is approximately what you would expect if was a “random set” — that is, the product of the measures of and .
Renormalization was the tool here too. Apparently the key to proving this result was to show that the renormalization map on the space of interval exchange maps is chaotic. I don’t know exactly what this means.
I have always had a soft spot for interval exchange maps, because I once heard a fascinating open problem and thought about it very hard with no success. Suppose you are given a polygonal but not necessarily convex room lined with mirrors and you switch a light on. Must it illuminate the whole room? (Assume that the light comes from a point source.) There is a very nice construction called Kafka’s study, which shows that the answer can be no in a room with a smooth boundary. To draw it, you begin by drawing an ellipse, cutting it in half along the line joining its two foci, which I’ll take to be horizontal, keeping only one half, and then creating a sort of mushroom shape with the half ellipse at the top and a curve that goes horizontally through the two foci but also dips down between the foci (to make the “stalk” of the mushroom). If a beam of light comes out of one focus and hits the boundary of the ellipse, then it bounces back to the other focus. From this it is easy to see that if you switch on a light in the stalk part of the room, then the two other bits that do not lie in the top half of the ellipse will remain dark. I think the idea behind the name was that Kafka could work in the side parts without being disturbed by noise from the stalk part.
Another way of thinking about this is as a billiards problem. If you fire off a billiards ball (infinitesimally small of course) from the stalk part of the room, then however much it bounces, it will never reach the side parts.
What about the polygonal case? If a room is polygonal and all the sides make an angle with the horizontal that’s a rational multiple of , then a billiard ball will only ever travel in one of a finite number of directions, so we can define a map from the set of pairs of the form (boundary point, possible direction from that boundary point) to itself, which, if you think about it for a bit, can be seen to be an interval exchange map.
Years ago I managed to prove to my own satisfaction the known (I’m pretty sure, though I don’t know enough about the area to know where to find it) result that for almost every direction you send a billiard ball out in the resulting orbit will be dense. However, once the angles stop being nice rational multiples of , the dynamical system becomes a rather unpleasant map that moves bits of the plane about while also applying affine transformations to them.
As a means of simplifying the problem, I decided to consider a natural 2D analogue of interval exchange maps. This time you take a square, chop it up into finitely many rectangles, and reassemble the rectangles in some other way into the square. That led to a question I spent a long time on and couldn’t answer. (This was probably in about 1989 or so.) Take a rectangle exchange map of the kind I’ve just described, and take a point in the square. Is it recurrent? That is, will its iterates necessarily come back arbitrarily close to the original point? In the 1D case the answer is yes, and I seem to remember that was a key lemma in the proof about dense orbits.
Note that I’m not asking whether almost all points are recurrent: that is an easy excercise (and a result of Poincaré). I really want them all.
Incidentally, a few years after I was obsessed with the billiards-in-polygons problem, a paper came out that purported to solve it. Imagine my surprise when the polygon in question had rational angles. It turned out that the paper did something like assuming that corners absorbed light, or something like that. Anyhow, as far as I know the following two questions are still open, but if not, then I’d be interested to be pointed to the appropriate literature.
1. If you have a light source that’s more like a real light in that light comes in all directions from everywhere in a non-empty open set, then must an arbitrary polygonal room be illuminated?
2. If you take a point in a polygonal room and send off a billiard ball, is it true that for almost every direction you might choose the trajectory of the ball will be dense? (As far as I know “almost every” could mean for every direction not belonging to some countable set.)
Moving on to the other two of Avila’s results, I’m going to say much less. The first one was a solution of the ten-martini problem, so called because Mark Kac offered ten martinis to whoever solved it. Unfortunately, he had died by the time Avila was in a position to claim them. I didn’t really understand the problem, but it was to do with the Schrödinger equation and boiled down to a problem in spectral theory, which Avila, remarkably, solved using dynamical systems.
The last problem was one that Etienne Ghys told us most people assume must be easy when they hear it for the first time, and often offer incorrect proofs. Maybe because he had said that I didn’t have any particular feeling that it should be easy, but perhaps you, dear reader, will.
It is known that a diffeomorphism on a manifold can be approximated (in ) by a diffeomorphism. Avila showed that if the diffeomorphism is volume preserving, then the one can be taken to be volume preserving as well. The proof was apparently very hard.
The main other thing I remember from the talk was that Ghys prepared a sequence of photos that flashed up in front of us in a seemingly endless sequence, of all Avila’s collaborators. The fact that he has so many is one of the remarkable things about him: he is apparently very generous with his ideas, a great illustration of how that kind of generosity can be hugely beneficial not just to the people who are on the receiving end but also to those who exhibit it.