I’m going to try the same exercise with Curt McMullen’s talk about Mirzakhani’s work that I did with Ofer Zeitouni’s about Hairer: that is, I’ll begin by seeing what I can remember if I don’t look at my notes. However, I remember disoncertingly little, and what I do remember is somewhat impressionistic.
The most concrete thing I remember (without being 100% sure I’ve got it right) is that one of Mirzakhani’s major results concerns counting closed geodesics in Riemann surfaces. A geodesic is roughly speaking a curve that feels like a straight line to an inhabitant of the surface. Another way of putting it is that if you take two points that are close together on a geodesic, then the part of the geodesic between those points is the shortest curve that joins those two points. (Hmm, on writing that I feel that I’ve made an elementary mistake of exposition, in that I have assumed that you know what a Riemann surface is, and then gone to a little trouble to say what a geodesic is, when not many people will know the former without also knowing the latter. To atone for that, let me add a link to the Wikipedia article on Riemann surfaces, though I’m afraid that article is not much good for the beginner. A beginner’s definition, not precise at all but perhaps adequate for the purposes of reading this post, is that a Riemann surface is a surface like a sphere or a torus, but with some very important extra structure that comes from the fact that each little patch of surface looks like a little patch of the complex plane.)
If you follow your nose inside a Riemann surface, then sometimes you get back to where you started and are pointing in the same direction. In that case, you follow your original path all over again and the geodesic is called closed. But sometimes that doesn’t happen.
We can further classify closed geodesics into two types: those that cross themselves and those that don’t. The ones that don’t are called simple. An example of a simple closed geodesic is a great circle on the surface of a sphere. Apparently, the problem of counting closed geodesics was pretty much solved, but the problem of counting simple closed geodesics was significantly harder. It is this problem that Mirzakhani solved. (I’m not quite sure what “solved” meant here — perhaps her work means that if someone gives you a Riemann surface, you can tell them how many simple closed geodesics it contains.)
The more I write, the more I realize that the counting must be up to some kind of equivalence, since otherwise it seems to me that there will almost certainly either be no simple closed geodesics or uncountably many. But I’ll have to wait to look at my notes to get more precise about that.
The other main thing I remember from the talk is that moduli spaces were a very important part of Mirzakhani’s work, which provided another nice thematic connection between the work of different medallists. Just as Avila studied whole families of dynamical systems, a moduli space is a whole family of Riemann surfaces. And in both cases the family is far more than merely a set of objects: it is a set with geometrical structure. For example, if you take all interval exchange maps that chop into five parts and permute them in a certain specified way, then each one is uniquely determined by the end points of the intervals other than and . So we can naturally associate with each one an element of the set
(Those include some degenerate examples.) This is a polyhedral subset of , so it has nice geometrical, topological and measure-theoretic structure, which allows one to talk about almost all interval exchange maps, or nowhere dense sets of interval exchange maps, and so on.
An example that people often give to demonstrate what a moduli space is (and I should say that my entire knowledge of this concept comes from my memory of editing a very nice article by David Ben-Zvi on the subject for the Princeton Companion to Mathematics — though obviously anything I say about them that is false is not his fault) is the space of all tori. If you are not used to Riemann surfaces, then you may think that there is just one torus up to isomorphism, but there you would be wrong. Topologically it is true, but we want an isomorphism of Riemann surfaces, and the maps that you are allowed to use are much more rigid. So for example if you take the complex plane and quotient out by , you get a torus that is not isomorphic to the torus you get if instead you quotient out by the triangular lattice. (Roughly speaking, the obvious attempt to define an isomorphism would involve shearing the plane, but shears are not holomorphic.)
If we quotient by two lattices, when will the results give isomorphic tori? If one is an expansion of the other, then they will, and if one is a rotation of the other, then they will again. From that we get that if two complex numbers generate a lattice, then the isomorphism type of the torus depends only on their ratio. So we have already reduced the family of tori to a single complex parameter. However, that isn’t the whole story as different complex parameters do not necessarily give rise to different tori. But it gives some idea that the tori form a “space” that itself has an interesting geometrical structure. For reasons I don’t fully understand, moduli spaces are very helpful in the study of Riemann surfaces, and are also extremely interesting objects in their own right.
OK that’s about it for what I remember. But before I look at my notes, I’d like to mention briefly one other connection with Avila, which is that Mirzakhani is also very interested in billiards in polygons, though this wasn’t mentioned in the laudatio.
Actually, that reminds me of one other thing, which is that one of Mirzakhani’s results is strongly reminiscent of famous results of Marina Ratner. Maybe I’ll be able to say more about that after looking at my notes.
OK, now I’ve looked at my notes I find that, as I thought, I had forgotten quite a bit.
One important detail is that Mirzakhani looked at surfaces of genus at least two (that is, surfaces with at least two “holes”, so not tori). This is important because it means that the metrics on them are hyperbolic. It turns out that the moduli space of Riemann surfaces of genus is a complex variety of complex dimension , and is also a symplectic orbifold. (An orbifold is a bit like a manifold but is allowed to have a few singularities. In the torus example, one of these singularities arises as a result of the fact that the triangular lattice has a symmetry — rotation by 60 degrees — that most lattices do not have.)
The moduli spaces are totally inhomogeneous. That is very important, but I don’t know what it means. (I can’t remember whether McMullen told us — probably he did.)
McMullen concentrated on three aspects of Mirzakhani’s work. The first was what I’ve already mentioned, namely counting simple closed geodesics. My feeling that there would be uncountably many of these unless one looked at equivalence classes somehow was based on the sphere and the torus, so maybe when the geometry becomes hyperbolic.
He told us that if is a Riemann surface of genus , then the the number of simple loops grows like . I can’t remember what the parameter means. I’ve written to indicate what is being counted.
It seems a bit silly not to try to find out what is going on here, so let me have a quick look at the citation.
Ah, that makes much more sense! stands for length. So the formula is an estimate for the number of simple loops of length at most . If you look at all closed geodesics (i.e., allowing self-crossing ones too) then the growth rate is .
This apparently led to a new proof of a famous conjecture of Witten — a formula for intersection numbers on the moduli space — which was originally proved by Kontsevich in 1992.
Another consequence is the result that the probability that a random simple loop in genus 2 cuts the surface into two pieces is 1/7.
The second major topic was complex geodesics in . I don’t know the precise definition, but I presume that the idea is that if you take a point in that is surrounded by a copy of an infinitesimally small part of the complex plane, then there is a unique way of continuing that “in the same direction” and getting what I presume is a Riemann surface that lives inside . So it would be a little bit like a 2D generalization of a geodesic but would also involve the complex structure. Ah, I see that I have written that a complex geodesic is a holomorphic isometry from the hyperbolic plane to , though I wonder whether that should be a local isometry — that is, that for each point in the hyperbolic plane there is a neighbourhood such that the restriction of the map to that neighbourhood is an isometry.
I’ve written that there are complex geodesics through every point in in every direction, and that they are called Teichmuller discs.
Apparently real geodesics are usually dense in . Sometimes they can be exotic shapes such as fractal cobwebs (whatever those are), defying classification. What about in two dimensions? Can we get some 2D analogue of fractal cobwebs? No we can’t. Mirzakhani and her coworkers showed that you always get an algebraic subvariety. This is strongly reminiscent of work of Margulis and Ratner.
What is remarkable about this result is that it is an analogue of the Margulis/Ratner results in a totally inhomogeneous situation, which was completely unexpected.
I’ve just cheated and looked at the citation again, because it seemed to be particularly important to get some idea of what “totally inhomogeneous” means. The answer is fairly simple. A homogeneous space is one where the geometry at every point is the same. To say that is totally inhomogeneous is to say that at no two points is the geometry the same. While looking for that, I also saw that Mirzakhani solved the simple-loop-counting problem by connecting it to a certain volume computation in the moduli space . So it was a definite case where looking at the entire family helps you to prove things about the individual members of the family.
The third aspect of Mirzakhani’s work that McMullen talked about concerned something called earthquake flow that was defined by Thurston. I thought I had some understanding of what this was when I was watching the talk, but can’t really remember now. On watching the explanation again, I find that I can understand part of what McMullen says (about deforming Riemann surfaces by cutting along closed geodesics and giving them a twist, and then doing something similar but with an entire “lamination” of closed geodesics), but I still don’t quite get how that leads to a flow. (If you want to try, then the video is here and the explanation starts at 25:24.)
The result is that the earthquake flow is ergodic and mixing, and this means something like that if you randomly apply earthquakes then you get all shapes of genus . Apparently, Mirzakhani established a measurable isomorphism between earthquake flow and horocycle flow, and this was a big surprise. Those are just words to me, but when I hear someone like Curt McMullen say that a result is very surprising, then I am impressed.