1. A pioneer on many frontiers
Many remarkable breakthroughs were acknowledged at this year’s International Congress of Mathematicians, but perhaps the most significant one for the intellectual community was not mathematical at all: for the first time in history, the Congress bestowed the highest award in mathematics — the Fields medal — on a female mathematician, Maryam Mirzakhani.
This was, in fact, only one of several historical landmarks this year. Mirzakhani is also the first Iranian to receive the Fields medal, Artur Avila is the first Brazilian, and Manjul Bharghava is the first recipient of Indian descent. But the conspicuous dearth of women in mathematics is, I think, the most glaring example of intellectual oppression that exists in any academic community.
It’s obviously a personal tragedy that so many women are pushed out of mathematics, and it’s also clear that the intellectual community is losing out because of it. Throughout history, female mathematicians who persisted through society’s myriad hurdles have made great contributions to mathematics. Names like Sophie Germain, Emmy Noether, and Sofia Kovalevskaya adorn some of the most important theorems from a time when women were not even permitted to participate officially in higher mathematical education, nor to hold full research positions.
There’s still a gaping gender disparity in mathematics, but perhaps the awarding of the Fields medal to Mirzakhani signals the beginning of change. In the future, girls will be able to look up to her as a role model. Minorities will feel encouraged that the circles of influence in mathematics, which have been tightly locked by pride for so long, are finally open to them. Beginning researchers will study her work and be inspired by what can be accomplished in the face of adversity.
On that note, last weekend I set aside some time to try and learn about Mirzakhani’s mathematics. I’ve been aware of this area, complex dynamics, ever since I took my freshman math class from Mirzakhani’s thesis advisor, but every time I looked at it in the past it seemed impenetrable to me. This time I received another refreshing lesson in humility, but eventually I started to understand something of what was going on. I will try to explain here one of the threads of the story. The goal is just to put her work into context, in a more comprehensible way than reading the literature.
There are already several great resources documenting Mirzakhani’s life and work. In particular, I thought this was a fantastic article. Here is also an interesting interview from a few years ago. Curt McMullen has given a very nice overview of her mathematical contributions.
But before we begin the mathematics, I wanted to highlight some pieces from the articles that stood out to me. I really like how Mirzakhani describes, “doing mathematics for me is like being on a long hike with no trail and no end in sight!” I’ve been asked a few times this summer alone to explain what pure mathematics is about, and I still find it impossible to explain. In most areas of research (especially here in industry), you start with problems that have clear practical motivations. But mathematics is different; mathematics is a hike. The purpose is not to go from point A to point B; the purpose is to enjoy what’s in between, and to savor things along the way that you didn’t anticipate when you started out. As Mirzakhani says, “the beauty of mathematics only shows itself to more patient followers.”
Another chord that resonated with me was McMullen’s comment that what distinguishes Mirzakhani from other Olympiad champions is her “ability to generate her own vision.” It seems like so much of my mathematical experience has been about performing certain party tricks, like following a tortuous lecture, recalling an intricate proof from memory, or producing the solution to a clever puzzle. The importance of vision and imagination is underemphasized in education.
Finally, I want to highlight this quote:
To her dismay, Mirzakhani did poorly in her mathematics class that year. Her math teacher didn’t think she was particularly talented, which undermined her confidence. “At that age, it’s so important what others see in you,” Mirzakhani said. “I lost my interest in math.”
In my time at Harvard, I saw many talented students quit mathematics. I thought about it several times myself, and I still do. To be fair, there are many good reasons not to want to pursue mathematics academia, but I don’t think discouragement is one of them. I often hear people say that mathematics is “beautiful,” but I can testify that the beauty is hard to see when you’re demoralized.
At such a time, in my senior year, a senior faculty member once sent me a line of encouragement. It was the only time that such a thing happened, but it felt so meaningful. This brand of kindness costs almost nothing, but can make such an impact on a young person’s life. It’s certainly easier to give than a Fields medal. What would the gender gap look like if we handed out encouragement more freely? I don’t know, but maybe we should find out.
2. What is moduli space?
Much of Mirzakhani’s breakthrough concerns the geometry of “moduli space.” In her thesis she introduced an approach to counting simple geodesics on hyperbolic space by relating them to volumes in moduli space, and more recently she has proved rigidity properties of orbits in moduli spaces. That’s cool if you’re used to thinking about such things; otherwise it’s more like cool story bro, what is this moduli space thingy anyway?
Very roughly speaking, a moduli space is a geometric object that parametrizes a certain class of things of interest. You can imagine a moduli space as a kind of map, in which each point of the map represents an entity. This might sound like a crazy thing if you haven’t seen it before, but it’s actually pretty familiar.
Suppose, as a crude example, that you were interested in modeling a pair of particles, perhaps gas particles, in space. Each particle has a position in , so the set of possible configurations for the pair of particles is parametrized by . Suppose you wanted to know more, like the velocities of each particle. Then there are three dimensions for each particle’s velocity, so the ensemble of the two particles and their velocities would be parametrized by . If you added an additional condition, say that the speed of one of the particles should be some fixed constant, then that constrains its velocity vector to lie on a sphere, and the resulting parameter space would be some sort of cylindrical object within . Every point of this cylinder thing would represent a different particle system.
The previous example wasn’t really a moduli space in any mathematical sense, but it illustrates the idea. We were interested in certain systems of gas particles, and we parametrized it by a geometric object, each of whose points represents a different possible system.
The most famous moduli space is , the “moduli space of genus curves.” (Technically, this only applies if .) This moduli space was “discovered” and studied by Riemann, and — in classic mathematical fashion — it took another century for mathematicians to say what a moduli space actually is. To be fair, the technical setup needed to give a proper definition is rather sophisticated, but at a first encounter it’s enough to know that a moduli space is a space that naturally parametrizes things, and you understand the moduli space in terms of what it parametrizes.
In this case, parametrizes “complex structures on a genus surface.” (A complex curve is a real surface, hence the confusing interchange of terminology.) Given a topological surface, there are many ways to turn it into a complex manifold, i.e. to give an atlas of charts between pieces of the surface and the complex plane, and these are what is parametrized by . A different way to think about the same thing is that there are many different complex curves having the same underlying topology, and parametrizes them. The moduli spaces that we’ll be looking at later aren’t quite the same as , but they are very related.
Example. Why is moduli space useful? I can’t resist giving an example close to my own research interests. Mathematicians like to know how to solve equations, and a particularly interesting class of equations takes the form
The modern point of view is that you should think of the set of solutions as forming a geometric object, which in this case called an elliptic curve. Then we can think interchangeably about solutions to the equation and points of the elliptic curve. It turns out that the set of complex-valued solutions to such an equation looks topologically like a genus 1 surface.
Number theorists are interested in knowing whether such an equation has a rational solution, i.e. a solution such that and are actually rational numbers. An even more refined question is whether or not it has a particular type of rational solution, say a “torsion point of order .” (This just means the usual notion of torsion element in a group, but the technical definition is unimportant.)
Luckily, there is a moduli space parameterizing the set of elliptic curves together with a torsion point of order ! Crucially, this moduli space is itself a curve, which we call , and the parametrization is refined in the sense that if a point on has rational coordinates, then the elliptic curve plus torsion point it parametrizes also has rational coordinates.
So the punchline is that if we want to know whether any elliptic curve (and there are infinitely many) has a rational torsion point of order , then we just have to figure out whether the particular curve has a rational point. Witness the power of moduli space: it has reduced an infinite calculation to a finite one!
To be fair, this gain comes at a price. The price is that even though is a curve and can in principle be described by one equation, it can be very hard to say what this equation is. So you can wind up in the bizarre situation of having to show that there is no solution to an equation, when you don’t even know the equation. As silly as it sounds, this is actually possible (in favorable circumstances). The advantage of the geometric approach is that it can furnish enough tools to reason indirectly about such questions.
It’s perhaps worth remarking that moduli spaces can quickly become very complicated, even if the objects that they describe are quite simple. We saw hints of this already when discussing parametrization of gas particles. There are moduli spaces called Hilbert schemes, which parametrize certain types of submanifolds in projective space. In principle the Hilbert schemes can be described in terms of equations, but even Hilbert schemes parameterizing fairly simple configurations can be so complex that it would require more particles than exist in the known universe to write down their equations. So understanding the geometry of moduli space is very difficult in general, and one has to settle for reasoning indirectly and achieving only partial understanding.
3. Billiards in Polygons
Billiards are a source of many interesting problems in dynamics. The setup is very simple: you have some billiard table, which is an enclosed plane region, and a billiard ball inside it. You shoot off the billiard ball in some direction and it travels at constant speed, reflecting (elastically) off the walls subject to the usual rule: angle of incidence equals angle of reflection. Figure 2 depicts some billiards paths in a rectangle.
So why is this interesting at all? It turns out that billiards can arise in modeling physical systems. Consider a simple example in which you have two particles confined to the unit interval , moving around with velocities . Assume that they collide with each other or the walls elastically, i.e. preserving momentum and energy. So in a collision, if their pre-collision velocities are and their post-collision velocities are , then conservation of momentum says that
and conservation of energy says that
If we rescale our coordinates appropriately, the second condition just reads that the initial and final velocity vectors have the same magnitude, and the first condition says that the inner product of the velocity vector with is constant. But the inner product measures the angle between the two vectors, so if you unwind what this all means then you will see that the collision rules for velocity are precisely the same as the reflection rules for a billiard wall in direction ! Similar considerations apply for collisions with the walls of the interval at and .
The conclusion is that the parameter space for the velocities of the system looks like a triangle, and in this parameter space a time evolution of the system corresponds to a billiard path! In other words, billiards can be interpreted as describing how a physical system evolves in time. If you thought that was meta, then you better buckle up.
There are many questions we can ask about billiards, but here are two basic ones, which are nevertheless extremely interesting and difficult.
- What can you say about closed paths, that is billiard trajectories that come back to themselves? These would correspond to systems of evolution that are periodic. If there are closed paths, one could go further and ask if there are infinitely many of them. If so, then how are they distributed, i.e. how does the quantity
grow with ?
- What can you say about dense orbits, that is billiard trajectories that run around chaotically and come arbitrarily close to every possible point? (Or phrased in terms of the physical model, can the system evolve in such a way that it essentially attains every possible state?) When you have dense orbits, you can ask the more refined question: are they equidistributed in the sense that the amount of time that they spend in a certain region of the billiard table is proportional to its area? (Does the system spend a “fair” amount of time in each state?)
3.2. Deforming billiard tables
The questions just posed are impossibly difficult in general. We will restrict our attention to billiard tables that are polygons, but the questions are completely open even for the class of triangles. However, they are tractable in some special cases, and that allows us to say something in more generality. How? The very interesting observation here is that even if you are interested in a question about a specific setting, it may still be useful to consider the question for other settings.
You may have already used this trick without thinking about it. Perhaps at some point you forgot why a matrix satisfies its own characteristic polynomial. This is obvious for diagonal matrices, though, and the diagonalizable matrices are dense in the space of all matrices (at least over the complex numbers). So it must be true for all matrices!
The point is that if the question is hard for your given polygon, perhaps you can deform it to a “nicer” polygon for which it’s easier to answer. Of course, you have to take care to perform the deformation in a way that preserves all the necessary structure involved.
In our case, we’re interested in questions about billiard trajectories, or straight lines, in our polygon, so we had better deform through transformations that preserve lines, i.e. through the group of linear transformations, . And if we want to further study questions of equi-distribution, then we had better further restrict ourselves to linear transformations that preserve area, i.e. .
The question is now: if we start with a polygon and deform it through , then what other kinds of polygons will we get? We’ll shortly see how to answer this question using moduli spaces.
4. Connections with moduli space
4.1. The Unfolding Trick
For special classes of polygons, there is a useful “unfolding trick” that relates the polygons with genus surfaces. Specifically, we’ll consider the rational polygons, whose angles are all rational multiples of .
The idea, which you might have seen already, is that instead of reflecting the billiard trajectory, you can keep the billiard trajectory a straight line and instead reflect the table. (See Figure 3.)
The rationality of the angles implies that the edges will eventually match up perfectly after only finitely many such reflections. If you then glue the corresponding edges, you will obtain a compact, genus surface (possibly with singular points, which we’ll ignore). The billiard path in the polygon is then equivalent to a path in the surface (see Figure 4). So we have translated questions about polygons into questions about surfaces, which opens us up to use the mathematical theory of surfaces. In order to see how this is useful, we should point out that the unfolding procedure gives more data than just a surface.
Whenever we have a plane polygon, we can regard it as a subset of , from which it acquires a natural complex structure. When we glue this polygon up to a surface, we get in this way a particular complex structure on the surface. So associated to each rational polygon we have not only a surface, but a surface plus a complex structure. As mentioned earlier, there is a moduli space parameterizing complex structures on a surface, so the unfolding trick takes a rational polygon and produces a point of .
In fact, because of the particular way that the complex structure was constructed, we get, in addition to the complex structure, the data of a “flat structure on the surface.” You see, the plane is a flat object and so it makes sense to talk about the notion of the “family of lines of angle ” laminating the plane, and hence the polygon. On a curved surface this doesn’t necessarily make sense, but because we glued up our surface from a polygon, we can transfer this “family of lines of angle ” to the surface. If you wanted to be more fancy, you could describe this in terms of Thurston’s notion of measured foliations.
Here is an equivalent way of capturing the same data. On there is the natural holomorphic -form . This restricts to a holomorphic -form on the polygon. Now it doesn’t necessarily descend to the surface when we glue up the polygon because we might have glued edges in an orientation-reversing way, but its square is a “quadratic differential” , which does always descend to the surface. (A quadratic differential is just a section of , the symmetric square of the holomorphic cotangent bundle on .)
The reason that these are the same data is not obvious but not too deep either: if you have a quadratic differential, you can essentially of integrate to get flat coordinates; conversely, if you have flat coordinates then you can form in a canonical fashion.
So the point is that if we start with a rational polygon, then we obtain a surface enriched with a complex structure and a quadratic differential . Now the key fact is that there is a moduli space of such objects! It is stratified by the vanishing datum of the quadratic differential. More precisely, if we let denote the unordered tuple of zero multiplicities of the quadratic differential, then there is a moduli space parameterizing pairs where is a surface equipped with complex structure, and is a quadratic differential whose zeros have multiplicities . Each rational polygon corresponds to a point of this moduli space .
Now let’s return to the question that motivated all this discussion. If we start with a (rational) polygon and deform it through , then what other kinds of polygons will we get? Well, the action extends to the moduli space in a natural way: for a pair , you pick a (not necessarily rational) plane polygon with parallel opposite edges representing this data, and act on it by . So in these terms, our question becomes: what are the orbits of a point under this action? The orbits consist of billiard tables having the same billiard path behavior, since they are related by linear transformations.
The whole group is quite a handful, so let’s start by considering some simpler subgroups. There are a few distinguished one-parameter subgroups of :
Here we think of indexing the time, and the matrices as a time-flow of linear transformations. The first is called the geodesic (or Teichmuller) flow, and the second is called the horocycle flow. We aim to study what we find when we start these flows at a point .
Now you see we’ve done something very meta. We started out asking about the time evolution of a mechanical system, and saw how it could be interpreted as billiards in a polygon. Then, to study billiards in a polygon, we proposed to deform the polygon through a family of polygons, in such a way that the properties of the billiards are preserved. But each polygon represents a point in this moduli space , so we have translated a question about billiard flows in all polygons into a question about flows in the moduli space . It’s like moduli within moduli! (Cue Inception soundtrack.)
4.2. Geodesic flow
At last, we can start describing some of the actual work of Mirzakhani. We have seen how understanding the game of billiards ties into understanding the geometry of the moduli space , especially its -orbits. In a recent series of papers Mirzakhani does just that.
Let’s focus on the geodesic flow first. This is so named because comes equipped with a natural metric, called the Teichmuller metric, and the orbits of the flow trace out what are called geodesics for this metric. So we are interested in geodesics on .
The geometry of is quite complicated, but there are some helpful analogies that we can use as crutches. For compact hyperbolic (negatively curved) manifolds, these questions have been studied and the answers are known (after much work).
Theorem 1 (Margulis) If is a compact hyperbolic manifold, then
where is a quantity called the entropy of the geodesic flow.
is trickier because it is not strictly negatively curved – its curvature is everywhere but not strictly negative – and it is not compact. However, it does have finite volume. So it is in some sense analogous to a compact hyperbolic manifold, and we might conjecture that the results are similar. In a paper with Eskin and Rafi, Mirzakhani confirmed this conjecture. Most of the results are rather technical even to state, but here is a special case of the simplest one.
Theorem 2 (Eskin, Mirzakhani, Rafi) If is connected, then
So how do you prove this? The (very, very) rough idea of the proof goes back to work of Eskin, Margulis, Masur, etc. As we mentioned, things are known for compact hyperbolic manifolds, but in the non-compact case the problem is that the geodesics might sort of escape off to infinity. It turns out that there is a kind of “force” pushing the geodesics away from infinity, but you have to quantify this somehow.
Here is the spirit of it. A non-compact hyperbolic space will have a “thin part” stretching off to infinity, which makes it non-compact, so imagine dividing your space into a “compact part” and a “thin part”. You want to construct a function on your space with special property that at any point in the thin part of the space, its average value at nearby points satisfies a kind of “exponentially subharmonic” inequality of the form
The idea is that if the right inequality is satified, then quantifies a kind of force that tries to push the path away from the thin part, which implies that most of the geodesics spending their time in the compact part, and then you can mimic the argument for compact hyperbolic spaces. Of course, the art is all in a careful choice of the function .
4.3. Horocycle flow
The geodesic flow can be nasty. You can obtain geodesics that have crazy Cantor-like behavior, which are dense but not equidistributed, and which “diverge” off to infinity. But it turns out that when you throw in the unipotent action from the horocycle flow as well, many of the pathologies disappear.
A very rough heuristic explanation is the chaotic behavior stems from exponential stability, which is hinted at in the form of the geodesic flow. Now, you might complain that we artificially made the geodesic flow exponential. But the point is that the geodesic flow is obtained by exponentiating the Lie algebra element while the horocycle flow is obtained by exponentiating the element . That is the nice thing about nilpotent matrices: even when you exponentiate them, their size grows only polynomially!
Here the analogy is that moduli space behaves somewhat like a homogeneous space, which is a space possessing a transitive group action. What that means is that the space has so many symmetries that every point looks locally the same as every other point. For instance, is homogeneous because you can translate any point to any other point, and the sphere is homogeneous because you can rotate any point to any other point. A cube has some symmetries, but not all points look alike; for instance, a corner point looks different from an point in the middle of a face.
There are celebrated results of Marina Ratner studying the closures of unipotently generated orbits on a particularly prominent class of homogeneous spaces, namely those arising as a quotient of a Lie group by a lattice. The first is a rigidity result, asserting that the closure is itself a homogeneous space.
Theorem 3 (Ratner’s orbit closure theorem) Let be a compact Lie group, be a homogeneous space, and let be a connected subgroup generated by unipotent elements. Then for any , the orbit closure is a homogeneous space of finite volume, i.e. for some closed subgroup .
The second of Ratner’s theorems is an equidistribution result, which we won’t state here. Although these theorems are deep and difficult to prove, the fact that they apply to Lie groups means that they are in some sense “just linear algebra.” The amazing thing is that Mirzakhani and Eskin were able to establish analogous results for , suggesting that the moduli space is somehow analogous to homogeneous spaces, even though they “look” nothing alike! Moduli space is in fact totally inhomogeneous, meaning that no two points look locally alike. But even though it is completely opposite of homogeneous space, in this sense, it turns out to have a similar rigidity property.
Theorem 4 (Eskin, Mirzakhani) If , then the orbit closure is a submanifold of .
The full theorem is slightly more precise; see the paper for details. This suggests that the orbits, which we can think of families of polygons having similar billiards properties, very nicely tile the moduli space .
5. Additional References