This is a particularly historic moment, since Mirzakhani is the first woman to win a Fields Medal.
Here’s what she did to be recognised as one of the greatest young mathematicians in the world.
Mirzakhani’s research is based on compact Riemann surfaces. These are geometric structures constructed by cleverly stretching, bending, and gluing together parts of the complex number plane: the geometric representation of the complex numbers, made up of sums of real and imaginary numbers. The imaginary numbers are represented as multiples of the square root of -1, written as i.
To see how this works, check out the image of the complex plane. There are dots drawn in at each of the points where both the real and imaginary parts of the complex number are integers, giving us a grid.
Imagine taking the plane, and rolling it up like a rug along the vertical imaginary axis, so that each point in the grid with a given real part is touching all the points with the same real part, but different integer imaginary parts. 1+i gets “glued” to 1+2i, 1+3i, 1+4i, and so on.
After we’ve rolled up our plane in this way, we end up with a cylindrical tube. Now, we can repeat this process, rolling up our tube along the horizontal axis just as we did above, gluing together points that have the same imaginary part. At the end of this process, we get a torus, or something that looks like a doughnut: one of our Riemann surfaces.
A torus winds up being a fairly simple type of Riemann surface, and there are many ways to stretch around and glue together the complex plane or parts of the plane that lead to much more complicated structures.
A classic result from topology shows that there are only a few options for the overall global structure of surfaces: a sphere, a bunch of doughnuts glued together, or a bunch of things that look like Mobius strips glued together. These structures can be twisted and stretched in many different kinds of ways.
That last group of surfaces, made up of Mobius strips, is extremely peculiar: they don’t have well defined “inside” or “outside” areas the way spheres and toruses do, they can’t be represented in three-dimensional space without getting jumbled up with themselves, and it’s possible to walk all the way around such a surface and come back to your starting point with your left and right sides reversed.
When we take the complex plane or part of the complex plane and stretch and glue it together to make a Riemann surface, we get something that looks like one of those first three types of surfaces: a sphere, a torus, or a bunch of toruses glued together, like the double torus.
Their exact geometric structures — their shapes — are determined by what parts of the complex plane are getting glued together. The surfaces can be twisted all over the place, stretched in different directions.
In the most interesting cases, when we’re working with a surface that has a broad structure of two or more doughnuts glued together, these geometries take on a non-Euclidean hyperbolic nature, in which strange things happen.
In the usual, flat plane, if you have a line and a point not on that line, there is exactly one line going through the point that is parallel to the original line. In a hyperbolic space, there are infinitely many parallel lines. Triangles on a hyperbolic plane have angles that add up to less than 180°.
On a hyperbolic surface, we don’t have straight lines, given the curvature of the surface. Instead, geometers consider “geodesics” when working in a non-Euclidian space. Geodesics are the natural analogue of lines: just as a line segment in a plane is the shortest path between two points, a geodesic on a Riemann surface is the curve with the shortest hyperbolic length between two points on the surface.
Mirzakhani’s doctoral research focused on a counting problem involving a special type of geodesic on Riemann surfaces. Because of the compact closed nature of surfaces, geodesics can sometimes loop around the surface and come back to their starting point, and these are called closed geodesics. Closed geodesics that don’t intersect themselves while tracing their path around the surface are called simple closed geodesics.
Mathematicians like counting things, and so Mirzakhani’s doctoral thesis answered the question “how many simple closed geodesics shorter than some given length can there be on a particular Riemann surface?”
Mirzakhani found a formula, based on the given length, that says roughly how many curves of this type can be on a particular surface. What’s truly amazing about her research is how she went about finding that formula.
As mentioned above, the interesting hyperbolic Riemann surfaces all have a broad global structure that looks like some number of toruses or doughnuts glued together. It’s possible to take two surfaces made out of the same number of doughnuts and deform by pulling or stretching on one surface to get the other surface, and so these surfaces are in some way related to each other. The number of doughnuts a surface is made out of is called the genus of the surface.
Another thing mathematicians like doing is taking a collection of mathematical structures that are related to each other, like Riemann surfaces of the same genus, and turning that collection into a mathematical structure on its own.
If we take all the Riemann surfaces of a particular genus, view those as points, and then define the “distance” between two surfaces to be, very roughly and loosely speaking, based on how much we have to do to stretch one surface to turn it into the other, we get a geometry of sorts on the entire collection of surfaces, called a moduli space.
Moduli spaces have extremely complicated and strange geometries, and mathematicians are still far from a full understanding of them. Mirzakhani, however, was able to use that strange geometry to answer her counting question. She showed that certain types of volume measurements in moduli spaces are related to the number of simple closed geodesics on a particular space.
What’s remarkable about Mirzakhani’s work is that she’s answering a question about a particular surface by looking at the moduli space of all surfaces. As the International Mathematical Union puts it in their press release on Mirzakhani’s award, while the number of these special curves “is a statement about a single, though arbitrary, hyperbolic structure on a surface, Mirzakhani proved it by considering all such structures simultaneously” by looking at the geometry of moduli spaces.
Mirzakhani’s later and current work goes into the structure of moduli space itself. She has solved similar kinds of counting problems involving geodesics, not on a particular surface, but within the complicated geometry of moduli space itself. Her work also involves various links between geodesics in moduli space and certain problems in the field of dynamical systems, in which mathematicians study how a system evolves in either an orderly or chaotic fashion.
Few mathematicians have been able to directly attack problems within the strange realm of moduli space with the same skill as Mirzakhani, and her Fields Medal testifies to her abilities in this largely unexplored area.
For more detail, check out the International Mathematical Union’s press release here.
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