The work of Maryam Mirzakhani
(Notes for the talk at the Women in Math day on 13 May 2024).
I am going to describe Mirzakhani’s work on the geometry of surfaces.
Classification of surfaces by genus
The trichotomy: metric of positive curvature, 0 curvature, negative curvature.
Counting simple geodesics
Sphere
Only one simple geodesic, up to homotopy.
Torus
Number of simple geodesics of length at most \(L\) = The number of integral points \((p,q)\) in a circle of radius \(L\) with \(\gcd(p,q) = 1\).
Grows roughly as \(L^2\).
Higher genus
Theorem (Mirzakhani) Fix a surface \(S\) of genus \(g\) with a metric of constant negative curvature \(-1\) (hyperbolic metric). Then the number of simple geodesics of length \(L\) on \(S\) is asympototic to \(L^{6g-6}\).
Remark The limit of the ratio (the asympototic constant) depends on the metric.
Remark If you remove the adjective simple, that is, allow crossings, then the counting problem had been solved before. Then the number is asymptotic to \(e^L/L\). So, a negligible portion of geodesics are closed geodesics.
Ideas in the proof
Mirzakhani’s insight is that the counting problem in higher genus is very similar to the counting problem in genus 1, if you interpret it correctly.
To do so, observe that integral points \((p,q)\) with \(\gcd(p,q) = 1\) lie in the \(\operatorname{SL}_2(\mathbf{Z})\) orbit of one such point, say \((1,0)\).
The same is true in higher genus. The group \(\operatorname{SL}_2(\mathbf{Z})\) needs to be replaced by the mapping class group \(\operatorname{Mod}(S)\) that, roughly speaking, captures the ways of breaking \(S\) and assembling it back. It is known that up to \(\operatorname{Mod}(S)\) action, there are only finitely many geodesics. So, we may fix a representative \(\gamma\), and count how many of its \(\operatorname{Mod}(S)\)-translates have length at most \(L\).
Next, for the torus, we interpreted a geodesics as points of \(\mathbf{R}^2\), and ultimately, used the area in \(\mathbf{R}^2\) to estimate their number. Similarly, geodesics in \(S\) can be interpreted as points in a space \(\operatorname{LM}(S)\), called the space of measured laminations on \(S\). It is a manifold of dimension \(6g-6\), and has a measure. It also has an integral structure, so it makes sense to talk about integral points in \(\operatorname{LM}(S)\) The problem of counting geodesics of length atmost \(L\) is then equivalent to counting integral points in \(\operatorname{LM}(S)\) in a ball of radius \(L\) , and since \(\operatorname{LM}(S)\) has dimension \(6g-6\), the count is proportional to \(L^{6g-6}\).
Connections
The space \(\operatorname{LM}(S)\) is closely connected to other spaces of fundamental importance. In particular, it is closely related to \(M_g\), the moduli space of algebraic curves of genus \(g\), a space that is very dear to me. Mirzakhani’s work shows that geodesic counting is related to the volumes of the moduli spaces \(M_g\). Witten had conjectured that these volumes were conjectured to satisfy complicated recurrence relations. Mirzakhani’s geodesic counting gives a new proof this conjecture.
Although the volumes of \(M_g\) have a non-algebraic definition, it is equal to a quantities defined in purely algebraic terms using intersection theory in algebraic geometry. Mirzakhani’s work gives a new perspective on these intersection numbers.
What is extraordinary in her work is how she bridges an array of different fields: topology, dynamics, algebraic and differential geometry, among others.
References
- The work of Maryam Mirzakhany by Curtis McMullen https://people.math.harvard.edu/~ctm/papers/home/text/papers/icm14/icm14.pdf
- Growth of the number of simple closed geodesics on hyperbolic surfaces by Maryam Mirzakhani, Annals of Mathematics, 2008.