# Maryam Mirzakhani

Maryam Mirzakhani is known for her work on moduli spaces of Riemann surfaces.  Some of her most cited work looks at the moduli space of a genus $g$ Riemann surface with $n$ geodesic boundary components.  In two of her papers, she computes the volume of these moduli spaces, with respect to the Weil-Petersson metric (see below).  In another, she provides a means for counting the number of simple closed geodesics of length at most $L$. Mirzakhani is also known for her work on billiards (see the review of her paper with Eskin and Mohammadi below), a subject closely related to moduli space questions.  Teichmüller theory and the geometry of moduli spaces are famously deep subjects.  Making progress requires mastering large areas of analysis, dynamical systems, differential geometry, algebraic geometry, and topology.  I can only appreciate Mirzakhani’s work superficially, as I have not mastered those subjects.   Instead, some reviews of her work are reproduced below.

Notes:
1.
One of her biggest projects, joint work with Eskin studying the action of SL(2,R) on moduli space, is not published yet.  So there is no item in MathSciNet for it.  You can read the latest version on the arXiv.

2. Mirzakhani published three papers as an undergraduate:  MR1366852MR1386951MR1615548.  The second of these is regularly cited by combinatorists. The third paper was in the Monthly.

3. I started writing this post back in March, when I was highlighting the work of some remarkable mathematicians.  It was delayed because describing her work is not simple: it is substantial and uses deep and difficult tools from several areas.  Her papers are quite well written, with accessible introductions.  However, the genius is in the details, which require real commitment to understand.  The video produced for the ICM where she won her Fields Medal allows her to present something of her work.  Amie Wilkinson describes Mirzakhani’s working style in this article in the NY Times.  In a recent blog post, Terry Tao comments on how Mirzakhani was able to see disparate mathematical results through the lens of the mathematics she was developing herself.

4. Thank you to Tom Ward who spotted an inequality that was reversed in the original version of this post.

MR2264808
Mirzakhani, Maryam
Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces.
Invent. Math. 167 (2007), no. 1, 179–222.
32G15 (14H15)

For $L:= (L_1,\dots , L_n)\in \Bbb{R}_+^n$ one can define the moduli space $M_{g,n}(L)=M_{g,n}(L_1,\dots ,L_n)$ of hyperbolic Riemann surfaces of genus $g$ with $n$ geodesic boundary components of lengths $L_1,\dots ,L_n$. $M_{g,n}(L)$ carries a symplectic form called the Weil-Petersson symplectic form. The main object of study of the paper under review is the volume $V_{g,n}(L)=V_{g,n}(L_1,\dots ,L_n)$ of $M_{g,n}(L)$ calculated with respect to the volume form associated to the Weil-Petersson symplectic form. The main result of this paper is an explicit recursive formula (Section 5) for $V_{g,n}(L_1,\dots ,L_n)$ that allows one to effectively calculate $V_{g,n}(L_1,\dots ,L_n)$ from bottom-up. As an application, the author proves the following polynomial behavior of $V_{g,n}(L)$ (Theorem 1.1): $$V_{g,n}(L_1,\dots ,L_n)=\sum_{\alpha=(\alpha_1,\dots ,\alpha_n)\in \Bbb{Z}_+^n,\, \sum_{i=1}^n\alpha_i\leq 3g-3+n}C_\alpha \prod_{i=1}^nL_i^{2\alpha_i},$$ where $C_\alpha$ are some rational multiples of $\pi^{6g-6+2n-\sum_{i=1}^n2\alpha_i}$. In another work [J. Amer. Math. Soc. 20 (2007), no. 1, 1–23 (electronic); MR2257394] the author gave another proof of this result using symplectic reduction techniques. She also found a relation between $C_\alpha$ and intersection numbers on moduli spaces of Riemann surfaces.
The author’s approach to the recursive formula for $V_{g,n}(L)$ begins with a generalization of McShane’s identity [G. McShane, Invent. Math. 132 (1998), no. 3, 607–632; MR1625712]. We summarize this result (Theorem 1.3) as follows. Let $X$ be a hyperbolic Riemann surface with $n$ geodesic boundary components $\beta_1,\dots ,\beta_n$ of lengths $L_1,\dots ,L_n$. Then the following holds: $$\sum_{\{\gamma_1,\gamma_2\}}\scr{D}(L_1,l_{\gamma_1}(X), l_{\gamma_2}(X))+\sum_{i=2}^n\sum_\gamma \scr{R}(L_1,L_i, l_\gamma(X))=L_1.\tag1$$ The ingredients of this formula are explained below:

• for a geodesic $\gamma\subset X$, its length is denoted by $l_\gamma(X)$;
• the first sum is over all unordered pairs of simple closed geodesics $\{\gamma_1, \gamma_2\}$ bounding a pair of pants with $\beta_1$;
• the second sum is over all simple closed geodesics $\gamma$ bounding a pair of pants with $\beta_1,\beta_i$.
• the functions $\scr{D}, \scr{R}\colon \Bbb{R}_+^3\to \Bbb{R}_+$ are defined in terms of the geometry of a pair of pants (see Section 3). These functions can be explicitly calculated (Lemma 3.1): $$\scr{D}(x,y,z)=2\log\left(\frac{e^{\frac{x}{2}}+e^{\frac{y+z}{2}}} {e^{\frac{-x}{2}}+e^{\frac{y+z}{2}}}\right),$$ $$\scr{R}(x,y,z)=x-\log \left(\frac{\cosh(\frac{y}{2})+\cosh (\frac{x+z}{2})}{\cosh(\frac{y}{2})+\cosh(\frac{x-z}{2})} \right).$$

The author’s proof of (1) is based on a detailed analysis of geodesics and pairs of pants on $X$, carried out in Sections 3 and 4.

To prove the recursive formula for $V_{g,n}(L)$ the author develops a method to integrate functions given in terms of hyperbolic lengths. Note that the functions involved in (1) are of this kind. The author finds a way to express integrals of such functions over moduli spaces of Riemann surfaces using Weil-Petersson volumes. This is Theorem 7.1. The recursive formula for $V_{g,n}(L)$ is then obtained by integrating (1) against the Weil-Petersson volume form and applying Theorem 7.1.

Reviewed by Hsian-Hua Tseng

MR2257394
Mirzakhani, Maryam
Weil-Petersson volumes and intersection theory on the moduli space of curves.
J. Amer. Math. Soc. 20 (2007), no. 1, 1–23.
14H15 (14N35 32G15)

For $b_1,\dots,b_n\in\Bbb{R}_+$, put $b=(b_1,\dots,b_n)$ and let $M_{g,n}(b)=M_{g,n}(b_1,\dots,b_n)$ be the moduli space of hyperbolic Riemann surfaces with geodesic boundary components of lengths $b_1,\dots,b_n$. On $M_{g,n}(b)$ there is a symplectic form called the Weil-Petersson symplectic form. Let $V_{g,n}(b)=V_{g,n}(b_1,\dots,b_n)$ denote the volume of $M_{g,n}(b)$ calculated using the volume form associated to the Weil-Petersson symplectic form. The paper under review presents an explicit relationship between the Weil-Petersson volume $V_{g,n} (b)$ of $M_{g,n} (b)$ and the intersection numbers of tautological classes on the moduli space of stable curves. To achieve this, the author expresses the compactified moduli space $\overline{M}_{g,n}(b)$ as a symplectic quotient, as follows: consider the following moduli space of bordered Riemann surfaces with marked points: $$\widehat{M_{g,n}}:=\{(X,p_1,\dots,p_n)\mid X\in\overline{M}_{g,n}(b_1,\dots,b_n),\ b_i\geq 0,\ p_i\in\tilde{\beta}_i\}.$$ Here $\tilde{\beta}_i$ is a parallel curve to the $i$-th boundary component $\beta_i\subset X$. The moduli space $\widehat{M_{g,n}}$ carries a symplectic form, naturally induced from the Weil-Petersson form. For a bordered Riemann surface $X$, denote by $l_{\beta_i}(X)$ the length of its $i$-th boundary component $\beta_i$. This gives a map $l^2/2\colon \widehat{M_{g,n}}\to \Bbb{R}_+^n$ defined by $$(l^2/2)(X, p_1,\dots,p_n) =(l_{\beta_1}(X)^2/2,\dots ,l_{\beta_n}(X)^2/2).$$ The author proves that $l^2/2$ is the moment map associated to the $T=(S^1)^n$ action on $\widehat{M_{g,n}}$ defined by rotating the points $p_1,\dots,p_n$, and that the symplectic quotient at value $(b_1,\dots,b_n)$ is symplectomorphic to $\overline{M}_{g,n}(b_1,\dots,b_n)$. Then, using the relationship between symplectic forms of reduced spaces at different values, she proves that the volume $V_{g,n}(b_1,\dots,b_n)$ is a polynomial in the $b_i$’s for $b_i$ sufficiently small, and the coefficients of this polynomial are explicitly given (see Theorem 4.4). In particular, the leading coefficients are, up to some prefactors, of the form $\int_{\overline{M}_{g,n}}\psi_1^{\alpha_1}\cdots\psi_n^{\alpha_n}$ with $\alpha_1+\dots +\alpha_n=3g-3+n$.

In a previous work [Invent. Math. 167 (2007), no. 1, 179–222; MR2264808], the author found a recursive formula for the volumes $V_{g,n}(b)$. A review of this formula is given in Section 5 of the paper under review. In the present paper the author uses this recursive formula and the result about coefficients of the volume polynomial to derive a proof of Witten’s conjecture (Kontsevich’s theorem) in the form of Virasoro constraints of a point. This is done by substituting the volume polynomials into the recursion, then extracting a recursion for the leading coefficients. By relating the coefficients with descendant integrals over $\overline{M}_{g,n}$, Virasoro constraints appear immediately.

Reviewed by Hsian-Hua Tseng

MR2415399
Mirzakhani, Maryam
Growth of the number of simple closed geodesics on hyperbolic surfaces.
Ann. of Math. (2) 168 (2008), no. 1, 97–125.
32G15

Let $X$ be a complete hyperbolic Riemann surface of genus $g$, with finite area and $n$ cusps. The paper under review studies the growth of the number $s_X(L)$ of simple closed geodesics of length at most $L$. In fact the author studies a more refined problem, as follows. Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ boundary components. The mapping class group ${\rm Mod}_{g,n}$ acts on the set of isotopy classes of simple closed curves on $S_{g,n}$, and each isotopy class of a simple closed curve contains a unique simple closed geodesic on $X$. For a simple closed geodesic $\gamma$ the author considers the following counting function: $$s_X(L,\gamma):=\#\{\alpha\in {\rm Mod}_{g,n}\cdot\gamma\,|\, l_\alpha(X)\leq L\},$$ where $l_\alpha(X)$ is the length of $\alpha$. Note that $s_X(L)=\sum_\gamma s_X(L,\gamma)$. The definition of this counting function can be extended to multi-curves $\gamma=\sum_{i=1}^k a_i\gamma_i$. Here, by definition, multi-curves $\gamma_i$’s are disjoint, essential, nonperipheral simple closed curves which are pairwise non-homotopic, and $a_i>0$. The length of a multi-curve $\gamma$ is defined to be $l_\gamma(X):=\sum_{i=1}^k a_il_{\gamma_i}(X)$. The first main result of this paper asserts that for a rational multi-curve $\gamma=\sum_i a_i\gamma_i$ (i.e. $a_i\in \Bbb{Q}$), the limit $n_\gamma(X):=\lim_{L\to\infty}\frac{s_X(L,\gamma)}{L^{6g-6+2n}}$ defines a continuous proper function $n_\gamma\colon\scr{M}_{g,n}\to\Bbb{R}_+$.

The main tool used in this paper is the space $\scr{ML}_{g,n}$ of compactly supported measured laminations on $S_{g,n}$. There is a length function $\scr{ML}_{g,n}\to \Bbb{R}_+$ induced by the hyperbolic metric $X$ on $S_{g,n}$. The Thurston measure $B(X):=\mu_{Th}(B_X)$ of the unit ball $B_X$ with respect to this length function defines a function $B\colon \scr{M}_{g,n}\to\Bbb{R}_+$. The author proves that this function $B$ is integrable with respect to the Weil-Petersson volume form. Set $b_{g,n}:=\int_{\scr{M}_{g,n}}B(X) dX$. The next main result of this paper states that for each rational multi-curve $\gamma$ there is a number $c(\gamma)\in \Bbb{Q}_{>0}$ such that $n_\gamma(X)=\frac{c(\gamma)B(X)}{b_{g,n}}$. The proofs of these statements rely heavily on a study of measures on $\scr{ML}_{g,n}$. In fact the two main results are direct consequences of a statement about the asymptotic behavior of some discrete measures. The second main result has the following corollary: for rational multi-curves $\gamma_1,\gamma_2$ we have $\lim_{L\to\infty}\frac{s_X(L, \gamma_1)}{s_X(L,\gamma_2)}=\frac{c(\gamma_1)}{c(\gamma_2)}$. This may be rephrased as saying that the relative frequencies of different types of simple closed curves on $X$ are universal rational numbers.

Reviewed by Hsian-Hua Tseng

MR3418528
Isolation, equidistribution, and orbit closures for the ${\rm SL}(2,\Bbb R)$ action on moduli space. (English summary)
Ann. of Math. (2) 182 (2015), no. 2, 673–721.
58D27 (22F10 32G15 37C85 37D40 60B15)

The results in this paper are analogous to the theory of unipotent flows and concern orbit closures and equidistribution for the $\textrm{SL}(2,\Bbb R)$-action on the moduli space of compact Riemann surfaces. Their number is such that a review can only give an impressionistic sampling. The proofs rely on the measure-classification theorem from [A. Eskin and M. Mirzakhani, “Invariant and stationary measures for the $\textrm{SL}(2,{\Bbb R})$ action on moduli space”, preprint, arXiv:1302.3320], which is a partial analogue of M. Ratner’s measure-classification theorem in the theory of unipotent flows [Ann. of Math. (2) 134 (1991), no. 3, 545–607; MR1135878], which was in turn motivated by the Raghunathan Conjecture [S. G. Dani, Invent. Math. 64 (1981), no. 2, 357–385; MR0629475; G. A. Margulis, in Number theory, trace formulas and discrete groups (Oslo, 1987), 377–398, Academic Press, Boston, MA, 1989; MR0993328]. The second major ingredient is the main technical result of this paper (Proposition 2.13), an isolation property of closed $\textrm{SL}(2,\Bbb R)$-invariant manifolds, whose proof takes up of Sections 4–10.

The proofs of the principal results are in Section 3; these are actually simpler than the proofs of the analogous results in the theory of unipotent flows, due in no small part to the fact (Proposition 2.16, a consequence of the isolation property) that there are at most countably many affine invariant submanifolds in each stratum, while unipotent flows may have continuous families of invariant manifolds (which involve the centralizer and normalizer of the acting group). The proof of the isolation property in turn is based on the recurrence properties of the $\textrm{SL}(2,\Bbb R)$-action from [J. S. Athreya, Geom. Dedicata 119 (2006), 121–140; MR2247652] and on the uniform hyperbolicity in compact sets of the Teichmüller geodesic flow [G. Forni, Ann. of Math. (2) 155 (2002), no. 1, 1–103 (Corollary 2.1); MR1888794].

Some terminology to give formal statements: $H(\alpha)$ denotes a stratum of Abelian differentials, i.e., the space of pairs $(M,\omega)$ where $M$ is a Riemann surface and $\omega$ is a holomorphic 1-form on $M$ whose zeros have multiplicities $\alpha_1\cdots\alpha_n$ with $\sum\alpha_i=\chi(M)\ge0$. The form $\omega$ defines a canonical flat metric on $M$ with cone points at the zeros of $\omega$, i.e., a flat surface or translation surface [A. Zorich, in Frontiers in number theory, physics, and geometry. I, 437–583, Springer, Berlin, 2006; MR2261104]. The space $H(\alpha)$ admits an action of $\textrm{SL}(2,\Bbb R)$ which generalizes the action of $\textrm{SL}(2,\Bbb R)$ on the space $\textrm{GL}(2,\Bbb R)/\textrm{SL}(2,\Bbb Z)$ of flat tori. A “unit hyperboloid” $H_1(\alpha)$ is defined as a subset of translation surfaces in $H(\alpha)$ of area one: $\frac i2\int_M\omega\wedge\overline\omega=1$.

The aforementioned measure-classification result of [A. Eskin and M. Mirzakhani, op. cit.] is: A probability measure on $H_1(\alpha)$ that is invariant under $P:=\left\{\left(\begin{array}{cc}*&*\\ 0&*\end{array}\right)\right\}\subset\textrm{SL}(2,\Bbb R)$ is $\textrm{SL}(2,\Bbb R)$-invariant and affine (i.e., supported on an immersed submanifold and compatible with Lebesgue measure in a particular way; the submanifold is then also said to be affine).

With $a_t:=\textrm{diag}(e^t,e^{-t})$ and $r_\theta:=\left(\begin{array}{cc}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right)$ we can now state the main isolation property of this paper.
If $\varnothing\subseteqq M\subset H_1(\alpha)$ is an affine invariant submanifold, then there is an $\textrm{SO}(2)$-invariant $f\colon H_1(\alpha)\to[1,\infty]$ such that:

• $M=f^{-1}(\infty)$.
• $f$ is bounded on compact subsets of $H_1(\alpha)\smallsetminus M$.
• $\overline{f^{-1}([1,\ell])}$ is compact for all $\ell$.
• $\exists b$ (depending only on the “complexity” of $M$) $\forall c\in(0,1)$ $\exists T>0$ $$(A_tf)(x):=\frac1{2\pi}\int_0^{2\pi}f(a_tr_\theta x)\,d\theta\le cf(x)+b$$ whenever $x\in H_1(\alpha)\smallsetminus M$ and $t>T$.
• There is $\sigma>1$ such that $\sigma^{-1}f(x)\le f(gx)\le\sigma f(x)$ for all $x\in H_1(\alpha)$ and $g\in\textrm{SL}(2,\Bbb R)$ near the identity.

Here is an overview of the many consequences derived here.

Orbit closures in $H_1(\alpha)$ are affine invariant submanifolds (the unipotent counterpart is in [M. Ratner, Duke Math. J. 63 (1991), no. 1, 235–280; MR1106945]) and any closed $P$-invariant subset of $H_1(\alpha)$ is a finite union of affine invariant manifolds.

The space of ergodic $P$-invariant probability measures on $H_1(\alpha)$ is weak*-compact (the unipotent counterpart is in [S. Mozes and N. A. Shah, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159; MR1314973]).

Equidistribution for sectors, random walks and Følner sets (the first of which implies that for any $x\in H_1(\alpha)$ there is a unique affine invariant manifold of minimal dimension that contains $x$); uniform versions of the equidistribution results (Theorems 2.7, 2.9) are analogous to [S. G. Dani and G. A. Margulis, in I. M. Gelʹfand Seminar, 91–137, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993 (Theorem 3); MR1237827], which plays a key role in applications of the theory.

Orbit counting in rational billiards: Let $N(Q,T)$ denote the number of cylinders of periodic trajectories of length at most $T$ for the billiard flow on a rational polygon $Q$. It is known that this grows quadratically [H. A. Masur, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 151–176; MR1053805; in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 215–228, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988; MR0955824]: $N(Q,e^t)e^{-2t}$ is bounded above and away from 0 for $t>1$. The uniform equidistribution result for sectors implies that
$$\frac1t\int_0^tN(Q,e^s)e^{-2s}\,ds\underset{t\to\infty}\rightarrow c,$$
where $c$ is the Siegel-Veech constant [W. A. Veech, Ann. of Math. (2) 148 (1998), no. 3, 895–944; MR1670061; A. Eskin, H. A. Masur and A. Zorich, Publ. Math. Inst. Hautes Études Sci. No. 97 (2003), 61–179; MR2010740] associated to the affine invariant submanifold $M = \textrm{SL}(2,\Bbb R)S$ with $S$ the flat surface obtained by unfolding $Q$. The authors find it natural to conjecture that in fact $N(Q,e^t)e^{-2t}\,ds\underset{t\to\infty}\rightarrow c$, but this seems beyond current methods (yet is known in special cases [A. Eskin, H. A. Masur and M. Schmoll, Duke Math. J. 118 (2003), no. 3, 427–463; MR1983037; A. Eskin, J. Marklof and D. W. Morris, Ergodic Theory Dynam. Systems 26 (2006), no. 1, 129–162; MR2201941; K. Calta and K. Wortman, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 379–398; MR2599885; M. Bainbridge, Geom. Funct. Anal. 20 (2010), no. 2, 299–356; MR2671280]).

Reviewed by Boris Hasselblatt