Maryam Mirzakhani

How to use this course

Mirzakhani’s career lasted barely fifteen years after her doctorate, yet it reshaped at least four areas: hyperbolic geometry, the intersection theory of moduli spaces, Teichmüller dynamics, and the probabilistic study of random surfaces. A biography that skips the mathematics misses the person, because the mathematics was where she lived. This course therefore alternates between narrative lessons (Parts I and III) and technical lessons (Part II).

Each technical lesson states its results precisely, sketches the key idea of the proof, and ends with exercises. Solutions are folded under each exercise. Readers who want only the life story can read Parts I and III, plus the opening paragraph of each lesson in Part II, which is written for a general reader.

Timeline

Tehran, 1977–1995

Maryam Mirzakhani was born in Tehran on 12 May 1977, less than two years before the revolution, and spent her early childhood during the Iran–Iraq war. When the war ended in 1988 she was eleven. She later credited that timing: the end of the war reopened opportunities, and she passed the entrance examination for the Farzanegan school for girls, run by Iran’s National Organization for Development of Exceptional Talents (NODET).

Her first ambition was to be a writer. She read everything she could find, and with her closest friend, Roya Beheshti, she haunted the bookshops of Tehran, buying books almost at random. Biographies of women such as Marie Curie and Helen Keller stayed with her. Mathematics did not come first: in her first year of middle school a teacher judged her untalented, and her confidence fell with her marks. A different teacher the following year changed that.

The Olympiad

In high school the two friends found a set of olympiad problems and worked on them for days. Mirzakhani solved three of the six. They then asked their principal to organise the kind of problem-solving class that boys’ schools already had. She agreed, and in 1994 both girls were selected for the Iranian team, the first girls to make it. At the International Mathematical Olympiad in Hong Kong, Mirzakhani scored 41 out of 42 and won gold. In Toronto in 1995 she scored a perfect 42 and became the first Iranian student to win two gold medals.

During these years she and Beheshti also co-wrote a book of elementary number theory problems in Persian. Beheshti also became a professional mathematician, working in algebraic geometry.

Sharif University, 1995–1999

She entered the Sharif University of Technology in 1995 with a fellowship from the Institute for Research in Fundamental Sciences (IPM). Sharif had a lively culture of problem seminars, and she published while still an undergraduate. One early paper constructed a small planar graph that is not 4-choosable, a contribution to list colouring that sharpened Voigt’s 1993 example. She also found a simpler proof of a theorem of Schur in linear algebra.

17 March 1998

Returning from a mathematics competition in Ahvaz, the bus carrying Sharif students left the road and fell into a ravine. Seven students and two drivers were killed. Mirzakhani was one of the survivors. Many of the dead were friends and fellow olympians; the loss marked the small community of Iranian mathematicians for years.

She graduated in 1999 and left for graduate school at Harvard.

Harvard and the thesis, 1999–2004

At Harvard she began attending Curtis McMullen’s seminar on hyperbolic geometry. She understood little at first, by her own account, but she kept coming and kept asking questions, and McMullen became her advisor. She was drawn to problems that are simple to state about surfaces and hard to answer.

Her 2004 thesis, Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves, solved three problems at once. It computed the Weil–Petersson volumes of moduli spaces by a recursion, gave a new proof of the Witten–Kontsevich theorem, and found the exact polynomial growth rate of simple closed geodesics on a hyperbolic surface. The work appeared in three papers, in Inventiones Mathematicae, the Journal of the American Mathematical Society and the Annals of Mathematics. A single thesis yielding papers in all three is extremely rare.

Her way of working was already visible. She thought in pictures, drawing on large sheets of paper spread across the floor, and she was content to stay with a problem for a long time. Her colleagues described her as slow in the way that good mathematicians are slow: unhurried, persistent, and unwilling to stop at a partial understanding. Part II explains what the thesis did.

Surfaces, Teichmüller space and moduli space

For the general reader. A surface such as a doughnut with two holes can be given many shapes. Mirzakhani studied not one shape but the space of all of them, called moduli space, and asked how measurements on a single surface relate to the geometry of that whole space.

Hyperbolic surfaces

Let \(S_{g,n}\) be an orientable surface of genus \(g\) with \(n\) punctures or boundary components and \(\chi = 2-2g-n<0\). By uniformization, every complex structure on \(S_{g,n}\) corresponds to a unique complete metric of constant curvature \(-1\). By Gauss–Bonnet, every such surface has area

In every free homotopy class of essential closed curves there is a unique closed geodesic. A geodesic is simple if it has no self-intersections.

Teichmüller space and Fenchel–Nielsen coordinates

The Teichmüller space \(\mathcal T_{g,n}\) is the space of hyperbolic metrics on \(S_{g,n}\) together with a marking, up to isotopy. Cut the surface along \(N = 3g-3+n\) disjoint simple closed geodesics \(\gamma_1,\dots,\gamma_N\); it falls apart into \(2g-2+n\) pairs of pants. A pair of pants with geodesic boundary is determined by its three boundary lengths. Rebuilding \(X\) requires the lengths \(\ell_i\) and the twist \(\tau_i\) with which each seam is reglued. This gives the Fenchel–Nielsen coordinates

Moduli space and the Weil–Petersson form

The mapping class group \(\mathrm{Mod}_{g,n}\), orientation-preserving diffeomorphisms modulo isotopy, acts properly discontinuously on \(\mathcal T_{g,n}\). The quotient \(\mathcal M_{g,n} = \mathcal T_{g,n}/\mathrm{Mod}_{g,n}\) is the moduli space of hyperbolic surfaces, an orbifold. It carries the Weil–Petersson Kähler metric, whose symplectic form has a strikingly simple expression.

The volume form is \(\omega_{WP}^N/N!\), and \(\mathcal M_{g,n}\) has finite volume. For surfaces with geodesic boundary components of fixed lengths \(L=(L_1,\dots,L_n)\), write \(\mathcal M_{g,n}(L)\) and \(V_{g,n}(L)=\operatorname{Vol}_{WP}\mathcal M_{g,n}(L)\); a cusp corresponds to \(L_i=0\).

Closed versus simple closed geodesics

Let \(c_X(L)\) count primitive closed geodesics of length at most \(L\) on a closed hyperbolic surface \(X\). The prime geodesic theorem of Huber and Margulis states

an exact analogue of the prime number theorem. Simple closed geodesics are far rarer: Birman and Series showed that their union is nowhere dense and has Hausdorff dimension 1, and McShane and Rivin proved polynomial upper and lower bounds. Finding the exact growth rate was the problem Mirzakhani solved (Lesson 07).

McShane’s identity and the volume recursion

For the general reader. Mirzakhani found a way to compute the volume of moduli space by summing a quantity over all simple closed curves on every surface at once. A strange identity about a single surface became a formula for the whole space.

McShane’s identity

The identity comes from geodesics shot from the cusp: almost all of them hit themselves or return to the cusp, and the ones that do so near a given simple geodesic \(\gamma\) fill a set of measure governed by \(\ell_\gamma\). Mirzakhani generalized this to every surface with geodesic boundary. With boundary \(\beta_1\) of length \(L_1\), define

The unfolding trick

The identity holds on every \(X\in\mathcal M_{g,n}(L)\), so it can be integrated over moduli space. The left side is a sum over a mapping class group orbit, and such sums unfold. For a simple closed curve \(\gamma\) and a function \(f\) on \(\mathbb R_{>0}\),

where \(c_\gamma\) is an explicit symmetry constant and \(V(S\setminus\gamma; x,x,L)\) is the volume of the moduli space of the cut surface, whose two new boundaries both have length \(x\). The factor \(x\) is the range \([0,x)\) of the twist parameter around \(\gamma\), and Wolpert’s formula makes the measure split as \(d\ell\,d\tau\) times the volume of what remains.

The recursion

Applying the trick to each term of the identity gives Mirzakhani’s recursion. Write \(K=\{2,\dots,n\}\) and

The base cases are \(V_{0,3}(L)=1\) and \(V_{1,1}(L) = \frac{1}{48}(L^2+4\pi^2)\), with the common convention that includes a factor \(\tfrac12\) for the elliptic involution of every one-holed torus. The recursion then gives, for example,

The terms decompose the surface the way a pair of pants containing \(\beta_1\) can sit inside it: removing the pants leaves a connected surface, two pieces, or a surface where two boundaries have merged. Eynard and Orantin showed in 2007 that the recursion is an instance of their topological recursion, for the spectral curve \(x=z^2,\ y=\sin(2\pi z)/4\pi\).

A new road to Witten–Kontsevich

For the general reader. In 1990 the physicist Edward Witten predicted that certain numbers attached to moduli spaces obey the equations of a classical wave hierarchy. Maxim Kontsevich proved this in 1992. Mirzakhani’s volume recursion gave an independent, purely geometric proof.

Intersection numbers

On the Deligne–Mumford compactification \(\overline{\mathcal M}_{g,n}\), let \(\psi_i\) be the first Chern class of the line bundle whose fibre is the cotangent line at the \(i\)-th marked point, and let \(\kappa_1\) be the first Mumford–Morita–Miller class; the Weil–Petersson form satisfies \([\omega_{WP}] = 2\pi^2\kappa_1\) (Wolpert). The intersection numbers

are the subject of Witten’s conjecture: their generating function \(F(t_0,t_1,\dots)\) satisfies the KdV hierarchy, equivalently the Virasoro constraints.

Volumes are intersection numbers

Mirzakhani realised \(\mathcal M_{g,n}(L)\) as a symplectic reduction of a torus action on a larger moduli space of surfaces with marked boundary points. The Duistermaat–Heckman theorem then expresses the volume through characteristic classes:

The top-degree part (\(m=0\)) of \(V_{g,n}\) is therefore the generating function of \(\psi\)-intersection numbers. Reading off the top-degree part of the recursion of Lesson 05 yields exactly the Virasoro constraints, and so Witten’s conjecture. Mulase and Safnuk made this derivation explicit in 2006.

Counting simple closed geodesics

For the general reader. On a hyperbolic surface the number of closed geodesics of length at most \(L\) grows exponentially, like the primes. If one keeps only the geodesics that never cross themselves, the growth becomes polynomial. Mirzakhani found the exact exponent, the exact constant, and proved that the proportions of different types of curves are rational numbers that do not depend on the shape of the surface.

Measured laminations

Thurston’s space \(\mathcal{ML}_{g,n}\) of measured geodesic laminations is a piecewise-linear manifold homeomorphic to \(\mathbb R^{6g-6+2n}\). Train-track charts give it a natural integral structure, whose integer points \(\mathcal{ML}_{g,n}(\mathbb Z)\) are the integral multicurves, and a \(\mathrm{Mod}_{g,n}\)-invariant Lebesgue-class measure \(\mu_{\mathrm{Th}}\). Hyperbolic length extends continuously and homogeneously to \(\ell_X:\mathcal{ML}_{g,n}\to\mathbb R_{\ge0}\). Define the Mirzakhani function

Idea of the proof

Fix \(\gamma\) and consider, for each \(X\), the counting measure \(\mu_L^X = L^{-(6g-6+2n)}\sum_{\alpha\in\mathrm{Mod}\cdot\gamma}\delta_{\alpha/L}\) on \(\mathcal{ML}_{g,n}\). Two ingredients identify its limit.

1. Ergodicity. Any weak limit is \(\mathrm{Mod}\)-invariant and absolutely continuous, and the action of \(\mathrm{Mod}_{g,n}\) on \(\mathcal{ML}_{g,n}\) is ergodic for \(\mu_{\mathrm{Th}}\) (Masur). So every limit is \(c\cdot\mu_{\mathrm{Th}}\) for some \(c\), and evaluating on \(\{\ell_X\le1\}\) gives \(s_X(L,\gamma)\sim c\,B(X)L^{6g-6+2n}\).
2. Integration. The constant \(c\) is the same for all \(X\). Integrating \(s_X(L,\gamma)\) over \(\mathcal M_{g,n}\) with the unfolding trick of Lesson 05 gives an explicit polynomial in \(L\) whose leading coefficient is \(c(\gamma)\). Comparing with \(c\int B = c\,b_{g,n}\) fixes \(c=c(\gamma)/b_{g,n}\).

The genus-two surprise

On a closed surface of genus 2 there are two types of simple closed curves: non-separating, and separating (cutting the surface into two one-holed tori). The theorem says their frequencies are in a fixed rational ratio on every genus-2 surface. The correct ratio is 48 non-separating curves for each separating one. Her original paper reported a ratio of 6; the error, by a factor of eight, came to light through later work, and Mark Bell confirmed the value 48 by large-scale computer experiments in 2019.

The earthquake flow is ergodic

For the general reader. Thurston introduced “earthquakes”: cut a surface along a curve, slide one side against the other, and reglue. Doing this continuously defines a flow on moduli space. Mirzakhani proved that this flow mixes moduli space thoroughly, by showing it secretly coincides with a much better understood flow.

Earthquakes

For a simple closed geodesic \(\gamma\), the twist deformation \(\mathrm{tw}^t_\gamma\) changes the Fenchel–Nielsen twist about \(\gamma\) by \(t\). Thurston extended this by continuity to any measured lamination \(\lambda\), defining the left earthquake \(\mathrm{tw}^t_\lambda X\). His earthquake theorem states that any two points of \(\mathcal T_{g}\) are joined by a unique left earthquake. On the bundle of unit-length laminations

the earthquake flow is \(E_t(X,\lambda) = (\mathrm{tw}^t_\lambda X,\lambda)\). It preserves a natural Lebesgue-class measure built from \(\mu_{\mathrm{Th}}\) and \(\omega_{WP}\).

The conjugacy

On the bundle \(Q^1\mathcal M_{g,n}\) of unit-area quadratic differentials, \(\mathrm{SL}(2,\mathbb R)\) acts, and the horocycle flow is \(h_t = \begin{pmatrix}1&t\\0&1\end{pmatrix}\). Masur and Veech proved it is ergodic and mixing for the Masur–Veech measure.

The map is built from Thurston’s horocyclic foliation: for generic \((X,\lambda)\) the lamination \(\lambda\) fills the surface, the complementary ideal triangles carry a canonical foliation by horocyclic arcs, and \(\lambda\) together with this foliation determines a quadratic differential. The point of the theorem is that a flow defined by hyperbolic geometry and one defined by flat geometry are the same dynamical system. As an application she obtained a second proof of the counting theorem through equidistribution of earthquake orbits.

The magic wand theorem

For the general reader. Take a polygon made of glass, glue its sides in pairs, and you get a translation surface. The group of linear maps of the plane acts on all such surfaces at once. Eskin, Mirzakhani and Mohammadi proved that the closure of any orbit of this action is a remarkably tame object. Problems about billiards in rational polygons, previously out of reach, became questions about these closures, which is why Anton Zorich called the result a magic wand.

Translation surfaces and Ratner’s theorem

A translation surface is a pair \((X,\omega)\) with \(\omega\) a non-zero holomorphic 1-form; equivalently a finite union of polygons with sides glued by translations. The space \(\mathcal H(\kappa)\) of such surfaces with zeros of orders \(\kappa=(k_1,\dots,k_s)\) is a stratum. Period coordinates \(\omega\mapsto(\int_{\gamma_i}\omega)\in H^1(X,Z(\omega);\mathbb C)\) give it an affine structure, and \(\mathrm{GL}^+(2,\mathbb R)\) acts by post-composition with the charts. The diagonal flow \(g_t=\mathrm{diag}(e^t,e^{-t})\) is the Teichmüller geodesic flow.

On homogeneous spaces \(G/\Gamma\), Ratner’s theorems say that orbit closures and invariant measures of unipotent flows are algebraic. Strata are not homogeneous, and orbit closures of \(g_t\) can be fractal. The question was whether the full \(\mathrm{SL}(2,\mathbb R)\) action is rigid anyway.

Filip showed in 2016 that every such submanifold is a quasi-projective algebraic variety. The proof of Eskin–Mirzakhani runs over two hundred pages and adapts the “exponential drift” method of Benoist and Quint to a setting with no ambient Lie group, using Oseledets theory for the Kontsevich–Zorich cocycle.

Applications

Billiards in a polygon whose angles are rational multiples of \(\pi\) unfold to the straight-line flow on a translation surface. The magic wand gives, for every rational polygon, quadratic growth of periodic billiard trajectories in a Cesàro-averaged sense, with constants determined by the orbit closure. It also settles questions about the wind-tree model of Ehrenfest and the illumination problem: in any rational polygon, a light source illuminates all but finitely many points (Lelièvre, Monteil and Weiss, 2016).

Random surfaces of large genus

For the general reader. Since moduli space has a finite volume, one can pick a surface “at random”. In her last years Mirzakhani showed what a typical surface of very large genus looks like, opening a probabilistic theory of geometry that is now one of the most active areas of the subject.

The Weil–Petersson model

Normalising \(\omega_{WP}^{3g-3}/(3g-3)!\) on \(\mathcal M_g\) gives a probability measure \(\mathbb P_g\). Mirzakhani’s integration formulas make the expected number of geodesics of a given type an explicit integral of volumes, and so large-genus asymptotics of \(V_{g,n}\) become geometric statements.

With Peter Zograf she proved the volume asymptotics \(V_g \sim C\,\frac{(2g-3)!\,(4\pi^2)^{2g-3}}{\sqrt g}\), and with Bram Petri that the lengths of short closed geodesics on a random surface converge to a Poisson point process with intensity \(\frac{\cosh t - 1}{t}\,dt\). Her spectral bound was the first of a sequence: \(\frac{3}{16}-\varepsilon\) by Wu–Xue and by Lipnowski–Wright (2021), and the optimal \(\frac14-\varepsilon\) by Anantharaman and Monk, building directly on her volume expansions.

Princeton, Stanford, Seoul, 2004–2014

After her doctorate she became a Clay Research Fellow and an assistant professor at Princeton, where she wrote the earthquake-flow paper and began her long collaboration with Alex Eskin. In 2008, at 31, she was appointed full professor at Stanford. She married Jan Vondrák, a Czech theoretical computer scientist, and they had a daughter, Anahita. Watching her mother cover big sheets with drawings and formulas, Anahita called the work “painting”.

Recognition followed quickly: the AMS Blumenthal Award in 2009, the Satter Prize in 2013, and the Clay Research Award in 2014. On 13 August 2014, at the opening of the International Congress of Mathematicians in Seoul, she received the Fields Medal for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces. She was the first woman and the first Iranian to receive it, in the eight decades since the medal was first awarded in 1936.

She disliked publicity and accepted it mainly for what it might mean to young women. Asked what doing mathematics was like, she compared it to solving a puzzle or “connecting the dots in a detective case”, and elsewhere to being lost in a jungle and building new tools to find a way out. She also compared it to writing a novel, where characters change as you get to know them.

Illness, death and legacy, 2013–2017

She was diagnosed with breast cancer in 2013, the year before the Fields Medal. She kept working through treatment. In 2016 the cancer spread to her liver and bones. She died on 14 July 2017 at the age of 40. Iran’s President Hassan Rouhani issued a statement of mourning, and several Iranian newspapers broke with convention by printing photographs of her without a headscarf.

Mathematical legacy

Papers continued to appear after her death, among them effective counting of simple closed geodesics with Eskin and Mohammadi, and the classification of full-rank orbit closures with Alex Wright. Every lesson in Part II now heads a living field: topological recursion and its links to Jackiw–Teitelboim gravity, where Saad, Shenker and Stanford showed in 2019 that Mirzakhani’s volumes compute the path integral; the classification of orbit closures; and the spectral theory of random surfaces.

Honours in her name

• Since 2018, her birthday, 12 May, is celebrated worldwide as the day of women in mathematics.
• The Breakthrough Prize Foundation awards the Maryam Mirzakhani New Frontiers Prize to women mathematicians early in their careers.
• Student societies, lecture series and fellowships bear her name, including the Mirzakhani Society for women and non-binary mathematics students at Oxford, which she visited in 2015.
• The documentary Secrets of the Surface (George Csicsery, 2020) presents her life and mathematics.

Connections

People linked to Maryam Mirzakhani across this collection. A name without a link has no page here yet. See Maryam Mirzakhani in the network of all 337 lives →

Teachers and advisors

• Curtis McMullen 2004

Influenced by

• Edward Witten
• Grigory Margulis
• Howard Masur
• Marina Ratner
• Maxim Kontsevich
• Oswald Teichmüller
• William Thurston

Colleagues and collaborators

• Alex Eskin 2006-2017
• Alex Wright
• Amir Mohammadi

Friends

• Roya Beheshti

Also mentioned in this life

• André Weil
• Pierre Deligne
• Paul Ehrenfest
• Carl Friedrich Gauss
• Felix Hausdorff
• Pierre-Simon Laplace
• Henri Lebesgue
• Sophus Lie
• Marie Curie
• David Mumford
• Siméon Denis Poisson
• Bernhard Riemann
• Issai Schur

Fields

• hyperbolic geometry 3 lives
• teichmüller theory 3 lives
• ergodic theory 7 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Maryam Mirzakhani, MacTutor History of Mathematics.
2. International Mathematical Union, The work of Maryam Mirzakhani, Fields Medal news release, 2014.
3. A. Wright, Maryam Mirzakhani (1977–2017), retrospective.
4. NPR, Maryam Mirzakhani, prize-winning mathematician, dies at 40, 15 July 2017.
5. E. Klarreich, “A tenacious explorer of abstract surfaces”, Quanta Magazine, 12 August 2014.

Mathematics

1. M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007).
2. M. Mirzakhani, Weil–Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007).
3. M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. 168 (2008).
4. M. Mirzakhani, Ergodic theory of the earthquake flow, Int. Math. Res. Not. (2008).
5. M. Mirzakhani, Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom. 94 (2013).
6. A. Eskin, M. Mirzakhani, A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL(2,ℝ) action on moduli space, Ann. of Math. 182 (2015).
7. A. Eskin, M. Mirzakhani, Invariant and stationary measures for the SL(2,ℝ) action on moduli space, Publ. Math. IHÉS 127 (2018).
8. A. Zorich, The magic wand theorem of A. Eskin and M. Mirzakhani, arXiv:1502.05654.
9. A. Wright, A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces, Bull. AMS (2020).
10. F. Arana-Herrera, Counting problems from the viewpoint of ergodic theory, Ergodic Theory Dynam. Systems (2025).
11. M. Bell, Experimental statistics for Mirzakhani’s theorem, arXiv:1910.08155.
12. C. McMullen, The mathematical work of Maryam Mirzakhani, Proc. ICM 2014.