Mathematics

Maryam Mirzakhani: The First Woman to Win the Fields Medal

The inspiring story of Maryam Mirzakhani, an Iranian mathematician who made groundbreaking contributions to the geometry of Riemann surfaces and moduli spaces, becoming the first woman to receive the Fields Medal.

Profiles Geometry

Growing Up in Tehran

Maryam Mirzakhani was born on 12 May 1977 in Tehran, Iran. She grew up during the Iran–Iraq War (1980–1988), a period of enormous hardship for the country. Despite the turmoil, she benefited from Iran's strong tradition of mathematical education.

As a child, Mirzakhani was not initially drawn to mathematics. She loved reading novels and dreamed of becoming a writer. It was only in middle school that her interest in mathematics ignited, partly through the encouragement of a teacher and partly through the intellectual challenge of competition problems.

Mathematical Competitions

Mirzakhani attended the Farzanegan School, a school for intellectually gifted girls run by Iran's National Organization for the Development of Exceptional Talents. There she developed her mathematical abilities rapidly.

In 1994, she became the first Iranian woman to compete in the International Mathematical Olympiad (IMO), winning a gold medal with a score of 41 out of 42. The following year, at the 1995 IMO in Toronto, she achieved a perfect score of 42/42, earning a second gold medal.

"You have to spend some energy and effort to see the beauty of math." — Maryam Mirzakhani

Education

Mirzakhani earned her Bachelor's degree in mathematics from the Sharif University of Technology in Tehran in 1999. She then moved to the United States for graduate study at Harvard University, where she worked under the supervision of Curtis McMullen, himself a Fields Medalist.

Her doctoral thesis (2004) was a remarkable piece of work that solved two longstanding problems simultaneously. The thesis contained results that would appear in three papers published in the top mathematics journals — Annals of Mathematics, Inventiones Mathematicae, and the Journal of the American Mathematical Society.

The Geometry of Riemann Surfaces

Mirzakhani's mathematical world was the geometry of Riemann surfaces — two-dimensional surfaces equipped with a complex structure. A central object in her work is the moduli space $\mathcal{M}_{g,n}$ , the space that parametrizes all Riemann surfaces of genus $g$ with $n$ marked points (up to isomorphism).

Simple Closed Geodesics

In her thesis, Mirzakhani studied the number of simple closed geodesics on a hyperbolic surface of genus $g$ . A geodesic on a hyperbolic surface is a curve of locally shortest length. A simple geodesic does not cross itself. She proved an asymptotic formula:

Theorem (Mirzakhani, 2004)

The number of simple closed geodesics of length at most $L$ on a hyperbolic surface $X$ of genus $g$ satisfies:

$s_X(L) \sim c_X \cdot L^{6g - 6} \quad \text{as } L \to \infty$

where $c_X > 0$ is a constant depending on the geometry of $X$ .

The exponent $6g - 6$ is precisely the dimension of the moduli space $\mathcal{M}_g$ , a deep connection that was not at all obvious.

Weil–Petersson Volumes

Mirzakhani developed a recursive formula for the Weil–Petersson volumes of moduli spaces. The Weil–Petersson metric is a natural Riemannian metric on $\mathcal{M}_{g,n}$ , and computing its total volume had been an important open problem. Her recursive formula expresses the volume $V_{g,n}(L_1, \ldots, L_n)$ — where the $L_i$ are the lengths of the boundary components — in terms of volumes of lower-dimensional moduli spaces.

This work had a remarkable consequence: a new proof of the Witten–Kontsevich theorem, which relates the intersection theory of $\mathcal{M}_{g,n}$ to integrable systems. Kontsevich had received the Fields Medal in 1998 partly for his proof of this theorem; Mirzakhani found a completely different proof using hyperbolic geometry.

Dynamics on Moduli Space

After her thesis, Mirzakhani turned to an even deeper area: the dynamics of the action of $\mathrm{SL}(2,\mathbb{R})$ on moduli spaces of Abelian differentials (flat surfaces). This work, carried out in collaboration with Alex Eskin and Amir Mohammadi, culminated in a result often described as the most important theorem in the field in decades.

The Magic Wand Theorem

The Eskin–Mirzakhani theorem (sometimes called the "Magic Wand theorem") classifies orbit closures of the $\mathrm{SL}(2,\mathbb{R})$ action on moduli spaces of translation surfaces:

Theorem (Eskin–Mirzakhani, 2013)

Every $\mathrm{SL}(2,\mathbb{R})$ -orbit closure in a stratum of Abelian differentials is an affine invariant submanifold, locally defined by linear equations with real coefficients in period coordinates.

The proof, which runs to over 200 pages, uses techniques from ergodic theory, algebraic geometry, and dynamical systems. It is a far-reaching analogue of the landmark theorems of Marina Ratner on unipotent flows on homogeneous spaces, adapted to the vastly more complex setting of moduli spaces.

The Fields Medal

In 2014, Maryam Mirzakhani was awarded the Fields Medal at the International Congress of Mathematicians in Seoul, South Korea. She was the first woman and the first Iranian to receive the prize.

The citation recognized her "outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."

"This is a great honor. I will be happy if it encourages young female scientists and mathematicians. I am sure there will be many more women winning this kind of award in coming years." — Maryam Mirzakhani, upon receiving the Fields Medal

Working Style

Mirzakhani's working style was distinctive. She would spread large sheets of paper on the floor and draw intricate diagrams, tracing the geometry of surfaces and geodesics. Her daughter, Anahita, once described her mother's work as "painting."

She was known for tackling deep, difficult problems with patience and persistence, sometimes working on a single question for years. Her approach combined geometric intuition with technical mastery of analysis and algebraic techniques.

Illness and Legacy

In 2013, Mirzakhani was diagnosed with breast cancer. Despite her illness, she continued to work on mathematics. The cancer eventually spread to her bones and liver.

Maryam Mirzakhani died on 14 July 2017 in Stanford, California, at the age of 40. Her death was mourned around the world. Iran's president broke with protocol to publish an unveiled photograph of her. The International Mathematical Union and mathematical societies worldwide paid tribute.

In her honor, the Maryam Mirzakhani New Frontiers Prize was established in 2019 by the Breakthrough Prize Foundation, awarded annually to outstanding early-career women mathematicians.

Mathematical Impact

Mirzakhani's work opened entirely new directions in the study of moduli spaces, Teichmüller theory, and geometric dynamics. Her ideas continue to inspire active research in:

Teichmüller dynamics and the classification of $\mathrm{SL}(2,\mathbb{R})$ -orbit closures
Counting problems in geometry, from simple geodesics to lattice points
Weil–Petersson geometry and intersection theory on moduli spaces
Flat surfaces and interval exchange transformations

Her mathematical legacy, combined with her role as a symbol of excellence and perseverance, ensures that her influence will be felt for generations.

"The beauty of mathematics only shows itself to more patient followers."

— Maryam Mirzakhani

References

Mirzakhani, M., "Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces," Inventiones Mathematicae, 167(1), 2007.
Eskin, A. and Mirzakhani, M., "Invariant and stationary measures for the SL(2,R) action on moduli space," Publications Mathématiques de l'IHÉS, 127(1), 2018.
Eskin, A., Mirzakhani, M. and Mohammadi, A., "Isolation, equidistribution, and orbit closures for the SL(2,R) action on moduli space," Annals of Mathematics, 182(2), 2015.
Wikipedia — Maryam Mirzakhani
Fields Medal 2014 — ICM Citation
Stanford Mathematics — Memorial
Quanta Magazine — "A Tenacious Explorer of Abstract Surfaces"

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