Maryam Mirzakhani (1977–2017) Pioneering Mathematician and Winner of the Fields Medal

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COMMENT OBITUARY Maryam Mirzakhani (1977–2017) Pioneering mathematician and winner of the Fields Medal.

aryam Mirzakhani was one of University of Chicago in Illinois, Mirzakhani the greatest mathematicians used this method to prove, for such spaces, of her generation. She made a version of a theorem about a group of Mmonumental contributions to the study of symmetric geometric objects known as the dynamics and geometry of mathemati- Lie groups. The theorem was proposed cal objects called Riemann surfaces. Just as by Marina Ratner, another leading math- impressive as her theorems was her ability to ematician in the field who also died in July, push a field in a new direction by always pro- aged 78. The proof — a monumental work

viding a fresh point of view. Her raw talent written up in a 200-page paper (A. Eskin MIRZAKHANI/CORBIS/GETTY MARYAM was rare, even among the most celebrated and M. Mirzakhani Preprint at https://arxiv. mathematicians, and she was known for org/abs/1302.3320; 2013) — tied together having a taste for difficult problems. disparate fields including geometry, topol- She became an icon without wanting to ogy and dynamical systems, and spawned be. She was the first woman and first Iranian a field of its own. It has been dubbed the to win the Fields Medal, considered the ‘magic wand’ theorem because it enabled highest honour in mathematics. For women, many previously intractable mathematical Mirzakhani was a role model, pursuing problems to be solved. a successful career in a male-dominated Despite the fame and attention she field. For Iran, she represented the country’s received, Mirzakhani remained humble and tradition of intellectualism. And for young grounded, always avoiding the spotlight. She scientists, she was a calming force that rose listened to the work of other mathematicians above the pressures of academia. She died with excitement and asked forward-looking aged 40 from breast cancer on 14 July. find the average of all such numbers questions that hinted at possible new direc- Mirzakhani was born in May 1977 in corresponding to points in the ‘moduli tions. At conferences, she could be found Tehran. She attended school there and twice space’ of Riemann surfaces: a ‘space’, or set, talking with graduate students and Fields won gold medals for Iran in the International of points, each of which represents one of medallists alike. She generously shared her Mathematical Olympiad. Being hailed as a the shapes a surface can take. Computing ideas with the community and helped others genius allowed her to pursue pure math- such an average requires one to calculate to further their careers. ematics — not an easy career choice for the ‘volume’, or size, of the space of Riemann I visited Maryam in December 2016. women in Iran. surfaces that contain a curve of a certain We walked from her home in Palo Alto, Mirzakhani gained a bachelor’s degree in length. A clever recursive formula for the California, to Stanford’s maths depart- mathematics in 1999 from the Sharif Univer- volumes of various moduli spaces solved ment to listen to a lecture by the Russian– sity of Technology in Tehran. She left to do the problem. The solution had several stun- French mathematician Mikhail Gromov. doctoral work in the United States and earned ning ramifications in seemingly distant Mirzakhani was diagnosed with cancer in her PhD in 2004 from Harvard University fields. For example, it offered a new proof 2013 and had already been treated for the in Cambridge, Massachusetts, under the of a famous theorem by the Russian–French illness, but by this time it had returned and supervision of Curtis McMullen. She turned mathematician Maxim Kontsevich, which spread, and she was in pain. We stopped down a junior fellowship there to become a had implications in quantum field theory. every few minutes along the walk so that she Clay Mathematics Institute research fellow In later work, Mirzakhani studied the could lie down on a bench to rest. Maryam at Princeton University in New Jersey. She dynamics of a billiard ball, or point mass, told me that she didn’t want to take long- became a full professor at Stanford University moving in a polygon. A ball moves in a term leave from work for her illness and in California in 2008, by which time she was straight line until it hits the edge of the poly- that she would like to continue her respon- considered a leader in the fields of hyperbolic gon; then it bounces back at the same angle sibilities as an editor of the Journal of the geometry, topology and dynamics. She stayed at which it hit. A mathematician could ask American Mathematical Society. I couldn’t at Stanford until her death. several questions about such a system. For resist telling her about the maths problems I Mirzakhani’s PhD concerned Riemann instance, is it possible for a ball to move was thinking about, and despite all that was surfaces. Picture a surface with several holes inside a given polygon in such a way that the going on in her life, she was happy to listen in it, like that of a pretzel or two doughnuts path it takes is eventually repeated — and, if and offer helpful insights. stuck together, and then imagine trying so, how many such paths are there, and what The mathematics community has lost one to wrap a rubber band around the surface do they look like? The problem of whether of its greatest minds much too early, and I without it overlapping itself. Mirzakhani a repeating path exists for a general polygon have lost a friend. ■ wanted to work out how many different is still unsolved. ways this can be done for a rubber band of In some cases, it is helpful to embed the Kasra Rafi is an associate professor in the a given length. space of certain billiard tables in a larger Department of Mathematics, University of She realized that she could flip the space in which every point is a surface Toronto, Toronto, Canada. He was a friend method. Instead of fixing a surface and that is locally either flat or cone-shaped. and collaborator of Maryam Mirzakhani. counting the number of curves, she could With Alex Eskin, a mathematician at the e-mail: [email protected]