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Maryam Mirzakhani, (born May 3, 1977, Tehran, Iran), Iranian mathematician who became (2014) the first woman and the first Iranian to be awarded a Fields Medal. The citation for her award recognized “her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.”

While a teenager, Mirzakhani won gold medals in the 1994 and 1995 International Mathematical Olympiads for high-school students, attaining a perfect score in 1995. In 1999 she received a B.Sc. degree in mathematics from the Sharif University of Technology in Tehran. Five years later she earned a Ph.D. from Harvard University for her dissertation Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves. Mirzakhani served (2004–08) as a Clay Mathematics Institute research fellow and an assistant professor of mathematics at Princeton University. In 2008 she became a professor at Stanford University.

Mirzakhani’s work focused on the study of hyperbolic surfaces by means of their moduli spaces. In hyperbolic space, in contrast with normal Euclidean space, Euclid’s fifth postulate (that one and only one line parallel to a given line can pass through a fixed point) does not hold. In non-Euclidean hyperbolic space, an infinite number of parallel lines can pass through such a fixed point. The sum of the angles of a triangle in hyperbolic space is less than 180°. In such a curved space, the shortest path between two points is known as a geodesic. For example, on a sphere the geodesic is a great circle. Mirzakhani’s research involved calculating the number of a certain type of geodesic, called simple closed geodesics, on hyperbolic surfaces.

Her technique involved considering the moduli spaces of the surfaces. In this case the modulus space is a collection of all Riemann spaces that have a certain characteristic. Mirzakhani found that a property of the modulus space corresponds to the number of simple closed geodesics of the hyperbolic surface.

Tehran

As a child growing up in Tehran, Mirzakhani had no intention of becoming a mathematician. Her chief goal was simply to read every book she could find. She also watched television biographies of famous women such as Marie Curie and Helen Keller, and later read “Lust for Life,” a novel about Vincent van Gogh. These stories instilled in her an undefined ambition to do something great with her life — become a writer, perhaps.

Mirzakhani finished elementary school just as the Iran-Iraq war was drawing to a close and opportunities were opening up for motivated students. She took a placement test that secured her a spot at the Farzanegan middle school for girls in Tehran, which is administered by Iran’s National Organization for Development of Exceptional Talents. “I think I was the lucky generation,” she said. “I was a teenager when things got more stable.”

In her first week at the new school, she made a lifelong friend, Roya Beheshti, who is now a mathematics professor at Washington University in St. Louis. As children, the two explored the bookstores that lined the crowded commercial street near their school. Browsing was discouraged, so they randomly chose books to buy. “Now, it sounds very strange,” Mirzakhani said. “But books were very cheap, so we would just buy them.”To her dismay, Mirzakhani did poorly in her mathematics class that year. Her math teacher didn’t think she was particularly talented, which undermined her confidence. At that age, “it’s so important what others see in you,” Mirzakhani said. “I lost my interest in math.”

The following year, Mirzakhani had a more encouraging teacher, however, and her performance improved enormously. “Starting from the second year, she was a star,” Beheshti said.

Mirzakhani went on to the Farzanegan high school for girls. There, she and Beheshti got hold of the questions from that year’s national competition to determine which high school students would go to the International Olympiad in Informatics, an annual programming competition for high school students. Mirzakhani and Beheshti worked on the problems for several days and managed to solve three out of six. Even though students at the competition must complete the exam in three hours, Mirzakhani was excited to be able to do any problems at all.

Eager to discover what they were capable of in similar competitions, Mirzakhani and Beheshti went to the principal of their school and demanded that she arrange for math problem-solving classes like the ones being taught at the comparable high school for boys. “The principal of the school was a very strong character,” Mirzakhani recalled. “If we really wanted something, she would make it happen.” The principal was undeterred by the fact that Iran’s International Mathematical Olympiad team had never fielded a girl, Mirzakhani said. “Her mindset was very positive and upbeat — that ‘you can do it, even though you’ll be the first one,’ ” Mirzakhani said. “I think that has influenced my life quite a lot.”

In 1994, when Mirzakhani was 17, she and Beheshti made the Iranian math Olympiad team. Mirzakhani’s score on the Olympiad test earned her a gold medal. The following year, she returned and achieved a perfect score. Having entered the competitions to discover what she could do, Mirzakhani emerged with a deep love of mathematics. “You have to spend some energy and effort to see the beauty of math,” she said.

Even today, said Anton Zorich of the Université Paris Diderot-Paris 7 in France, Mirzakhani gives “the impression of a 17-year-old girl who is absolutely excited by all the mathematics that happens around her.”

Harvard

Gold medals at the mathematical Olympiad don’t always translate into success in mathematics research, McMullen observed. “In these contests, someone has carefully crafted a problem with a clever solution, but in research, maybe the problem doesn’t have a solution at all.” Unlike many Olympiad high-scorers, he said, Mirzakhani “has the ability to generate her own vision.”

After completing an undergraduate degree in mathematics at Sharif University in Tehran in 1999, Mirzakhani went to graduate school at Harvard University, where she started attending McMullen’s seminar. At first, she didn’t understand much of what he was talking about but was captivated by the beauty of the subject, hyperbolic geometry. She started going to McMullen’s office and peppering him with questions, scribbling down notes in Farsi.

“She had a sort of daring imagination,” recalled McMullen, a 1998 Fields medalist. “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, ‘Is it right?’ I was always very flattered that she thought I would know.”

Mirzakhani became fascinated with hyperbolic surfaces — doughnut-shaped surfaces with two or more holes that have a non-standard geometry which, roughly speaking, gives each point on the surface a saddle shape. Hyperbolic doughnuts can’t be constructed in ordinary space; they exist in an abstract sense, in which distances and angles are measured according to a particular set of equations. An imaginary creature living on a surface governed by such equations would experience each point as a saddle point.

It turns out that each many-holed doughnut can be given a hyperbolic structure in infinitely many ways — with fat doughnut rings, narrow ones, or any combination of the two. In the century and a half since such hyperbolic surfaces were discovered, they have become some of the central objects in geometry, with connections to many branches of mathematics and even physics.

But when Mirzakhani started graduate school, some of the simplest questions about such surfaces were unanswered. One concerned straight lines, or “geodesics,” on a hyperbolic surface. Even a curved surface can have a notion of a “straight” line segment: it’s simply the shortest path between two points. On a hyperbolic surface, some geodesics are infinitely long, like straight lines in the plane, but others close up into a loop, like the great circles on a sphere.

The number of closed geodesics of a given length on a hyperbolic surface grows exponentially as the length of the geodesics grows. Most of these geodesics cut across themselves many times before closing up smoothly, but a tiny proportion of them, called “simple” geodesics, never intersect themselves. Simple geodesics are “the key object to unlocking the structure and geometry of the whole surface,” Farb said.

Yet mathematicians couldn’t pin down just how many simple closed geodesics of a given length a hyperbolic surface can have. Among closed geodesic loops, the simple ones are “miracles that [effectively] happen zero percent of the time,” Farb said. For that reason, counting them accurately is incredibly difficult: “If you have a little bit of an error, you’ve missed it,” he said.

In her doctoral thesis, completed in 2004, Mirzakhani answered this question, developing a formula for how the number of simple geodesics of length L grows as Lgets larger. Along the way, she built connections to two other major research questions, solving both. One concerned a formula for the volume of the so-called “moduli” space — the set of all possible hyperbolic structures on a given surface. The other was a surprising new proof of an old conjecture proposed by the physicistEdward Witten of the Institute for Advanced Study in Princeton, N.J., about certain topological measurements of moduli spaces related to string theory. Witten’s conjecture is so difficult that the first mathematician to prove it — Maxim Kontsevichof the Institut des Hautes études Scientifiques, near Paris — was awarded a Fields Medal in 1998 in part for that work.

Farb said that solving each of these problems “would have been an event, and connecting them would have been an event.” Mirzakhani did both.

Mirzakhani’s thesis resulted in three papers published in the three top journals of mathematics: Annals of Mathematics, Inventiones Mathematicae and Journal of the American Mathematical Society. The majority of mathematicians will never produce something as good, Farb said — “and that’s what she did in her thesis.”

‘A Titanic Work’

Mirzakhani likes to describe herself as slow. Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. “Months or years later, you see very different aspects” of a problem, she said. There are problems she has been thinking about for more than a decade. “And still there’s not much I can do about them,” she said.

Mirzakhani doesn’t feel intimidated by mathematicians who knock down one problem after another. “I don’t get easily disappointed,” she said. “I’m quite confident, in some sense.”

Her slow and steady approach also applies to other areas of her life. One day while she was a graduate student at Harvard, her future husband, then a graduate student at the Massachusetts Institute of Technology, learned this lesson about Mirzakhani when the two went for a run. “She’s very petite, and I was in good shape, so I thought I’d do well, and at first, I was ahead,” recalled Jan Vondrak, who is now a theoretical computer scientist at IBM Almaden Research Center in San Jose, Calif. “But she never slows down. After half an hour, I was done, but she was still running at the same pace.”

As she thinks about mathematics, Mirzakhani constantly doodles, drawing surfaces and other images related to her research. “She has these huge pieces of paper on the floor and spends hours and hours drawing what look to me like the same picture over and over,” Vondrak said, adding that papers and books are scattered haphazardly about her home office. “I have no idea how she can work like this, but it works out in the end,” he said. Perhaps, he speculates, that is because “the problems she is working on are so abstract and complicated, she can’t afford to make logical steps one by one but has to make big jumps.”

Mirzakhani is the first woman to win a Fields Medal. The gender imbalance in mathematics is long-standing and pervasive, and the Fields Medal, in particular, is ill-suited to the career arcs of many female mathematicians. It is restricted to mathematicians younger than 40, focusing on the very years during which many women dial back their careers to raise children.

Mirzakhani feels certain, however, that there will be many more female Fields medalists in the future. “There are really many great female mathematicians doing great things,” she said.

In the meantime, while she feels greatly honored to have been awarded a Fields Medal, she has no desire to be the face of women in mathematics, she said. Her ambitious teenage self would have been overjoyed by the award, she said, but today, she is eager to deflect attention from her achievements so she can focus on research.

Mirzakhani has big plans for the next chapters of her mathematical story. She has started working with Wright to try to develop a complete list of the kinds of sets that translation surface orbits can fill up. Such a classification would be a “magic wand” for understanding billiards and translation surfaces, Zorich has written.

It’s no small task, but Mirzakhani has learned over the years to think big. “You have to ignore low-hanging fruit, which is a little tricky,” she said. “I’m not sure if it’s the best way of doing things, actually — you’re torturing yourself along the way.” But she enjoys it, she said. “Life isn’t supposed to be easy.”

Thomas Lin contributed reporting from Stanford, Calif.

This article is part of a five-part series on the 2014 Fields Medal and Nevanlinna Prize winners, reprinted with permission from Quanta Magazine, an editorially independent division of SimonsFoundation.org whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

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Clay Mathematics Institute Research Fellow 2004

Harvard Junior Fellowship Harvard University, 2003

Merit fellowship Harvard University, 2003

IPM Fellowship The Institute for theoretical Physics and Mathematics, Tehran, Iran, 1995-1999

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