Monday, July 17, 2017

A00758 - Maryam Mirzakhani, First Woman to Win a Fields Medal


Maryam Mirzakhani was awarded a Fields Medal in 2014. CreditStanford University

Maryam Mirzakhani, an Iranian mathematician who was the only woman ever to win a Fields Medal, the most prestigious honor in mathematics, died on Friday. She was 40.
The cause was breast cancer, said Stanford University, where she was a professor. The university did not say where she died.
Her death is “a big loss and shock to the mathematical community worldwide,” said Peter C. Sarnak, a mathematician at Princeton University and the Institute for Advanced Study.
The Fields Medal, established in 1936, is often described as the Nobel Prizeof mathematics. But unlike the Nobels, the Fields are bestowed only on people aged 40 or younger, not just to honor their accomplishments but also to predict future mathematical triumphs. The Fields are awarded every four years, with up to four mathematicians chosen at a time.
“She was in the midst of doing fantastic work,” Dr. Sarnak said. “Not only did she solve many problems; in solving problems, she developed tools that are now the bread and butter of people working in the field.”
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Dr. Mirzakhani was one of four Fields winners in 2014, at the International Congress of Mathematicians in South Korea. Until then, all 52 recipients had been men. She was also the only Iranian ever to win the award.
President Hassan Rouhani of Iran released a statement expressing “great grief and sorrow.”
He wrote, “The unparalleled excellence of the creative scientist and humble person that echoed Iran’s name in scientific circles around the world was a turning point in introducing Iranian women and youth on their way to conquer the summits of pride and various international stages.”
Dr. Mirzakhani’s mathematics looked at the interplay of dynamics and geometry, in some ways a more complicated version of billiards, with balls bouncing from one side to another of a rectangular billiards table eternally.
A ball’s path can sometimes be a repeating pattern. A simple example is a ball that hits a side at a right angle. It would then bounce back and forth in a line forever, never moving to any other part of the table.


But if a ball bounced at an angle, its trajectory would be more intricate, often covering the entire table.
“You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union to profile the 2014 Fields winners. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.”

Maryam Mirzakhani: A Tenacious Explorer of Abstract Surfaces Video by Quanta Magazine

In work with Alex Eskin of the University of Chicago, Dr. Mirzakhani examined billiards tables of more complicated shapes, and in fact considered the dynamics of balls bouncing around all possible tables that fit certain criteria.
It was a challenging problem that had been attacked by many prominent mathematicians. That included Curtis T. McMullen, her thesis adviser at Harvard and also a Fields medalist, who had solved a special case. But no one had a good idea of the path toward a more encompassing solution.
Amie Wilkinson, a mathematics professor at the University of Chicago, recalled sitting in on a meeting with Dr. Mirzakhani and Dr. Eskin. Whereas Dr. Eskin tended to be pessimistic, seeing all the potential pitfalls that could scuttle a proof, Dr. Mirzakhani was the opposite.
“Just pushing and pushing and pushing,” Dr. Wilkinson said. “Completely optimistic the whole time.’’
After a decade of work, Dr. Mirzakhani and Dr. Eskin proved not the original problem that they had set out to solve but a slightly different one.
“When these trajectories unwind,’’ Dr. Wilkinson said, “they reveal deep properties about numbers and geometry.”
Dr. Sarnak said that though Dr. Mirzakhani wrote relatively few papers, she was still a game changer. “I’m sure in the long run, she would have had many more of these decisive papers,” he said.


The front pages of Iranian newspapers on Sunday with pictures of Dr. Mirzakhani. Some news outlets took the unusual step of running a picture of her without a head covering. CreditAtta Kenare/Agence France-Presse — Getty Images

In addition to being mathematically talented, “she was a person who thought deeply from the ground up,” he said.
“That’s always the mark of someone who makes a permanent contribution,” he added.
In an interview in 2014 with Quanta Magazine, published by the Simons Foundation, Dr. Mirzakhani, who described herself as a “slow” mathematician, acknowledged her tendency to take the harder path.
“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.”
Maryam Mirzakhani was born on May 3, 1977, in Tehran. As a child, she read voraciously and wanted to become a writer. Iran was at war with Iraq at the time, but the war ended as she entered middle school.
“I think I was the lucky generation,” she said in the Fields video, “because I was a teenager when things became more stable.”
In high school, she was a member of the Iranian team at the International Mathematical Olympiad. She won a gold medal in the olympiad in 1994, and the next year won another gold medal, with a perfect score.
After completing a bachelor’s degree at Sharif University of Technology in Tehran in 1999, she attended graduate school at Harvard, completing her doctorate in 2004. She then became a professor at Princeton before moving to Stanford in 2008.
Survivors include her husband, Jan Vondrák, who is also a mathematics professor at Stanford, and a daughter, Anahita.
Dr. Mirzakhani often dived into her math research by doodling on vast pieces of paper sprawled on the floor, with equations at the edges. Her daughter described it as “painting.”
“It is like being lost in a jungle,” Dr. Mirzakhani said, “and trying to use all the knowledge that you can gather to come up with some new tricks — and with some luck you might find a way out.”


Winners of the 2014 Fields Medal in mathematics, from left: Maryam Mirzakhani, Artur Avila, Manjul Bhargava and Martin Hairer. CreditInternational Mathematical Union

An Iranian mathematician is the first woman ever to receive a Fields Medal, often considered to be mathematics’ equivalent of the Nobel Prize.
The recipient, Maryam Mirzakhani, a professor at Stanford, was one of four winners honored on Wednesday at the International Congress of Mathematicians in Seoul, South Korea.
The Fields Medal is given every four years, and several can be awarded at once. The other recipients this year are Artur Avila of the National Institute of Pure and Applied Mathematics in Brazil and the National Center for Scientific Research in France; Manjul Bhargava of Princeton University; and Martin Hairer of the University of Warwick in England.
The 52 medalists from previous years were all men.
“This is a great honor. I will be happy if it encourages young female scientists and mathematicians,” Dr. Mirzakhani was quoted as saying in a Stanford news release on Tuesday. “I am sure there will be many more women winning this kind of award in coming years.”
Ingrid Daubechies, a professor of mathematics at Duke and president of the International Mathematical Union, called the news “a great joy” in an email.
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“All researchers in mathematics will tell you that there is no difference between the math done by a woman or a man, and of course the decision of the Fields Medal committee is based only on the results of each candidate,” she wrote. “That said, I bet the vast majority of the mathematicians in the world will be happy that it will no longer be possible to say that ‘the Fields Medal has always been awarded only to men.’ ”
Much of the research by Dr. Mirzakhani, who was born in Tehran in 1977, has involved the behavior of dynamical systems. There are no exact mathematical solutions for many dynamical systems, even simple ones.
“What Maryam discovered is that in another regime, the dynamical orbits are tightly constrained to follow algebraic laws,” said Curtis T. McMullen, a professor at Harvard who was Dr. Mirzakhani’s doctoral adviser. “These dynamical systems describe surfaces with many handles, like pretzels, whose shape is evolving over time by twisting and stretching in a precise way. They are related to billiards on tables that are not rectangular but still polygonal, like the regular octagon.”
Dr. Avila, 35, investigated a different area of dynamical systems, including an understanding of fractals. Dr. Bhargava, 40, was recognized for new methods in the geometry of numbers, especially prime numbers, and Dr. Hairer, 38, made advances in the study of the effect of random noise on partial differential equations, capturing the effect of turbulence on ocean currents or the flow of air around airplane wings.
While women have reached parity in many academic fields, mathematics is still dominated by men, who earn about 70 percent of the doctoral degrees. The disparity is even more striking at the highest echelons. Since 2003, the Norwegian Academy of Science and Letters has awarded the Abel Prize, recognizing outstanding mathematicians with a monetary award of about $1 million; all 14 recipients so far are men. No woman has won the Wolf Prize in Mathematics, another prestigious award.
The Fields Medal was conceived by John Charles Fields, a Canadian mathematician, “in recognition of work already done” and as “an encouragement for further achievement.” Judges have interpreted the terms of the Fields trust to mean that the award should usually be limited to mathematicians age 40 or younger.
Dr. McMullen, himself a Fields medalist, did not speculate on why it had taken so long for a woman to be recognized. “I would prefer to look forward and celebrate this occasion,” he said, “and see it as a sign of positive trends in society and in science.”
Maryam Mirzakhani (Persianمریم میرزاخانی‎‎‎; 3 May 1977 – 14 July 2017) was an Iranian[7][8][1] mathematician and a professor of mathematics at Stanford University.[9][10][11] Her research topics include Teichmüller theoryhyperbolic geometryergodic theory, and symplectic geometry.[1]
On 13 August 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics.[12][13] The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".[14]
On 14 July 2017 Mirzakhani passed away at the age of 40, after being diagnosed with breast cancer.

Early life and education

Mirzakhani was born on 3 May 1977 in Tehran, Iran.[15] She attended Farzanegan School there, part of the National Organization for Development of Exceptional Talents.
In 1994, Mirzakhani achieved the gold medal level in the International Mathematical Olympiad, the first female Iranian student to do so. In the 1995 International Mathematical Olympiad, she became the first Iranian student to achieve a perfect score and to win two gold medals.[16][17][18]
She obtained her BSc in mathematics (1999) from Sharif University of Technology in Tehran. She went to the United States for graduate work, earning a PhD from Harvard University in 2004, where she worked under the supervision of the Fields Medalist Curtis McMullen.


Mirzakhani was a 2004 research fellow of the Clay Mathematics Institute and a professor at Princeton University.[19] In 2008 she became a professor at Stanford University.[20][14]

Research work

File:Remise de la médaille Fields à Maryam Mirzakhani.webm
Maryam Mirzakhani. August 2014
Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Mirzakhani discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevichon the intersection numbers of tautological classes on moduli space,[9] as well as an asymptotic formula for the growth of the number of simple closed geodesics on a compact hyperbolic surface, generalizing the theorem of the three geodesics for spherical surfaces.[21] Her subsequent work focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.[22]
Most recently as of 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal.[23][24] The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990s.[24] The International Mathematical Union said in its press release that, "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."[24]
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".[25]The award was made in Seoul at the International Congress of Mathematicians on 13 August.[26]
At the time of the award, Jordan Ellenberg explained her research to a popular audience:
... [Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn't directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables ... This isn't the kind of thing you do to win at pool, but it's the kind of thing you do to win a Fields Medal. And it's what you need to do in order to expose the dynamics at the heart of geometry; for there's no question that they're there.[27]
In 2014, President Hassan Rouhani of Iran congratulated her for winning the topmost world mathematics prize.[28]
Mirzakhani has an Erdős number of 3.[29]

Personal life

Mirzakhani was married to Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who is an associate professor at Stanford University;[30]their daughter is named Anahita.[31]
Mirzakhani described herself as a "slow" mathematician, saying that
You have to spend some energy and effort to see the beauty of math.
To solve problems, Mirzakhani would draw doodles on sheets of paper, and write mathematical formulas around the drawings. Her daughter described her mother's work as "painting".
I don’t have any particular recipe [for developing new proofs]... It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.
— Maryam Mirzakhani, [32]


Mirzakhani was diagnosed with breast cancer in 2013.[33] After four years, it spread to her bone marrow.[34] Mirzakhani died from breast cancer on 14 July 2017 at the age of 40.[32][35][36]

Awards and honors


Maryam Mirzakhani(born May 3, 1977TehranIran—died July 15, 2017), 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.
The mathematics section of the National Academy of Sciences lists 104 members. Just four are women. As recently as June, that number was six.
Marina Ratner and Maryam Mirzakhani could not have been more different, in personality and in background. Dr. Ratner was a Soviet Union-born Jew who ended up at the University of California, Berkeley, by way of Israel. She had a heart attack at 78 at her home in early July.
Success came relatively late in her career, in her 50s, when she produced her most famous results, known as Ratner’s Theorems. They turned out to be surprisingly and broadly applicable, with many elegant uses.
In the early 1990s, when I was a graduate student at Berkeley, a professor tried to persuade Dr. Ratner to be my thesis adviser. She wouldn’t consider it: She believed that, years earlier, she had failed her first and only doctoral student and didn’t want another.
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Dr. Mirzakhani was a young superstar from Iran who worked nearby at Stanford University. Just 40 when she died of cancer in July, she was the first woman to receive the prestigious Fields Medal.
I first heard about Dr. Mirzakhani when, as a graduate student, she proved a new formula describing the curves on certain abstract surfaces, an insight that turned out to have profound consequences — offering, for example, a new proof of a famous conjecture in physics about quantum gravity.
I was inspired by both women and their patient assaults on deeply difficult problems. Their work was closely related and is connected to some of the oldest questions in mathematics.
Maryam Mirzakhani in 2014, the year she won the Fields Medal.CreditSeoul ICM 2014, via Agence France-Presse — Getty Images
The ancient Greeks were fascinated by the Platonic solid — a three-dimensional shape that can be constructed by gluing together identical flat pieces in a uniform fashion. The pieces must be regular polygons, with all sides the same length and all angles equal. For example, a cube is a Platonic solid made of six squares.
Early philosophers wondered how many Platonic solids there were. The definition appears to allow for infinite possibilities, yet, remarkably, there are only five such solids, a fact whose proof is credited to the early Greek mathematician Theaetetus. The paring of the seemingly limitless to a finite number is a case of what mathematicians call rigidity.
Something that is rigid cannot be deformed or bent without destroying its essential nature. Like Platonic solids, rigid objects are typically rare, and sometimes theoretical objects can be so rigid they don’t exist — mathematical unicorns.
In common usage, rigidity connotes inflexibility, usually negatively. Diamonds, however, owe their strength to the rigidity of their molecular structure. Controlled rigidity — that is, flexing only along certain directions — allows suspension bridges to survive high winds.
Dr. Ratner and Dr. Mirzakhani were experts in this more subtle form of rigidity. They worked to characterize shapes preserved by motions of space.
One example is a mathematical model called the Koch snowflake, which displays a repeating pattern of triangles along its edges. The edge of this snowflake will look the same at whatever scale it is viewed.
The snowflake is fundamentally unchanged by rescaling; other mathematical objects remain the same under different types of motions. The shape of a ball, for example, is not changed when it is spun.
A Koch snowflake. via Wikimedia Commons
Dr. Ratner and Dr. Mirzakhani studied shapes that are preserved under more sophisticated types of motions, and in higher dimensional spaces.
In Dr. Ratner’s case, that motion was of a shearing type, similar to a strong wind high in the atmosphere. Dr. Mirzakhani, with my colleague Alex Eskin, focused on shearing, stretching and compressing.
These mathematicians proved that the only possible preserved shapes in this case are, unlike the snowflake, very regular and smooth, like the surface of a ball.
The consequences are far-reaching: Dr. Ratner’s results yielded a tool that researchers have turned to a wide variety of uses, like illumining properties in sequences of numbers and describing the essential building blocks in algebraic geometry.
The work of Dr. Mirzakhani and Dr. Eskin has similarly been called the “magic wand theorem” for its multitude of uses, including an application to something called the wind-tree model.


More than a century ago, physicists attempting to describe the process of diffusion imagined an infinite forest of regularly spaced identical and rectangular trees. The wind blows through this bizarre forest, bouncing off the trees as light reflects off a mirror.
Dr. Mirzakhani and Dr. Eskin did not themselves explore the wind-tree model, but other mathematicians used their magic wand theorem to prove that a broad universality exists in these forests: Once the number of sides to each tree is fixed, the wind will explore the forest at the same fundamental rate, regardless of the actual shape of the tree.
There are other talented women exploring fundamental questions like these, but why are there not more? In 2015, women accounted for only 14 percent of the tenured positions in Ph.D.-granting math departments in the United States. That is up from 9 percent two decades earlier.
Dr. Ratner’s theorems are some of the most important in the past half-century, but she never quite received the recognition she deserved. That is partly because her best work came late in her career, and partly because of how she worked — always alone, without collaborators or graduate students to spread her reputation.
Berkeley did not even put out a news release when she died.
By contrast, Dr. Mirzakhani’s work, two decades later, was immediately recognized and acclaimed. Word of her death spread quickly — it was front-page news in Iran. Perhaps that is a sign of progress.
I first met Dr. Mirzakhani in 2004. She was finishing her Ph.D. at Harvard. I was a professor at Northwestern, pregnant with my second child.
Given her reputation, I expected to meet a fearless warrior with a single-minded focus. I was quite disarmed when the conversation turned to being a mathematician and a mother.
“How do you do it?” she asked. That such a mind could be preoccupied with such a question points, I think, to the obstacles women still face in climbing to math’s upper echelons.
At Harvard, the number of tenured women research mathematicians is currently zero. At my institution, the University of Chicago, until 2011 only one woman had ever held such a position.
We are only gradually joining the ranks, in what might be called a “trickle up” fashion.
Students often tell me that my presence on the faculty convinces them that women belong in mathematics. Though she would have shrugged it off, I was similarly inspired by Dr. Ratner.
I hope I played this role for Dr. Mirzakhani. And for all of her reticence about being famous, Dr. Mirzakhani has inspired an entire generation of younger women.
There are a surprising number of social pressures against becoming a mathematician. When you’re in the minority, it takes extra strength and toughness to persist. Dr. Ratner and Dr. Mirzakhani had both.
For the inspiration they provide, but above all for the beauty of their mathematics, we celebrate their lives.

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