Sunday, July 16, 2017

Untimely Death: Maryam Mirzakhani

Maryam Mirzakhani (May 3, 1977 to July 15, 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry.

On August 13, 2014, Mirzakhani became both the first woman and the first Iranian honored with the Fields Medal, the most prestigious award in mathematics. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces."

Early Life and Education

Mirzakhani was born on 3 May 1977 in Tehran, Iran. She attended Farzanegan School there, part of the National Organization for Development of Exceptional Talents.

In 1994, Mirzakhani won a gold medal 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.

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. In 2008 she became a professor at Stanford University.

Research Work

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 Kontsevich on the intersection numbers of tautological classes on moduli space, 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. 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.

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. 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. 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."

Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces". The award was made in Seoul at the International Congress of Mathematicians on 13 August.

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.

President Hassan Rouhani of Iran congratulated her.

Personal Life

Mirzakhani was married to Jan Vondrák, a Czech theoretical computer scientist and applied mathematician who is an associate professor at Stanford University; they had a daughter named Anahita. Mirzakhani was diagnosed with breast cancer in 2013. After four years, it spread to her bone marrow. She died on 15 July 2017.

Major Awards and Honors

  • AMS Blumenthal Award 2009
  • The 2013 AMS Ruth Lyttle Satter Prize in Mathematics. "Presented every two years by the American Mathematical Society, the Satter Prize recognizes an outstanding contribution to mathematics research by a woman in the preceding six years. The prize was awarded on Thursday, 10 January 2013, at the Joint Mathematics Meetings in San Diego
  • Clay Research Award 2014
  • Fields Medal 2014


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