Mirzakhani was a 2004 research fellow of the
Clay Mathematics Institute and a professor at
Princeton University. In 2009, she became a professor at
Stanford University.
Research work , August 2014 Mirzakhani made several contributions to the theory of
moduli spaces of
Riemann surfaces. Mirzakhani's early work solved the problem of counting simple closed
geodesics on
hyperbolic Riemann surfaces by finding a relationship to volume calculations on moduli space. Geodesics are the natural generalization of the idea of a "
straight line" to "
curved spaces". Slightly more formally, a curve is a geodesic if no slight deformation can make it shorter. Closed geodesics are geodesics which are also closed curves—that is, they are curves that close up into loops. A closed geodesic is
simple if it does not cross itself. A previous result, known as the "
prime number theorem for geodesics", established that the number of closed geodesics of length less than L grows exponentially with L – it is asymptotic to e^L/L. However, the analogous counting problem for simple closed geodesics remained open, despite being "the key object to unlocking the structure and geometry of the whole surface," according to University of Chicago topologist
Benson Farb. Mirzakhani's 2004 PhD thesis solved this problem, showing that the number of simple closed geodesics of length less than L is polynomial in L. Explicitly, it is asymptotic to cL^{6g-6}, where g is the
genus (roughly, the number of "holes") and c is a constant depending on the hyperbolic structure. This result can be seen as a generalization of the
theorem of the three geodesics for
spherical surfaces. Mirzakhani solved this counting problem by relating it to the problem of computing volumes in
moduli space—a space whose points correspond to different complex structures on a surface genus g. In her thesis, Mirzakhani found a volume formula for the moduli space of bordered Riemann surfaces of genus g with n geodesic boundary components. From this formula followed the counting for simple closed geodesics mentioned above, as well as a number of other results. 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. 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. One can construct a simple earthquake map by cutting a surface along a finite number of disjoint simple closed geodesics, sliding the edges of each of these cuts past each other by some amount, and closing the surface back up. One can imagine the surface being cut by
strike-slip faults. An earthquake is a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic, one puts a
measure on them. In 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.
Awarding of Fields Medal ,
Martin Hairer (at back), Maryam Mirzakhani (with her daughter) and
Manjul Bhargava at the ICM 2014 in Seoul 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: In 2014, President
Hassan Rouhani of Iran congratulated her for winning the award. == Personal life ==