Maryam Mirzakhani

Maryam Mirzakhani ( fa, مریم میرزاخانی, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. In 2005, as a result of her research, she was honored in ''Popular Science's'' fourth annual "Brilliant 10" in which she was acknowledged as one of the top 10 young minds who have pushed their fields in innovative directions.

On 13 August 2014, Mirzakhani was honored with the Fields Medal, the most prestigious award in mathematics, becoming the first Iranian to be honored with the award and the first of only two women to date. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces".

On 14 July 2017, Mirzakhani died of breast cancer at the age of 40.

Early life and education

Mirzakhani was born on 12 May 1977 in Tehran, Iran. As a child, she attended Tehran Farzanegan School, part of the National Organization for Development of Exceptional Talents (NODET). In her junior and senior years of high school, she won the gold medal for mathematics in the Iranian National Olympiad, thus allowing her to bypass the national college entrance exam. In 1994, Mirzakhani became the first Iranian woman to win a gold medal at the International Mathematical Olympiad in Hong Kong, scoring 41 out of 42 points. The following year, in Toronto, she became the first Iranian to achieve the full score and to win two gold medals in the International Mathematical Olympiad. Later in her life, she collaborated with friend, colleague, and Olympiad silver medalist, Roya Beheshti Zavareh ( fa, رؤیا بهشتی زواره), on their book Elementary Number Theory, Challenging Problems','' (in Persian) which was published in 1999. Mirzakhani and Zavareh together were the first women to compete in the Iranian National Mathematical Olympiad and won gold and silver medals in 1995, respectively.

On 17 March 1998, after attending a conference consisting of gifted individuals and former Olympiad competitors, Mirzakhani and Zavareh, along with other attendees, boarded a bus in Ahvaz en route to Tehran. The bus was involved in an accident wherein it fell off a cliff, killing seven of the passengers—all Sharif University students. This incident is widely considered a national tragedy in Iran. Mirzakhani and Zavareh were two of the few survivors.

In 1999, she obtained a Bachelor of Science in mathematics from the Sharif University of Technology. During her time there, she received recognition from the American Mathematical Society for her work in developing a simple proof of the theorem of Schur. She then went to the United States for graduate work, earning a PhD in 2004 from Harvard University, where she worked under the supervision of the Fields Medalist, Curtis T. McMullen. At Harvard, she is said to have been "distinguished by determination and relentless questioning". She used to take her class notes in her native language Persian.

Career

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

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 $cL^$, 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 cut 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

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