Filmed in Canada, Iran, and the United States, Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics’ highest prize, the Fields Medal.
Mirzakhani’s contributions are explained in the film by leading mathematicians and illustrated by animated sequences. Her mathematical colleagues from around the world, as well as former teachers, classmates, and students in Iran today, convey the deep impact of her achievements. The path of her education, success on Iran’s Math Olympiad team, and her brilliant work, make Mirzakhani an ideal role model for girls looking toward careers in science and mathematics.
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Maryam Mirzakhani مریم میرزاخانی, (1977 – 2017) grew up in Tehran, Iran. She attended Tehran Farzanegan School, part of the National Organization for Development of Exceptional Talents (NODET). In 1994, Mirzakhani earned a gold medal at the International Mathematical Olympiad (IMO), the first female Iranian student to do so. In the 1995 IMO, she became the first Iranian student to achieve a perfect score and to win two gold medals. She obtained her BSc in mathematics in 1999 from the Sharif University of Technology. She then went to the United States for graduate work, earning her PhD in 2004 from Harvard University, where she worked under the supervision of Fields medalist Curtis T. McMullen. At Harvard she is said to have been "distinguished by ... determination and relentless questioning," despite not being a native English speaker, and took her class notes in Persian.
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.
Her research topics included Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, she discovered a formula expressing the volume of the moduli space of surfaces of type (g,n) with given boundary lengths as a polynomial in those lengths. 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.
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. 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.”
In 2014, she became both the first woman and the first Iranian to be honored with the Fields Medal. The award committee cited her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces."
Maryam Mirzakhani was married to Jan Vondrák, a Czech mathematician and eventual colleague. They had one child. Mirzakhani died of breast cancer in 2017 at the age of 40.
Despite her every effort to avoid public attention, Mirzakhani’s achievements brought her accolades and publicity. She was widely revered in Iran. After her death, Iranian president Hassan Rouhani said "unprecedented brilliance of this creative scientist and modest human being, who made Iran's name resonate in the world's scientific forums, was a turning point in showing the great will of Iranian women and young people on the path towards reaching the peaks of glory and in various international arenas." And Sharif University of Technology announced that its faculty of mathematics will be renamed "Mirzakhani."
Mirzakhani’s discoveries in the areas of counting simple closed curves on hyperbolic surfaces, spaces of surfaces, and the trajectories of billiard balls have provided insights in mathematical physics, and opened the door for expansion in a number of areas. Her depth of insight in mathematics was matched by her depth as a caring and generous human being who exhibited humility even as she achieved the highest honors. Our film reflects this unique combination of qualities.