Centre International de Recherche Scientifique
Centre International de Recherche Scientifique
mmirzakh
Professor of mathematics at Stanford University, USA
Teichmüller Theory, Ergodic Theory
Maryam Mirzakhani has made stunning advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new frontiers in this area. Her insights have integrated methods from diverse fields, such as algebraic geometry, topology and probability theory.
In hyperbolic geometry, Mirzakhani established asymptotic formulas and statistics for the number of simple closed geodesics on a Riemann surface of genus g. She next used these results to give a new and completely unexpected proof of Witten’s conjecture, a formula for characteristic classes for the moduli spaces of Riemann surfaces with marked points.
In dynamics, she found a remarkable new construction that bridges the holomorphic and symplectic aspects of moduli space, and used it to show that Thurston’s earthquake flow is ergodic and mixing.
Most recently, in the complex realm, Mirzakhani and her coworkers produced the long sought-after proof of the conjecture that – while the closure of a real geodesic in moduli space can be a fractal cobweb, defying classification – the closure of a complex geodesic is always an algebraic subvariety.
Her work has revealed that the rigidity theory of homogeneous spaces (developed by Margulis, Ratner and others) has a definite resonance in the highly inhomogeneous, but equally fundamental realm of moduli spaces, where many developments are still unfolding
2014 Fields Medal for her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces (She is the first woman to win the prize).
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