Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Maryam 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 tautology classes on moduli space,[5] as well as an asymptotic formula for the growth of the number of simple closed geodesics on a compact hyperbolic surface.[17] Her subsequent work has 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.[18]
17 Mirzakhani, Maryam (2008). "Growth of the number of simple closed geodesics on hyperbolic surfaces". Annals of Mathematics 168 (1): 97–125. doi:10.4007/annals.2008.168.97. MR 2415399. Zbl 1177.37036. 論文PDF annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p03.pdf Growth of the number of simple closed geodesics on hyperbolic surfaces Annals of Mathematics, 168 (2008), 97–125
www.mathunion.org/fileadmin/IMU/Prizes/2014/news_release_mirzakhani.pdf The Work of Maryam Mirzakhani - International Mathematical Union 抜粋 Riemann knew that these deformations depend on 6g - 6 parameters or "moduli", meaning that the "moduli space" of Riemann surfaces of genus g has dimension 6g - 6. However, this says nothing about the global structure of moduli space, which is extremely complicated and still very mysterious. Moduli space has a very intricate geometry of its own, and dierent ways of looking at Riemann surfaces lead to dierent insights into its geometry and structure. For example, thinking of Riemann surfaces as algebraic curves leads to the conclusion that moduli space itself is an algebraic object called an algebraic variety. In Mirzakhani's proof of her counting result for simple closed geodesics, another structure on moduli space enters, a so-called symplectic structure, which, in particular, allows one to measure volumes (though not lengths). Generalizing earlier work of G. McShane, Mirzakhani establishes a link between the volume calculations on moduli space and the counting problem for simple closed geodesics on a single surface. She calculates certain volumes in moduli space and then deduces the counting result for simple closed geodesics from this calculation. つづく
This point of view led Mirzakhani to new insights into other questions about moduli space. One consequence was a new and unexpected proof of a conjecture of Edward Witten (a 1990 Fields Medalist), one of the leadingfigures in string theory. Moduli space has many special loci inside it that correspond to Riemann surfaces with particular properties, and these loci can intersect. For suitably chosen loci, these intersections have physical interpretations. Based on physical intuition and calculations that were not entirely rigorous, Witten made a conjecture about these intersections that grabbed the attention of mathematicians. Maxim Kontsevich (a 1998 Fields Medalist) proved Witten's conjecture through a direct verication in 1992. Fifteen years later, Mirzakhani's work linked Witten's deep conjecture about moduli space to elementary counting problems of geodesics on individual surfaces.
ja.wikipedia.org/wiki/AdS/CFT%E5%AF%BE%E5%BF%9C 理論物理学では、AdS/CFT対応（−たいおう、anti-de Sitter/conformal field theory correspondence）は、マルダセーナ双対(Maldacena duality)あるいはゲージ／重力双対(gauge/gravity duality)とも呼ばれ、2つの物理理論の種類の間の関係を予言するものである。
