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
