Moduli spaces from a random flat and hyperbolic surfaces perspective – MOST

Geometry and dynamics in the moduli space of Abelian differentials proved to be extremely efficient in the study of surface foliations, billiards in polygons and in certain mathematical models of statistical and solid state physics. Major contributions of A. Avila, A. Eskin, C. McMullen, M. Mirzakhani, M. Kontsevich, A. Okounkov, J.-C. Yoccoz, to mention only Fields Medal and Breakthrough Prize winners, made geometry and dynamics in the moduli spaces one of the most active areas of modern mathematics. Moduli spaces of Riemann surfaces and related moduli spaces of Abelian and quadratic differentials are parametrized by the genus g of the surface. Considering all associated hyperbolic metrics (respectively all flat metrics associated to Abelian or quadratic differentials) at once, one observes diversity of geometric properties getting more and more sophisticated when genus grows. However, “most of” hyperbolic (respectively flat) metrics, on the contrary, progressively share certain similarity of geometric properties converging in some rigorous sense. Here the notion of “most of” has explicit quantitative meaning, for example, in terms of the finite Weil-Petersson measure on the moduli space. Global characteristics of the moduli spaces, like intersection numbers of ?-classes were traditionally studied through algebra-geometric tools, where all formulae are exact, but difficult to manipulate in large genus. We have discovered that certain important characteristics admit simple uniform large genus approximate asymptotic formulae. The project aims to study large genus asymptotic geometry and dynamics of moduli spaces and of related objects from probabilistic and asymptotic perspectives. This will provide applications to enumerative geometry, combinatorics and dynamics, including count of meanders, solution of Arnold’s problem on random interval exchange permutations, asymptotics of Lyapunov exponents and of diffusion rates of Ehrenfest wind-tree billiards.