Résume | The Teichmuller geodesic flow in the moduli space of Abelian and
quadratic differentials is a powerful tool in the study of measured
foliations on surfaces, of billiards in rational polygons, of
interval exchange transformations, to name only some applications.
To obtain qualitative information based on ergodicity of the
Teichmuller geodesic flow (like diffusion rate of Ehrensfest
billiard or the error term in ergodic averages of interval
exchanges) one has to know how to normalize the finite invariant
measure for the Teichmuller geodesic flow. The total measure of the
moduli space of Abelian or quadratic differentials is called the
Masur-Veech volume of the corresponding space. One of the
approaches to evaluation of the Masur-Veech volume is through count
of square-tiled surfaces, analogous to count of integer points in a
ball of huge radius $R$.
In this talk I will present our approach to count of Masur-Veech
volumes and of square-tiled surfaces. I will also explain the
relations between this count and Mirzakhani's count of simple
closed geodesic multicurves on hyperbolic surfaces. I will
illustrate how this count allows to count meanders on surfaces of
any genus g. This is a joint work with V. Delecroix, E. Goujard and P. Zograf. |