Professor Anton Zorich:
I plan to give a panorama of geometry and dynamics of moduli spaces and discuss certain applications of this subject. The course would be informal: I plan to omit most of the proofs in order to concentrate on conceptual ideas and especially on ties between various facets of the discussed fields. This would allow to describe almost everything from scratch: I do not assume any particular knowledge of dynamics, geometry or combinatorics.
I plan to start with flat surfaces, similar to those which arise from the study of billiards in rational polygons. This would lead us to the moduli space of Abelian differentials and to the natural action of the group GL(2,R) on this space. I plan to present the Magic Wand Theorem of Eskin-Mirzakhani-Mohammadi and the complement to it due to Filip. I will introduce the notion of Lyapunov exponents and Eskin-Kontsevich-Zorich formula for the sum of Lyapunov exponents of the Teichmuller flow. As an illustration of all this technique we will compute the diffusion rate of the Ehrenfest wind-tree billiard and of its relatives.
We will proceed with the fascinating world of hyperbolic surfaces. I plan to give an idea of how Maryam Mirzakhani has counted simple closed geodesic multicurves on closed hyperbolic surfaces. We will discuss in this context Witten-Kontsevich correlators, mention Witten's conjecture, Fenhel-Nielsen coordinates and symplectic structure of the moduli space of hyperbolic surfaces. If time allows, we will discuss train-tracks.
Finally, in the last part of the course I plan to bridge flat and hyperbolic worlds. Using the count of metric ribbon graphs by Kontsevich and Norbury, we will compute, following Delecroix-Goujard-Zorich, Masur-Veech volume of the moduli space of quadratic differentials through count of square-tiled surfaces. We will conclude with recent spectacular results of Amol Aggarwal on large genus asymptotics of Masur-Veech volumes and of Witten-Kontsevich correlators. As an application I will describe (following Delecroix-Goujard-Zorich) random geodesic multicurves on surfaces of large genera and random square-tiled surfaces of large genus. If time allows, we will count meanders on a surface of any genus.
Anton Zorich is Distinguished Professor of Mathematics at Université Paris Diderot - Paris 7. He is a former member of the Institut Universitaire de France. His research lies on the border between dynamical systems, geometry and topology. He often performs computer experiments which sometimes lead to conjectures proved years or decades later. He usually works in collaboration; often with Alex Eskin and Maxim Kontsevich. He was an invited speaker at the International Congress of Mathematicians in Madrid in 2006.