Prof. Anton Zorich
Université Paris Cité
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.
In this course I plan to discuss square-tiled surfaces and interval exchange transformations from unusual angle, focusing on their geometric and combinatorial aspects. As a teaser I will describe a solution of an old problem of Vladimir Arnold on asymptotic statistics of transitive permutations among random interval exchange permutations.
We will see that the solution of this seemingly naive combinatorial problem involves Masur-Veech volumes of strata in the moduli space of Abelian differentials, and leads, in turn, to further challenging conjectures such as asymptotic equidistribution of integer points on closed long Teichmuller horocycles in the moduli spaces of Abelian differentials and reduction of certain sums to rational linear combinations of multiple zeta values.
Arnold's problem would also lead us to square-tiled surfaces and to arithmetic Teichmuller discs. As an application we will count oriented meanders on a surface of arbitrary genus g. (The Arnold's problem and the problem on asymptotic count of oriented meanders were very recently solved by V.Delecroix, E.Goujard, P.Zograf and myself.)
En route, we will discuss canonical aspects of interval exchange transformations like Rauzy-Veech induction, Abelian differentials corresponding to suspensions over an interval exchange, relation to the Teichmuller geodesic flow, the induced cocycle (Kontsevich - Zorich cocycle) in the Hodge bundle over this flow, and the Lyapunov exponents of this cocycle.
I plan to alternate theoretical lectures and practical exercises treating numerous concrete examples. One of the goals of the course is to present certain number of open problems and conjectures. I do not assume any knowledge of the subjects mentioned above: I plan to tell all necessary facts on the way.