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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
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.
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.
Lecture 1. Arnold's Problem on Interval Exchange Permutations
Lecture 2. Thursday, Sep 29. Arnold's Problem on Interval Exchange Permutations. Part 2
Lecture 3. Tuesday, Oct 11. Explicit representatives of components of strata
Lecture 4. Tuesday, October 13, 2022. Explicit computations
Lecture 5. Tuesday, November 1, 2022. Teichmuller geodesic flow and an idea of renormalization.
Lecture 6. Thursday, November 3, 2022. Rauzy--Veech induction as a renormalization.
Lecture 7. Tuesday, November 8, 2022.Rauzy--Veech induction in details.Continued fractions and cutting sequences as a particular case of the Rauzy--Veech induction.
Lecture 8. Thursday, November 10, 2022. Lyapunov exponents of the Teichmuller geodesic flow and of the Kontsevich-Zorich cocycle. General facts.
Lecture 9. Tuesday, November 15, 2022. Cocycles and subbundles related to the Kontsevich--Zorich cocycle
Lecture 10. Thursday, November 17, 2022. Formula for the sum of the Lyapunov exponents of the Kontsevich-Zorich cocycle
Lecture 11.Tuesday, November 22, 2022. Diffusion rate of Ehrenfest windtree billiards and Lyapunov exponents of the Teichmuller flow
Lecture 12.Thursday, November 24, 2022. Sum of Lyapunov exponents of the Kontsevich--Zorich cocycle: outline of the proof