Random multicurves on surfaces of large genus and random square-tiled surfaces of large genus

Abstract

It is common in mathematics to study decompositions of compound objects into primitive blocks. For example, the Erdos-Kac Theorem describes the decomposition of a random large integer number into prime factors. There are theorems describing the decomposition of a random permutation of a large number of elements into disjoint cycles.

I will present our formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula performed by Aggarwal and the uniform large genus asymptotics of intersection numbers of psi-classes on the moduli spaces of complex curves proved by Aggarwal allowed us to describe the decomposition of a random square-tiled surface of large genus into maximal horizontal cylinders. Our results imply, in particular, that with a probability which tends to 1, as genus grows, all "corners" of a random square-tiled surface live on the same horizontal and on the same vertical critical leave.

Maryam Mirzakhani has ingeniously computed frequencies of simple closed multi-geodesics of any topological type on a hyperbolic surface. Developing the results of Mirzakhani we give a detailed portrait of a random hyperbolic multi-geodesics (random multicure) on a Riemann surface of large genus.

Speaker

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.