Universality Phenomena in Geometry and Dynamics of Moduli spaces

Introduction

Geometry and dynamics in the moduli spaces proved to be extremely efficient in the study of surface foliations, billiards in polygons, and in mathematical models of statistical and solid state physics like Ehrenfest billiards or Novikov's problem on electron transport. Ideas of study of surface dynamics through geometry of moduli spaces originate in works of Thurston, Masur, and Veech. The area is flourishing ever since.

Contributions to the Field

Contributions of notable mathematicians such as Avila, Eskin, McMullen, Mirzakhani, Kontsevich, Okounkov, and Yoccoz, to mention only Fields Medal and Breakthrough Prize winners, made geometry and dynamics in the moduli spaces one of the most active areas of modern mathematics.

Moduli Spaces Overview

Moduli spaces of Riemann surfaces and related moduli spaces of Abelian differentials are parametrized by a genus ( g ) of the surface. Considering all associated hyperbolic (respectively flat) metrics at once, one observes more and more sophisticated diversity of geometric properties when genus grows.

However, most metrics, on the contrary, progressively share certain similarities. Here the notion of “most of” has explicit quantitative meaning, for example, in terms of the Weil-Petersson measure.

Global Characteristics

Global characteristics of the moduli spaces, such as:

1. Weil-Petersson volumes
2. Masur-Veech volumes
3. Siegel-Veech constants
4. Intersection numbers of ( \psi )-classes

were traditionally studied through algebra-geometric tools, where all formulae are exact but difficult to manipulate in large genus. Most of these quantities admit simple uniform large genus approximate asymptotic formulae.

Project Goals

The project aims to study large genus asymptotic geometry and dynamics of moduli spaces and of related objects from probabilistic and asymptotic perspectives. This will provide important applications to:

• Enumerative geometry
• Combinatorics
• Dynamics

Including:

• Count of meanders in all genera
• Solution of Arnold’s problem on statistics of random interval exchange permutations
• Asymptotics of Lyapunov exponents
• Diffusion rates of Ehrenfest billiards