A. Zorich - Counting simple closed geodesics and volumes of moduli spaces (Part 1)

In  the  first two lectures I will try to tell (or, rather, to  give  an  idea)  of  how  Maryam Mirzakhani has counted simple  closed  geodesics on hyperbolic surfaces. I plan to briefly  mention her

Multicurves have played a fundamental role in the study of mapping class groups of surfaces since the work of Dehn. A beautiful method of describing such systems on the n-punctured disk is given by

A major challenge in dynamics on moduli spaces is to understand the behavior of the horocycle flow. We will motivate this problem and discuss what is known and what is not known about it, focusing on

1. Symbolic dynamics: Arnoux - Rauzy words and Rauzy gasket 2. Topology: Arnoux - Yoccoz example and its generalization 3. Novikov’s problem: how dynamics meets topology and together they help to

The first part of the course will be devoted to some of the classical results about mapping class groups of finite-type surfaces. Topics may include: generation by twists, Nielsen-Thurston

In these lectures we summarized results on the cohomological equation for translation flows on translation surfaces (myself, Marmi, Moussa and Yoccoz, Marmi and Yoccoz) and apply these results to the

We will give the proof of Mirzakhani's theorem that the earthquake flow and Teichmuller unipotent flow are measurably isomorphic. We will assume some familiarity with quadratic differentials, but no

We discuss the problem of finding the minimal dilatation among pseudo-Anosov mapping classes on a fixed closed surface. In particular, for every nonorientable closed surface of even genus, we consider

Mathematicians have long understood periodic trajectories on the square billiard table. In the present work, we describe periodic trajectories on the regular pentagon – their geometry, symbolic

1. Symbolic dynamics: Arnoux - Rauzy words and Rauzy gasket 2. Topology: Arnoux - Yoccoz example and its generalization 3. Novikov’s problem: how dynamics meets topology and together they help to

Effective convergence of ergodic averages and cusp excursions of geodesics on moduli spaces We survey some applications of effective convergence of ergodic averages to the analysis of cusp ex