**Tentative Schedule**

First week

**Javier Aramayona (Madrid)** *MCG and infinite MCG*

*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 classification,*

*abelianization, isomorphic rigidity, geometry of combinatorial models.*

*In the second part we will explore some aspects of "big" mapping class*

*groups, highlighting the analogies and differences with their finite-type*

*counterparts, notably around isomorphic rigidity, abelianization, and*

*geometry of combinatorial models.*

**Bertrand Deroin (CNRS) ***Monodromy of algebraic families of curves*

*The mini-course will focus on the properties of the monodromies of algebraic families of curves defined over the complex numbers. One of the goal will be to prove the irreducibility of those representations for locally varying families (Shiga). If time permit we will see how to apply this to prove the geometric Shafarevich and Mordell conjecture. The material that will be developed along the lectures are *

- analytic structure of Teichmüller spaces

- theory of Kleinian groups

- Bers embedding

- b-groups

- Mumford compactness criterion

- Imayoshi-Shiga finiteness theorem.

**Pascal Hubert (Marseille) & Sasha Skripchenko (Moscow)**

*Rauzy gasket, Arnoux-Yoccoz interval exchange map, Novikov's problem*

*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 physics*

*4. Lyapunov exponents for the Rauzy gasket: what do we know about them*

*5. Multidimensional fraction algorithms: why do they care*

*6. Open problem session (sometimes, say, more than 30 years open!)*

**Christopher Leininger (Urbana-Champaign) ***Teichmüller spaces and pseudo-Anosov homeomorphism*

*I will start by describing the Teichmuller space of a surface of finite type from the perspective of both hyperbolic and complex structures and the action of the mapping class group on it. Then I will describe Thurston's compactification of Teichmuller space, and state his classification theorem. After that, I will focus on pseudo-Anosov homeomorphisms, describe a little bit about their dynamics, and discuss the geometry/dynamics of the associated mapping tori.*

**Carlos Matheus (CNRS) ***Square tiled surfaces*

*a) basic definitions and examples*

*b) strata and genus*

*c) reduced and primitive origamis, SL(2,R) action, Veech groups*

*d) automorphisms and affine homeomorphisms *

*e) homology of origamis *

*f) Kontsevich-Zorich cocycle*

*g) Lyapunov exponents of the Wollmilchsau *

Second Week

**Simion Filip (Harvard)** *K3 surfaces and Dynamics*

*K3 surfaces provide a meeting ground for geometry (algebraic, differential), arithmetic, and dynamics. I hope to discuss:*

- Basic definitions and examples

- Geometry (algebraic, differential, etc.) of complex surfaces

- Torelli theorems for K3 surfaces

- Dynamics on K3s (Cantat, McMullen)

- Analogies with flat surfaces

- (time permitting) Integral-affine structures

**Giovanni Forni (Maryland)** *Cohomological equation and Ruelle resonnences*

*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 asymptotic of correlations for pseudo-Anosov maps, which were recently obtained by a direct method by Faure, Gouezel and Lanneau. In this vein, we consider the generalization of this asymptotic to generic Teichmueller orbits (pseudo-Anosov maps correspond to periodic Teichmueller orbits) and to (partially hyperbolic) automorphisms of Heisenberg nilmanifolds (from results on the cohomological equation due to L. Flaminio and myself).*

**John Smillie (Warwick)& ****Barak Weiss (Tel Aviv)** *Horocycle dynamics*

*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 the genus 2 case. Specific topics to be covered include:*

* SL_2(R) orbit closures and invariant measures in genus 2.

* Quantitative nondivergence.

* The structure of minimal sets.

* Rel and real-rel, and their interaction with the horocycle flow

* Horizontal data diagrams and other invariants for horocycle invariant measures.

* Classification of measures and orbit-closures in the eigenform loci.

* Recent and not-so-recent examples of unexpected measures and orbit-closures.

**Alex Wright (Stanford)** *Earthquakes and counting*

*We will discuss Mirzakhani's work relating earthquake flow and Teichmuller horocycle flow, her recent work on counting mapping class group orbits of (non-simple) curves on surfaces, and possibly also related recent work of Erlandsson, Rafi, and Souto.*

* *

**Anton Zorich (Paris 7)** *Counting simple closed geodesics and volumes of moduli spaces*

*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 count of Weil-Peterson volumes and her proof of Witten's conjecture, but only on the level of some key ideas.*

In the last lecture I plan to show how ideas of Mirzakhani work in counting problems related to flat surfaces, namely, in computation of Masur-Veech volumes and in counting meanders.

* *