Abstracts of the coursesTentative Schedule
First week
Javier Aramayona (Madrid) MCG and infinite MCG
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
Pascal Hubert (Marseille) & Sasha Skripchenko (Moscow) 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.
a) basic definitions and examples
Simion Filip (Harvard) K3 surfaces and Dynamics K3 surfaces provide a meeting ground for geometry (algebraic, differential), arithmetic, and dynamics. I hope to discuss:
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: Alex Wright (Stanford) Mirzakhani's work on Earthquakes 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 familiarity with earthquakes, and the first lecture will be devoted to preliminaries. The second lecture will cover the proof, and the final lecture additional connections such as the link between Weil-Petersson and Masur-Veech volumes. If time allows, we will mention Mirzakhani's recent result on counting mapping class group orbits, which relies on her work on earthquake flow.
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.
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