| Introduction to Teichmüller spaces |
| Xavier Buff |
| Université Paul Sabatier (Francia) |
| Abstract |
| Mini cours 1: Teichmüller spaces |
| I will introduce the notion of Teichmüller spaces and the basic objects to study them: quasiconformal maps and extremal maps. I will prove that Teichmüller spaces are contractible. I will define the analytic structure on Teichmüller spaces. I will focus on the case of finite dimensional Teichmüller spaces and study the Teichmüller metric and its geodesics. |
| Mini cours 2: Surface homeomorphisms |
| aaI will study the classification of homeomorphisms of surfaces up to homotopy. I will define the types of homeomorphisms: periodic, reducible, pseudo-Anosov. I will then prove the classification theorem of homeomorphisms of compact surfaces: every orientation preserving homeomorphism of a compact oriented surface is homotopic to a periodic homeomorphism, a reducible homeomorphism or a pseudo-Anosov homeomorphism. |
| Mini cours 3: Dynamics of rational maps |
| I will present various applications of Teichmüller space to the dynamics of rational maps, including the Sullivan no wandering domain theorem, the topological characterization of postcritically finite rational maps due to Thursont and the Epstein deformation space associated to a rational map. |