Teichmüller Theory

Teichmüller spaces are important objects appearing in different areas of mathematics such as, e.g., geometry of 3-manifolds and dynamical systems. They have been studied by many famous mathematicians including Ahlfors, Thurston etc. The main goal of the course is to give an accessible introduction to the theory and to prove some of the basic but highly nontrivial results about their geometric structure. During the course we introduce quasiconformal mappings and discuss their properties, prove a few classical results in hyperbolic geometry of Riemann surfaces, define and investigate the metric structure (at least one of many) on the Teichmüller spaces. If time permits, we also cover some applications (e.g., in dynamical systems).