Course Identification

Basic group theory and topology. An elementary knowledge of differential geometry and complex analysis is also recommended. 

This course is an introduction to Riemann surfaces and Mapping Class Groups. Specifically, we will be mainly concerned with the geometric properties of (closed) Riemann surfaces and their groups of diffeomorphisms. Special emphasis will be put on the Teichmüller space, and the final goal is to prove Thurston’s classification of diffeomorphisms of hyperbolic surfaces. 

Synopsis:

1) Preliminaries and the Torus case
– introduction to Riemann surfaces
– geometries on surfaces
– homoeomorphisms of the torus
– Teichmüller space of the torus and quadratic differentials

2) Hyperbolic surfaces
– curves and geodesics
– homotopies and isotopies
– Mapping Class Groups
– Dehn–Lickorish Theorem

3) Teichmüller space
– Fenchel–Nielsen coordinates
– Teichmüller metric

4) Thurston classification
– compactification of the Teichmüller space
– geodesic laminations
– classification of diffeomorphisms 

Upon successful completion of the course, the student will be familiar with various elements in the theory of Riemann surfaces and mapping class groups.