Teichmüller space, mapping class groups, and translation surfaces

Abstract

Topological surfaces are classified by their number of holes, but a single surface can support many distinct geometric structures. Among these, hyperbolic structures play a central role. The space that parameterises all such hyperbolic structures is called the Teichmüller space.

In the first hour, I will give an introduction to these spaces and to mapping class groups, which capture the symmetries of a surface.

In the second hour, I will introduce translation surfaces and explain their connections to Teichmüller theory. I will then present an important system of coordinates on the space parameterising translation surfaces, the period coordinates, and discuss results characterising the periods that can arise.