Title: Moduli space of Riemann surfaces and Teichmueller dynamics
Speaker: Prof. Dawei Chen (Boston College)
An abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon. Varying the shape of the polygon induces an SL(2,R)-action on the moduli space of differentials, called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics from the viewpoint of algebraic geometry.
In the first lecture I will introduce basic definitions and properties of moduli spaces of Riemann surfaces, of abelian differentials and the SL(2,R)-action.
In the second lecture I will introduce several examples of special SL(2,R)-orbits, including Hurwitz spaces of torus coverings and Teichmueller curves. Their study is related to the classical Hurwitz counting problem from a combinatorial viewpoint.
The third lecture will focus on the interplay between dynamical properties of SL(2,R)-orbits and intersection theory on moduli space. If time allows, I will also explain some recent breakthroughs as well as open problems in this field.