An Abelian differential defines a flat structure on the underlying Riemann surface, such that it can be realized as a plane polygon with edges suitably identified. Varying the shape of the polygon induces a GL(2,R)-action on the moduli space of Abelian differentials, which is called Teichmueller dynamics. In this lecture series, I will give an elementary introduction to Teichmueller dynamics, with a focus on a beautiful interplay between algebraic geometry, combinatorics, dynamics, and number theory.
In the first lecture I will introduce basic properties of moduli spaces of Riemann surfaces and Abelian differentials as well as the GL(2,R)-action. In the second lecture I will introduce several examples of special GL(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 a correspondence between dynamical invariants of GL(2,R)-orbits and intersection theory on moduli spaces. The fourth lecture will be an overview on some recent breakthroughs, e.g. the Fields Medal work of Mirzakhani, as well as open problems in this field.