Title: Teichmüller dynamics and the moduli space of curves     Abstract: Integrating a differential on a Riemann surface allows the pair to be expressed as a collection of polygons in the plane with parallel side identifications. The action of GL(2,R) on the plane extends naturally to these polygons, and the orbits of the action, originally considered for their dynamical importance, have unexpected algebraic properties. In this talk, we'll introduce these ideas and discuss ways that this new perspective can be applied to questions on the birational geometry of moduli spaces of curves.