 Abstract : An Abelian differential defines a flat metric such that the underlying Riemann surface can be realized as a polygon whose edges are pairwise identified via translation. Varying the shape of such polygons induces a GL(2,R)-action on the moduli space of Abelian differentials, called Teichmueller dynamics, whose study has provided fascinating results and opened new avenues to many fields in mathematics. In the first lecture I will give an accessible introduction to this subject, with a focus on its connections to billiard dynamics, enumerative geometry, and arithmetic geometry. In the remaining lectures I will explain how to use techniques in algebraic geometry to study various problems in Teichmueller dynamics, including compactification, birational geometry, cycle class computation, etc. A number of recent developments and open problems will be mentioned. |
.jpg)
