The Nielsen Realization Problem
The Nielsen Realization Problem

Will Cavendish, Princeton University
Fine Hall 314
The purpose of this talk will be to describe some problems in topology whose solutions rely on geometric techniques. I will introduce the mapping class group of a space $X$, which is the group of homeomorphisms of $X$ up to isotopy. I will then outline Steve Kerckhoff's solution to the Nielsen realization problem, which shows that any finite subgroup of $Mod_g$, the mapping class group of a genus $g$ surface, can be realized as the automorphism group of a Riemann surface of genus $g$. The proof relies heavily on an understanding of the geometry of Teichmüller space $T_g$, and ties in to many natural geometric questions about this space which I hope to describe.