Character Varieties and 3-manifolds

The SL(2,C) character variety of a finite volume hyperbolic 3-manifold M can be thought of as the moduli space of hyperbolic structures on M. This affine algebraic set encodes much of the topology of M, but many basic properties of this set are unknown.  I will introduce the character variety and discuss several ways in which the geometry of the character variety is related to the topology of M and the structure of the fundamental group of M.