Title: Character Varieties and Hyperbolic 3-manifolds

Abstract: The SL(2,C) character variety of a finite volume hyperbolic 3-manifold is a complex algebraic set, X(M). This moduli space encodes a great deal of topological data about M. The canonical component of X(M) is a deformation space of M, as seen in Thurston's hyperbolic Dehn surgery theorem. Character varieties have been key ingredients in many aspects of modern topology, including the proof of the cyclic surgery theorem. They can also be used to detect essential surfaces in manifolds and splittings of their fundamental groups.

I will introduce this set and discuss the relationship between the topology of M and the algebraic geometric structure of X(M). I will focus on the case where M has one cusp, where the canonical component of X(M) is a curve.