Uniqueness of Dehn Surgery on Knots

Dehn surgery is an operation to construct 3-manifolds from framed knots. Although Dehn surgery is ubiquitous in low-dimensional topology, it remains unclear how unique the description of a 3-manifold as a Dehn surgery along a knot may be. Recently, there has been substantial progress on such questions using tools ranging from gauge theory to hyperbolic geometry. I will give an elementary introduction to Dehn surgery and discuss how ideas from the angle-deformation theory of hyperbolic 3-manifolds are useful. In particular, I will discuss recent joint work with Marc Kegel.