A lot is known about compact surfaces. The Classification Theorem for Closed Surfaces describes their topology, and Riemann Uniformization describes their geometry. These are not easy theorems by any means, but understanding either of their full proofs would be reasonable projects for senior undergraduates. Closed three manifolds are a little more complicated. Prime Decomposition describes their topology and Thurston Geometrization describes the geometry of their prime pieces. The proof of the first is hard, but doable for a first or second year graduate student. The proof of the second won Grigori Perelman a Fields Medal in 2006. It is perhaps surprising then that each closed hyperbolic surface has a large infinite family of distinct hyperbolic structures while closed hyperbolic three manifolds have only one. The latter fact is known as the Mostow Rigidity Theorem, which sits at the center of much of the push and pull of the rigidity of three (and higher!) dimensional manifolds and the flexibility of surfaces and will thus be the star of the talk.

Don't worry if you've never seen these particular results (or, in fact, hyperbolic structures at all), the talk should be fairly self-contained. I don't plan to delve too far into the technical details of any particular proof or machinery, but rather to give a sense of why Mostow Rigidity is exciting and where it fits in the broader theory of low-dimensional geometric topology. Hyperbolic pants will have an unexpected cameo appearance.