3-manifolds after Perelman: topology, geometry, and effective rigidity

Our understanding of 3-manifolds has advanced considerably since Perelman’s proof of Thurston’s Geometrization Conjecture, with many original approaches to the conjecture becoming new exciting theorems in their own right. In the hyperbolic setting, however, determining precisely how a topological description of a 3-manifold determines its geometry has remained a central challenge. I’ll review examples and describe some progress toward elucidating geometric information.