The classification of hyperbolic 3-manifolds

The space of complete hyperbolic structures on a fixed 3-manifold has a rich and mysterious structure, which has been studied using methods of complex analysis, dynamics, topology and geometry. A basic open   question in this area has been Thurston's "ending lamination conjecture", which states that a hyperbolic structure on a 3-manifold is completely determined by "end invariants" that describe asymptotic properties of its ends. Recently in joint work with J. Brock and R. Canary we were able to prove this conjecture (in the incompressible-boundary case), using in an essential way the combinatorial structure of the set of simple closed curves on a surface. This gives a complete classification of the points in the deformation space, although finer questions remain unanswered. I will give an overview of the structure of this field and our current state of knowledge.