Mathematical Conversations

Powerful theorems of Thurston, Perelman, and Mostow tell us that almost every 3-dimensional manifold admits a hyperbolic metric, and that this metric is unique. Thus, in principle, there is a 1-to-1 correspondence between a combinatorial description of a 3-manifold and its geometry. For instance, we should be able to inspect a diagram of a knot and predict the volume of the complement. Even better, we should be able to build a dictionary between combinatorial features and geometric measurements. I will describe a bit of what is known and unknown in this vein.