From `entopology' to morphology in hyperbolic geometry

Like the work of the entomologist, the classication of hyperbolic 3-manifolds with finitely generated fundamental group involves sorting a lot of pretty ugly looking bugs. Mathematically, this classication reduces to Thurston's *Ending Lamination Conjecture*; our proof (with Canary and Minsky) produced an array of tools for understanding how a topological specification of a 3-manifold relates to its geometry. Once such specication, the *Heegaard splitting*, seems ripe for exploration along these lines, yet a complete picture is still under development. I'll review the state of things, discuss some key examples, and discuss how this may help us understand the geometry of a `random 3-manifold.'