Much of the mathematical appeal of Thurston's work on the geometrization of 3-manifolds lies in its hands-on nature: synthetic and combinatorial constructions involving geodesic loops, simplicial surfaces, and quasi geodesics gain their power from rigidity theorems due to Mostow and Sullivan that guarantee that rough geometric estimates ensure spot-on control. But considerable analytic work of Ahflors, Bers, and others, lies at the foundation.

Recently new clues have emerged to an analytic framework for understanding Thurston's intuition and conjectures. In this talk, I will describe some historical context and recent developments that use Graham and Witten's notion of "renormalized volume", as elaborated by Krasnov and Schlenker, to provide a satisfying analytic explanation for the connection between volumes of hyperbolic 3-manifolds and Weil-Petersson lengths of geodesics in moduli space. I'll discuss applications to Weil-Petersson geometry as well as some new results. This talk describes joint work with Ken Bromberg and Martin Bridgeman.