Rice University Department of Mathematics Colloquium

Volume, Renormalized

4:00 pm Thursday, January 25, 2018
Jeff Brock

Abstract:Much of the appeal of Thurston's work on the geometrization of 3-manifolds lies in its hands-on simplicity: synthetic and combinatorial constructions involving geodesic loops, simplicial surfaces, and coarse "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 and Bers lies at the foundation, and only recently has work of Graham and Witten been found to give clues to an analytic framework for understanding Thurston's intuition and conjectures. In this talk I will describe history 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 that fiber over the circle and Weil-Petersson lengths of closed geodesics on moduli space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This talk describes joint work with Ken Bromberg and Martin Bridgeman.

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