“Volumes of hyperbolic 3-manifolds”
David Gabai, Princeton University

Abstract:
As part of his revolutionary work on hyperbolic geometry in the 1970's, Thurston generalizing work of Jorgensen and Gromov, showed that that the set of volumes of complete finite volume hyperbolic 3-manifolds is closed and well ordered. Recently, Robert Meyerhoff and Peter Milley and the speaker showed that the Weeks manifold is the unique lowest volume closed orientable one, culminating a 30+ year effort by many mathematicians using a wide variety of techniques. In particular, we make use of work of Agol - Dunfield which relies on Perelman's work on Ricci flow. This lecture will survey these developments and discuss various open problems.