**David Gabai**

Princeton University, USA

July 2, 2012

**Volumes of hyperbolic 3-manifolds**:
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 geometric and topological techniques. This lecture will survey these and other developments and discuss some of the many open problems in the area.