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 - Dunfiled
  which relies on Perelman's work on Ricci flow.  This lecture will
  survey these developments and discuss various open problems.