9/15/2006

Larry Guth
Stanford University

Volumes of balls in large Riemannian manifolds

In the 80's, Gromov made several conjectures about the  volumes of balls in Riemannian manifolds.  The spirit of the  conjectures is that if a Riemannian manifold is "large", then it  should contain a unit ball whose volume is not too small.  For  example, if you take the standard metric on the n-sphere and  increase it pointwise to form a new metric, then Gromov's  conjecture implies that the new metric should contain a unit ball  whose volume is bounded below by a constant c(n).  I proved some  of the conjectures, including this one.  I will explain the  conjectures and give some context, and then I will try to say  something about the proof.