Gromov's compactness theorem for metric spaces asserts that every

uniformly compact sequence of metric spaces has a subsequence which

converges in the Gromov-Hausdorff sense to a compact metric space. I will

show in this talk that if one replaces the Hausdorff distance appearing in

Gromov's theorem by the flat distance then every sequence of oriented

k-dimensional Riemannian manifolds with a uniform bound on diameter and

volume has a subsequence which converges in this new distance to a

countably k-rectifiable metric space. I will then sketch some applications

of this theorem.

The new distance mentioned above was first introduced and studied by

Christina Sormani and myself. I will explain the basic properties of this

distance and its relationship with other distances.