Filling Riemannian Manifolds with Integral Current Spaces

After defining Riemannian manifolds and Integral Current Spaces I will define the Filling Volume of their boundaries.  This is a notion first introduced by Gromov and then adapted in joint work with Wenger and with Portegies.  I will present sequences of Riemannian manifolds whose volumes remain bounded uniformly away from 0 but whose filling volumes converge to 0.