|Workshop on Geometric Functionals: Analysis and Applications|
|Topic:||Filling metric spaces|
|Affiliation:||University of Toronto; Member, School of Mathematics|
|Date:||Friday, March 8|
|Time/Room:||11:30am - 12:30pm/Simonyi Hall 101|
Abstract: Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain smallconstant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry. Guth asked if one can generalize his result in two directions: first, from closed manifolds to n-dimensional metric spaces using the Hausdorff measure instead of volume, and, then, to metric spaces of any dimension using n-dimensional Hausdorffcontent instead of Hausdorff measure. If true, such a result will immediately lead to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss Guth's theorem and an ongoing joint project towards resolution of Guth's problem.