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 small

constant 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 Hausdorff

content 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.