**Nonlinear Analysis and PDEs**Goals of these seminars are to discuss techniques that are used nonlinear problems arising in applied mathematics, physics or differential geometry.

**Alexander Nabutovsky, University of Toronto**

*Filling metric spaces*

**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 a much stronger and more general result holds true: Is there a constant e(m)>o such that each compact metric space with m-dimensional Hausdorff content less than e(m) always has (m-1)-dimensional Uryson width less than 1? Note that here the dimension of the metric space is not assumed to be m, and is allowed to be arbitrary. Such a result immediately leads to interesting new inequalities even for closed Riemannian manifolds. In my talk I am are going to discuss a joint project with Yevgeny Liokumovich, Boris Lishak and Regina Rotman towards the positive resolution of Guth's problem.

These events are sponsored by the Initiative for the Theoretical Sciences. For more information go to https://www.math.csi.cuny.edu/~mlucia/GCactivities.html.

Those participating in the Nonlinear Analysis and PDE seminar may also be interested in the Geometric Analysis Seminar which meets Tuesdays in the same room 6496 starting at 3pm.