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Geometry-Topology Seminar

Thursday, April 4, 2019 - 5:45pm

Alex Nabutovsky

University of Toronto and IAS


University of Pennsylvania


This is the second of two talks in our seminar today, sponsored jointly with Bryn Mawr, Haverford and Temple

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.