Bounds on Urysohn width

Abstract

The Urysohn d-width of a metric space quantifies how closely it can be approximated by a d-dimensional simplicial complex. Namely, the d-width of a space is at most w if it admits a continuous map to a d-complex with all fibers of diameter at most w. This notion was introduced in the context of dimension theory, used in approximation theory, appeared in the work of Gromov on systolic geometry, and nowadays it is a metric invariant of independent interest. The main results of this thesis establish bounds on the width, relating local and global geometry of Riemannian manifolds in two contexts. One of them is bounding the global width of a manifold in terms of the width of its unit balls. The other one is waist-like inequalities, when a manifold is sliced into a family of (singular) surfaces, and the global width is related to the supremal width of the slices.