Algebra/Topology Seminar 30112020

Speaker: Alexey Balitskiy

Title: Urysohn width in various contexts

Abstract: The Urysohn d-width of a riemannian manifold quantifies how well it can be approximated by a d-dimensional simplicial complex. I will review how this notion helps in systolic geometry (following Gromov's line of thought) and combinatorial topology. In particular, I hope to explain briefly the negative answer to a question of Larry Guth, showing that the width can behave counterintuitively: it can happen that an n-manifold is "essentially n-dimensional" (that is, of substantial (n-1)-width) but all its unit balls are "almost 1-dimensional" (that is, of small 1-width).