Any metric space can be thickened in a canonical way using the theory of optimal transport. This yields a family of thickenings that capture local connectivity properties of the space. I will introduce this theory and explain some of its applications to combinatorics and discrete geometry. In particular, this can be used to prove Borsuk--Ulam type results, study the structure of zeros of trigonometric polynomials, or illuminate convexity properties of circle actions on Euclidean space.

This is joint work with Henry Adams and Johnathan Bush. Speaker(s): Florian Frick (Carnegie Mellon University)