Title: The Persistent Topology of Optimal Transport Based Metric Thickenings
Speaker: Qingsong Wang (Ohio State University)
Abstract: A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the p-Vietoris–Rips and p-Čech metric thickenings for any p between 1 and infinity, which include all measures on X whose p-diameter or p-radius is bounded from above, equipped with an optimal transport metric. These families recover the previously studied Vietoris–Rips and Čech metric thickenings when p is infinity. As our main contribution, we prove a stability theorem for the persistent homology of p-Vietoris–Rips and p-Čech metric thickenings, which is novel even in the case p is infinity. In the specific case p equals 2, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2-Vietoris–Rips thickenings of the sphere as the scale increases. This is joint work with Henry Adams, Facundo Mémoli and Michael Moy.
