Henry Adams - Bridging applied and geometric topology

38th Annual Geometric Topology Workshop (Online), June 15-17, 2021

Henry Adams, Colorado State University
Title: Bridging applied and geometric topology
Abstract: I will advertise open questions in applied topology for which tools from geometric topology are relevant. If a point cloud is sampled from a manifold, then as more samples are drawn, the persistent homology of the Vietoris-Rips complex of the point cloud converges to the persistent homology of the Vietoris-Rips complex of the manifold. But what are the homotopy types of Vietoris-Rips complexes of manifolds? Essentially nothing is known, except that as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. I will survey emerging connections between Vietoris-Rips complexes of manifolds and the filling radius (which Gromov used to prove the systolic inequality in 1983), thick-thin decompositions, and sweepouts. This is joint work with Baris Coskunuzer.