Title: Vietoris-Rips complexes of circles, ellipses, and higher-dimensional spheres
Speaker: Henry Adams (Colorado State)
Abstract: Given a metric space X and a distance threshold r > 0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Janko Latschev proves an analogous theorem for sufficiently dense samplings. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applied topology and persistent homology. We describe how as r 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. These homotopy types are connected to cyclic and centrally symmetric polytopes and orbitopes. Interestingly, Latschev's result fails for the ellipse with larger r values, and certain Vietoris-Rips complexes of ellipses contain ephemeral summands in their persistent homology modules. We argue that infinite Vietoris-Rips complexes should be equipped with a different topology: an optimal transport or Wasserstein metric thickening the metric on X. Using this new metric, we describe the first change in homotopy type (as r increases) of Vietoris-Rips complexes of higher-dimensional spheres. Joint work with Michal Adamaszek, Florian Frick, and Samadwara Reddy.
Seminar URL: https://research.math.osu.edu/tgda/tgda-seminar.html