DMS Topology Seminar

Speaker: Henry Adams 

Title:  An introduction to Vietoris-Rips complexes

Abstract: I will give an introduction to Vietoris-Rips complexes, Vietoris-Rips metric thickenings, and their uses in applied and computational topology. So, this talk will provide a complementary perspective to Krystyna Kuperberg's talk last week, though attendance last week is not assumed, since I wasn't there! If a dataset is sampled from some unknown underlying space (say a manifold), then as more and more samples are drawn, the Vietoris-Rips persistent homology of the dataset converges to the Vietoris-Rips persistent homology of the manifold. But little is known about Vietoris-Rips complexes of manifolds. An exception is the case of the circle: I will describe how as the scale parameter increases, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, ..., until finally they are contractible. Much less is known about Vietoris-Rips complexes of spheres. Throughout my talk, I'll advertise some open questions.