Title: Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations
Speaker: Fabian Roll (TU Munich)
Speaker's URL: https://www.roll.science
Abstract: Geometric complexes are the main tool in topological data analysis to replace a point cloud with a combinatorial model that is suitable for computations. In this talk, we will focus on the Vietoris-Rips complex, which arises directly from the metric structure of the underlying space. After explaining this construction, I will mention the original application to geometric group theory, and discuss its prominent use nowadays in the context of persistent homology. For the latter I will touch on an application to the study of COVID-19 genetic evolution data and the outstanding performance on genetic distances of Ripser, which is the current state of the art software to compute Vietoris-Rips persistent homology.
Motivated by this computational observation, I will present a generalization of a well-known result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. By considering the notion of geodesic defect, I will explain how to extend this result to general metric spaces in a way that is also compatible with the Vietoris-Rips filtration. Further, I will explain how the Vietoris-Rips complexes of a finite tree metric space collapse to the corresponding subforests. Finally, I will explain how to relate these collapses to modern computational methods by showing that they are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data. This is joint work with Ulrich Bauer.
URL associated with Seminar: https://tgda.osu.edu/activities/tdga-seminar/