Ulrich Bauer - Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Rips Filtrations

Explore a comprehensive lecture on Gromov hyperbolicity, geodesic defect, and apparent pairs in Rips filtrations. Delve into the computational aspects of persistent homology for Vietoris-Rips filtrations, examining the generalization of Eliyahu Rips' result on the contractibility of Vietoris-Rips complexes of geodesic spaces. Investigate the concept of geodesic defect and its application to general metric spaces, compatible with Rips filtration. Learn about the collapse of Vietoris-Rips complexes to corresponding subforests for finite tree metrics. Discover the connection between these collapses and the apparent pairs gradient, an algorithmic optimization used in Ripser, and understand its impact on performance with tree-like metric data. Cover topics such as filter versions, geodesic defect definitions and theorems, generic and general tree metrics, collapsibility lemma, and engage in a summary and discussion of the presented concepts.