Title: From discrete to persistent homotopy groups

Ayres Hall 112

Abstract: As one of the main tools in topological data analysis, persistent homology has been used to extract both geometric and topological features of datasets. Motivated by the fact that homotopy in general contains more information than homology, we study notions of persistent homotopy groups of compact metric spaces, together with their stability properties in the Gromov-Hausdorff sense. Our construction in degree one makes use of the discrete homotopy theory, which was established by Berestovskii and Plaut and further developed by Wilkins. Under fairly mild assumptions on the spaces, we proved that the classical fundamental group has an underlying tree-like structure (i.e. a dendrogram) and an associated ultrametric. We finally describe the notion of persistent rational homotopy groups, which is easier to handle but still contains additional information compared to persistent homology.