Geometric interpretations of persistent homology

Abstract: Persistent homology as a parameterized version of homology appeared around the turn of the millennium. It was motivated by the desire to analyze and compare geometric shapes. While homology counts the number of holes in a space, persistent homology is typically expected to detect the sizes of holes at different scales. Nowadays, persistent homology is the workhorse of topological data analysis with substantial theoretical foundations in algebra, topology, etc. In this talk we will provide a survey of results interpreting parts of persistence homology in terms of properties of underlying spaces. The properties in question include homology of a space, shortest 1-dimensional homology (loop) basis of a geodesic space, locally shortest loops, systole, subspaces of contraction, proximity properties, rigidity of critical simplices, and local minima of the distance function.