Abstract:
In this talk, I will give an introduction to topological data analysis (TDA), with an emphasis on the notion of persistence diagrams. These objects, arising in algebraic topology, provide a concise, quantitative way to visualise the homological information carried by filtrations of topological spaces. In TDA, filtrations are often built from data sets using Vietoris-Rips complexes or similar constructions.
One can study persistence diagrams from a geometric point of view, by equipping the space of diagrams with metrics inspired by optimal transport. I will discuss this connection and what it reveals about the metric structure of spaces of persistence diagrams.