Mathematics

Persistent homology is one of the main tools of topological data analysis and has had many successes in applications. Multi-persistence is a natural extension of the idea of persistence but has not yet enjoyed the same success. The main obstacle is the increased complexity of the underlying algebraic structure. In this talk, I will introduce the idea of multi-persistence and show why it is a natural object to consider. I will give an overview of the challenges introduced by the multi-persistence that need to be solved before one can successfully apply it in TDA. Finally, I will present recent works that contribute to two of the challenges. Firstly, looking at the modules that can be obtained by projecting a d-dimensional module to a (d-1)-dimensional module, we can gain some understanding of the structure and complexity of multi-persistence. Secondly, we can obtain geometrically interesting bifiltrations by using the k-fold cover, defined as the union of intersections of k-balls with varying radii r.