In topological data analysis, one uses persistent homology and its dual notion persistent cohomology to study the evolution of (co)homology across a filtration. Compared with homology, cohomology is enriched with a graded ring structure given by the cup product operation. In this talk, we utilize the cup product operation to define a new invariant for the persistent cohomology ring, called the persistent cup-length function, which is able to extract and encode additional information across a filtration, compared to the persistent (co)homology vector space. The persistent cup-length function is a lifted version of the standard invariant: the cup-length of a cohomology ring, which is the maximum number of cocycles having non-zero cup product. We show that the persistent cup-length function is stable under suitable interleaving-type distances, and we devise a polynomial time algorithm for its computation.
Ling Zhou, The Ohio State University (Columbus, OH, US)Authors: Marco Contessoto, Facundo Mémoli, Anastasios Stefanou and Ling Zhou
2022 AWM Research Symposium
Women in Computational Topology