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

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.

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