Some persistent cohomology invariants and an axiomatic version of persistent homology

Author(s)

Marco Antônio de Freitas Contessoto

• Document type: Doctoral Thesis
• Press: São José do Rio Preto. 2022-01-21.
• Institution: Universidade Estadual Paulista (Unesp). Instituto de Biociências Letras e Ciências Exatas. São José do Rio Preto
• Defense date: 2021-12-15
• Advisor: Alice Kimie Miwa Libardi; Roberto Facundo Mémoli Techera

Abstract

In this work we find two chapters focussing on two of the most important tools for topological data analysis: persistent homology and persistent cohomology. The approaches given to these two tools are very different in nature and objectives. With numerous applications in the most varied areas, the persistent homology has already proved to be a very powerful tool, but there are no study about an axiomatic approach of it. We define persistent versions of the Eilenberg-Sttenrod axioms, with which we can develop and construct its properties. To conclude, we prove a uniqueness theorem, showing the full characterization of our theory through these axioms. Considering the dual tool of the previous one, we have the persistence cohomology. Much studied in recent articles, cohomology comes as an alternative form, faster and with the same efficiency than persistence homology, since due to the dualities, we have similar constructions. However, very little addressed in these works, the ring structure that is gained by working with cohomology did not have relevant development in TDA. In this work, we will define two invariants totally related to this ring structure, which arises through the cup products. We will calculate several examples of these invariants, showing situations in which they are able to give us more information than the old tools. (AU)

• FAPESP's process: 17/25675-1 - Persistent cohomology
• Grantee: Marco Antonio de Freitas Contessoto
• Support Opportunities: Scholarships in Brazil - Doctorate