A Study of Persistent Homology through Persistence Modules

Abstract

Persistent homology is a tool used in topological data analysis, providing a multiscale approach. Persistence modules provide an algebraic approach to persistent homology. Their decomposition into interval modules is studied using quivers and Gabriel's theorem, as well as generalisations of this result. It is shown that if such a decomposition exists, it is unique up to isomorphism. Several sufficient conditions for such a decomposition to exist are put forward. Afterwards, the interleaving distance is defined. It is used to prove the stability theorem, which justifies the use of persistent homology in topological data analysis.