Betti tables of multiparameter persistence modules

Date

2024-8-26

Author

Geven, Berat

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People still try to search for new effective methods to analyze the given data. Topological data analysis (TDA) is just one way for doing this, and multiparameter persistent homology is currently one of the favorite areas of TDA. Moreover, Betti tables is an important invariant in multiparameter persistent homology. The aim of this thesis is to give a survey about Betti tables also known as multigraded Betti numbers. In this thesis, we elaborate the article On the Support of the Betti Tables of Multiparameter Persistent Homology Modules, which gives a bound to support of the Betti tables of some special multiparameter persistence modules. After giving the necessary background about the topic, we give an equivalent definition for Betti tables of a persistence module by using Koszul complex, and we construct Koszul complexes of persistence modules iteratively. Then, a proof of bounding the support of the Betti tables of the given persistence module by using the critical cells of a discrete gradient vector field is given. Lastly, for the case n = 2 this result is strengthened.

Subject Keywords

Betti tables, Persistent homology, Koszul complex, Discrete morse theory, Multiparameter persistence

URI

https://hdl.handle.net/11511/111071

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Graduate School of Natural and Applied Sciences, Thesis