Betti tables of multiparameter persistence modules

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Tezin Türü: Yüksek Lisans

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2024

Tezin Dili: İngilizce

Öğrenci: BERAT GEVEN

Danışman: Semra Pamuk

Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu

Özet:

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.