H′-splittings Of 3-manifolds And Persistent Homology For Hypergraphs

This thesis consists of two parts.We first introduce the H^′-splitting of three dimensional manifolds.For a compact connected surface F properly embedded in a compact connected orientable 3-manifold M,if F decomposes M into two handle bodies H₁and H₂,then H₁∪_FH₂is called an H^′-splitting for M.Clearly,when M is closed,this is just the Heegaard splitting for M.When M is with boundary,the H^′-splitting is different from the Heegaard splitting for M.In this thesis,we first show that any compact connected orientable 3-manifold admits an H^′-splitting.Then,we generalize Casson-Gordon theoren on weakly reducible Heegaard splitting to the H^′-splitting case.Then,we introduce persistent homology.As a generalization of simplicial complex ho-mology,Wu and others develop the theory of hypergraphs,embedding homology and persistent embedding homology on hypergraphs.To get a better understanding of embedding homology,in this thesis,we define a new homology（?）,serving as an complement to the embedding homol-ogy.The second main result of this thesis is an algorithm for computing persistent embedded homology and（?） for hypergraphs.This algorithm is a generalization of classical algorithm on simplicial complex.Finally,we apply persistent homology on data analysis.In this thesis,we use Vietoris Rips complex and persistent homology to analyse the folding process of peptide Aβ（1-28）.We mod-ify classical persistent homology to localized weighted persistent homology.Persistent barcode numbers are computed and reduced to two dimensions by principal component analysis,result-ing in geometric visualization of the folding paths and stable conformation states.By different combination of localizations and dimensions,one obtains persistent barcodes in other potentially interesting coordinate systems and reveals new metastable states.We show this by comparing results of dimension one and dimension two barcodes.