WEIGHTED TOPOLOGICAL DATA ANALYSIS

Abstract

In the thesis, we develop the theory of weighted persistent homology. We study further properties and applications of weighted homology and persistent homology, such as the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. We also generalize the combinatorial Laplace operator of Horak and Jost by introducing a weighted coboundary operator induced by a weight function. In a subsequent chapter, we develop and study the theory of weighted fundamental groups of weighted simplicial complexes. Finally, we also study Forman's discrete Morse theory in the context of weighted homology. The results presented in this thesis are based on joint works with the author’s supervisor Professor Wu Jie, and/or Prof. Xia Kelin, Dr. Ren Shiquan.