Application of Algebraic Topology in Learning and Data Science

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Shamsaddini, Vahid | 2022

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 55387 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir
Abstract:
This thesis has provided an overview of the theoretical justification for the discrete Morse theory، persistent homology، and applications. Once the primary definitions of complexes have been established in the first chapter، we have explored the Discrete morse theory in detail in the second chapter. Then it is used to prove that disconnectivity is an eva- sive feature of graphs. Furthermore، some theorems related to discrete morse theory have been used to reduce the time complexity of ho- mology algorithms. Next chapter، we try to establish the foundation of persistent homology، and then we explore stability theorems of persis- tent homology. Finally، an application chapter has been added، which studies the application of persistent homology in industry and data sci- ence as well as its application in pure mathematics
Keywords:
Discrete Morse Theory ; Data Science ; Computational Algebraic Topology ; Algebraic Connectivity ; Persistent Homology