Being able to describe the shape of data is of paramount importance to the fields of biology, physics, chemistry, pharmaceutics, etc. Therefore, in recent years, scientists from the TDA community have been applying advanced mathematical tools to decode the topological structures of data. Methods such as persistent homology, path homology, and de Rham-Hodge theory have become the main workhorse of TDA, which pioneered new branches in algebraic topology and differential geometry. Later, various topological Laplacians such as graph Laplacian, Hodge Laplacian, sheaf Laplacian, and Dirac Laplacian are proposed to preserve topological invariants and geometric shapes simultaneously. However, such Laplacians fail to extract the topological and geometric deformations when one introduces the filtration parameters in. Therefore, we proposed a new topological Laplacians called persistent Laplacians to fully recover the topological persistence and homotopic shape evolution during filtration.It is worth mentioning that persistent Laplacians are insensitive to asymmetry or directed relations, which limits their power to preserve the directional information of structures in practical applications. Therefore, we proposed persistent path Laplacians to overcome this issue. Similar to the persistent Laplacians, one can also extract the topological persistence and geometric deformations during filtration from the persistent path Laplacians by calculating their harmonic and non-harmonic spectra. In addition, the persistent path Laplacians are constructed on the directed graphs or network, which address the importance of directional representation in datasets such as gene regulation datasets in biology.Versatile mathematical tools have been playing an essential role in various biological applications. Since the first COVID-19 case was reported in December 2019, researchers worldwide have been pursuing scientific endeavors in the SARS-CoV-2 projects. Instead of designing promising vaccines and antibody therapies that required wet lab resources, we proposed a new mathematical-AI model called TopNetmAb to systematically analyze the mutation-induced impacts on the SARS-CoV-2 infectivity, vaccines, and antibody drugs. In this dissertation, the topological data analysis (including the persistent Laplacians mentioned above), artificial intelligence, various network models, and genomics analysis are all included in our SARS-CoV-2-related projects to provide comprehensive representations for the understanding of the transmission and evolution of SARS-CoV-2.
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