Yu, Matthew2023-06-052023-06-052023-06-052023-05-26http://hdl.handle.net/10012/19523This thesis aims to highlight aspects of mathematics and physics that arise in topological field theories. We will consider invertible and noninvertible topological theories. In the former case, we compute the classification of these invertible theories which arise as the trivial bulk of some anomalous theory one dimension lower. The computation tools used here were conceived in algebraic topology and this work aims to develop these techniques for applications to physical theories. In the latter case, to study such theories in low dimensions we develop part of the theory of fusion 2-categories. Using techniques here allow us to classify noninvertible phases up to equivalence.enDoctoral Thesis