Fusion categories and topological quantum field theory

A fusion category is a mathematical structure modelling interacting particle systems. I will study fusion categories, and develop the first explicit description of all low dimensional examples. My work has major implications for the development of topological quantum field theory. I will prove new results about 3 dimensional topological quantum field theory, with numerous applications to topology. These mathematical theories successfully model fractional quantum Hall materials, which are the substrate for the proposed topological quantum computing architecture. My project contributes to the development of operator algebra theory, and the new mathematical fields of quantum algebra and quantum topology and has applications to physical models.