Fusion Categories

Abstract: A fusion category is roughly a collection of abstract “objects” that is closed under direct sums, tensor products, and duals. They capture symmetries that are more general than those arising from groups. A basic example is the category of finite-dimensional vector spaces: direct sums, tensor products, and duals of finite-dimensional vector spaces are again finite-dimensional vector spaces. Similarly, the representation category of a finite group is a fusion category. Fusion categories arise naturally in many areas of modern mathematics, including representation theory, operator algebras, topology, and mathematical physics.

In physics, fusion categories provide a mathematical framework for describing topological phases of matter, quantum computation, and topological quantum field theories. Understanding the structure and classification of fusion categories is therefore an active area of research connecting pure mathematics with theoretical physics.

In this project, we will explore the basic theory of fusion categories and investigate selected research problems. The specific direction can be tailored to the student's background and interests. If time permits, there will also be an opportunity to develop the results into a research paper.