Tensor Categories and Applications

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This project is devoted to study of tensor categories, which are abstract mathematical objects expressing an intuitive notion of symmetry in precise mathematical language. These objects naturally appear in various areas of mathematics. In addition, tensor categories are relevant to some areas of physics, such as condensed matter physics, where some special types of tensor categories are used to describe anyons -- quasi-particles that are governed by statistics different from bosonic or fermionic type. Tensor categories also appear in theoretical computer science as a framework for topological quantum computation, one potential framework for design of a quantum computer. The aim of this project is to advance knowledge about tensor categories. The project addresses the following areas: development of general theory with an eye on specific classification theorems; applications of general theory of tensor categories to other areas of mathematics; and search for new interesting examples of tensor categories. More specifically, the work includes the following projects: study of semisimple tensor categories with specific Grothendieck rings; study of possible actions (or, equivalently, of module categories) of well-known tensor categories constructed via the Wess-Zumino-Witten model; study of symmetric tensor categories in positive characteristic; search for new examples of tensor categories using theory of Kazhdan-Lusztig cells and Soergel bimodules; and search for applications of tensor categories in the theory of vertex algebras and finite W-algebras.