Subfactors and Tensor Categories

One of the oldest and most important concepts in mathematics and science is the notion of symmetry, which describes how an object or system can be rearranged or permuted in interesting ways, such as in the rotations of a cube or the shuffling of a deck of cards. Mathematicians have studied symmetry for hundreds of years using the language of group theory, but in the past 40 years have investigated a more expansive notion of symmetry, known as quantum symmetry, which arises in several core areas of mathematical research, including operator algebras, quantum field theory, and representation theory. These fields are inherently vital research areas, and enjoy fruitful interplay with physics, computer science, and technology. For instance, mathematicians were inspired by physics to study abstract notions of quantum symmetry, and develop models that were later found to describe the physical properties of certain exotic materials. Within this broader context of the mathematics of quantum symmetry, this project aims to discover and study novel examples of quantum groups from analytical, algebraic, and topological points of view, including certain classes of examples of interest in condensed matter physics. The project will contribute to Indiana University’s broad emphasis on quantum science and create research opportunities for graduate students and postdocs. This project considers quantum symmetry in a variety of contexts, focusing on example-driven problems about von Neumann subfactors and tensor categories. The main emphasis is on problems that can be approached using skein theoretic or planar algebraic methods. The project uses techniques developed in the operator algebraic subfactor community applied to questions in other areas. More specifically, the proposed work uses module and bimodule categories to study subfactors and their generalizations, uses skein theory to understand families of tensor categories, and studies local topological field theories related to tensor categories. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.