Subfactors, Tensor Categories, and Higher Dimensional Algebra

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Symmetries are movements of an object or system back onto itself. For example, shuffling a deck of cards or rotating an atom around its nucleus. Over the last 200 years, Group Theory has been spectacularly successful in using symmetry to study a wide variety of problems in mathematics, physics, chemistry, and engineering. More recently, starting in the 1980s, mathematicians and mathematical physicists have developed a theory of Quantum Groups and Quantum Symmetry which adapt ideas from Group Theory to more general settings appearing in quantum physics and related mathematics. The first spectacular application of what we now call Group Theory was Galois' use of symmetries of number systems to explain why there's a generalization of the familiar quadratic formula to solve equations using third or fourth powers, but no such formula for solving equations with fifth powers. Galois's number systems are parallel to von Neumann factors developed to study quantum mechanics, and several of the first spectacular applications of what we would know call Quantum Groups or Quantum Symmetries appeared in Jones's study of subfactors. More recently Quantum Symmetry has become central to certain approaches to constructing quantum computers using so-called topological phases of matter. The main techniques used in this project involve higher dimensional algebra. Here the idea is that we typically write mathematical symbols on a line when doing calculations, but in many settings it is better to use the whole plane to draw pictures, or to do the calculation on "paper" in more interesting shapes like circles or spheres. Sometimes the geometry allows for insight into the mathematics, for example you can "rotate" a calculation taking place in the plane. The goal of this project is to use higher dimensional algebra to study questions in quantum symmetry. The project will also contribute to the development of the US workforce through the training of Ph.D. and high school students.In more technical language, we use planar algebras, topological quantum field theory, and higher categories to study questions coming from von Neumann subfactors and tensor categories. These include very example-driven questions like constructing and studying exceptional small index subfactors, as well as more theoretical questions like understanding the structure of the 3-category of fusion categories (where subfactors appear as 1-morphisms). More specifically, this project focuses on four closely related research programs. First, we will develop a purely algebraic analogue of the author's previous work on the classification of small index subfactors. Second we will use module categories more systematically to approach analytic questions about reconstructing subfactors from their standard invariants. Third, we will use skein theoretic approaches and Jones's planar algebras to study tensor categories, including non-semisimple tensor categories. Finally, we study tensor categories and braided tensor categories using higher-categories and local topological quantum field theories building on the Lurie-Baez-Dolan cobordism hypothesis.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.