CAREER: Subfactors, Tensor Categories, and Local Topological Field Theory

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One of the main threads in mathematics is the study of the symmetries of an object. Another is the study of groups, which are very important throughout mathematics and science. The collection of all classical symmetries of an object forms a group, and in quantum settings, a similar role is played by a generalized version of groups called quantum groups or tensor categories. These again appear both in science (e.g. phases of matter) and mathematics (e.g. knot theory). One of the earliest appearances of groups was in the study of Galois theory, where one looks at symmetries of a number system that do not move a subsystem, and in parallel one of the earliest appearances of quantum groups was in a similar Galois theory for von Neumann factors. The research portion of this project focuses on the study of quantum groups and subfactors. One important technique used to study such quantum groups is called "higher dimensional algebra." The education and outreach portion of this project extends the investigator's decade-long involvement in high school education at summer math programs and more recently in education of high school math teachers. He will develop new courses and research projects for high school students at the Canada/USA Mathcamp based on his research, he will start a program for awarding dissertation writing fellowships for graduate students with an established record of extraordinary education and outreach, with a particular emphasis on underrepresented groups, he will start a math club for mathematics education majors, and he will supervise undergraduate research projects. This research project uses techniques from higher category theory and topology to better understand the theory of tensor categories and specific examples coming from the theory of subfactors. This will be done through two major research programs. The first is to study the structure of small index subfactors that the investigator classified in prior work and to prove other classification results related to bimodules and intermediate subfactors. The second program is to understand local topological field theories with values the 3-category of tensor categories and related higher categories using the insight given by the Baez-Dolan-Hopkins-Lurie cobordism hypothesis.