CAREER: Cohomology, classification, and constructions of tensor categories

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The concept of symmetry has been fundamental in physics since the ancient Greeks. Mathematically, we make the notion of symmetry precise via group theory. Fundamental objects in quantum physics include abstract things like von Neumann algebras and quantum field theories. In the past several decades, it has become clear that fundamental objects in quantum physics have symmetries that are best described by certain generalizations of groups, called tensor categories. This project will provide mathematicians, physicists, and other scientists invested in quantum science with a deeper understanding of how tensor categories are structured, new examples of tensor categories, and their concrete data related to these examples. The PI will look for interesting examples via classification and constructions, which would be useful not only to the theory of tensor categories itself but also to related areas of mathematics, such as topological quantum field theory and the study of vertex algebras. The educational component of this project will establish networks of diverse voices in quantum symmetries research and in the mathematical community more broadly. The PI will organize collaborative research workshops at Indiana University (IU) for junior mathematicians start a 4-week long summer program at IU for underrepresented incoming graduate students. In more detail, the PI will study the cohomology of tensor categories, classify modular categories, and find new tensor categories via constructions, which would be useful not only to the theory of tensor categories itself but also to related areas of math, such as topological quantum field theory and the study of vertex operator algebras. In the non-semisimple setting, the PI will incorporate geometric techniques in the study of the cohomology, via support varieties, to learn about the structure of these categories. The PI will utilize Lie theoretic techniques to construct Hopf algebras and Nichols algebras in some symmetric tensor categories in positive characteristic with the aim of advancing their classification. To deepen the understanding of the structure of fusion and modular categories, the PI will focus on perfect fusion categories and on the classification program for "small" fusion categories. Studying perfect fusion categories will enable mathematicians to understand weakly integral fusion categories and would yield, for example, insights into the detection of universal anyons by experimental physicists. In addition, the PI will investigate the effect of some constructions such as gauging and zesting in different settings.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.