Homotopical Methods in Higher Representation Theory

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The study of symmetry is a fundamental tool for understanding the world around us, as well as the mathematical frameworks we use to describe it. As we push the boundaries of our understanding of mathematics and explore the frontiers of theoretical physics, it is becoming increasingly clear that a more refined notion of symmetry is required for the next generation of theories. The mathematical exploration and development of these types of "higher" symmetries reveals structure and form in any system where they are present, reducing degrees of uncertainty in these system and bringing progress within reach. While a great deal of progress has been made developing these higher symmetries, the razor's edge of current understanding has revealed that the most exciting applications will require a more flexible theory than has been previously developed. This project will fill this knowledge gap by providing a robust approach to higher symmetries, capable of describing more complex phenomena and supplying the requisite framework to facilitate the most pressing applications. This award will also support the training and professional development of graduate students and postdocs. This project utilizes homotopical methods to expand existing technology in the field of higher representation theory. The primary aim is to expand the notion of categorified quantum group from its additive framework to a more robust and homotopically flavored theory that is better adapted for solving the most pressing open problems in the area. In particular, such a framework will shed light on tensor products of higher representations, lead to a richer higher representation theory that allows for infinite dimensional representations, and provide direct applications to stable refinements in link homology. This award will also support the training and professional development of graduate students and postdocs.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.