New Topologically Inspired Directions in Higher Representation Theory

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The interaction and interplay between geometric and algebraic techniques have fueled many exciting developments throughout mathematics and theoretical physics. Inherent within many of these deep connections is the notion of symmetry. As we push the boundaries of mathematics to unexplored frontiers, it is becoming increasingly clear that a more refined notion of symmetry is required for the next generation of theories. This project leverages cutting-edge developments in geometry to uncover new algebraic structures governing these new "higher" symmetries. The project will also provide research training opportunities for students.Leveraging new insights gained through the interplay between low-dimensional topology and representation theory, this project details a program to achieve results not apparent through one specialization alone. The PI will introduce fundamentally new tools in higher representation theory inspired by advances in emerging areas of topology, including bordered Heegaard-Floer theory, odd link homologies, non-semisimple TQFT, and spectrafication (or stable homotopy refinements of link homologies). Transporting and reinterpreting these recent advances through a representation-theoretic lens will illuminate new connections and allow for extensions that are not only useful in advancing the field of representation theory but can also be back-fed into the study of low-dimensional topology and surrounding fields.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.