Research

My research employs the mathematical perspective of 'categorification' to uncover a hidden layer of structure living above much of modern mathematics and theoretical physics. I access this higher level structure to advance our understanding of fundamental symmetries and how they can manifest in theoretical models. Much of my work advanced the theory of quantum groups, defining 'categorified quantum groups', and applying them to problems in low-dimensional topology and representation theory. More recently, my focus has turned towards applying these techniques in topological quantum computing and exactly solvable models.

This research has been funded by the National Science Foundation, the Alfred P. Sloan Foundation, the Simons Foundation, and the Army Research Office award on Advancing Quantum Information through Categorification. I just completed an NSF funded Focused Collaboration Grant on Categorifying Quantum 3-Manifold Invariants. I am currently directing the Simons Collaboration on New Structures in Low-dimensional Topology.