My research explores the rich interface between mathematics and theoretical physics through the lens of categorification—a powerful framework that lifts classical algebraic structures to higher categorical analogues. This perspective reveals hidden layers of symmetry and structure, enabling new insights across representation theory, low-dimensional topology, and quantum computation.

Categorification of quantum groups and applications to low-dimensional topology

A central thread in my work has been the categorification of quantum groups, where I helped define and develop the theory of categorified quantum groups and their connections to link invariants and representation theory. These ideas have reshaped how we understand structures underlying knot homologies and have provided new tools for studying 3-manifold invariants.

Advancing topological quantum computation leveraging new techniques from low-dimensional toplogy

More recently, I have turned toward topological quantum computation, leveraging techniques from topological quantum field theory (TQFT) and non-semisimple representation theory to explore new models for fault-tolerant quantum information processing. This includes work on non-unitary TQFTs, modular categories, and the computational potential of exotic anyonic systems.

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.    We 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.