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.
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.
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.
Several news outlets covered our recent paper, Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory, including Scientific American, Physics World, Live Science, Spektrum, and Al Jazeera
Our paper A quantum algorithm for Khovanov homology was featured in Nature News on April 10th, 2025
Graduate Students Sung Kim and Filippo Iulianelli present at the AMS Western Section to be held at California Polytechnic State University, San Luis Obispo, May 3-4, 2025
My work on the ArXiv
Online Lectures
Presentations from MSRI
Derived super equivalences from odd categorified quantum groups, Braids in Representation Theory and Algebraic Combinatorics, ICERM, February 17th
Bordered Heegaard-Floer homology, category O, and higher representation theory, Algebraic and Geometric Categorification, Banff International Research Station
Extended graphical calculus for categorified sl(2), Low-Dimensional Topology and Categorification, Stony Brook UniversityJune 21-25, 2010