Research statement (Fall 2026)

I work on algebra and representation theory within mathematical physics. Specifically conformal field theory and vertex operator algebra representation theory.

My current research consists of deriving differential equations satisfied by matrix coefficients of products of twisted intertwining operators and q-traces of products of twisted intertwining operators. I then use these differential equations to show that their solutions converge absolutely in appropriate regions and analytically extend to larger regions. The proof of such convergence and analytic extension is a necessary step in the vertex-operator-algebra-representation-theoretic construction of orbifold conformal field theory, specifically to show existence of genus-zero and genus-one correlation functions. 

The method I choose to use works with the full vertex operator algebra and its twisted representation theory, as opposed to a method focused on proving properties about fixed-point subalgebras. I have been successful at proving convergence in certain cases that the fixed-point-subalgebra method has not (yet) been able to prove. 

The convergence and analytic extension are also a necessary step to construct the G-crossed (modular) tensor categories using the method of Jishen Du and Yi-Zhi Huang (see [DH26]). My current work contributes to Huang's program to construct orbifold conformal field theories (and related structures) using the twisted representation theory of vertex operator algebras. 

Things I am doing at the moment include: 

• Deriving differential equations for twisted intertwining operators among modules twisted by automorphisms in an abelian group. (The abelian group is general; it can contain automorphisms of infinite order and non-semisimple automorphisms.)

• Solving the twisted KZ equations in certain cases (involving automorphisms of infinite order!) and proving regularity of the solutions.

• Deriving new twisted KZ equations beyond (1 g g)-type intertwining operators. 

• Deriving differential equations for products of intertwining operators among modules twisted by automorphisms in a general finite group. (The finite group is general; need not be abelian, can be simple. Very hard) 

• Deriving differential equations for q-traces of (1 g g)-type twisted intertwining operators. (Including certain cases of automorphisms of infinite order.) Trying to extend to more cases.