Research Interests
My research interests center around the rapidly developing theory of vertex operator algebras(VOAs) and their generalizations. In particular, I am interested in representation theoretic aspects of the theory, and applications of vertex algebras in combinatorics and q-series.
Vertex operator algebras are increasingly popular algebraic structures with many applications in representation theory, conformal field theory, finite group theory, and other fields. Vertex algebra theory can be thought of as mathematically nonclassical in the same way string theory is a nonclassical physical notion. Vertex algebras were first introduced by Borcherds in 1986 as an algebraic interpretation of the chiral algebras physicists use in string theory. He went on to use these objects to prove the Conway-Norton conjecture, which earned him the Fields Medal in 1998. Frenkel, Lepowsky, and Meurman introduced the notion of a vertex operator algebra, which they used to construct the largest sporadic finite simple group, the Monster \(M\), as the group of symmetries of a certain vertex operator algebra \(V^{\natural}\), which is also a graded vector space. Before this, the construction of M was by way of a highly nonassociative algebra \(B\), the Griess algebra, of dimension 196,883. While \(V^{\natural}\) is infinite dimensional, this additional structure makes it a more natural space than \(B\) with which to associate \(M\). Furthermore, we have
$$\sum_{n\geq-1}(\text{dim }V^{\natural}_n)q^n=J(q)=q^{-1}+0+196,884q+\dots$$
where \( J(q)\) is the classical modular form!