# Research Area: Algebra and Number Theory

### Quasigroups and Loops

Anytime the associative law (xy)z = x(yz) fails, we enter the realm of nonassociative mathematics. Traditionally the subject is split into two areas: nonassociative algebras (such as Lie and Jordan algebras), and quasigroups and loops (including parts of the theory of latin squares).

At DU, we mostly focus on quasigroups and loops. A quasigroup is a set with binary operation * for which the equation x*y=z has a unique solution whenever the other two variables are specified. Loops are quasigroups with an identity element.

Numerous techniques are used in loop theory, borrowing from group theory, combinatorics, universal algebra, and automated deduction. The investigation often focuses on a particular variety of loops, such as Moufang loops (satisfying the identity ((xy)x)z = x(y(xz))).

### Vertex Operator Algebras

Vertex operator algebras (VOAs) arise naturally in the context of representation theory of various infinite dimensional Lie algebras. In physics, they are certain algebraic structures underlying two-dimensional conformal field theories. They were first defined mathematically by Richard Borcherds (1986) in his proof of the Moonshine conjecture, and in the last 35 years they have found applications in a diverse range of subjects including finite groups, representation theory, combinatorics, number theory, and geometry.

Our focus at DU is on problems based on the representation theory of W-algebras and affine Kac-Moody algebras, geometric methods, tensor category theory, and number theoretic aspects.

### Integer Partitions

Integer partitions are simply the number of ways of writing a positive integer as a sum of positive integers. While so simple to describe, they are a fundamental combinatorial object and have profound connections to various other branches of mathematics such as algebra, number theory, topology and mathematical physics.

Our current focus is on investigating integer partition identities of Rogers-Ramanujan-type, especially with regards to their relations to representation theory of affine Kac-Moody algebras and vertex operator algebras.