# 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.

### Group Theory and Representation Theory of Finite Groups

Group theory is an abstract branch of mathematics motivated by the study of symmetries of an object, such as those coming from nature, art, communication networks, or any other place that symmetry might play a role. Representations are structure-preserving maps from an abstract group to a matrix group, which create a theory helpful for understanding the structure of a group and the symmetries it represents by allowing us to use the full power of linear algebra.

Schaeffer Fry’s research involves the irreducible representations of finite groups of Lie type, sometimes called finite reductive groups, which are analogues of Lie groups over finite fields. The Classification of Finite Simple Groups tells us that “most” finite simple groups can be obtained from these groups, so understanding their structure and representation theory is a crucial step toward many problems in the theory of finite groups and their representations. She has been particularly interested in problems concerning local-global conjectures in character theory and problems involving the action of Galois automorphisms on characters.