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My research lies in the intersection of nonassociative algebras, combinatorics, and universal algebra. I primarily work with quasigroups, a type of nonassociative algebra. While the study of nonassociative algebras dates back to the time of Euler, the development of quasigroup theory in the twentieth century came from a variety of areas. Some motivators include web geometry, alternative algebras, and universal algebra.
What is a quasigroup?
A set Q equipped with a binary operation * satisfies the Latin square property if for all x, y, z in Q, knowing any two arguments in the equation x*y=z uniquely specifies the third. The ability to solve for the third argument given any two arguments give rise to right and left divisions. Formally, we capture the algebraic structure in the following definition.
An equational quasigroup is a set Q equipped with three binary operations multiplication *, right division /, and left division \ such that for all x, y in Q, the following equations are satisfied:
(SL) x*(x\y) = y = (y/x)*x (SR)
(IL) x\(x*y) = y = (y*x)/x. (IR)
Groups are examples of quasigroups. As the binary multiplication of a quasigroup need not be associative, we may view quasigroups as the nonassociative analogue of groups. However, this analogy paints a pale portrait of the rich and vast landscape of quasigroup theory. The integers under subtraction form a quasigroup with a truly nonassociative multiplication. Quandles are examples of right-distributive, idempotent right-quasigroups.
See my research statement for more details on my current and past projects!
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