Research

In noncommutative algebra, we study mathematical structures (called rings) whose multiplication operation is not necessarily commutative. Such mathematical structures are actually ubiquitous.

If you have taken linear algebra, then you have seen that the set of all 2 ✕ 2 matrices with entries in R has a natural multiplication, but for two matrices A and B, the products AB and BA are different, in general. One way to understand this phenomenon is that A and B can be viewed as linear transformations R² → R², where multiplication of matrices corresponds to composition of functions. And, of course, the order in which you compose functions matters (your day turns out very differently if you first put on your shoes and then your socks vs. first putting on your socks and then your shoes).

Noncommutative rings also arise naturally in quantum mechanics (the position and momentum operators do not commute), conformal field theory (where symmetries are captured by so-called quantum groups), the study of Lie algebras (the universal enveloping algebra a Lie algebra is not necessarily commutative), and the study of differential operators (multiplication by t and differentiation d/dt do not commute).

I study noncommutative rings using both ring-theoretic and homological techniques. I also study Hopf algebras and weak Hopf algebras, and their actions on rings (this is part of what is known as quantum symmetry). If you’d like to learn more about some of my favorite parts of noncommutative algebra, I can recommend the following:

• An Invitation to Noncommutative Algebra by Chelsea Walton
• Symmetries of Algebras: A Guide for Newcomers a forthcoming book by Chelsea Walton
• Invariant Theory of Artin–Schelter Regular Algebras: A Survey by Ellen Kirkman
• Artin–Schelter Regular Algebras by Daniel Rogalski
• An Introduction to Noncommutative Projective Geometry by Daniel Rogalski

Consider n-dimensional affine space F₃ⁿ over the field with three elements. A capset in F₃ⁿ is a collection of points in which no three points are collinear. Finding the largest possible capset in F₃ⁿ is a very difficult problem (in fact, the exact answer is not known when n > 6). One interesting and useful fact is: although the definition of a capset is geometric, there is an equivalent arithmetic definition. A subset of F₃ⁿ is a capset if and only if it contains no three-term arithmetic progressions. Hence, techniques of combinatorial number theory can (and have) been used to study capsets.

I learned about capsets as an undergraduate at Lafayette College’s REU. Since then, I have advised several student projects of my own related to capsets. It turns out that the affine geometric space F₃⁴ can be visualized using the deck of cards from the game SET. My REU mentor Liz McMahon (together with her family) wrote a great book on the mathematics of SET.

• The Joy of SET by Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon.