Webs arise in many contexts. My undergraduate research has focused on their combinatorial and representation-theoretic aspects. Broadly speaking, webs provide a visual way to study semisimple Lie algebras and their quantum representations. I focus on sl_n webs, where much is known for n=2 and n=3, but the general theory is not as well established and thus remains a highly active area of research.

In combinatorics, webs are naturally indexed by famous objects such as Young tableaux and noncrossing matchings. In a recent paper I wrote with Professor Tymoczko and two other students, we demonstrate that a well-studied operation on rectangular standard Young tableaux, called evacuation, corresponds to an operation on sl_n web graphs, called reflection. Our result generalizes previous work by Patrias and Pechenik for n=2 and n=3 to all n. 

Circulant graphs have been extensively studied for their ability to generate quantum error codes (QECCs) with high minimum distance, a parameter related to the number of errors a code can correct. In this project, we generated QECCs from the adjacency matrices of multidimensional circulant graphs (MDCs), a generalization of circulant graphs in multiple coordinates. We explored the graph-theoretic properties of these new graphs, establishing isomorphism characteristics within a broader class of vertex-transitive graphs. Using these results, we reduced the search space for new QECCs and showed, with an exhaustive computer search, that MDCs can produce comparable codes to circulant graphs. We discovered two new 0-dimensional QECCs of lengths 77 and 90 with respective minimum distances 19 and 22, improving upon previous best-known values by one.  We published these results in the journal Discrete Mathematics. Utilizing a similar approach, we were able to construct five new ternary QECCs with the best known parameters at the time of discovery; however, better constructions have been presented since.  

In 2022, Defant introduced toric promotion as a cyclic analogue of Schützenberger's famous promotion operator. Toric promotion acts on the labeling of a chosen simple graph by a series of involutions. A natural question is how toric promotion behaves under certain graph operations. In this project,  I analyzed the orbits of toric promotion under the bridge sum operation, which joins two graphs by adding an edge between a vertex of each graph. Defant described the orbit size of toric promotion on trees and showed that it does not depend on the initial labeling; I showed an analogous result for complete graphs. I described the orbit length of toric promotion on the bridge sum of two complete graphs and showed that it is independent of the initial labeling. I described the orbit length of toric promotion on the bridge sum of a complete graph with any tree, and showed that it is also independent of the initial labeling. Finally, I showed that the orbit length of toric promotion on the bridge sum of either a tree or a complete graph with any simple graph does not depend on the restriction of the initial labeling to the tree or complete subgraph. I submitted these results for publication in the Electronic Journal of Combinatorics. I will be presenting these results at the 2026 Joint Mathematics Meetings.