Kai Mawhinney

Thesis Advisor: Dr. Andrés R. Vindas-Meléndez (Harvey Mudd College - Department of Mathematics)

Second Reader: TBD.

Contact: kmawhinney@hmc.edu

Abstract:

This thesis will be aimed at open problems in matroid theory and extremal combinatorics. I am interested in cycle systems of matroids, which are collections of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. A well-known conjecture of Richard Stanley (the h-vector conjecture) holds for the class of matroids that admit cycle systems, so it is natural to ask for a description of this class. A result of the Texas State REU 2025 demonstrated that all matroids with cycle systems are regular. I improve upon this result by showing that no matroid with a cycle system can contain the matroid R10 as a minor. As a consequence, I show that all matroids with cycle systems can be made with 1- ,2- , and 3-sums of graphic and cographic matroids.

Further topics may include West's stack-sorting map and its relation to the combinatorics of permutations and Ehrhart theory.