The Squish Map and the \(SL_2(\mathbb{C})\) Double-Dimer Model

Spring 2021 – Present

Mentor: Dr. Ben Young

We exhibit a measure-preserving map between the dimer model on the honeycomb graph, and the \(SL_2(\mathbb{C})\) double dimer model on a coarser honeycomb graph. We compute the most interesting special case of this map, related to plane partition enumeration with 2-periodic weights, and give some curious conjectural formulae which arise as special cases.

Tilings of Benzels on the Hexagon Grid

Spring 2023 – Present

Collaborators: Colin Defant, Rupert Li, Jim Propp, and Ben Young

Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and prove some additional results that he left tacit. In order to solve these problems, we first transfer benzels into the square grid. One of our primary tools, which we combine with several new ideas, is a bijection (rediscovered by Stanton and White and often attributed to them although it is considerably older) between k-ribbon tableaux of certain skew shapes and certain k-tuples of Young tableaux.

Automorphisms and Characters of Finite Groups

Fall 2018 – Spring 2019

Mentor: Dr. Mandi Schaeffer Fry

Investigated questions in representation theory regarding the symplectic group of degree four. Used tools provided by group representations, irreducibility, CG-modules, group algebras, Maschke’s theorem, Schur’s lemma, and character theory.

Sorting Permutations in the Presence of an Adversary

Summer 2018

Research Experience for Undergraduates at Boise State University

Mentor: Dr. Marion Scheepers

Answered questions about game theory and sorting algorithms presented in the context of sorting ciliate DNA. Used tools provided by linear algebra, graph theory, game theory, and the combinatorics of permutations to analyze sorting related problems. Gave weekly progress update presentations to the REU community throughout the summer.

Quarto

Spring 2018 – Fall 2018

Mentor: Dr. Diane Davis

An independent research project focused on the mathematical possibilities presented by the two-player impartial game Quarto, using tools provided by game theory, linear algebra, and abstract algebra including group actions and binary vector fields.