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

Spring 2021 – Spring 2024

Mentor: Dr. Ben Young

Published: Electronic Journal of Combinatorics

A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a “coarser” plane partition, made of cubes of side length 2. Indeed, there are two such approximations obtained by “rounding up” or “rounding down” to the nearest cube. We relate this coarsening (or downsampling) operation to the squish map introduced by the second author in earlier work. We exhibit a related measure-preserving map between the dimer model on the honeycomb graph, and the \(SL_2\) double dimer model on a coarser honeycomb graph; we compute the most interesting special case of this map, related to plane partition q-enumeration with 2-periodic weights. As an application, we specialize the weights to be certain roots of unity, obtain novel generating functions (some known, some new, and some conjectural) that (-1)-enumerate certain classes of pairs of plane partitions according to how their dimer configurations interact.

Tilings of Benzels via Generalized Compression

Spring 2023 – Present

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

Preprint: Arxiv

Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain \(k\)-ribbon tilings to \((k−1)\)-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp’s conjectures and recast several others as problems about perfect matchings.

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.