Research

Graduate Research

My master's project at Colorado State looked at Schur-Q functions, which are symmetric functions that form a basis of an interesting subalgebra of the algebra of symmetric functions. I used a recent combinatorial formula for decomposing a skew Schur-Q function in the non-skew basis to examine new cases in the question of when two skew Schur-Q functions are equal. I (virtually) presented a poster at FPSAC 2022, and an extended abstract was published in the conference proceedings. The arXiv version of the paper can be found here: Inequality of a Class of Near-Ribbon Schur-Q Functions.

My doctoral project at Colorado State involved drawing new connections between chromatic (quasi-)symmetric functions and the combinatorics of the cohomology ring of Hessenberg varieties. In particular, I found bijections between known polynomial bases of these cohomology rings and sets of P-tableaux, with the goal of generalizing these bijections to find bases in trickier cases. I also created a higher Specht basis for two cases of Hessenberg varieties, giving a more explicit view of the structure of the underlying symmetric group action. You can find the arXiv version of the paper here: Higher Specht bases and q-series for the cohomology rings of certain Hessenberg varieties.

In addition, I worked with Maria Gillespie and Joseph Pappe to answer a combinatorial question about when the chromatic quasisymmetric function is symmetric. We found that, when G is a tree, the chromatic quasisymmetric function is never symmetric. Further, we found a family of graphs (inspired by unit interval graphs) for which the chromatic quasisymmetric function is always symmetric. When is the chromatic quasisymmetric function symmetric?

Graduate Research Workshop in Combinatorics

In 2024, I attended GRWC and collaborated on two projects! The first project involves Kohnert diagrams - an arrangement of cells in a 2-dimensional grid - and Kohnert moves - an operation which sends one diagram to another. The associated Kohnert poset and Kohnert polynomials are relevant for Schubert calculus. We studied the structure of the Kohnert poset for "northeast" diagrams, which generalize the lock diagrams which mirror key diagrams. You can find our paper here: Kohnert posets and polynomials of northeast diagrams.

I also worked on a project looking at expansions of the chromatic symmetric function into the Schur basis. Gasharov showed that if G is the incomparability graph of a (3+1)-free poset, then the symmetric function was Schur positive, and we tried extending these results to other claw-free families of graphs. While I am no longer working on this project, I am happy to get you in contact with someone who is!

Undergraduate Research

While at Willamette University, I completed an REU in geometric graph theory with my advisor, Josh Laison, and two other students. Inspired by a paper by Frédéric Maire on the intersection graphs of the maximal rectangles of a polyomino, we looked at how to generalize this construction to more general polygons. Eventually, this project resulted in a publication in Graphs and Combinatorics: Intersection Graphs of Maximal Sub-Polygons of k-Lizards.

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