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. A preprint of the full paper can be found here: Inequality of a Class of Near-Ribbon Schur-Q Functions.

My doctoral project at Colorado State involves drawing new connections between chromatic (quasi-)symmetric functions and the combinatorics of the cohomology ring of Hessenberg varieties. In particular, I am finding new bijections between known polynomial bases of these cohomology rings and certain tableaux, with the goal of generalizing these bijections to find bases in trickier cases. Stay tuned for a paper with my results!

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 important, and we studied a subclass of these objects called lock diagrams (essentially right-justified Kohnert diagrams).

The second project looked 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 are extending these results to other claw-free families of graphs.

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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