Research
My primary research interest is in algebraic combinatorics. I like thinking about polynomials--my favorite approaches to studying them involve combining the insights offered by abstract structures with computational methods. I am currently working on understanding homological invariants of certain ideals that arise in Schubert calculus.
Publications:
Representations from matrix varieties, and filtered RSK (with Abigail Price and Alexander Yong).
We describe a "filtered" generalization of RSK and use it to prove a combinatorial rule for the multiplicities of irreducible representations in the coordinate rings of a class of matrix varieties we call "bicrystalline". We show that matrix Schubert varieties and their unions are bicrystalline. Our rule generalizes the classical Cauchy identity, Littlewood-Richardson rule, and the Knutson-Miller Hilbert series formula for matrix Schubert varieties.
We prove a sharp lower bound on the number of terms in an element of the reduced Gröbner basis of a Schubert determinantal ideal. We give three applications, including a pattern-avoidance characterization of toric matrix Schubert varieties that complements work of Escobar-Mészáros.
Combinatorial commutative algebra rules (with Alexander Yong).
We describe a general algorithm for generating sets of "hieroglyphs" that compute the (multi)degree of a given variety.
Schubert determinantal ideals are Hilbertian (with Alexander Yong).
We prove that the Hilbert functions of Schubert determinantal ideals are polynomial using a simple bound on the Castelnuovo-Mumford regularity of these ideals.
A hypergraph characterization of nearly complete intersections (with Chiara Bondi, Courtney Gibbons, Yuye Ke, Spencer Martin, and Shrunal Pothagoni), 2021. In: Miller, C., Striuli, J., Witt, E.E. (eds) Women in Commutative Algebra. Association for Women in Mathematics Series, vol 29. Springer, Cham.
We characterize a family of ideals called nearly complete ideals by representing them as certain hypergraphs. We use this characterization to describe some features of their Betti numbers.