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 various ideals stable under large group actions, particularly those that arise in Schubert calculus.
Publications:
RSK as a linear operator (with Alexander Yong).
We study the Robinson-Schensted-Knuth correspondence (RSK) as the transition operator between the monomial and bitableau bases of polynomial functions on the space of mxn complex-valued matrices. Our results characterize the diagonalizability of RSK_{m, n, d} (the restriction of RSK to degree-d homogeneous polynomials) and give formulas for its trace and determinant more efficient than the naïve ones. We also give several conjectures and open problems for future research.
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. Slides and video from a (virtual) talk I gave at the Michigan State University Combinatorics and Graph Theory seminar are available.
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.