Research
My research lies at the interface of commutative algebra, algebraic combinatorics, representation theory, and algebraic geometry. I utilize tools from topological, tropical, and algebraic combinatorics to investigate homological aspects of algebras with a group action or some other underlying combinatorial structure. Recently I have been especially interested in combinatorial Koszul algebras, Grobner degenerations for determinantal varieties such as positroid varieties, and interactions between free resolutions and tropical and discrete combinatorics.
Determinantal Varieties
Recently I have been interested in understanding certain Grobner degenerations and standard monomial theory for positroid varieties and torus characters for cyclic Demazure modules introduced by Lam.
I also have studied homological aspects of determinantal facet ideals, whose origins lie in algebraic statistics. In ongoing work, I am utilizing similar techniques to investigate matrix Schubert varieties.
Combinatorial Koszul algebras
Koszul algebras and their associated Koszul duals are quadratic algebras with rich structure, especially when a group acts on them and/or they arise in a combinatorial context.
In my work, I investigate combinatorial Koszul algebras such as Veronese subrings and Orlik-Solomon algebras of supersolvable matroids.
With my Summer 2023 REU students, we explored the S_n representations for the Koszul duals of two Koszul algebras arising from the Boolean lattice: the S_n-fixed order complex of the Boolean lattice, and the Chow ring of the Boolean matroid.
Tropical Combinatorics
I enjoy using techniques from commutative algebra to better understand ideals and algebras arising from generalizations of tropical hyperplane arrangements and triangulations of Postnikov's type A root polytopes.
With an honors thesis student in academic year '22-'23, we wrote a package which can be used to study homological aspects of monomial ideals arising from tropical hyperplane arrangements.
Polarization of monomial ideals
Polarization is ubiquitous technique for deforming an arbitrary monomial ideal to a squarefree one which is homologically indistinguishable.
I have results linking polarizations to shellable balls, Schur modules of hook partitions, and triangulations of products of simplices. With my Summer 2022 REU students, we generalized some of this work and began investigating which polarizations were smooth points on the the corresponding Hilbert scheme.