A top-degree basis for the Garsia-Haiman module. (2025) In progress.
Torus Equivariant Cohomology of Delta-Springer Fibers. (2025) In progress.
Delta-Springer Modules, Descent Bases, and Higher Specht Bases. (2025) In progress. With M. Hanada. We introduce a basis of Garsia-Stanton descent monomials for Griffin's Delta-Springer modules, simultaneously generalizing the descent basis for the Garsia-Procesi module, as well as the Haglund-Rhoades-Shimozono generalized coinvariant algebras. We also give a conjectural higher Specht basis, generalizing the R_n,k construction of Gillespie-Rhoades, and prove our result for two row partitions. arXiv.
Equivariant Cohomology and Orbit Harmonics. (2024) With T. Matsumura, B. Rhoades, Submitted. We study the moduli space X_{n,k,d} of n-tuples of lines in k-dimensional complex space that form a d-dimensional subspace. We give a quotient presentation of its torus-equivariant cohomology ring, using the method of orbit harmonics. arXiv. FPSAC 2025 extended abstract.
A Descent Basis for the Garsia-Procesi Module. (2024) With E. Carlsson, Advances in Mathematics. We introduce a basis of Garsia-Stanton descent monomials for the Garsia-Procesi module, and show that the combinatorial construction yields a new formula for the modified Hall-Littlewood polynomials. We also discuss the subtle connection to Haglund's combinatorial formula at q = 0. arXiv.
(Basis of the Garsia-Haiman module V_l) The Garsia-Haiman module is well known to carry the Frobenius character of the modified Macdonald polynomial, and has dimension n!. The only known proof of this is given by Haiman in his 2001 paper via the polygraph theorem, and no explicit basis is known. I am working on a direct combinatorial construction of a basis, inspired by ideas from Carlsson-Oblomkov and affine Springer fibres. This would lead to a simple proof of the n! theorem.
(Schur expansion of the Macdonald polynomial) The resolution of the Macdonald positivity conjecture implies that the modified Macdonald polynomial has a N(q,t)-Schur expansion. However, only special cases are known. The construction of a basis above may shed new light on the elusive Schur expansion.
(Geometric Model and Generalizations for Diagonal Coinvariants and their Submodules) Haiman proved that the diagonal coinvariant algebra DRn has complex dimension (n+1)^(n-1), and has Frobenius character given by \nabla(en). The corresponding geometric object is the (n,n+1)-affine Springer fiber. How far can we generalize in these three directions, and is there a unified picture that explains this phenomenon?
(Orbit Harmonics) Goresky-MacPherson give a construction of linear subspace arrangements Z living in H_2^T(X), and show that Spec(H*_T(X)) = C[Z] when H*(X) is generated in degree 2. It is often the case that for a natural group action G on X^T, this isomorphism can be upgraded to an isomorphism of G-modules. If H*(X) is not generated in degree 2, how far can this be pushed?
(Hessenberg Varieties) Regular semisimple Hessenberg varieties are GKM varieties, and their cohomology ring is usually realized as a module over the torus fixed points. Is there a polynomial ring representation? Regular semisimple Hessenberg varieties have H^*(X) not generated in degree 2, leading to complications.
Summer School on Algebraic Combinatorics at KIAS with Thomas Lam, Anton Mellit, Mark Shimozono, Syu Kato, Donghyun Kim, and Melissa Sherman-Bennett, July 28-31, 2025.
LAWRGe: Schubert Calculus and Quantum Integrability at USC with Allen Knutson and Paul Zinn-Justin, June 6-10, 2022.
Quantized symplectic singularities and applications to Lie theory at MIT with Ivan Loseu, June 13-17, 2022.
WARTHOG: Infinite dimensional methods in Commutative Algebra at University of Oregon with Andrew Snowden, June 26-30, 2022.