Delta-Springer Modules, Descent Bases, and Higher Specht Bases. (2025) In progress. With M. Hanada.
A top-degree basis for the Garsia-Haiman module. (2025) In progress.
Torus Equivariant Cohomology of Delta-Springer Fibers. (2025) In progress.
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.
(Descent Basis of R_n,l,s) In his thesis, Griffin introduced a family of modules R_n,l,s, which unified the generalized coinvariant algebra R_n,k of Haglund-Rhoades-Shimozono, and the Garsia-Procesi module R_l. Meyer found a generalization of the Garsia-Stanton descent monomials for the R_n,k case, and I discovered the basis for the R_l case. I have conjectured a basis for the entire R_n,l,s module that specializes to the two bases above, and am working on extending other results from the R_n,k setting to all of R_n,l,s.
(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 by manipulating the geometry of the Hilbert Scheme, 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.
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.