Research subjects:
My research focuses on the intersection of algebraic combinatorics and enumerative algebraic geometry. The core principle of algebraic combinatorics is that interesting enumeration properties are often shadows of rich algebraic and geometric structures. Enumerative results are often derived from these structures, but they can also be used to identify them. My work embodies both types of connections, relating symmetric function theory and the combinatorics of Coxeter groups to the geometry and representation theory of classical Lie groups.
Publications:
An up-to-date list of my publications is available on Google Scholar or in my CV: HamakerCV. Almost all of these papers are also available on arxiv.org. Here is a list of selected publicationsf:
Characters of local and regular permutation statistics (2022+) with Brendon Rhoades
We introduce a novel family of permutation statistics called regular that contains almost all reasonable pattern counting statistics previously in the literature and show their expected value, when restricted to fixed conjugacy class, is a polynomial in the cycle type. Since regular statistics are closed under multiplication, our results also extend to higher moments. As a consequence, show the expected value of many regular statistics on a sequence of cycle types depends only the limiting proportion of fixed points and the variance only on the proportions of fixed points and two cycles. Our proofs rely critically on representation theory of the symmetric group and a novel basis for the ring of symmetric functions.
Grobner geometry of Schubert polynomials through ice (2022) with Oliver Pechenik and Anna Weigandt
One definition of Schubert polynomials is that they are the multidegrees of matrix Schubert varieties, which are affine varieties indexed by permutations and defined by certain rank conditions on a generic matrix. With respect to a term order that chooses the anti diagonal of a matrix, Knutson and Miller showed the defining ideals of these varieties have initial ideals that are unions of coordinate subspaces, recovering the pipe dream formula for Schubert polynomials. With respect to a diagonal term order, the same result cannot hold in general, but Knutson, Miller and Yong showed it does hold for vexillary permutations. We bypass a fundamental obstruction in the Knutson-Miller-Yong work to extend the diagonal term order results to a larger family of permutations. In addition, we conjecture (later proved by Klein and Weigandt) that the analogous degeneration for diagonal term orders of arbitrary matrix Schubert varieties recovers the bumpless pipe dream formula.
Doppelgängers: bijections of plane partitions (2020) with Rebecca Patrias, Oliver Pechenik and Nathan Williams
Two posets are called doppelgängers if they have the same order polynomial. Using a K-theoretic analogue of jeu de taquin, we prove that the rectangle and shifted trapezoid of appropriate dimensions are doppelgängers. As a special case, this gives the first combinatorial proof of Proctor’s identity that n x n x n plane partitions and 2n x 2n x 2n symmetric, self-complementary plane partitions are equinumerous. Our result is a special case of a more general phenomenon for the coincidental types, and we present it in this generality.
Schur P-positivity and involution Stanley symmetric functions (2019) with Eric Marberg and Brendan Pawlowski
Bruhat order, when restricted to involutions in the symmetric group, has many of the combinatorial properties as for all permutations. There is a geometric interpretation in terms of orthogonal group orbits in the complete flag variety, an involution weak order and involution reduced words. In previous joint work, we introduced involution Stanley symmetric functions, which can be used to enumerate involution reduced words. In this paper, we give two proofs that these symmetric functions are Schur-P positive, hence that involution reduced words can be enumerated using shifted standard Young tableaux. The first proof is based on analogues of the Lascoux-Schutzenberger transition equations (introduced in the subsequent paper) and Pfaffian formulas for the cohomology classes of certain Grassmannian analogues. The second is based on an insertion algorithm.
Transition formulas for involution Schubert polynomials (2018) with Eric Marberg and Brendan Pawlowski
Bruhat order, when restricted to involutions or fixed-point-free (FPF) involutions in the symmetric group, has many of the combinatorial properties as for all permutations. There is a geometric interpretation in terms of orthogonal group orbits and symplectic group orbits in the complete flag variety, with cohomology representatives given by the (FPF) involution Schubert polynomials introduced by Wyser and Yong. In this paper, we introduce recurrences for these polynomials analogous to the transition equations for Schubert polynomials, and demonstrate a combinatorial realization akin to the Little bijection. Our recurrences are demonstrated in a purely combinatorial fashion, and it remains an important open problem to interpret them geometrically.
Collaborators:
I’ve had the pleasure of collaborating with a diverse and talented group of mathematicians. Collaborators include:
Niels Bonneux^, Sara Billey, Michael Coopman^, Jonathan Fang*, Adam Gregory^, Adam Keilthy*, Eric Marberg, Alejandro Morales, Igor Pak, Rebecca Patrias^, Brendan Pawlowski, Oliver Pechenik, Victor Reiner, Brendon Rhoades, Austin Roberts, Bruce Sagan, Luis Serrano, John Stembridge, Marco Stevens^, Justin Troyka, Vincent Vatter, Anna Weigandt, Lilly Webster*, Nathan Williams, Benjamin Young, Yinuo Zhang*, Shuqi Zhou*
* indicates undergraduate coauthor, ^ indicates graduate student coauthor at the time of writing
Mentoring:
I am currently advising three PhD students:
Adam Gregory (2024 expected)
Joshua Arroyo (2025 expected)
Michael Coopman (2025 expected)
At the University of Florida, I’ve worked on research projects with three undergraduates: Hugh Dennin, Jonathan Fang and Raymond Ying
Prior to arriving at UF, I had the pleasure of working with undergraduates at three REUs.
University of Minnesota 2015: Adam Keilthy, Lilly Webster, Yinuo Zhang, Shuqi Zhou
University of Washington 2016: Thomas Browning, Max Hopkins, Zander Kelley
University of Michigan 2018: Ruilin Shi