Postdoctoral Assistant Professor Department of Mathematics University of Michigan Email: firstinital + lastname at umich.edu Pronouns: he, him, his

Hi! I am an NSF Research Fellow and IBL Research Assistant Professor at the University of Michigan. I received my Ph.D. from Harvard University, where my advisor was Lauren Williams. Before that, I was an undergraduate at NC State University. Here is my CV.

I'm interested in problems related to Lie and quiver representation theory, especially from combinatorial, geometric, and categorical viewpoints. My work has especially focused on Coxeter groups, infinite root systems, and the geometry of flag varieties.

Papers and preprints

1. Extended Weak Order for the Rank 3 Universal Coxeter Group (with C. Defant, P. Hersh, J. McCammond, T. McConville, and D. Speyer).
   [ArXiv]
   We prove several of Dyer's conjectures on biclosed sets in the case of the rank 3 universal Coxeter group. We do so by using hyperbolic geometry to show that the rank 3 universal Coxeter arrangement is a clean arrangement.
2. Torsion classes for affine preprojective algebras.
   [Thesis] [FPSAC]
   We prove that the extended weak order of an affine Weyl group is a lattice quotient of the lattice of torsion classes for an affine preprojective algebra. In particular, this gives a parametrization of torsion pairs in the category of nilpotent modules for an affine preprojective algebra. It also implies the existence of Cambrian quotients for the extended weak order of affine Weyl groups which describe the exchange graphs of affine-type cluster algebras. This is one of the main results of my thesis.
3. Appendix to A note on Combinatorial Invariance of Kazhdan–Lusztig polynomials, by F. Esposito and M. Marietti (with C. Gaetz). Bulletin of the London Mathematical Society, Volume 57, Issue 8, 2025.
   [ArXiv] [Journal]
   The main paper gives a conjecture on the combinatorial structure of intervals in the symmetric group (in terms of double shortcuts) which would imply the combinatorial invariance conjecture. In the appendix, we prove this conjecture (or rather, a variant in terms of double hypercubes) in the case of elementary intervals.
4. The Affine Tamari Lattice (with C. Defant).
   [ArXiv]
   We introduce the affine Tamari lattice, which is a quotient of the type-\(\widetilde{A}\) extended weak order modeling the cluster theory and representation theory of the oriented cycle quiver. We show that it admits descriptions analogous to the classical Tamari lattice, for instance using translation-invariant binary trees and 312-avoiding translation-invariant total orders. Among other things, we describe its rowmotion and maximal chains, the latter of which we use to describe the maximal green sequences for the completed path algebra of an oriented cycle quiver.
5. The BBDVW Conjecture for Kazhdan-Lusztig polynomials of lower intervals (with C. Gaetz).
   [ArXiv]
   Blundell, Buesing, Davies, Veličković, and Williamson conjectured a recurrence for Kazhdan-Lusztig polynomials in the symmetric group with the assistance of machine learning. We prove their conjecture in the case of lower intervals.
6. On two notions of total positivity for generalized partial flag varieties of classical Lie types (with J. Boretsky, C. Eur, and J. Gao).
   [ArXiv] [FPSAC]
   We compare Lusztig's locus of total positivity for partial flag varieties with the locus of positivity of the Plücker coordinates. We characterize the partial flag varieties of types B and C for which these two notions of positivity coincide.
7. Oriented matroid structures on rank 3 root systems (with K. Tung).
   [ArXiv]
   We prove a conjecture of Dyer and Wang that any rank 3 root system has a unique oriented matroid structure compatible with its root system structure. Our result is conditional on the conjecture that every rank 3 Coxeter arrangement is clean.
8. Extended weak order for the affine symmetric group.
   [ArXiv] [FPSAC]
   We describe the lattice of translation-invariant total orderings of the integers, which is a combinatorially natural extension of the type-\(\widetilde{A}\) extended weak order. We show it is a profinite semidistributive lattice and give a diagrammatic interpretation of its canonical join representations using arc diagrams. These properties descend to extended weak order, where they also have a geometric interpretation in terms of shards. The article is written as an introduction for the combinatorialist and emphasizes some aspects of extended weak order that are only visible for infinite lattices, such as widely generated elements (a useful notion we introduce) and the role of profiniteness.
9. On combinatorial invariance of parabolic Kazhdan–Lusztig polynomials (with C. Gaetz). Selecta Mathematica, Volume 31, Article 51, 2025.
   [ArXiv] [Journal]
   We show that several conjectures on the combinatorial invariance of parabolic KL polynomials are in fact equivalent to the classical combinatorial invariance conjecture for KL polynomials.
10. Bender–Knuth Billiards in Coxeter Groups (with C. Defant, E. Hodges, N. Kravitz, and M. Lee). Forum of Mathematics, Sigma, Volume 13, e7, 2025.
    [ArXiv] [Journal]
    We introduce non-invertible Bender–Knuth toggles on a Coxeter group, which generalize Bender–Knuth involutions on linear extensions of a poset. These form a dynamical system which in some cases can be realized as billiards in a Coxeter arrangement. The construction depends on a choice of convex set in the Coxeter group; we describe many cases where every trajectory is drawn into this convex set, including Coxeter groups that are finite, rank 3, right-angled, or of type \(\widetilde{A}\) or \(\widetilde{C}\). We call these groups futuristic; we also show that the remaining affine Coxeter groups are not futuristic (in fact, they are ancient).
11. Affine extended weak order is a lattice (with D. Speyer).
    [ArXiv] [FPSAC]
    We prove that the extended weak order introduced by Matthew Dyer, which is the poset of biclosed sets in a positive root system, is a complete lattice for affine Weyl groups. This is a special case of a decades-old conjecture of Dyer that this lattice property is true for any Coxeter group. To prove it, we introduce the notion of a clean hyperplane arrangement, which is defined by the property that its regions can be computed using only codimension 2 data. We show that root poset order ideals in a finite or rank 3 untwisted affine root system are clean. We prove that Dyer's two main conjectures on biclosed sets reduce to finding sufficiently many clean subarrangements of rank 3 Coxeter arrangements.
12. Combinatorial invariance for Kazhdan–Lusztig \(R\)-polynomials of elementary intervals (with C. Gaetz). Mathematische Annalen, Volume 392, pages 3299-3317, 2025.
    [ArXiv] [FPSAC] [Journal] [View]
    An interval in Bruhat order is called simple if the roots connecting its minimal element to its atoms are linearly independent. We prove that if two Bruhat intervals in the symmetric group are isomorphic, and they are isomorphic to a simple interval, then their \(R\)-polynomials are equal. In particular, this holds for intervals isomorphic to a lower interval in Bruhat order, so this generalizes the result of Brenti that two isomorphic lower intervals in the symmetric group have the same \(R\)-polynomial. Our proof uses hypercube decompositions, which were introduced by Blundell, Buesing, Davies, Veličković, and Williamson and discovered with the assistance of techniques from machine learning.
13. Combinatorial descriptions of biclosed sets in affine type (with D. Speyer). Combinatorial Theory, Volume 4, Issue 2, 2024.
    [ArXiv] [FPSAC] [Journal]
    We give a parametrization of biclosed sets in an affine root system in terms of faces of the Coxeter fan (equivalently, the permutahedron) of its associated finite root system. In classical types, we give an explicit combinatorial description of biclosed sets as total orders of the integers, analogous to the description of affine Weyl groups as permutation groups.
14. Channels, Billiards, and Perfect Matching 2-Divisibility (with R. Liu). Electronic Journal of Combinatorics, Volume 28, Issue 2, 2021.
    [ArXiv] [FPSAC] [Journal]
    We give a lower bound on the power of 2 dividing the number of perfect matchings of a planar graph using vertex sets called channels. We build many tools for computing the channels of various graphs. As a result, we recover many results from the literature and apply our technique to a conjecture of Pachter. In nice situations, we can identify channels with billiard trajectories in the graph, which lets us use the theory of arithmetic billiards to explain the power of 2 appearing in the number of domino tilings of a rectangle.

Other things

In the 2021-2022 school year, I co-organized the Trivial Notions graduate seminar. More information can be found here.

In Summer 2021, I taught a tutorial called "Quantum Mechanics for the Mathematically-Minded". You can find the course notes here.