Publications/preprints (ordered by first online appearance)

1. Homology character of the parabolic coset poset. (with Matthieu Josuat-Vergès)
   Motivated by the analogy with the Coxeter complex on one side, and parking functions on the other side, we study the poset of parabolic cosets in a finite Coxeter group. We show that this poset is Cohen-Macaulay, and get an explicit formula for the character of its (unique) nonzero homology group in terms of the Möbius function of the intersection lattice. This homology character becomes a positive element of the parabolic Burnside ring (in its natural basis) after tensoring with the sign character. The coefficients of this character essentially encode the colored h-vector of the positive chamber complex (following Bastidas, Hohlweg, and Saliola, this complex is defined by taking Weyl chambers that lie on the positive side of a generic hyperplane). Roughly speaking, tensoring by the sign character on one side corresponds to the transformation going from the f-vector to the h-vector on the other side.

   • Available as an arXiv preprint at arxiv:2509.11905.
     
     

2. Deformations of restricted reflection arrangements. (with Olivier Bernardi)
   We construct free, non-constant multiplicities and free Shi-like deformations for all restrictions of Weyl arrangements A_W. Both are given in terms of root-theoretic data of W and our proofs are case-free. In type-A we give a bijective proof of resulting product formulas for the number of regions of the deformations, while in the way proving a wide generalization of Joyal’s bijection between labeled trees and functions.

   • An extended abstract of this work has been accepted as a talk for FPSAC 2025.

3. Generalized Laplacian tilings and root zonotopes. (with Alex McDonough)
   Given any two equi-oriented collections A and B of vectors in R^r, we construct a family of tilings of R^r. The tiles themselves are related to the bases of A, while their relative frequency are related to bases of B. Our construction gives a common generalization to Penrose tilings as well as zonotopal space tilings associated to regular matroids.

   • An extended abstract of this work has been accepted as a poster for FPSAC 2025.

4. Counting unicellular maps under cyclic symmetries.
   We count unicellular maps (matchings of the edges of a 2n-gon) of arbitrary genus with respect to the 2n-rotation symmetries of the polygon. An associated generating function that keeps track of the number of symmetric vertices of the resulting map generalizes the celebrated Harer-Zagier formula.

   • Appeared in Sém. Lothar. Combin. 91B.102 (2024), 9 pp.

   • This work was accepted as a poster for FPSAC 2024.

5. Cluster parking functions.  (with Matthieu Josuat-Vergès)
   The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product (over noncrossing partitions). There is a natural structure of simplicial complex on these objects, and our main goal is to show that it has the homotopy type of a (pure) wedge of spheres. The unique nonzero homology group (as a representation of the underlying reflection group) is a sign-twisted parking representation, which is the same as Gordon's quotient of diagonal coinvariants. Along the way, we prove some properties of the poset of parking functions. We also provide a long list of remaining open problems.

   • To appear in Bull. Belg. Math. Soc. Simon Stevin; available at arxiv:2402.03052.

6. Two classes of posets with real-rooted chain polynomials. (with Christos A. Athanasiadis and Katerina Kalampogia-Evangelinou)
   The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to finite Coxeter groups, are shown to have this property. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set {1,2,…,n} which have ascents at specified positions is shown to be real-rooted, hence log-concave and unimodal, and a good estimate for the location of the peak is deduced.

   • Appeared in Electron. J. Combin. 31 (2024), no.4; also available at arxiv:2307.04839.

7. Recursions and proofs in Cataland. (with Matthieu Josuat-Vergès)
   We give the first type-independent proof of the Kreweras-style formulas for the enumeration of noncrossing partitions in a real reflection group W, with respect to  parabolic type. This answers a central open question in Coxeter-Catalan combinatorics, originally asked by Athanasiadis-Reiner in 2003, special cases of which have been open even longer. Our proof also covers the m-Fuss version of the problem, as well as similar Loday-style formulas for the refined-by-type enumeration of faces of the m-cluster complex of W. It proceeds by developing a family of combinatorial recursions that completely determine the enumeration and proving their algebraic counterparts.

   • Appeared in Sém. Lothar. Combin. 89B.75 (2023), 12 pp.

   • This work was accepted as a talk (video and slides) for FPSAC 2023.

8. Hurwitz numbers for reflection groups III: Uniform formulas. (with Joel Brewster Lewis and Alejandro Morales)
   We give uniform formulas for the number of full reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the formula for the genus-0 Hurwitz numbers. This paper is the culmination of a series of three.

   • Appeared in J. Lond. Math. Soc. 111 (2025), no. 3; also available at arxiv:2308.04751.

   • An extended abstract of this work (that encompasses all three parts of the series) was accepted as a talk (video and slides) for FPSAC 2021.

9. The generalized cluster complex: refined enumeration of faces and related parking spaces. (with Matthieu Josuat-Vergès)
   The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. As consequences of our enumeration of faces, we completely explain some observations by Fomin and Reading about f-vectors, we obtain identities that essentially refine the transformation between f-vectors and h-vectors, and introduce new kinds of parking spaces that we call cluster parking spaces.

   • Appeared in SIGMA 19 (2023), 069; also available at arxiv:2209.12540.

10. Counting nearest faraway flats for Coxeter chambers.
    In a finite Coxeter group W and with two given conjugacy classes of parabolic subgroups [X] and [Y], we count those parabolic subgroups of W in [Y] that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of W in [X]. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a new geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group W.

    • Appeared in J. Comb. Algebra 8 (2024), no. 1/2, pp. 121–146; also available at arxiv:2209.06201.

11. Hurwitz Orbits on Reflection Factorizations of Parabolic Quasi-Coxeter Elements. (with Joel Brewster Lewis)
    We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.

    • Appeared in the Electron. J. Combin. 31 (2024), no.1; also available at arxiv:2209.00774.

12. Hurwitz numbers for reflection groups II: Parabolic Quasi-Coxeter elements. (with Joel Brewster Lewis and Alejandro Morales)
    We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections Red_W(g) of reduced reflection factorizations of g and RGS(W,g) of the relative generating sets of g. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size #Red_W(g) with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-0 Hurwitz numbers.

    • Appeared in Journal of Algebra 641 (2024), pp. 648-715; also available at arxiv:2209.00066.

    • An extended abstract of this work (that encompasses all three parts of the series) was accepted as a talk (video and slides) for FPSAC 2021.

13. Hurwitz numbers for reflection groups I: Generatingfunctionology. (with Joel Brewster Lewis and Alejandro Morales)
    The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus 0). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group W is full if the factors generate the whole group W. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family G(m,p,n) of complex reflection groups in terms of the generating functions of the symmetric group Sn and the cyclic group of order m/p. As a corollary, we obtain leading-term formulas which count minimum-length full reflection factorizations of an arbitrary element in G(m,p,n) in terms of the Hurwitz numbers of genus 0 and 1 and number-theoretic functions. We also study the structural properties of such generating functions for any complex reflection group; in particular, we show via representation-theoretic methods that they can by expressed as finite sums of exponentials of the variable.

    • Appeared in Enumerative Combinatorics and Applications, 2:3 (2022) Article S2R20; also available at arxiv:2112.03427.

    • An extended abstract of this work (that encompasses all three parts of the series) was accepted as a talk (video and slides) for FPSAC 2021.

14. Counting chains in the noncrossing partition lattice via the W-Laplacian. (with Guillaume Chapuy)
    We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the W-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

    • Appeared in Journal of Algebra 602, 381-404 (2022); also available at arxiv:2109.04341.

    • Here are the slides from a seminar talk at the UMass Discrete Math seminar with its video recording, and another, extended version, from a seminar talk at Ruhr Universität Bochum.

15. Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups. (with Guillaume Chapuy)
    We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group W, in terms of the spectrum of an associated operator, the W-Laplacian. This covers in particular all finite Coxeter groups. The results of this paper include generalizations of the Matrix Tree and Matrix Forest theorems to reflection groups, and cover reduced (shortest length) as well as arbitrary length factorizations.
    Our formulas are relative to a choice of weighting system that consists of n free scalar parameters and is defined in terms of a tower of parabolic subgroups. To study such systems we introduce (a class of) variants of the Jucys-Murphy elements for every group, from which we define a new notion of `tower equivalence' of virtual characters. A main technical point is to prove the tower equivalence between virtual characters naturally appearing in the problem, and exterior products of the reflection representation of W.
    Finally we study how this W-Laplacian matrix we introduce can be used in other problems in Coxeter combinatorics. We explain how it defines analogues of trees for W and how it relates them to Coxeter factorizations, we give new numerological identities between the Coxeter number of W and those of its parabolic subgroups, and finally, when W is a Weyl group, we produce a new, explicit formula for the volume of the corresponding root zonotope.

    • Appeared in Proc. Lond. Math. Soc. 126 (2023), no. 1, 129-191; also available at arxiv:2012.04519.

    • Here are the slides from conference talks at Sydney and Bad Boll and various other seminar talks.

    • An extended abstract of this work was accepted for FPSAC 2020.

    • Here are the slides of a contributed talk for the virtual conference (Polytop)ics: Recent advances on polytopes, organized by the Max Planck Institute; they cover the calculation of volumes of root zonotopes (section 8.3). A shorter version also appeared as a lightning talk at the Geometry and Combinatorics from Root Systems Workshop, part of the Combinatorial Algebraic Geometry program at ICERM.

16. On enumerating factorizations in reflection groups.
    We describe an approach,via Malle's permutation Ψ on the set of irreducible characters Irr(W), that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements.

    • Appeared in Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 359-385; also available at arxiv:1811.06566.

    • An extended abstract of this work was accepted as a talk (slides) for FPSAC 2019.

    • Here are the slides from a seminar talk at TU Kaiserslautern and the Einstein Workshop on Algebraic Combinatorics.

17. Cyclic sieving for reduced reflection factorizations of the Coxeter element.
    In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice NC(W), associated to a well-generated complex reflection group W. We use this framework to prove, in a unified way, various instances of the cyclic sieving phenomenon on the set of reduced reflection factorizations of the Coxeter element. These include in particular the conjectures of N. Williams on the actions Pro and Twist.

    • Appeared in Sém. Lothar. Combin. 80B Art. 86 (2018), 12 pp.

    • This project was accepted as a talk for FPSAC 2018; here are the slides.

18. Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element.
    In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice NC(W) associated to a well-generated complex reflection group W. A chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of NC(W) and the fibers of a finite quasi-homogeneous morphism, the LL map.
    We consider a variant of the LL map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of NC(W). In particular, we extend work of Bessis and Ripoll and enumerate the so-called "primitive factorizations" of the Coxeter element c. That is, length additive factorizations of the form c=w·t1⋯tk, where w belongs to a given conjugacy class and the ti's are reflections.

    • Appeared in Math. Ann. 391 (2025), no. 3, pp. 3459–3500; also available at arxiv:1808.10395.

    • Here are slides from a talk at Amiens and an older, slightly different version for MIT.

19. The Hilbert scheme of 11 points in A3 is irreducible. (with Joachim Jelisiejew, Bernt Ivar Utstøl Nødland, and Zach Teitler)
    We prove that the Hilbert scheme of 11 points on a smooth threefold is irreducible. In the course of the proof, we present several known and new techniques for producing curves on the Hilbert scheme.

    • Appeared in Combinatorial Algebraic Geometry, Fields Institute Communications; also available at arxiv:1701.03089.

    • A project initiated during the event "Apprenticeship weeks" at the Fields Institute under the supervision of Greg Smith and Bernd Sturmfels.

19. Applications of geometric techniques in Coxeter-Catalan combinatorics.
In the seminal work [Bes 15], Bessis gave a geometric interpretation of the noncrossing lattice NC(W) associated to a well-generated complex reflection group W. He used it as a combinatorial recipe to construct the universal covering space of the arrangement complement V\⋃H, and to show that it is contractible, hence proving the K(π,1) conjecture.
Bessis' work however relies on a few properties of NC(W) that are only known via case by case verification. In particular, it depends on the numerological coincidence between the number of chains in NC(W) and the degree of a finite morphism, the LL map.
We propose a (partially conjectural) approach that refines Bessis' work and transforms the numerological coincidence into a corollary. Furthermore, we consider a variant of the LL map and apply it to the study of finer enumerative properties of NC(W). In particular, we extend work of Bessis and Ripoll and enumerate the so-called ``primitive factorizations" of the Coxeter element c. That is, length additive factorizations of the form c=w·t1⋯tk, where w belongs to a prescribed conjugacy class and the ti's are reflections.

• My PhD thesis, available here, was defended in August 2017. A big part of it is still otherwise unpublished and should be of interest to the community. In particular, it contains [Chapters 5-7] a retelling of the geometry in David Bessis' seminal work (filling in some gaps where necessary) and a (partially incomplete) new approach [Chapter 8] for the proof of the trivialization theorem (and hence also the dual braid presentation of B(W)). The latter is both uniform and does not rely on the numerological coincidence between the degree of the LL map and the number of saturated chains in the noncrossing lattice NC(W).

• Here are the slides from my thesis defense. The first half tells part of the singularity theory behind this story focusing on polynomials and their monodromy.

• The Swallow's tail featured on top, as well as in my thesis, is Salvador Dalí's last painting. Dalí once described René Thom's theory of catastrophes as "the most beautiful aesthetic theory in the world". Catastrophes are known as perestroikas in Russia and as singularities in the US.
  The semi-universal deformations of simple singularities give rise to the discriminant hypersurfaces of (simply-laced) reflection groups. For (part of) the mathematical story behind this, going all the way back to Hilbert's 13th problem, have a look at the Introduction of my thesis.

20. Parking space conjectures.
A prominent line of research in Coxeter combinatorics has been for a better understanding of the noncrossing lattice NC(W), associated to a reflection group W. In [ARR15], Armstrong, Reiner and Rhoades, defined two new Parking Spaces, an isomorphism between which would give uniform proofs and understanding to many a combinatorial formulae. The purpose of this report is to describe a rephrasing of their Main Conjecture, due to Gordon and Ripoll [GR12], in terms of the geometric framework for NC(W), introduced by Bessis in [Bes15].

• This paper, available here, was my oral exam paper to proceed to (PhD) candidacy in Minnesota. It sets up background from Armstrong, Rhoades, and Reiner's, Parking Spaces paper and gives a geometric rephrasing —due to Gordon and Ripoll, but otherwise unpublished— of their main conjecture. It contains little original work, but the presentation might still be useful.

• I'm worried that, as far as opening lines go, I'll never be able to do better than with this.