Welcome to the web page of Theo Douvropoulos!
Ⓒ 2017 Salvador Dali, Fundacio Gala-Salvador Dali, Artists Rights Society
(1983, Oil on canvas, 73 x 92,2 cm).
You can find me at:
Lederle Graduate Research Tower,
University of Massachusetts,
Room: LGRT 1546
Lederle Graduate Research Tower,
University of Massachusetts
710 N. Pleasant Street
Amherst, MA 01003-9305, USA
I am a Marshall H. Stone Visiting Assistant Professor at the Mathematics and Statistics Department of the University of Massachusetts at Amherst.
If you are looking for the mathematician who got a PhD at the University of Minnesota, under the supervision of Vic Reiner and was a postdoc at IRIF working with Guillaume Chapuy, you have found me! You can call me 'Theo'.
I was in the job market during the Fall of 2019. If you are interested in my work, check out my (now slightly outdated) research statement.
If you are looking for my cousin, the physicist Theodosios G. Douvropoulos, you should look here.
My research interests
I enjoy thinking about combinatorial phenomena that have generalizations in the world of reflection groups; what is usually called, Coxeter Combinatorics. This is a diverse world, whose residents -the complex reflection groups- albeit objects easy to grasp, can nonetheless reveal deep connections between algebraic, topological and geometric concepts.
Hurwitz numbers for reflection groups.
We give formulas for the number of transitive reflection factorizations of a parabolic quasi-Coxeter element in a Weyl group or complex reflection group, generalizing the Hurwitz formulas for the symmetric group.
with Joel Brewster Lewis and Alejandro Morales.
Counting chains in the noncrossing partition lattice via the W-Laplacian.
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.
with Guillaume Chapuy.
Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups.
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.
with Guillaume Chapuy.
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.
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.
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. 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.
The Hilbert scheme of 11 points in A3 is irreducible
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.
with Joachim Jelisiejew, Bernt Ivar Utstøl Nødland, and Zach Teitler.
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; maybe you've never thought of a polynomial like this.
- The Swallow's tail featured on the top of this page, 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 (how appropriate) 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.
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.