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Research

I am interested in the combinatorics and geometry of Coxeter groups, and of real and complex reflection groups, as well as their related structures (braid groups, arrangement of hyperplanes, root systems...)

I am currently a postdoctoral research fellow at the Faculty of Mathematics of University of Vienna (Austria), working with Christian Krattenthaler and the Algebraic Combinatorics Group.

Before that (Sep 2010 -- Aug 2013) I was a postdoctoral research fellow at LaCIM (Laboratoire de Combinatoire et Informatique Mathématique), UQÀM (Université du Québec à Montréal). I was working in particular with Christophe Hohlweg.

Formerly I was a PhD student at the DMA-ENS (Department of Mathematics and Applications of the École Normale Supérieure in Paris), under the supervision of David Bessis.


Papers      —      Thesis      —      Conferences      —      Seminar talks


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Papers and preprints

→ Direct link to my arXiv page, my Google Scholar page.

Connectivity properties of factorization posets in generated groups, with Henri Mühle (arXiv preprint)
We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain total order of the generators that is compatible with the chosen element.
On non-conjugate Coxeter elements in well-generated reflection groups, with Victor Reiner and Christian Stump , Mathematische Zeitschrift 285 (2017), Issue 3–4, 1041-1062 (arXiv version, journal version)
Given an irreducible well-generated complex reflection group W with Coxeter number h, we show that the class of regular elements of order h form a single orbit in W under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element c is h-regular if and only if there exists a simple system S of reflections such that c is the product of the generators in S. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of W associated to different regular elements of order h are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of W on the conjugacy classes of h-regular elements. Finally, we extend several of these properties to regular elements of arbitrary order. We show that the action of reflection automorphisms also preserves, and is transitive on, the set of regular elements of a given order d, and we study the action of the Galois group on conjugacy classes of d-regular elements.
On non-conjugate Coxeter elements in well-generated reflection groups (extended abstract) (pdf)
Short version of the one above, accepted for a talk at the conference SFCA/FPSAC 2015 in Daejon, Korea, and published in the DMTCS Proceedings of the conference.
On the Limit Set of Root Systems of Coxeter Groups acting on Lorentzian spaces, with Christophe Hohlweg and Jean-Philippe Préaux (arXiv preprint)
[submitted]
The notion of limit roots of a Coxeter group W was recently introduced (see the two papers below): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space, W admits a faithful representation as a discrete reflection group of isometries on a hyperbolic space; the accumulation set of any of its orbits is then classically called the limit set of W. In this article we show that the set of limit roots of a Coxeter group W acting on a Lorentzian space is equal to the limit set of W seen as a discrete reflection group of hyperbolic isometries. We aim for this article to be as self-contained as possible in order to be accessible to the community familiar with reflection groups and root systems and to the community familiar with discrete subgroups of isometries in hyperbolic geometry.
Imaginary cones and limit roots of infinite Coxeter groups, with Matthew Dyer and Christophe Hohlweg, Mathematische Zeitschrift 284 (2016), Issue 3–4, 715–780 (arXiv version, journal version)
Let W be an infinite Coxeter group. We continue in this article the study of the set E of limit points of "normalized" roots (representing the directions of the roots) of a root system of W (see arXiv:1112.5415). In this article we study the close relations of the set E with the imaginary cone studied by the first author (see arXiv:1210.5206), which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W-action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.
Asymptotical behaviour of roots of infinite Coxeter groups, with Christophe Hohlweg and Jean-Philippe Labbé, Canadian Journal of Mathematics 66 (2014), 323-353 (arXiv version, journal version)
Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain that this subset can be built from the intersection with Q of the lines passing through two positive roots, and we establish that it is dense in E.
Asymptotical behaviour of roots of infinite Coxeter groups I (extended abstract) (pdf)
Short version of the one above, accepted for a talk at the conference SFCA/FPSAC 2012 in Nagoya, Japan, and published in the DMTCS Proceedings of the conference.
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element, Journal of Algebraic Combinatorics 36, Issue 4 (2012), Pages 649-673(arXiv version, published version)
When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NCP_W as a generalised Fuss-Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map LL, describing the factorisations of its discriminant and its Jacobian. As byproducts, we generalise a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorisations of a Coxeter element of W.

[Several versions of slides related to the results of this paper are available below in Sections Conferences and Seminar talks.]
Submaximal factorisations of a Coxeter element in complex reflection groups (extended abstract) (pdf)
This article is an extended abstract of the precedent, presented at the conference FPSAC 2011 in Reykjavik (Iceland), and published in the DMTCS Proceedings of the Conference.
Discriminants and Jacobians of virtual reflection groups (arXiv preprint)
Let A be a polynomial algebra with complex coefficients. Let B be a finite extension ring of A which is also a polynomial algebra. We describe the factorisation of the Jacobian J of the extension into irreducibles. We also introduce the notion of a well-ramified extension and define its discriminant polynomial D. In the particular case where A is the ring of invariants of B under the action of a group (i.e., a Galois extension), this framework corresponds to the classical invariant theory of complex reflection groups. In the more general case of a well-ramified extension, we explain how the pair (D,J) behaves similarly to a Galois extension. This work can be viewed as the first step towards a possible invariant theory of ``virtual reflection groups''.
Orbites d'Hurwitz des factorisations primitives d'un élément de Coxeter, Journal of Algebra 323 (2010), 1432-1453. (arXiv version; published version. N.B.: in French)
Hurwitz orbits of primitive factorisations of a Coxeter element
I study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a reflection group W. It is known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. I prove a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice associated to W. The proof uses the properties of the Lyashko-Looijenga covering and the geometry of the discriminant of W.


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PhD thesis and previous works

Thesis :

I defended my PhD thesis — Reflection groups, geometry of the discriminant and noncrossing partitions — on July 9th 2010 in Paris.

Jury:
David Bessis (thesis advisor), ENS.
Cédric Bonnafé, Université de Franche-Comté.
Frédéric Chapoton (referee), Université Lyon 1.
Patrick Dehornoy, Université de Caen.
Christian Krattenthaler (referee), Universität Wien.
François Loeser, ENS.
Jean Michel, Université Paris 7.

The manuscript and the slides (in French) of the defense are available.

Abstract.
    When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. This structure appears in several algebraic setups (dual braid monoid, cluster algebras...). Many combinatorial properties of NCP_W are proved case-by-case, using the classification of reflection groups. It is the case for Chapoton's formula, expressing the number of multichains of a given length in the lattice NCP_W, in terms of the invariant degrees of W. This thesis work is motivated by the search for a geometric explanation of this formula, which could lead to a uniform understanding of the connections between the combinatorics of NCP_W and the invariant theory of W.
    The starting point is to use the Lyashko-Looijenga covering (LL), based on the geometry of the discriminant of W. In the first chapter, some topological constructions of Bessis are refined, allowing to relate the fibers of LL with block factorisations of a Coxeter element. Then we prove a transitivity property for the Hurwitz action of the braid group B_n on certain factorisations. Chapter 2 is devoted to certain finite polynomial extensions, and to properties about their Jacobians and discriminants. In Chapter 3, these results are applied to the extension defined by the covering LL. We deduce — with a case-free proof — formulas for the number of submaximal factorisations of a Coxeter element in W, in terms of the homogeneous degrees of the irreducible components of the discriminant and Jacobian for LL.

Keywords: complex reflection groups, noncrossing partitions, Fuss-Catalan numbers, Chapoton's formula, Lyashko-Looijenga covering, factorisations of a Coxeter element.

Some previous work (during my Master):

Groupes de réflexions, groupes de tresses, structures de Garside. (pdf)
Introduction to my field of research (October 2006). In French.
Groupes de réflexions et structures de Garside (with David Bessis). (pdf)
Thesis project (October 2006) (summary here). In French.
Propriété de treillis dans les groupes de réflexions réels finis, based on Brady-Watt. (pdf)
Master's thesis (under the supervision of David Bessis) : essentially I clarified a recent proof of Brady-Watt about a lattice property of certain intervals in reflection groups. (March-September 2006). In French.

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Conference talks, and other conferences attended

(list of conferences I attended or gave a talk in, and conferences I plan to attend — for seminars' talks, see Section seminars below)

Algebraic and Geometric Combinatorics of Reflection Groups (Spring School / Workshop)
Montréal, Canada, May 29 to June 9, 2017.
Sage Days 79 (combinatorics and discrete geometry)
Jerusalem, Israel, November 21-24, 2016.
Finite Chevalley groups, reflection groups and braid groups: A conference in honour of Professor Jean Michel
EPFL, Lausanne, Switzerland, 21-23 September 2016.
77th Séminaire Lotharingien de Combinatoire
Strobl (Austria), 11-14 September 2016.
Bridges Finland 2016 (Mathematics, Music, Art, Architecture, Education, Culture )
Jyväskylä, Finland, August 9-13, 2016.
Imaginary Conference 2016 (Shaping the future of athematics communication)
Berlin, July 20-23, 2016.
Algebraic Combinatorics and Group Actions
Talk: Factorisations of a group element, Hurwitz action and shellability
Herstmonceux Castle, UK, 11-15 July 2016.
FPSAC 2016
28th International Conference on Formal Power Series and Algebraic Combinatorics, Vancouver, Canada 4-8 July 2016.
76th Séminaire Lotharingien de Combinatoire
Talk: Factorisations of a group element, Hurwitz action and shellability
Domaine Saint-Jacques, Ottrott (France), 3-6 April 2016.
Combinatorial Algebra meets Algebraic Combinatorics 2016
Talk: Factorisations of a group element, Hurwitz action and shellability
London, Ontario (Canada), 22-24 January 2016.
Algorithmic and Enumerative Combinatorics Summer School 2015
RISC, Hagenberg (Austria), 27-31 July 2015.
FPSAC 2015
Talk: Coxeter elements in well-generated complex reflection groups
27th International Conference on Formal Power Series and Algebraic Combinatorics, Daejeon, South Korea, July 6-10, 2015.
Chevalley groups, Coxeter groups and Artin-Tits groups
Conference for François Digne's retirement, April 1-3, 2015, Amiens.
Sage Days 64: Algebraic Combinatorics
Talk (together with J.-P. Labbé): Computing and displaying infinite root systems
UC Davis (US), 17-20 March 2015
73rd Séminaire Lotharingien de Combinatoire
Talk: Coxeter elements in well-generated reflection groups (slides)
Strobl (Austria), 8-10 September 2014.
Algorithmic and Enumerative Combinatorics Summer School 2014
RISC, Hagenberg (Austria), 18-22 August 2014.
FPSAC 2014
26th International Conference on Formal Power Series and Algebraic Combinatorics, Chicago, 29 June - 3 July 2014.
Workshop "Noncrossing partitions in representation theory"
Bielefeld (Germany), 12-14 June 2014.
Sage Days 57
Sage-Combinat Days in Cernay (France), 6-12 April 2014.
72nd Séminaire Lotharingien de Combinatoire
Lyon (France), 23-26 March 2014.
New perspectives in hyperplane and reflection arrangements
Talk: Complex reflection arrangements and factorisations of a Coxeter element
Workshop in Bochum (Germany), 10th February 2014.
Winter School in Lie Theory
CRM, Montreal (Canada), 6-17 January 2014. Winter school included in the CRM thematic semester New Directions in Lie Theory.
Recent Trends in Algebraic and Geometric Combinatorics
Madrid (Spain), 27-29 November 2013.
Combin' à Tours
Talk: Limit points of root systems of infinite Coxeter groups (slides)
Workshop of the ANR ACORT Algebraic Combinatorics in Representation Theory, Tours (France), 3-5 July 2013.
FPSAC 2013
25th International Conference on Formal Power Series and Algebraic Combinatorics, Paris, 24-28 June 2013.
Sage days 49: Free and Practical Software for (Algebraic) Combinatorics
Workshop on Sage, satellite event of FPSAC, Paris, 17-21 June 2013.
Algebra, Combinatorics and Representation Theory
Conference in honor of the 60th birthday of Andrei Zelevinsky, Northeastern University (Boston, MA), 24-28 April 2013.
Rational Catalan combinatorics
Workshop at the American Institute of Mathematics (Palo Alto, Californie), 17-21 December 2012.
CMS Winter Meeting, session of Algebraic Combinatorics
Talk : Limit points of root systems of infinite Coxeter groups (slides)
Montreal, 7-10 December 2012.
Coxeter Groups meet Convex Geometry
[conference I co-organized]
Mini-courses / Workshop, at LaCIM - UQÀM, 13-22 August 2012.
FPSAC 2012
24th International Conference on Formal Power Series and Algebraic Combinatorics, Nagoya, Japan, July 30–August 3, 2012.
Summer School on Algebraic and Enumerative Combinatorics
S. Miguel de Seide, Portugal, 2-13 July 2012.
Sage Days 38
Workshop on the mathematics software package Sage, CRM, Montreal, 7-11 May 2012.
Combinatorial Algebra meets Algebraic Combinatorics
Ninth Annual Meeting, LaCIM-UQÀM, Montreal, Quebec, 20-22 January 2012.
2011 CMS Winter Meeting
Canadian Mathematical Society seasonal meeting, Toronto, Ontario, 10-12 December 2011.
FPSAC 2011
Talk: Submaximal factorisations of a Coxeter element in complex reflection groups
23rd International Conference on Formal Power Series and Algebraic Combinatorics, Reykjavik, Iceland, 13-17 June 2011.
2011 Spring Eastern Sectional Meeting of the AMS, Special session Combinatorics of Coxeter Groups
Talk: Geometrical enumeration of certain factorisations of a Coxeter element in finite reflection groups
College of the Holy Cross, Worcester, MA, 9-10 April 2011.
Combinatorial Algebra meets Algebraic Combinatorics
Talk: Factorizations of a Coxeter element and discriminant of a reflection group (slides)
Lakehead University, Thunder Bay (Ontario), January 21-23, 2011.
Colloquium on Surfaces and Representations
Talk: Discriminant of a reflection group and factorisations of a Coxeter element (slides)
Team SAG, Université de Sherbrooke, October 6-9, 2010.
LaCIM 2010
Université du Québec à Montréal, August 29-31, 2010.
Journées Garside
Talk: Factorisations of the Garside element in the dual braid monoids (slides)
University of Caen Basse-Normandie, June 30th - July 1st 2010.
Artin-Tits groups, automorphisms and other related topics
Bourgogne University, Dijon, March 4-5, 2010.
Tresses in Pau (Braids in Pau)
Pau University, October 5-8, 2009.
Arrangements of hyperplanes, Mathematical Society of Japan (MSJ) Seasonal Institute (SI)
Hokkaido University, Sapporo (Japan), August 1-13, 2009.
Instructional workshop in association with the programme "Algebraic Lie Theory"
Newton Institute, Cambridge, January 12-23, 2009.
Hecke algebras, groups and geometry
Marseille, C.I.R.M., October 13-17, 2008.
Braids in Paris,
Paris, September 17-20, 2008.
Braids, knots, and applications
Montpellier, June 9-11, 2008.
Thompson's Groups: New Developments and Interfaces
Marseille, C.I.R.M., June 2-6, 2008.
LieGrits Workshop, final worshop of the research network "Flags, Quivers and Invariant Theory in Lie Representation Theory"
Mathematical Institute, University of Oxford, January 3-9, 2008.
"CLUSE" mathematical school, Group theory
Messigny-et-Vantoux, October 28th - November 1st, 2007.
Braids, groups and manifolds in Toulouse
Toulouse, September 5-8, 2007.
Knots, hyperplane arrangements and Coxeter groups
Marseille, C.I.R.M, June 4-8, 2007.
Around Broué's conjectures
Marseille, C.I.R.M., May 28th - June 1st, 2007.
Groups 007 ; week 4 : Combinatorial, algorithmic and cryptographic aspects of Group Theory
Marseille, C.I.R.M., February 26th - March 2nd, 2007.
Geometrical and cohomological group theory : rigidity and deformations
Marseille, C.I.R.M., April 18-22, 2006.

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Seminar talks

Factorisations dans un groupe, action d'Hurwitz, épluchabilité et ordre compatible
at the LaCIM combinatorics seminar, Montreal, 26th January 2016.
Coxeter elements in well-generated complex reflection groups (slides)
at the AG Algebra, Geometrie und Computeralgebra in TU Kaiserslautern, 21st May 2015.
What is a Coxeter element? (slides, in French)
at the LaCIM combinatorics seminar, Montreal, 11th July 2014.
Chains in the noncrossing partition lattice of a reflection group (slides)
Universität Wien, Arbeitsgemeinschaft Diskrete Mathematik, 19th November 2013.
Limit roots and imaginary cone in root systems of Coxeter groups
at the LaCIM combinatorics seminar, Montreal, 12th April 2013.
Limit points of root systems in infinite Coxeter groups
Several talks on these works in France, November 2012 (details below). Here are some slides (in French):
           long version (~1h30)        shorter version (~1h).
 
Asymptotical behaviour of roots of infinite Coxeter groups
at the Ottawa-Carleton algebra seminar, 21st March 2012.
Asymptotical behaviour of roots of infinite Coxeter groups
at the seminar SAG in University of Sherbrooke, 16th March 2012.
The braid group and generalizations
at the seminar CIRGET-LaCIM in Montreal, 9th February 2012.
Factorizations of a Coxeter element and discriminant of a reflection group (slides)
at the Combinatorics seminar at University of Minnesota, Minneapolis, 22nd April 2011.
The noncrossing partition lattice of a finite reflection group
at the Student Combinatorics seminar at University of Minnesota, Minneapolis, 21st April 2011.
Noncrossing partition lattice and discriminant of a reflection group
Applied Algebra Seminar, York University (ON), 17th January 2011.
Treillis des partitions non-croisées et discriminant d'un groupe de réflexion
at the LaCIM combinatorics seminar, 22nd October 2010.
Reflection groups, geometry of the discriminant and noncrossing partition (slides)
Thesis defense, 9th July 2010.
Discriminants d'un groupe de réflexion et factorisations d'un élément de Coxeter (slides, in French)
10th June 2010 at the seminar Algebra and Number Theory in Besançon.
Le morphisme de Lyashko-Looijenga : un groupe de réflexion virtuel ? (slides, in French)
25th February 2010 at the séminaire Chevalley.
Action d'Hurwitz sur certaines factorisations d'un élément de Coxeter
12th May 2009 in the seminar Algebra and Geometry of Laboratoire de Mathématiques Nicolas Oresme (LMNO) in Caen.
Groupes de réflexions et groupes de Coxeter : de l'autre côté des miroirs (slides, in French)
11th May 2009, at the ENS, expository talk for the conference day Mathématiques en mouvement of the Fondation Sciences Mathématiques de Paris.
Groupes de réflexions et treillis des partitions non-croisées
6th January 2009 in the seminar of PhD students in algebra and geometry of DMA.
Action d'Hurwitz sur les factorisations par blocs d'un élément de Coxeter
18th December 2008 at the séminaire Chevalley.
Combinatoire du treillis des partitions non-croisées généralisées
14th April 2008 at the workshop of PhD students in finite groups of Institut de Mathématiques de Jussieu.
Mieux comprendre les groupes de réflexions complexes finis (slides, in French)
18th March 2008, brief exposition of my subject of research.
Groupes de réflexions complexes et monoïde dual de tresses
2nd May 2007, expository talk for the maths students seminar at the ENS.
Propriété de treillis dans les groupes de réflexions réels finis, d'après Brady-Watt
28th March 2007 in the seminar of group theory of Amiens.
Propriété de treillis dans les groupes de réflexions réels finis
20th October 2006 at the workshop for PhD students in finite groups of Institut de Mathématiques de Jussieu.

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