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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 DMAENS
(Department of Mathematics and Applications of the École Normale
Supérieure in Paris), under the supervision
of David Bessis.
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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: Hurwitzconnectivity 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 nonconjugate Coxeter elements in wellgenerated reflection
groups, with Victor Reiner
and Christian Stump , Mathematische Zeitschrift 285 (2017), Issue 3–4, 10411062 (arXiv
version, journal version)
 Given an irreducible wellgenerated 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 hregular 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
hregular 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 dregular elements.
 On nonconjugate Coxeter elements in wellgenerated 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 JeanPhilippe
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 selfcontained
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 Waction 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 JeanPhilippe
Labbé, Canadian Journal of Mathematics 66 (2014),
323353 (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.

LyashkoLooijenga morphisms and submaximal factorisations of a
Coxeter element, Journal
of Algebraic Combinatorics 36, Issue 4
(2012), Pages 649673(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 ngon. A formula (for which the only
known proofs are casebycase) expresses the number of multichains of
a given length in NCP_W as a generalised FussCatalan number,
depending on the invariant degrees of W. We describe how to
understand some specifications of this formula in a casefree way,
using an interpretation of the chains of NCP_W as fibers of a
LyashkoLooijenga 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 wellramified
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 wellramified 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), 14321453. (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 noncrossing partitions lattice associated to W. The proof uses
the properties of the LyashkoLooijenga 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 FrancheComté.
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 wellgenerated 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 ngon. This
structure appears in several algebraic setups (dual
braid monoid, cluster algebras...). Many combinatorial properties of
NCP_W are proved casebycase, 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
LyashkoLooijenga 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 casefree
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,
FussCatalan numbers, Chapoton's formula, LyashkoLooijenga 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 BradyWatt. (pdf)
 Master's thesis (under the supervision of David Bessis) : essentially I
clarified a recent proof of BradyWatt about a lattice property of
certain intervals in reflection groups. (MarchSeptember 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 2124, 2016.
 Finite Chevalley groups, reflection groups and braid groups: A conference in honour of Professor Jean Michel
 EPFL, Lausanne, Switzerland, 2123 September 2016.
 77th Séminaire Lotharingien de Combinatoire
 Strobl (Austria), 1114 September 2016.
 Bridges Finland 2016 (Mathematics, Music, Art, Architecture, Education, Culture )
 Jyväskylä, Finland, August 913, 2016.
 Imaginary Conference 2016 (Shaping the future of athematics communication)
 Berlin, July 2023, 2016.
 Algebraic Combinatorics and Group Actions
Talk: Factorisations of a group element,
Hurwitz action and shellability
 Herstmonceux Castle, UK, 1115 July 2016.
 FPSAC 2016
 28th International Conference on Formal Power Series and Algebraic Combinatorics, Vancouver, Canada
48 July 2016.
 76th Séminaire Lotharingien de Combinatoire
Talk: Factorisations of a group element,
Hurwitz action and shellability
 Domaine SaintJacques, Ottrott (France), 36 April 2016.
 Combinatorial Algebra meets
Algebraic Combinatorics 2016
Talk: Factorisations of a group element, Hurwitz action and shellability

London, Ontario (Canada), 2224 January 2016.
 Algorithmic and Enumerative Combinatorics
Summer School 2015
 RISC, Hagenberg (Austria), 2731 July 2015.
 FPSAC 2015
Talk: Coxeter elements in wellgenerated complex reflection groups
 27th International Conference on Formal Power Series and Algebraic Combinatorics, Daejeon, South Korea, July 610, 2015.
 Chevalley groups, Coxeter groups and ArtinTits groups

Conference for François Digne's retirement, April 13, 2015, Amiens.

Sage Days 64: Algebraic Combinatorics
Talk (together with J.P. Labbé): Computing and displaying infinite root systems
 UC Davis (US), 1720 March 2015
 73rd Séminaire Lotharingien de Combinatoire
Talk: Coxeter elements in wellgenerated reflection groups (slides)
 Strobl (Austria), 810 September 2014.
 Algorithmic and Enumerative Combinatorics
Summer School 2014
 RISC, Hagenberg (Austria), 1822 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), 1214 June 2014.
 Sage Days 57
 SageCombinat Days in Cernay (France), 612 April 2014.
 72nd Séminaire Lotharingien de Combinatoire
 Lyon (France), 2326 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), 617 January 2014. Winter school included in the CRM thematic semester New Directions in Lie Theory.
 Recent Trends in Algebraic and Geometric Combinatorics
 Madrid (Spain), 2729 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), 35 July 2013.
 FPSAC 2013
 25th International Conference on Formal Power Series and Algebraic Combinatorics, Paris, 2428 June 2013.
 Sage days 49: Free and Practical Software for (Algebraic) Combinatorics
 Workshop on Sage, satellite event of FPSAC, Paris, 1721 June 2013.
 Algebra, Combinatorics and Representation Theory
 Conference in honor of the 60th birthday of Andrei Zelevinsky, Northeastern University (Boston, MA), 2428 April 2013.
 Rational Catalan combinatorics

Workshop at the American Institute of Mathematics (Palo Alto, Californie), 1721 December 2012.
 CMS Winter Meeting, session of Algebraic Combinatorics
Talk : Limit points of root systems of
infinite Coxeter groups (slides)
 Montreal, 710 December 2012.
 Coxeter Groups meet
Convex Geometry
[conference I coorganized]

Minicourses / Workshop, at LaCIM  UQÀM, 1322 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, 213 July 2012.
 Sage Days 38
 Workshop on the mathematics software package Sage, CRM,
Montreal, 711 May 2012.
 Combinatorial Algebra
meets Algebraic Combinatorics
 Ninth Annual Meeting, LaCIMUQÀM, Montreal, Quebec, 2022 January
2012.
 2011 CMS Winter
Meeting
 Canadian Mathematical Society seasonal meeting, Toronto, Ontario,
1012 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, 1317 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, 910 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 2123, 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 69, 2010.
 LaCIM 2010
 Université du Québec à Montréal, August 2931, 2010.
 Journées
Garside
Talk: Factorisations of the Garside element in the dual braid
monoids (slides)
 University of Caen BasseNormandie, June 30th  July 1st 2010.
 ArtinTits
groups, automorphisms and other related topics
 Bourgogne University, Dijon, March 45, 2010.
 Tresses
in Pau (Braids in Pau)
 Pau University, October 58, 2009.
 Arrangements
of hyperplanes, Mathematical Society of Japan (MSJ) Seasonal
Institute (SI)
 Hokkaido University, Sapporo (Japan), August 113, 2009.
 Instructional
workshop in association with the programme "Algebraic Lie Theory"
 Newton Institute, Cambridge, January 1223, 2009.
 Hecke algebras, groups and geometry
 Marseille, C.I.R.M., October 1317, 2008.
 Braids in
Paris,
 Paris, September 1720, 2008.
 Braids,
knots, and applications
 Montpellier, June 911, 2008.
 Thompson's
Groups: New Developments and Interfaces
 Marseille, C.I.R.M., June 26, 2008.
 LieGrits
Workshop, final worshop of the research network "Flags, Quivers and
Invariant Theory in Lie Representation Theory"
 Mathematical Institute, University of Oxford, January 39, 2008.
 "CLUSE"
mathematical school, Group theory

MessignyetVantoux, October 28th  November 1st, 2007.
 Braids, groups and
manifolds in Toulouse
 Toulouse, September 58, 2007.
 Knots, hyperplane arrangements and Coxeter groups
 Marseille, C.I.R.M, June 48, 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 1822, 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 wellgenerated 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).
 Marseille, December 3rd,
at séminaire
Algèbre, Dynamique et Topologie
 Montpellier, November 29th, at
séminaire
Topologies
 Strasbourg, November 19th, at séminaire Quantique
 Lyon, November 15th, at séminaire d'algèbre
 Palaiseau, November 12th,
at GT
Combi du LIX
 Paris, November 8th,
at séminaire
Chevalley
 Caen, November 6th, at séminaire Algèbre et géométrie
 Asymptotical behaviour of roots of infinite Coxeter
groups

at
the OttawaCarleton
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 CIRGETLaCIM 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 noncroisées 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 LyashkoLooijenga : 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 noncroisées
 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 noncroisées 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 BradyWatt

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.