David Bessis’s research while affiliated with Ecole Normale Supérieure de Paris and other places

We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups.

Garside groupoids, as recently introduced by Krammer, generalise Garside groups. A weak Garside group is a group that is equivalent as a category to a Garside groupoid. We show that any periodic loop in a Garside groupoid \CG may be viewed as a Garside element for a certain Garside structure on another Garside groupoid \CG_m, which is equivalent as a category to \CG. As a consequence, the centraliser of a periodic element in a weak Garside group is a weak Garside group. Our main tool is the notion of divided Garside categories, an analog for Garside categories of B\"okstedt-Hsiang-Madsen's subdivisions of Connes' cyclic category. This tool is used in our separate proof of the $K(\pi,1)$ property for complex reflection arrangements

Let V be a finite dimensional complex vector space and W\subseteq \GL(V) be a finite complex reflection group. Let V^{\reg} be the complement in V of the reflecting hyperplanes. We prove that V^{\reg} is a $K(\pi,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complexified real reflection groups. The complexified real case follows from a theorem of Deligne and, after contributions by Nakamura and Orlik-Solomon, only six exceptional cases remained open. In addition to solving this six cases, our approach is applicable to most previously known cases, including complexified real groups for which we obtain a new proof, based on new geometric objects. We also address a number of questions about \pi_1(W\cq V^{\reg}), the braid group of W. This includes a description of periodic elements in terms of a braid analog of Springer's theory of regular elements.

Let V be a finite dimensional complex vector space and W\subset \GL(V) be a finite complex reflection group. Let V^{\reg} be the complement in V of the reflecting hyperplanes. A classical conjecture predicts that V^{\reg} is a K(pi,1) space. When W is a complexified real reflection group, the conjecture follows from a theorem of Deligne. Our main result validates the conjecture for duality (or, equivalently, well-generated) complex reflection groups. This includes the complexified real case (but our proof is new) and new cases not previously known. We also address a number of questions about \pi_1(W\cq V^{\reg}), the braid group of W.

We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group G(e,e,r). For the particular case e=2 (resp. r=2), our lattice coincides with the lattice of simple elements for the type Dn (resp. I2(e)) dual braid monoid. Using this lattice, we construct a Garside structure for the braid group B(e,e,r). As a corollary, one may solve the word and conjugacy problems in this group.

We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group G(e,e,r). For the particular case e=2 (resp. r=2), our lattice coincides with the lattice of simple elements for the type $D_n$ (resp. $I_2(e)$) dual braid monoid. Using this lattice, we construct a Garside structure for the braid group B(e,e,r). As a corollary, one may solve the word and conjugacy problems in this group.

We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin groups of spherical type. Conjecturally, an analog construction should be available for arbitrary Artin groups and for braid groups of well-generated complex reflection groups. This article continues the exploration of the theory of Artin groups and generalised braid groups from the new point of view introduced by Birman-Ko-Lee in [BKL] for the classical braid group on n strings. In [B1], we generalised their construction to Artin groups of spherical type. In the current article, we study the case of the free group, which is the Artin group associated with the universal Coxeter group. The formal analogs of the main statements in [B1] turn out to be elementary consequences of classical material (some of which was known to Hurwitz and Artin). In an attempt to interpolate some recent generalisations of the dual monoid construction (by Digne for the Artin group of type Ãn, [D]; by Corran and the author for the braid group of the complex reflection group G(e, e, n), [BC]), we propose two conjectures describing properties of a generalised

We give presentations for the braid groups associated with the complex reflection groups $G_{24}$ and $G_{27}$. For the cases of $G_{29}$, $G_{31}$, $G_{33}$ and $G_{34}$, we give (strongly supported) conjectures. These presentations were obtained with VKCURVE, a GAP package implementing Van Kampen's method.

We give a new presentation of the braid group B of the complex reflection group G(e,e,r) which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this presentation gives rise to a Garside structure for B with Garside element a kind of generalised Coxeter element, and hence obtain solutions to the word and conjugacy problems for B.

We give a detailed account of the classical Van Kampen method for computing presentations of fundamental groups of complements of complex algebraic curves, and of a variant of this method, working with arbitrary projections (even with vertical asymptotes).