Prolog is a very unusual programming language, developed by Alain Colmerauer in one of the buildings on the way to the CIRM in Luminy. It is not fundamentally iterative in the way that, for example, GAP and Magma are. Instead it operates by taking a list of axioms as input, and responds at the command line to queries asking the language to achieve particular goals. It gained some notoriety by beating contestants on the game show Jeopardy in 2011. It is also the worlds fastest sudoku solver. I will describe some recent Prolog investigations to search for new simple Lie algebras over the field GF(2). We were able to discover some new examples in dimensions 15 and 31 and extrapolate from these to construct two new infinite families of simple Lie algebras. This is joint work with David Cushing and George Stagg.

Soit $n$ un entier et $Out(F_n)$ le groupe des automorphismes extérieurs d'un groupe libre non abélien. Soit $[g]$ une classe de conjugaison de $F_n$ et $F \in Out(F_n)$. La classe $[g]$ est à croissance exponentielle si la longueur (pour une base fixée de $F_n$) de $F^m([g])$ croît exponentiellement avec $m$. Nous construisons un espace topologique compact, appelé espace de courants relatifs, sur lequel $F$ agît par homéomorphisme et qui permet de traduire la croissance exponentielle en des termes dynamiques. Nous donnerons des applications algébriques de ces résultats dynamiques, qui imposent des contraintes sur la structure des sous-groupes de $Out(F_n)$.

In this talk we will introduce a family of groups called Hecke groups that generalize the modular group SL(2,Z). Named after Erich Hecke, These groups are at the crossroad between hyperbolic (and flat) geometry, number theory, and dynamics. We will explain the interaction between these aspects while asking the (open) question of determining the elements of the boundary at infinity of the hyperbolic plane that represent the cusp of the quotient surface.

On the interior of an n-dimensional proper convex cone there exists a canonical Riemannian metric, the Cheng-Yau metric. This metric is invariant with respect to linear automorphisms of the cone and decomposes in a direct product of a 1-dimensional radial and a (n-1)-dimensional transversal component. The corresponding family of (n-1)-dimensional submanifolds is the homothetic family of hyperbolic affine spheres which are asymptotic to the boundary of the cone. For 3-dimensional cones the affine sphere is a Riemann surface which can be uniformized by a global complex coordinate, defined either on the unit disk or the whole complex plane, depending on the conformal type of the surface. In this coordinate, the cubic form of the affine sphere can be represented by a cubic holomorphic differential. This yields a correspondence between holomorphic functions on the disk or on the plane and convex 3-dimensional cones.

On établira dans cet exposé la classification complète des motifs de variétés projectives homogènes sous l'action d'un groupe semisimple, à coefficients finis. Le critère obtenu met en jeu une nouvelle famille d'invariants motiviques, les traces de Tate. Nous illustrerons ce critère avec de nombreux exemples et en déduirons ensuite une généralisation de la caractérisation de l'équivalence motivique des groupes semisimples en fonction de leurs p-indices de Tits supérieurs.

We elaborate on various ways to pass to the limit a given family of finite particle systems, either by mean field limit, deriving the Vlasov equation, or by hydrodynamic or graph limit, obtaining the Euler equation. We provide convergence estimates. We also show how to pass from Liouville to Vlasov or to Euler by taking adequate moments. Our results encompass and generalize a number of known results of the literature. As a surprising consequence of our analysis, we show that, under appropriate regularity assumptions, solutions of any quasilinear PDE can be approximated by the solutions of finite particle systems. This is a work in collaboration with Thierry Paul.

In the end of the 19th century, Klein introduced a notion of geometric structure on a manifold associated to a group action on a model space. This notion includes the notion of euclidean, hyperbolic structures, but also flat conformal and projective structures on manifolds. We will consider some examples of geometric structures on surfaces and see how one can associate to such a structure a representation of a surface group. In particular the space of hyperbolic structures on a surface corresponds to a connected component of the space of representations of the correspongind surface groups. Finally we will see that one can draw a similar picture for convex projective structures on a surface.

Dans cet exposé je montrerai comment faire agir proprement discontinûment un groupe de Coxeter (infini) sur un espace affine R^N. Il s'agit d'une réinterprétation et généralisation des "espaces-temps de Margulis", dont j'esquisserai l'histoire en lien avec la conjecture d'Auslander ("le groupe fondamental d'une variété affine compacte est virtuellement résoluble"). Travail commun avec J.Danciger et F.Kassel.