La plupart des systèmes intégrables classiques admettent une complexification naturelle, où les tores de Liouville deviennent des tores algébriques complexes (variétés abéliennes). Le fait d'introduire un temps complexe et des coordonnées complexes nous fait découvrir les structures algébriques et géometriques sous-jacentes et plusieurs constructions classiques (linéarisation, coordonnées action-angle, solutions explicites, symétries cachées, ...) deviennent explicites, et pour des classes d'exemples même systématiques. A partir de la notion générale d'intégrabilité (au sens de Liouville) sur une variété de Poisson (réelle ou complexe) et après quelques rappels sur les variétés abéliennes, les systèmes algébriquement intégrables seront introduits et leurs propriétés de base seront illustrées sur quelques exemples simples. Pour finir, le critère de Kowalevski-Painlevé sera expliqué et illustré par deux exemples.

We build some examples of taut transversely affine foliations on some closed manifolds which are surface bundles over the circle. These examples illustrate how such foliations can be various. Under some restrictive hypothesis Nakayama describes in [5] such foliations which have no compact leaf. We extend Nakayama's theorem for transversely affine foliations without Reeb component. Also we give an example among these transversely affine foliations without non-trivial transversely projective deformation.

These are the notes of an introductory lecture series on convexity properties of the moment map, equivariant cohomology and conjugation spaces.

This short note is devoted to relationships between the cohomology theory of Koszul-Vinberg algebras and some related topics. Besides being useful for the study of some short exact sequences of KV algebras and of short exact sequences of modules over KV algebras, the so-called KV cohomology is useful to study the deformations of these KV algebras as well as to study the Dirac reductions of Poisson manifolds. We briefly discuss these items.

Let $(N,\alpha)$ be a compact contact manifold and $(N \times {\mathbb R}$, $d(e^t\alpha))$ its symplectization. We show that the group $G$ which is the identity component in the group of symplectic diffeomorphisms $\phi$ of $(N\times {\mathbb R}, d(e^t\alpha))$ that cover diffeomorphisms $\underline {\phi}$ of $ N\times S^1$ is simple, by showing that $G$ is isomorphic to the kernel of the Calabi homomorphism of the associated locally conformal symplectic structure.

In this paper, we develop one of the questions raised by the author in the mini-course he gave at the conference Geometry and Physics V held at the University Cheikh Anta Diop, Dakar in May 2007). Let $\Pi$ be a Poisson tensor on a manifold $M.$ We suppose that it is decomposable in a neighborhood $U$ of a point $m,$ i.e. we have $\Pi=X\wedge Y$ on $U$ where $X$ and $Y$ are two vector fields. We will exhibit examples where every Poisson tensor near enough $\Pi$ seems to be also decomposable in a neighborhood of a point which can be chosen arbitrarily near $m$; and this works even if $M$ has a big dimension. This idea is a consequence of a cohomology calculation which can be interesting by itself.

We shall study the equivalence problem for ordinary differential equations with respect to the affine transformation group $A( 2,{\mathbb R})$.

This paper contains a characterization of Reeb vector fields of K-contact forms in terms of J-holomorphic embeddings into the tangent unit sphere bundle. A consequence of this characterization is that these vector fields are critical points of a volume and an energy functionals defined on the set of unit vector fields. Reeb vector fields on closed, K-contact Einstein manifolds are absolute minimizers for the energy functional with a mean curvature correction. On odd-dimensional Einstein manifolds of positive sectional curvature, these unit vector fields are characterized by their minimizing property. It is also proved that any closed flat contact manifold admits a parallelization by three critical unit vector fields, one parallel (hence minimizing), the other two are Reeb vector fields of contact forms, not Killing and not minimizers of any of the volume or the energy functionals.

Un champ de vecteurs linéaire sur un groupe de Lie est défini par un groupe à un paramètre d'automorphismes. On obtient un système linéaire en ajoutant des champs contrôlés invariants à gauche. Ayala et Hacibekiroglu ont étudié l'observabilité d'un tel système lorsque la sortie est un morphisme de groupes de Lie. On montre qu'aucun système linéaire observable n'existe sur un groupe semi-simple, et on donne des conditions nécessaires pour qu'un tel système existe sur un groupe de Lie quelconque.

In these Lecture Notes of a mini-course delivered in the " Séminaire Itinérant de Géometrie et Physique Mathématique, " Geometry and Physics V" at the University Cheikh Anta Diop, Dakar in May 2007, we introduce the group of symplectic diffeomorphisms, the main results on its algebraic structure and on some of its local and global properties. This survey culminates with the most recent results on Hofer geometry, the definitions of the groups of symplectic and hamiltonian homeomorphisms, and the introduction to the $C^0$ symplectic topology.

Transversal generalized complex structures provide a framework unifying both transversely holomorphic foliations and generalized complex geometry. In this paper, we give characterizations of transversal generalized complex structures. Moreover, a natural extension of the basic Dolbeault cohomology is obtained.