Intersection de sous-groupes et de sous-variétés I

We study the intersection of a subvariety X of an abelian variety A over with the union of all the algebraic subgroups of A of given dimension d. Our main result states that if we remove a suitable exceptional subset from X and if d is small enough then the intersection enjoys a Northcott-like property: the points of bounded height on it form a finite set. The condition on d involves only the dimension of X and the structure of A up to isogenies. We show how it can be weakened if we assume certain conjectures in the direction of an abelian version of Lehmer's problem. The theorem is especially meaningful when X is a curve since it is then possible to bound the height and hence to prove finiteness of the set under consideration. This generalises the result of E. Viada on powers of elliptic curves and is analogous to work of E. Bombieri, D. Masser and U. Zannier on tori, whose general strategy we follow.     

Inégalités de Brascamp-Lieb et convexité

This note gives an elementary proof of an inequality of Brascamp-Lieb and of a new dual inequality that has numerous applications to convexity: lower estimates of volumes of projections, of exterior volume ratio and MM^* -estimate in the non symmetric case.     

Lemmes de multiplicités et constante de Seshadri

Résumé  On démontre dans cet article un raffinement des lemmes de multiplcités de Philippon, essentiellement dans le cas particulier où l’on dérive dans toutes les directions. L’amélioration est rendue possible grace à un point de vue plus géométrique et notamment par l’apparition nouvelle dans ce contexte de la notion de constante de Seshadri.     

Les épreuves répétées et les formules approchées de Laplace et de Charlier

Sans résumé     

Remplissage et surfaces de révolution

The filling function of a Riemannian manifold is either linear or at least quadratic. This fact was originally discovered by M. Gromov in 1985. We address the question of the existence of further obstructions. We give a partial answer: every superadditive and superquadratic function is asymptotic to the filling function of a surface of revolution. A function which furthermore satisfies a natural second-order differential inequation is equal to a filling function.     

Sur les réseaux de courbes et de surfaces algébriques

Sans résumé     

Caractérisation et existence de structures de Dirac multiplicatives

We define the product of two Dirac manifolds and introduce the notion of a Dirac–Lie group of Poisson type. This notion is equivalent to that of multiplicative Dirac structure and any real simply-connected Lie group carries a no trivial multiplicative Dirac structure when its dimension is at least 2. To cite this article: A. Affane, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Opérateurs de Dirac et équations de Maxwell

Sans résumé     

Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg

In this paper, we prove dispersive and Strichartz inequalities on the Heisenberg group. The proof involves the analysis of Besov-type spaces on the Heisenberg group.

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Le travail du troisième auteur est partiellement finacé par la NNSF de Chine.     

Sur la série de dirichlet et la série de facultés

Sans résumé Extrait d'une lettre à M. N. E.N?rlund.