We will first introduce the basic concepts pertaining to Kobayashi pseudo-distances and hyperbolic complex spaces, including Brody’s theorem and the Ahlfors-Schwarz lemma. One of the main goals of the theory is to understand conditions under which a given algebraic variety is Kobayashi hyperbolic. This leads to the introduction of jet spaces and jet metrics, and provides a strong link between the existence of entire curves and the existence of global algebraic differential equations.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness, structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness, structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

A nonsingular holomorphic foliation of codimension on a complex manifold is locally given by the level sets of a holomorphic submersion to the Euclidean space . If is a Stein manifold, there also exist plenty of global foliations of this form, so long as there are no topological obstructions. More precisely, if then any -tuple of pointwise linearly independent (1,0)-forms can be continuously deformed to a -tuple of differentials where is a holomorphic submersion of to . Such a submersion always exists if is no more than the integer part of . More generally, if is a complex vector subbundle of the tangent bundle such that is a flat bundle, then is homotopic (through complex vector subbundles of ) to an integrable subbundle, i.e., to the tangent bundle of a nonsingular holomorphic foliation on . I will prove these results and discuss open problems, the most interesting one of them being related to a conjecture of Bogomolov.
Mot(s) clés libre(s) : Grenoble, école d'été, mathématiques, institut fourier, summer school, holomorphic foliations

A nonsingular holomorphic foliation of codimension on a complex manifold is locally given by the level sets of a holomorphic submersion to the Euclidean space . If is a Stein manifold, there also exist plenty of global foliations of this form, so long as there are no topological obstructions. More precisely, if then any -tuple of pointwise linearly independent (1,0)-forms can be continuously deformed to a -tuple of differentials where is a holomorphic submersion of to . Such a submersion always exists if is no more than the integer part of . More generally, if is a complex vector subbundle of the tangent bundle such that is a flat bundle, then is homotopic (through complex vector subbundles of ) to an integrable subbundle, i.e., to the tangent bundle of a nonsingular holomorphic foliation on . I will prove these results and discuss open problems, the most interesting one of them being related to a conjecture of Bogomolov.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, applications, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

A nonsingular holomorphic foliation of codimension on a complex manifold is locally given by the level sets of a holomorphic submersion to the Euclidean space . If is a Stein manifold, there also exist plenty of global foliations of this form, so long as there are no topological obstructions. More precisely, if then any -tuple of pointwise linearly independent (1,0)-forms can be continuously deformed to a -tuple of differentials where is a holomorphic submersion of to . Such a submersion always exists if is no more than the integer part of . More generally, if is a complex vector subbundle of the tangent bundle such that is a flat bundle, then is homotopic (through complex vector subbundles of ) to an integrable subbundle, i.e., to the tangent bundle of a nonsingular holomorphic foliation on . I will prove these results and discuss open problems, the most interesting one of them being related to a conjecture of Bogomolov.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, holomorphic foliations, COURBES PSEUDOHOLOMORPHES

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation