In these lectures, Gromov{Witten theory will be introduced, assuming only basic moduli theory covered in the rst week of the School. Then the Crepant Transformation Conjecture will be explained. Some examples, with emphasis on the projective/global cases, will be given. Note: The construction of virtual fundamental class, which forms the foundation of the GW theory, will be given in Jun Li's concurrent lectures and will not be explained here.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

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

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

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

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

In this introductory course I will present the basic notions, both local and global, using classical examples. I will explain statements in connection with the resolution of singularities with for instance the singular Frobenius Theorem or the Liouvilian integration. I will also present some open questions which I will motivate by examples. Dans ce cours introductif je m’attacherai à présenter les notions de base tant locales que globales au travers d’exemples classiques. J’aborderai des énoncés liés à la résolution des singularités avec par exemple le théorème de Frobenius singulier ou l’intégration Liouvillienne. Je présenterai aussi quelques problèmes ouverts que je motiverai encore au travers d’exemples.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

In this introductory course I will present the basic notions, both local and global, using classical examples. I will explain statements in connection with the resolution of singularities with for instance the singular Frobenius Theorem or the Liouvilian integration. I will also present some open questions which I will motivate by examples. Dans ce cours introductif je m’attacherai à présenter les notions de base tant locales que globales au travers d’exemples classiques. J’aborderai des énoncés liés à la résolution des singularités avec par exemple le théorème de Frobenius singulier ou l’intégration Liouvillienne. Je présenterai aussi quelques problèmes ouverts que je motiverai encore au travers d’exemples.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

In this introductory course I will present the basic notions, both local and global, using classical examples. I will explain statements in connection with the resolution of singularities with for instance the singular Frobenius Theorem or the Liouvilian integration. I will also present some open questions which I will motivate by examples. Dans ce cours introductif je m’attacherai à présenter les notions de base tant locales que globales au travers d’exemples classiques. J’aborderai des énoncés liés à la résolution des singularités avec par exemple le théorème de Frobenius singulier ou l’intégration Liouvillienne. Je présenterai aussi quelques problèmes ouverts que je motiverai encore au travers d’exemples.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

L’utilisation de méthodes de théorie des faisceaux (Kashiwara-Schapira)a été dévelopée ces dernières années par Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara et Schapira. Nous essaierons d’en donner un aperçu à la fois pour démontrer des résultats classiques, comme la conjecture d’Arnold, et pour des résultats nouveaux. The use of methods from the Sheaf Theory (Kashiwara-Schapira) was developped recently by Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara and Schapira. We will try to give an insight of that, in order to prove classical results, such as the Arnold conjecture, and to obtain new results.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES