This series of lectures is dedicated to recent results concerning the existence of holomorphic curves on the surfaces of class VII. The first lecture will be an introduction to the Donaldson theory. We will present the fundamental notions and some important results in the theory, explaining ideas of the proofs. In the second lecture we will present the theory of holomorphic fiber bundles on complex surfaces, the stability notion, moduli spaces and the Kobayashi-Hitschin correspondence that links moduli spaces of stable fiber bundles (defined in the fram of complex geometry) to moduli spaces of instantons (defined in the frame of the Donaldson theory). In the last two lectures we will prove the existence of holomorphic curves on minimal surfaces of class VII with b2=1 or 2 and we will present the general strategy and the last results obtained in the general case.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

This series of lectures is dedicated to recent results concerning the existence of holomorphic curves on the surfaces of class VII. The first lecture will be an introduction to the Donaldson theory. We will present the fundamental notions and some important results in the theory, explaining ideas of the proofs. In the second lecture we will present the theory of holomorphic fiber bundles on complex surfaces, the stability notion, moduli spaces and the Kobayashi-Hitschin correspondence that links moduli spaces of stable fiber bundles (defined in the fram of complex geometry) to moduli spaces of instantons (defined in the frame of the Donaldson theory). In the last two lectures we will prove the existence of holomorphic curves on minimal surfaces of class VII with b2=1 or 2 and we will present the general strategy and the last results obtained in the general case.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

In order to control locally a space-­‐time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-­‐like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-­‐posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

In order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

In order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

We will focus in our lectures on the following : 1. J-complex discs in almost complex manifolds : general properties. Linearization and compactness. Gromov’s method : the Fredholm alternative for the d-bar operator. Attaching a complex disc to a Lagrangian manifold. Application : exotic symplectic structures. Hulls of totally real manifolds : Alexander’s theorem. 2. Real surfaces in (almost) complex surfaces. Filling real 2-spheres by a Levi-flat hypersurface (Bedford -Gaveau-Gromov theorem). Some applications. Symplectic and contact structures. Reeb foliation and the Weinsten conjecture. Hofer’s proof of the Weinstein conjecture. 3. J-complex lines and hyperbolicity. The KAM theory and Moser’s stability theorem for entire J-complex curves in tori. Global deformation and Bangert’s theorem.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

We will focus in our lectures on the following : 1. J-complex discs in almost complex manifolds : general properties. Linearization and compactness. Gromov’s method : the Fredholm alternative for the d-bar operator. Attaching a complex disc to a Lagrangian manifold. Application : exotic symplectic structures. Hulls of totally real manifolds : Alexander’s theorem. 2. Real surfaces in (almost) complex surfaces. Filling real 2-spheres by a Levi-flat hypersurface (Bedford -Gaveau-Gromov theorem). Some applications. Symplectic and contact structures. Reeb foliation and the Weinsten conjecture. Hofer’s proof of the Weinstein conjecture. 3. J-complex lines and hyperbolicity. The KAM theory and Moser’s stability theorem for entire J-complex curves in tori. Global deformation and Bangert’s theorem.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

I will explain how one can formulate and formalize the Gupta Bleuler framework for the Quantization of the electromagnetic field in an algebraic manner so that it works on globally hyperbolic space times. I will then discuss a construction of physical representations that works without the "spectral gap assumption" in the case of absense of zero energy resonances. These can be excluded by topologocial restrictions at infinity. This is based on joint work with Felix Finster.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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

The counting function associated to the moduli space of stable pairs on a 3-fold X is conjectured to give the Laurent expansion of a rational function. For toric X , this conjecture can be proven by a careful grouping of the box con gurations appearing in the stable pairs equivariant descendent vertex. I will describe this approach and then say a little about how it might also be used to study the Donaldson{Thomas vertex. This talk presents joint work with Rahul Pandharipande.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, summer school, Gromov-Witten, isntitut fourier