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

Quantum K-theory is as quantum cohomology a generalisation of the classical coho- mology algebra of a variety X . In this talk I will explain the connection between the geometry of the moduli space of stable maps, in particular rational connectedness properties, and the computation of structure constants for X a rational homogeneous space. This is based on a joint work with A. Buch, P.-E. Chaput and L. Mihalcea
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

The topic concerns relations among the kappa classes in the tautological ring of the moduli space of genus g curves. After a discussion of classical constructions in Wick form, we derive an explicit set of relations obtained from the virtual geometry of the moduli space of stable quotients. In a series of steps, the stable quotient relations are transformed to simpler and simpler forms. Our nal result establishes a previously conjectural set of tautological relations proposed a decade ago by Faber{Zagier.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

In this talk I will explain the idea of the Laplace transform that connects a counting problem in the A-model side with a recursion formula based on complex analysis in the B-model side, using a concrete example. The talk is based on my joint work with Chapman, Penkava and Safnuk.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric analysis, metric geometry, topology

We will give an introduction to Donaldson-Thomas theory and some basic tools and computations. In the last lecture, we hope to explain some aspects of the proof of the GW/DT correspondence for toric threefolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten