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

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

I will discuss the initial value problem for the Einstein equations and present results concerning the existence and asymptotic behavior of spacetimes, when the initial data set is assumed to be T2 symmetric and satisfies weak regularity conditions so that the spacetimes may exhibit impulsive gravitational waves and shock waves. This lecture is based on papers written over the period 2004—2014 and available at philippelefloch.org.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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

The first two lectures will present the fundamental results of symplectic topology : basic definitions, Moser’s lemma, normal forms of the symplectic structure near symplectic and Lagrangian submanifolds, characterization of Hamiltonian fibrations over any CW-complex. The third course will give the application of quantum homology to the splitting of the rational cohomology ring of any Hamiltonian fibration over S2, a generalization of a result of Deligne in the algebraic case and of Kirwan in the toric case. The fourth course will give the application of the quantum homology of a Lagrangian submanifold to the proof of the triviality of the monodromy of a weakly exact Lagrangian submanifold in any symplectic manifold.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

TBA
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, institut fourier, summer school, geometric measure theory, calculus of variation

The study of the asymptotic behavior of higher spin fields has proven to be a key point in understanding the stability properties of the Einstein equations. Penrose derived in the 60s the asymptotic behavior of these higher spin fields from a representation by Hertz potentiels satisfying a wave equation and a decay Ansatz for the solutions of the wave equation. The purpose of this talk is to perform the construction by Penrose in the context of the Cauchy problem on Minkowski space -­time for Maxwell fields and linearized gravity. Considering a Cauchy problem for Maxwell fields and linearized gravity with data in weighted Sobolev spaces, a Hertz potential is build from a generalization of the de Rham complex to arbitrary spin. The asymptotic behavior of these higher spin fields is then derived from the asymptotic behavior of the solutions of the wave equation.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

The method of « pseudoholomorphic » curves proved itself to be extremely useful in different fields. In symplectic topology, for instance Gromov’s Nonsqueezing Theorem, Arnold’s conjecture and the Floer homology, the Gromov-Witten invariants. In complex analysis and geometry, for instane polynomial hulls of totally real surfaces, envelopes of meromorphy, holomorphic foliations. We shall develop the theory of complex curves in almost complex manifolds and discuss some of these applications in our lectures.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

After recalling some features (and the value of) the invariant ``Ricci calculus'' of pseudo-­‐Riemannian geometry, we look at conformal rescaling from an elementary perspective. The idea of conformal covariance is visited and some covariant/invariant equations from physics are recovered in this framework. Motivated by the need to develop a more effective approach to such problems we are led into the idea of conformal geometry and a conformally invariant calculus; this``tractor calculus'' is then developed explicitly. We will discuss how to calculate using this, and touch on applications to the construction of conformal invariants and conformally invariant differential operators. The second part of the course is concerned with the application of conformal geometry and tractor calculus for the treatment of conformal compactification and the geometry of conformal infinity. The link with Friedrich’s conformal field equations will be made. As part of this part we also dedicate some time to the general problem of treating hypersurfaces in a conformal manifold, and in particular arrive at a conformal Gauss equation. Finally we show how these tools maybe applied to treat aspects of the asymptotic analysis of boundary problems on conformally compact manifolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis