Tri :
Date
Editeur
Auteur
Titre


Sergiu Klainerman  Remarks on the stability of Kerr for axisymetryc perturbations
/ Fanny Bastien
/ 01062014
/ Canalu.fr
Klainerman Sergiu
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TBA Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Jérémie Joudioux  Hertz potentials and the decay of higher spin fields
/ Fanny Bastien
/ 03072014
/ Canalu.fr
Joudioux Jérémie
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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

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Rod Gover  An introduction to conformal geometry and tractor calculus (Part 4)
/ Fanny Bastien
/ 25062014
/ Canalu.fr
Gover Rod
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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

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Rod Gover  An introduction to conformal geometry and tractor calculus (Part 3)
/ Fanny Bastien
/ 25062014
/ Canalu.fr
Gover Rod
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Voir le résumé
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

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Rod Gover  An introduction to conformal geometry and tractor calculus (Part 2)
/ Fanny Bastien
/ 24062014
/ Canalu.fr
Gover Rod
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Voir le résumé
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

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Rod Gover  An introduction to conformal geometry and tractor calculus (Part 1)
/ Fanny Bastien
/ 23062014
/ Canalu.fr
Gover Rod
Voir le résumé
Voir le résumé
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

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Christian Gérard  Introduction to field theory on curved spacetimes (Part 1)
/ Fanny Bastien
/ 20062014
/ Canalu.fr
Gérard Christian
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The aim of these lectures is to give an introduction to quantum field theory on curved spacetimes, from the point of view of partial differential equations and microlocal analysis. I will concentrate on free fields and quasifree states, and say very little on interacting fields or perturbative renormalization. I will start by describing the necessary algebraic background, namely CCR and CAR algebras, and the notion of quasifree states, with their basic properties and characterizations. I will then introduce the notion of globally hyperbolic spacetimes, and its importance for classical field theory (advanced and retarded fundamental solutions, unique solvability of the Cauchy problem). Using these results I will explain the algebraic quantization of the two main examples of quantum fields ona manifold, namely the KleinGordon (bosonic) and Dirac (fermionic) fields.In the second part of the lectures I will discuss the important notion of Hadamardstates , which are substitutes in curved spacetimes for the vacuum state in Minkowskispacetime. I will explain its original motivation, related to the definition of therenormalized stressenergy tensor in a quantum field theory. I will then describethe modern characterization of Hadamard states, by the wavefront set of their twopointfunctions, and prove the famous Radzikowski theorem , using the DuistermaatHörmander notion of distinguished parametrices . If time allows, I will also describe the quantization of gauge fields, using as example the Maxwell field. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Semyon Dyatlov  Spectral gaps for normally hyperbolic trapping
/ Fanny Bastien
/ 30062014
/ Canalu.fr
Dyatlov Semyon
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Motivated by wave decay for Kerr and Kerr de Sitter black holes, we study spectral gaps for codimension 2 normally hyperbolic trapped sets with smooth stable/unstable foliations. Using semiclassical defect measures, we recover the gap of Wunsch Zworski and Nonnenmacher Zworski in our case. Under the stronger assumptions of r normal hyperbolicity and pinching, we discover further spectral gaps, meaning that resonances are stratified into bands by decay rates. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Claudio Dappiaggi  On the role of asymptotic structures in the construction of quantum states for free field theories on curved backgrounds
/ Fanny Bastien
/ 03072014
/ Canalu.fr
Dappiaggi Claudio
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In the algebraic approach to quantum field theory on curved backgrounds, there exists a special class of quantum states for free fields, called of Hadamard form. These are of particular relevance since they yield finite quantum fluctuations of all observables and they can be used to implement interactions at a perturbative level. Although their existence is guaranteed on all globally hyperbolic spacetimes, for long time only few explicit examples were known. A way to bypass this problem exists on those manifolds which possess a null conformal boundary, such as all asymptotically flat spacetimes. In this talk we shall discuss this construction in particular for massless, conformally coupled scalar fields and for linearised gravity. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Mihalis Dafermos  The "inside story" of black hole stability
/ Fanny Bastien
/ 30062014
/ Canalu.fr
Dafermos Mihalis
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Voir le résumé
TBA Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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