Tri :
Date
Editeur
Auteur
Titre


Christian Gérard  Introduction to field theory on curved spacetimes (Part 4)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
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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Christian Gérard  Introduction to field theory on curved spacetimes (Part 3)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
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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Christian Gérard  Introduction to field theory on curved spacetimes (Part 2)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
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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Christian Gérard  Construction of Hadamard states for Klein‐Gordon fields
/ Fanny Bastien
/ Canalu.fr
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Voir le résumé
we will review a new construction of Hadamard states for quantized Klein‐Gordon fields on curved spacetimes, relying on pseudo differential calculus on a Cauchy surface. We also present some work in progress where Hadamard states are constructed from traces of Klein‐Gordon fields on a characteristic cone. (Joint work with Michal Wrochna). Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, calculus of variation, asymptotic analysis

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Camillo De Lellis  Center manifolds and regularity of areaminimizing currents (Part 4)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
areaminimizing. In fact the typical singularity of a 2dimensional
areaminimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Camillo De Lellis  Center manifolds and regularity of areaminimizing currents (Part 3)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
areaminimizing. In fact the typical singularity of a 2dimensional
areaminimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Camillo De Lellis  Center manifolds and regularity of areaminimizing currents (Part 1)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
A celebrated
theorem of Almgren shows that every integer rectifiable current which
minimizes (locally) the area is a smooth submanifold except for a
singular set of codimension at most 2. Almgren’s theorem is sharp in
codimension higher than 1, because holomorphic subvarieties of Cn are
areaminimizing. In fact the typical singularity of a 2dimensional
areaminimizing current is modelled by branch points of holomorphic
curves. These singularities are rather difficult to analyze because they
might be very high order phenomena. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Bruno Lévy  A numerical algorithm for L2 semidiscrete optimal transport in 3D
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Andrew Lorent  The AvilesGiga functional: past and present
/ Fanny Bastien
/ Canalu.fr
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Voir le résumé
indisponible Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Andras Vasy  Microlocal analysis and wave propagation (Part 4)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
Voir le résumé
In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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