Tri :
Date
Editeur
Auteur
Titre


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
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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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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Alexander Gorodnik  Diophantine approximation and flows on homogeneous spaces (Part 1)
/ Fanny Bastien
/ 24062013
/ Canalu.fr
Gorodnik Alexander
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Voir le résumé
The
fundamental problem in the theory of Diophantine approximation is to
understand how well points in the Euclidean space can be approximated by
rational vectors with given bounds on denominators. It turns out that
Diophantine properties of points can be encoded using flows on
homogeneous spaces, and in this course we explain how to use techniques
from the theory of dynamical systems to address some of questions in
Diophantine approximation. In particular, we give a dynamical proof of
Khinchin’s theorem and discuss Sprindzuk’s question regarding
Diophantine approximation with dependent quantities, which was solved
using nondivergence properties of unipotent flows. In conclusion we
explore the problem of Diophantine approximation on more general
algebraic varieties. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

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Alessandro Giacomini  Free discontinuity problems and Robin boundary conditions
/ 30062015
/ Canalu.fr
Giacomini Alessandro
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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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Christian Gérard  Introduction to field theory on curved spacetimes (Part 1)
/ Fanny Bastien
/ 20062014
/ Canalu.fr
Gérard Christian
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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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Joseph Fu  Integral geometric regularity (Part 5)
/ Fanny Bastien
/ 24062015
/ Canalu.fr
Fu Joseph
Voir le résumé
Voir le résumé
In the original form given by Blaschke
in the 1930s, the famous Principal Kinematic Formula expresses the Euler
characteristic of the intersection of two sufficiently regular objects
in euclidean space, integrated over the space of all possible relative
positions, in terms of geometric invariants associated to each of them
individually. It is natural to wonder about the precise regularity
needed for this to work. The question turns on the existence of the
normal cycle of such an object A, i.e. an integral current that stands
in for its manifolds of unit normals if A is too irregular for the
latter to exist in a literal sense. Despite significant recent progress,
a comprehensive understanding of this construction remains maddeningly
elusive. In these lectures we will discuss both of these aspects. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Joseph Fu  Integral geometric regularity (Part 4)
/ Fanny Bastien
/ 24062015
/ Canalu.fr
Fu Joseph
Voir le résumé
Voir le résumé
In the original form given by Blaschke
in the 1930s, the famous Principal Kinematic Formula expresses the Euler
characteristic of the intersection of two sufficiently regular objects
in euclidean space, integrated over the space of all possible relative
positions, in terms of geometric invariants associated to each of them
individually. It is natural to wonder about the precise regularity
needed for this to work. The question turns on the existence of the
normal cycle of such an object A, i.e. an integral current that stands
in for its manifolds of unit normals if A is too irregular for the
latter to exist in a literal sense. Despite significant recent progress,
a comprehensive understanding of this construction remains maddeningly
elusive. In these lectures we will discuss both of these aspects. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Joseph Fu  Integral geometric regularity (Part 3)
/ Fanny Bastien
/ 24062015
/ Canalu.fr
Fu Joseph
Voir le résumé
Voir le résumé
In the original form given by Blaschke
in the 1930s, the famous Principal Kinematic Formula expresses the Euler
characteristic of the intersection of two sufficiently regular objects
in euclidean space, integrated over the space of all possible relative
positions, in terms of geometric invariants associated to each of them
individually. It is natural to wonder about the precise regularity
needed for this to work. The question turns on the existence of the
normal cycle of such an object A, i.e. an integral current that stands
in for its manifolds of unit normals if A is too irregular for the
latter to exist in a literal sense. Despite significant recent progress,
a comprehensive understanding of this construction remains maddeningly
elusive. In these lectures we will discuss both of these aspects. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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Joseph Fu  Integral geometric regularity (Part 2)
/ Fanny Bastien
/ 23062015
/ Canalu.fr
Fu Joseph
Voir le résumé
Voir le résumé
In the original form given by Blaschke
in the 1930s, the famous Principal Kinematic Formula expresses the Euler
characteristic of the intersection of two sufficiently regular objects
in euclidean space, integrated over the space of all possible relative
positions, in terms of geometric invariants associated to each of them
individually. It is natural to wonder about the precise regularity
needed for this to work. The question turns on the existence of the
normal cycle of such an object A, i.e. an integral current that stands
in for its manifolds of unit normals if A is too irregular for the
latter to exist in a literal sense. Despite significant recent progress,
a comprehensive understanding of this construction remains maddeningly
elusive. In these lectures we will discuss both of these aspects. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

Accéder à la ressource

