Tri :
Date
Editeur
Auteur
Titre


Alexander Gorodnik  Diophantine approximation and flows on homogeneous spaces (Part 2)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
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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Alexander Gorodnik  Diophantine approximation and flows on homogeneous spaces (Part 3)
/ Fanny Bastien
/ Canalu.fr
Voir le résumé
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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Andras Vasy  Microlocal analysis and wave propagation (Part 2)
/ 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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Andras Vasy  Microlocal analysis and wave propagation (Part 3)
/ 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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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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Andrew Lorent  The AvilesGiga functional: past and present
/ 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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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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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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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

Accéder à la ressource


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

Accéder à la ressource

