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Documents  Hilgert, Joachim | enregistrements trouvés : 14

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Research talks;Partial Differential Equations;Geometry;Mathematical Physics

In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as (potentially) normally hyperbolic trapping, as well as the role of resonances. In this lecture I will describe a framework for the Fredholm analysis of non-elliptic problems both on manifolds without boundary and manifolds with boundary, with a view towards wave propagation on Kerr-de-Sitter spaces, which is the key analytic ingredient for showing the stability of black holes (see Peter Hintz' lecture). This lecture focuses on the general setup such as microlocal ellipticity, real principal type propagation, radial points ...

35A21 ; 35A27 ; 35B34 ; 35B40 ; 58J40 ; 58J47 ; 83C35 ; 83C57

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Geometry

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou. I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between ...

37D50 ; 53D25 ; 37D20 ; 35B34 ; 35P25

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- 357 p.
ISBN

Banach center publications , 0055

Localisation : Salle des périodiques 1er étage

géométrie # analyse # groupe de Lie fini # groupe de Lie infini # groupe topologique # algèbre de Lie # théorie de la représentation # groupe d'homotopie # opérateur de Toeplitz # espace de Hardy # structure de Jordan # noyau de Berezin

17-06 ; 22-06 ; 58-06

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

Recently David Borthwick discovered through numerical calculations surprising chain structures in the resonance spectrum of certain Schottky surfaces. In this talk we will see that theses resonance chains have the same origin as the resonance chains in the classical and quantum mechanical spectrum of the three disk system and we will see that they are related to a clustering in the length spectrum. Finally the existence of these chains will be proven for three funneled Schottky surfaces in a certain geometrical limit in the Teichmüller space. Joint work with S. Barkhofen and F. Faure. Recently David Borthwick discovered through numerical calculations surprising chain structures in the resonance spectrum of certain Schottky surfaces. In this talk we will see that theses resonance chains have the same origin as the resonance chains in the classical and quantum mechanical spectrum of the three disk system and we will see that they are related to a clustering in the length spectrum. Finally the existence of these chains will be ...

35P25 ; 58J50 ; 81Q05

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Topology

I will explain how one can get a complete description of the correlation spectrum of a Morse-Smale flow in terms of the Lyapunov exponents and of the periods of the flow. I will also discuss the relation of these results with differential topology.
This a joint work with Nguyen Viet Dang (Université Lyon 1).

37D15 ; 58J51 ; 37D40

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Research talks;Partial Differential Equations;Number Theory;Mathematical Physics

Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer operators characterize Maass cusp forms. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. The Fredholm determinant of the fast transfer operators equals the Selberg zeta function, which yields that the zeros of the Selberg zeta function (among which are the resonances) are determined by the eigenfunctions with eigenvalue 1 of the fast transfer operators. It is a natural question how the eigenspaces of these two types of transfer operators are related to each other.
Over the last few years I developed (partly jointly with coauthors) dual 'slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for certain hyperbolic surfaces/orbifolds L \ H with cusps (both of finite and infinite area; arithmetic and non-arithmetic).
Both types of transfer operators arise from discretizations of the geodesic flow on L \ H. The eigenfunctions with eigenvalue 1 of slow transfer ...

37C30 ; 11F03 ; 37D40

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Algebraic and Complex Geometry

Eisenstein series are the natural analog of ”plane waves” for hyperbolic manifolds of infinite volume. These non-$L^2$ eigenfunctions of the Laplacian parametrize the continuous spectrum. In this talk we will discuss the structure of nodal sets and domains for surfaces. Upper and lower bounds on the number of intersections of nodal lines with ”generic” real analytic curves will be given, together with similar bounds on the number of nodal domains inside the convex core. The results are based on equidistribution theorems for restriction of Eisenstein series to curves that bear some similarity with the so-called ”QER” results for compact manifolds. Eisenstein series are the natural analog of ”plane waves” for hyperbolic manifolds of infinite volume. These non-$L^2$ eigenfunctions of the Laplacian parametrize the continuous spectrum. In this talk we will discuss the structure of nodal sets and domains for surfaces. Upper and lower bounds on the number of intersections of nodal lines with ”generic” real analytic curves will be given, together with similar bounds on the number of nodal ...

58J50 ; 58J51 ; 35J05

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Research talks;Algebraic and Complex Geometry

For any symmetric space $X$ of noncompact type, its quotients by torsion-free discrete isometry groups $\Gamma$ are locally symmetric spaces. One problem is to understand the geometry and analysis, especially the spectral theory, and interaction between them of such spaces. Two classes of infinite groups $\Gamma$ have been extensively studied:
$(1) \Gamma$ is a lattice, and hence $\Gamma$ $\backslash$ $X$ has finite volume.
$(2) X$ is of rank $1$, for example, when $X$ is the real hyperbolic space, $\Gamma$ is geometrically finite and $\Gamma$ $\backslash$ $X$ has infinite volume.
When $\Gamma$ is a nonuniform lattice in case $(1)$ or any group in case $(2)$, compactification of $\Gamma$ $\backslash$ $X$ and its boundary play an important role in the geometric scattering theory of $\Gamma$ $\backslash$ $X$. When $X$ is of rank at least $2$, quotients of $X$ of finite volume have also been extensively studied. There has been a lot of recent interest and work to understand quotients $\Gamma$ $\backslash$ $X$ of infinite volume. For example, there are some generalizations of convex cocompact groups, but no generalizations yet of geometrically finite groups. They are related to the notion of thin groups. One naturally expects that these locally symmetric spaces should have real analytic compactifications with corners (with codimension equal to the rank), and their boundary should also be used to parametrize the continuous spectrum and to understand the geometrically scattering theory. These compactifications also provide a natural class of manifolds with corners. In this talk, I will describe some questions, open problems and results.
For any symmetric space $X$ of noncompact type, its quotients by torsion-free discrete isometry groups $\Gamma$ are locally symmetric spaces. One problem is to understand the geometry and analysis, especially the spectral theory, and interaction between them of such spaces. Two classes of infinite groups $\Gamma$ have been extensively studied:
$(1) \Gamma$ is a lattice, and hence $\Gamma$ $\backslash$ $X$ has finite volume.
$(2) X$ is of rank ...

53C35 ; 58J50

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Research talks;Partial Differential Equations;Mathematical Physics

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will discuss the geometry of these black holes, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this talk, I will emphasize geometric aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.
In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will discuss the geometry of these black holes, and then talk about the stability question for these black holes in the initial value fo...

35B40 ; 58J47 ; 83C05 ; 83C35 ; 83C57

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Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that may improve this estimate. This will be presented on a simple toy model, partially expanding maps. Work with Tobias Weich. Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that ...

37C30 ; 37D20 ; 58J50

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Research talks;Partial Differential Equations;Mathematical Physics

After a brief introduction to the spectral theory of hyperbolic surfaces, we will focus on the problem of understanding the asymptotic distribution of resonances for hyperbolic surfaces. The theory of open quantum chaotic systems has inspired several interesting conjectures about this distribution. We will highlight the recent theoretical progress towards these conjectures, and present some of the latest numerical evidence.

58J50 ; 11F72 ; 11M36

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ISBN 978-3-540-56954-1

Lecture notes in mathematics , 1552

Localisation : Collection 1er étage

espace homogène ordonné # géométrie et topologie des groupes de Lie # invariant de cône # radical cocompact # semi groupe de Lie # semi groupe de Ol'Shanskii # semi groupe de compression

22A15 ; 22A25 ; 22E30 ; 22E46 ; 53C30

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- x; 744 p.
ISBN 978-0-387-84793-1

Springer monographs in mathematics

Localisation : Ouvrage RdC (HILG)

groupe de Lie # algèbre de Lie # groupe de matrices # groupe réductif # variétés # fonction exponentielle # sous-groupe normal # groupe compact # forme réelle # décomposition de cartan # groupe abélien # groupe conforme # espace de Hilbert # produit tenseur

22Exx ; 22FXX ; 22E15 ; 22-01 ; 17B05 ; 53C30

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- 645 p.
ISBN 978-0-19-853569-0

Oxford mathematical monographs

Localisation : Ouvrage RdC (HILG)

cone # corps convexe # groupe de Lie # semi groupe

11Hxx ; 22-02

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