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Videothèque  | enregistrements trouvés : 68

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Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

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Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

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Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

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A general introduction to the state of the art in counting of rational and integral points on varieties, using various analytic methods with the Brauer-Manin obstruction.

14G05 ; 14F22

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A general introduction to the state of the art in counting of rational and integral points on varieties, using various analytic methods with the Brauer-Manin obstruction.

14G05 ; 14F22

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We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval [0, N], in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximately like exponential of a Brownian motion with a drift. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random potential is multiplied by a factor 1/ √N.
We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval [0, N], in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximately like exponential of a Brownian motion with a drift. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random ...

60B20 ; 65F15

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Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these cohomological tools.

20G10 ; 11E08 ; 11E72

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A general introduction to the state of the art in counting of rational and integral points on varieties, using various analytic methods with the Brauer-Manin obstruction.

14G05 ; 14F22

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Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.
Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties ...

81Q50 ; 37N20 ; 35P20 ; 58J51 ; 58J50 ; 37D40

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Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.
Given a quantum Hamiltonian, I will explain how the dynamical properties of the underlying classical Hamiltonian affect the behaviour of quantum eigenstates in the semiclassical limit. I will mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. I will introduce tools from microlocal analysis and show how to use them in order to illustrate the classical-quantum correspondance and to compare properties ...

81Q50 ; 37N20 ; 35P20 ; 58J51 ; 58J50 ; 37D40

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The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles.
These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof. Some progress has been made on some of these, but lots remains to be done, and open problems will be mentioned.

After the lectures a few references regarding the spectrum of the magnetic Schrödinger operator were suggested to me.
See the bibiography below.

Thank Alix Deleporte, Frédéric Faure, Stéphane Nonnenmacher and others for discussions relative to the magnetic Weyl law.
The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured that these quasi-particles had fractional statistics, i.e. a behavior ...

81Sxx ; 81V35 ; 81V70

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We present a lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic eld H on each edge, and the vertex values of the Lagrange multiplier p (used to enforce the solenoidality of the magnetic induction B = µH). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called ”first kind N´ed´elec” elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions. Hints on a whole family of elements will also be given.
We present a lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic eld H on each edge, and the vertex values of the Lagrange multiplier p (used to ...

65N30 ; 65N12

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Recently there has been a surge in interest in cut, or unfitted, finite element methods. In this class of methods typically the computational mesh is independent of the geometry. Interfaces and boundaries are allowed to cut through the mesh in a very general fashion. Constraints on the boundaries such as boundary or transmission conditions are typically imposed weakly using Nitsche’s method. In this talk we will discuss how these ideas can be combined in a fruitful way with the idea of hybridization, where additional degrees of freedom are added on the interfaces to further improve the decoupling of the systems, allowing for static condensation of interior unknowns. In the first part of the talk we will discuss how hybridization can be combined with the classical cut finite element method, using standard H1 -conforming finite elements in each subdomain, leading to a robust method allowing for the integration of polytopal geometries, where the subdomains are independent of the underlying mesh. This leads to a framework where it is easy to integrate multiscale features such as strongly varying coefficients, or multidimensional coupling, as in flow in fractured domains. Some examples of such applications will be given. In the second part of the talk we will focus on the Hybridized High Order Method (HHO) and show how cut techniques can be introduced in this context. The HHO is a recently introduced nonconforming method that allows for arbitrary order discretization of diffusive problems on polytopal meshes. HHO methods have hybrid unknowns, made of polynomials in the mesh elements and on the faces, without any continuity requirement. They rely on high-order local reconstructions, which are used to build consistent Galerkin contributions and appropriate stabilization terms designed to preserve the high-order approximation properties of the local reconstructions. Here we will show how cut element techniques can be introduced as a tool for the handling of (possibly curved) interfaces or boundaries that are allowed to cut through the polytopal mesh. In this context the cut element method plays the role of a local interface model, where the associated degrees of freedom are eliminated in the static condensation step. Issues of robustness and accuracy will be discussed and illustrated by some numerical examples.
Recently there has been a surge in interest in cut, or unfitted, finite element methods. In this class of methods typically the computational mesh is independent of the geometry. Interfaces and boundaries are allowed to cut through the mesh in a very general fashion. Constraints on the boundaries such as boundary or transmission conditions are typically imposed weakly using Nitsche’s method. In this talk we will discuss how these ideas can be ...

65N30 ; 34A38

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The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. This is a joint work with Clément Cancès (Lille) and Stella Krell (Nice).
The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is ...

65M08 ; 65M12

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This talk will be devoted to the usage of new discretization schemes on polyhedral meshes in an industrial context. These discretizations called CDO [1, 2] (Compatible Discrete Operator) or Hybrid High Order [3,4] (HHO) schemes have been recently implemented in Code Saturne [5]. Code Saturne is an open-source code developed at EDF R&D aiming at simulating single-phase flows. First, the advantages of robust polyhedral discretizations will be recalled. Then, the underpinning principles of CDO schemes will be presented as well as some applications: diffusion equations, transport problems, groundwater flows or the discretization of the Stokes equations. High Performance Computing (HPC) aspects will be also discussed as it is an essential feature in an industrial context either to address complex and large computational domains or to get a quick answer. Some highlights on the main outlooks will be given to conclude.
This talk will be devoted to the usage of new discretization schemes on polyhedral meshes in an industrial context. These discretizations called CDO [1, 2] (Compatible Discrete Operator) or Hybrid High Order [3,4] (HHO) schemes have been recently implemented in Code Saturne [5]. Code Saturne is an open-source code developed at EDF R&D aiming at simulating single-phase flows. First, the advantages of robust polyhedral discretizations will be ...

65Nxx ; 65N50 ; 76S05

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This talk is a continuation of ‘Understanding the growth of Laplace eigenfunctions’. We explain the method of geodesic beams in detail and review the development of these techniques in the setting of defect measures. We then describe the tools and give example applications in concrete geometric settings.

58C40 ; 35P20

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