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# Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 200

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## Automorphisms of curve and pants complexes in profinite content Funar, Louis | CIRM H

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Pants complexes of large surfaces were proved to be vigid by Margalit. We will consider convergence completions of curve and pants complexes and show that some weak four of rigidity holds for the latter. Some key tools come from the geometry of Deligne Mumford compactification of moduli spaces of curves with level structures.

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## Local systems and Satake duality Fock, Vladimir | CIRM H

Multi angle

Satake duality is an isomorphism between the algebra of characters of a simple Lie group G and the Hecke algebra of the affine Lie group corresponding to the Langlands dual to G. We will suggest a (conjectural) generalisation of this isomorphism replacing the characters by functions on local systems on surfaces and Hecke algebra by the algebra on cells on local systems with values in an affine Lie group.

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## Rigidity in rank one: dynamics and geometry - lecture 3 Gogolev, Andrey | CIRM H

Multi angle

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz. A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e ...

37D20

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## Some new dynamical applications of smooth parametrizations for C∞ systems - lecture 1 Burguet, David | CIRM H

Multi angle

Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers. Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...

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## Some new dynamical applications of smooth parametrizations for C∞ systems - lecture 2 Burguet, David | CIRM H

Multi angle

Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers. Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...

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## Some new dynamical applications of smooth parametrizations for C∞ systems - lecture 3 Burguet, David | CIRM H

Multi angle

Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers. Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...

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## Ergodic optimization - lecture 1 Contreras, Gonzalo | CIRM H

Multi angle

We will show the proof that for generic Lipschitz functions on an expanding map there is a unique maximizing measure, and it is supported on a periodic orbit.

37D35

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## Ergodic optimization - lecture 2 Contreras, Gonzalo | CIRM H

Multi angle

We will show the proof that for generic Lipschitz functions on an expanding map there is a unique maximizing measure, and it is supported on a periodic orbit.

37D35

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## Ergodic optimization - lecture 3 Contreras, Gonzalo | CIRM H

Multi angle

We will show the proof that for generic Lipschitz functions on an expanding map there is a unique maximizing measure, and it is supported on a periodic orbit.

37D35

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## Semiclassical behaviour of quantum eigenstates - lecture 1 Rivière, Gabriel | CIRM H

Multi angle

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 ...

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## Willmore stability and conformal rigidity of minimal surfaces in $\mathbb{S}^{n}$ Kusner, Rob | CIRM H

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Partial Differential Equations;Geometry

A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the equilateral minimal 2-torus in $\mathbb{S}^{5}$ have this spectral gap, and each is embedded by first eigenfunctions, so both are "persistently” $W$-stable. On the other hand, we discovered the equilateral torus has nontrivial third variation (with vanishing second variation) of $W$, and thus is not a $W$-minimizer (though it is the $W$-minimizer if we fix the conformal type!). This is evidence the Willmore Conjecture holds in every codimension. Another result concerns higher genus minimal surfaces (such as those constructed by Lawson and those by Karcher-Pinkall-Sterling) in $\mathbb{S}^{3}$ which Choe-Soret showed are embedded by first eigenfunctions: we show their first eigenspaces are always 4-dimensional, and that this implies each is (up to Möbius transformations of $\mathbb{S}^{n}$) the unique $W$-minimizer in its conformal class. (Some analogous results hold for free boundary minimal surfaces in the unit ball $\mathbb{B}^{n}$....). This is joint work with Peng Wang. A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the ...

53C42

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## Mathematical topics around fractional quantization - lecture 2 Rougerie, Nicolas | CIRM H

Multi angle

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.

Thanks to 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 ...

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## Virtual element approximation of magnetostatic Marini, Donatella | CIRM H

Multi angle

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 ...

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## Combining cut element methods and hybridization Burman, Erik | CIRM H

Multi angle

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 ...

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## Nonlinear free energy diminishing schemes for convection-diffusion equations: convergence and long time behaviour Chainais-Hillairet, Claire | CIRM H

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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 ...

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## Polyhedral discretizations for industrial applications Bonelle, Jérôme | CIRM H

Multi angle

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 ...

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## Semiclassical behaviour of quantum eigenstates - lecture 2 Rivière, Gabriel | CIRM H

Multi angle

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 ...

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## Minimal Maslov number of $R$-spaces canonically embedded in Einstein-Kähler $C$-spaces Ohnita, Yoshihiro | CIRM H

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Geometry

An $R$-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each $R$-space has the canonical embedding into a Kähler $C$-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is an invariant under Hamiltonian isotopies and very fundamental to the study of the Floer homology for intersections of Lagrangian submanifolds. In this talk we provide a Lie theoretic formula for the minimal Maslov number of $R$-spaces canonically embedded in Einstein-Kähler $C$-spaces and discuss several examples of the calculation by the formula. An $R$-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each $R$-space has the canonical embedding into a Kähler $C$-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is an invariant under Hamiltonian isotopies and very fundamental to the ...

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## Singularities of Teichmüller harmonic map flow Rupflin, Melanie | CIRM H

Multi angle

Partial Differential Equations;Geometry

We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.

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## (In)efficiency in mean field games Cardaliaguet, Pierre | CIRM H

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Control Theory and Optimization

Mean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In this joint work with Catherine Rainer (U. Brest), we study the efficiency of Nash MFG equilibria: Namely, we compare the social cost of a MFG equilibrium with the minimal cost a global planner can achieve. We find a structure condition on the game under which there exists efficient MFG equilibria and, in case this condition is not fulfilled, quantify how inefficient MFG equilibria are. Mean field games (MFG) are dynamic games with infinitely many infinitesimal agents. In this joint work with Catherine Rainer (U. Brest), we study the efficiency of Nash MFG equilibria: Namely, we compare the social cost of a MFG equilibrium with the minimal cost a global planner can achieve. We find a structure condition on the game under which there exists efficient MFG equilibria and, in case this condition is not fulfilled, quantify how ...

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