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Documents  Wunsch, Jared | enregistrements trouvés : 12

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Research talks

In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Using the description of concentration, we obtain quantitative improvements on the known bounds in a wide variety of settings. In this talk we will discuss a new geodesic beam approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of $L^{2}$ mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along ...

35P20 ; 58J50 ; 53C22 ; 53C40 ; 53C21

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Outreach;Mathematics Education and Popularization of Mathematics

Directrice de recherche CNRS au DMA, UMR 8553 (équipe Analyse)
Directrice Adjoint Scientifique à l'Insmi, en charge de la politique de sites (Institut des Sciences Mathématiques et de leurs Interactions - CNRS)
Adjointe Déléguée Scientifique Référente au CNRS

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Research talks;Partial Differential Equations;Geometry

37D40 ; 58J50

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Research talks

In the 80’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (”Ruelle resonances”). For a geodesic flow on a strictly negative curvature Riemannian manifold, following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no added quantization procedure. We will discuss consequences for the zeros of dynamical zeta functions. This shows that the problematic of classical chaos and quantum chaos are closely related. Joint work with Masato Tsujii. In the 80’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (”Ruelle resonances”). For a geodesic flow on a strictly negative curvature Riemannian manifold, following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no ...

37D20 ; 37D35 ; 81Q50 ; 81S10

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Research talks

We will describe the construction of complete non-compact Ricci-flat manifolds of dimension 7 and 8 with holonomy $G_{2}$ and Spin(7) respectively. The examples we consider all have non-maximal volume growth and an asymptotic geometry, so-called ALC geometry, that generalises to higher dimension the asymptotic geometry of 4-dimensional ALF hyperkähler metrics. The interest in these metrics is motivated by the study of codimension 1 collapse of compact manifolds with exceptional holonomy. The constructions we will describe are based on the study of adiabatic limits of ALC metrics on principal Seifert circle fibrations over asymptotically conical orbifolds, cohomogeneity one techniques and the desingularisation of ALC spaces with isolated conical singularities. The talk is partially based on joint work with Mark Haskins and Johannes Nordstrm. We will describe the construction of complete non-compact Ricci-flat manifolds of dimension 7 and 8 with holonomy $G_{2}$ and Spin(7) respectively. The examples we consider all have non-maximal volume growth and an asymptotic geometry, so-called ALC geometry, that generalises to higher dimension the asymptotic geometry of 4-dimensional ALF hyperkähler metrics. The interest in these metrics is motivated by the study of codimension 1 collapse of ...

53C10 ; 53C25 ; 53C29 ; 53C80

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Research talks

I will describe joint work with Dietrich Häfner and Andràs Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. Our proof uses a detailed description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy. I will describe joint work with Dietrich Häfner and Andràs Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. Our ...

35B35 ; 35C20 ; 83C05

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Research talks

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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Research talks;Analysis and its Applications;Partial Differential Equations

In this somewhat speculative talk I will briefly describe recent results with Xuwen Zhu on the boundary behaviour of the Weil-Petersson metric (on the moduli space of Riemann surfaces) and ongoing work with Jesse Gell-Redman on the associated Laplacian. I will then describe what I think happens for the wave equation in this context and what needs to be done to prove it.

30F60 ; 32G15 ; 35L05

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Research talks;Partial Differential Equations;Geometry;Mathematical Physics

Polyakov’s formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple local quantities. Such a formula is well known in the case of closed surfaces (Osgood, Philips, & Sarnak 1988) and surfaces with smooth boundary (Alvarez 1983). Due to the abstract nature of the definition of the zeta-regularized determinant of the Laplacian, it is typically impossible to compute an explicit formula. Nonetheless, Kokotov (genus one Kokotov & Klochko 2007, arbitrary genus Kokotov 2013) demonstrated such a formula for polyhedral surfaces ! I will discuss joint work with Clara Aldana concerning the zeta regularized determinant of the Laplacian on Euclidean domains with corners. We determine a Polyakov formula which expresses the dependence of the determinant on the opening angle at a corner. Our ultimate goal is to determine an explicit formula, in the spirit of Kokotov’s results, for the determinant on polygonal domains. Polyakov’s formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple local quantities. Such a formula is well known in the case of closed surfaces (Osgood, Philips, & Sarnak 1988) and surfaces with smooth boundary (Alvarez 1983). Due to the abstract nature of the definition of the zeta-regularized determinant of the Laplacian, it is typically impossible to compute an ...

35K08 ; 58C40 ; 58J52

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Research talks;Partial Differential Equations;Mathematical Physics

We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances to the chord lengths and reflectivity coefficients for the ray dynamics and hence give a quantum version of the Sabine law from acoustics. We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characterization of the resonance free regions in terms of dynamical quantities. In particular, we relate the imaginary part of resonances to the chord lengths and reflectivity ...

35P25 ; 35B34 ; 35Q40

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Research talks;Analysis and its Applications;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Geometry

In joint work with Luc Hillairet, we show that the Laplacian associated with the generic finite area triangle in hyperbolic plane with one vertex of angle zero has no positive Neumann eigenvalues. This is the first evidence for the Phillips-Sarnak philosophy that does not depend on a multiplicity hypothesis. The proof is based an a method that we call asymptotic separation of variables.

58J50 ; 35P05 ; 11F72

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- vi; 135 p.
ISBN 978-2-85629-367-6

Astérisque , 0351

Localisation : Périodique 1er étage

équation d'onde # coin # ensemble de fronts d'onde # diffraction

35A18 ; 35A21 ; 35B65 ; 35L05 ; 35L67 ; 58J47

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