Titres et résumés

Mini-Course

Mathieu Lewin : Lieb-Thirring inequalities: old and new.

Talks

Jack Borthwick: Scattering of Dirac fields near an extremal Kerr-de Sitter black hole.

A black hole is said to be extremal when two or more of its horizons coincide. After making sense of this statement in the Kerr-de Sitter family, which constitutes an analytical model for a rotating black hole in a universe with positive cosmological constant, we shall discuss the scattering of massive Dirac fields on an appropriate region of the spacetime. Our focus will be on the difficulties encountered due to the cumulated effects of the rotation, the cosmological constant and the « double » black hole horizon, as well as their resolution, in the construction of an analytical scattering theory.

Hakim Boumaza: Algebraic approach of the Anderson localization for quasi-one-dimensional models.

In this talk I will present several ergodic families of random operators in dimension one for which we want to prove Anderson localization. These operators are either discrete and acting on l^2(Z), or continuous and acting on L^2(R). First we discuss how to reduce the question of Anderson localization in dimension 1 to the study of an algebraic object, the Fürstenberg group. For this purpose, we present typical objects of the dimension 1: the transfer matrices, the Lyapunov exponents and a bit of Kotani theory. Then we present the tools from Lie group theory which allows to prove the required properties of the Fürstenberg group in different settings : discrete or continuous, scalar or matrix valued, Schrödinger type with Anderson potential, unitary or Dirac type operators.

Benoît Douçot: Topological electrostatics.

Two-dimensional electron gases under a strong magnetic field have tremendously expanded our understanding of many-body physics, with the discovery of integer and fractional quantum Hall effects, together with chiral edge states, fractional excitations, anyons. Another striking effect is the strong coupling between charge and spin/valley degrees of freedom, which takes place near integer filling M of the magnetic Landau levels. More precisely, because of the large energy gap associated to cyclotron motion, any slow spatial variation of the spin background induces a variation of the electronic density proportional to the topological density of the spin background. Minimizing Coulomb energy leads to an exotic class of two-dimensional crystals, which exhibit a periodic non-collinear spin texture called a Skyrmion lattice.

Lisette Jager: Weyl quantization on Wiener space : some positivity results.

A pseudodifferential calculus or quantization associates, with a function called "symbol'', a linear operator. This generalizes the differential operators.  The most widely used pseudodifferential calculus is the Weyl calculus, which associates, with a symbol f defined on R^{2n}, an operator acting on functions defined on R^n.
A Weyl calculus exists in an infinite dimensional setting. The space R^n is replaced by a couple consisting of an infinite dimensional Hilbert space H (to use scalar products or symplectic technics) and a probability space B, canonically associated with H and called a Wiener extension of H.  In this frame there exist symbol classes, a Calderòn Vaillancourt theorem has been proved, we have composition theorems (in the shape of asymptotic expansions in powers of a small parameter h).
In this talk we expose some results about positivity and non positivity. Some symbols with radial properties yield a positive operator. As in the finite dimensional case,  positive symbols do not necessarily give positive operators, but
a Gårding inequality is available. The Flandrin conjecture is no more valid in this case than in the finite dimensional case.

Gabriel Lando: The role of tunneling in the ionization of atoms by intense and ultrashort laser pulses.

The dimensionality curse forces most real-world applications of quantum theory to employ approximations in order to render numerical computations feasible. Among the crudest lies the Truncated Wigner Approximation (TWA), which evolves purely classically. In this talk I will make use of the TWA's classicality in the reverse direction: Instead of simplifying the numerical treatment of a complex system, I will use the TWA to bring forward the complexity of a seemingly trivial one. More specifically, I will consider the case of a simple 1-dimensional hydrogen atom interacting with an ultrashort and intense laser pulse. By comparing full quantum and TWA calculations, I will present numerical evidence to strongly doubt the "quantumness" of the ionization mechanism for this particular system. Since "tunneling ionization" is the first step in a plethora of models in strong-field physics, this simple analysis should help resolve a couple of inconsistencies regarding some experiments routinely performed during the last 10 years.

Thibaud Lemoine: The master field on the torus.

In this talk, I will define the two-dimensional Euclidean Yang-Mills measure in a probabilistic framework. It depends in particular on a compact group, called the structure group, and I will describe the limit of this measure when the structure group is taken as a classical group of rank N, when N goes to infinity. This limit, called master field, was conjectured to exist for any surface by Singer in 95, but the first rigorous construction was done on the plane by Lévy in the early 2010s. I will briefly explain how A. Dahlqvist and I extended his construction to the torus.

Cyril Letrouit: Propagation des singularités pour les ondes sous-elliptiques.

On considère l'équation des ondes où le Laplacien est remplacé par un sous-Laplacien, c'est-à-dire une "somme de carrés de Hörmander" hypoelliptique, et le but est de décrire la propagation des singularités dans ces équations d'ondes. Le phénomène principal que l'on décrit est la propagation de singularités le long de courbes dites "anormales" à n'importe quelle vitesse entre 0 et 1. Ce résultat général étend une idée due à R. Melrose, et on l'illustrera sur un exemple, le "cas Martinet", suivant un travail avec Y. Colin de Verdière. Nos résultats illustrent une correspondance classique/quantique entre géométrie sous-Riemannienne du côté classique, et opérateurs hypoelliptiques du côté quantique. Cette correspondance est aussi utile pour interpréter des résultats de contrôle d'ondes hypoelliptiques et de théorie spectrale des opérateurs hypoelliptiques.

Faedi Loulidi: Measurement incompatibility vs. Bell non-locality.

Incompatibility of quantum measurements and Bell inequality violations are two key features of quantum theory. It is known that the violation of the CHSH inequality is equivalent to the incompatibility of two binary quantum measurements. We explore when this phenomenon generalizes to other Bell inequalities and multiple binary measurements. We introduce tensor norms which fully characterize measurement incompatibility and violations of a general Bell inequality. By comparing these norms, we characterize the scenarios where the two manifestations of quantumness are equivalent.

Laura Monk: Geometry and spectrum of random hyperbolic surfaces.

The aim of this talk is to present recent progress in the spectral theory of hyperbolic surfaces, which was achieved by bringing a new probabilistic viewpoint to this classic field. Studying random hyperbolic surfaces allows to prove statements true for most surfaces rather than every single one of them, and hence to exclude some known pathological examples. I will present estimates on the density of eigenvalues and the spectral gap of typical surfaces in the Weil--Petersson random setting. These results are proven using the Selberg trace formula, together with geometric information on closed geodesics on random surfaces.

Pierre Pujol: From topological textures in magnetic systems to topological insulators in electronic systems.

Magnetic topological textures, such as Skyrmions, are a very active subject of research nowadays. When these magnetic degrees of freedom are coupled to electronic degrees of freedom, one can obtain the emergence of a very interesting quantum phenomena, another very active subject of research: Chern insulators. In this presentation we will illustrate these phenomena through various examples that relate them.

Nicolas Raymond: Semiclassical analysis of the Neumann Laplacian with constant magnetic field in three dimensions.

This talk deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of the domain in the direction of the field. In particular, we prove that they are simple under generic assumptions and we are led to revisit the two-term expansion of the lowest eigenvalue obtained by Helffer and Morame in 2004.
Joint (ongoing) work with Frédéric Hérau.

Simona Rota-Nodari: The Dirac-Klein-Gordon system in the strong coupling limit.

In this talk, I will present a recent result on the behaviour of the solutions of a Dirac-Klein-Gordon system in the limit of strong coupling and large masses of the Klein-Gordon fields. I will prove convergence of the solutions to the system to those of a cubic non-linear Dirac equation. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. This is a joint work with J. Lampart, L. Le Treust and J. Sabin.