| 11h15 - 12h15 | Jimmy Lamboley
(Paris-6) |
About optimal
shapes for the Dirichlet-Laplace operator Abstract: We are interested in optimal estimates of the eigenvalues (or functions of eigenvalues) of the Laplace operator with Dirichlet boundary conditions, involving geometrical informations of the considered domains. We will briefly review some well-known results on this topic, and then focus on new existence and regularity results for problems involving the perimeter of the domain. We will conclude with some remarks, results, and open problems when domains are assumed to be convex. |
| Déjeuner | ||
| 14h - 15h | Nikhil Savale
(Cologne) |
A Gutzwiller
type trace formula for the magnetic Dirac operator Abstract: For manifolds including metric-contact manifolds with non-resonant Reeb flow, we prove a Gutzwiller type trace formula for the associated magnetic Dirac operator involving contributions from Reeb orbits on the base. The method combines the use of almost analytic continuations and local index theory. The construction of appropriate microlocal weight/trapping functions then allows extension of the formula to large time. As an application, we prove a semiclassical limit formula for the eta invariant of the Dirac operator. |
| 15h15 - 16h15 | Sébastien
Breteaux (Metz) |
Quantum Mean
Field Asymptotics and Multiscale Analysis Abstract: In this work, we study how multiscale analysis and quantum mean field asymptotics can be brought together. In particular we study when a sequence of one-particle density matrices has a limit with two components : one classical and one quantum. The introduction of “separating quantization for a family” provides a simple criterion to check when those two types of limit are well separated. We give examples of explicit computations of such limits, and how to check that the separating assumption is satisfied. This is joint work with Z.Ammari and F.Nier. |
| 11h15 - 12h15 | Thomas Letendre (ENS Lyon) |
Sous-variétés nodales aléatoires : caractéristique d’Euler et volume moyens Résumé: Dans cet exposé, on s’intéressera à une sous-variété aléatoire Z(L) dans une variété riemannienne M, obtenue comme lieu d’annulation d’une combinaison linéaire aléatoire de fonctions propres du laplacien associées à des valeurs propres inférieures à L. Je présenterai deux résultats qui donnent les asymptotiques du volume moyen et de la caractéristique d’Euler moyenne de Z(L) lorsque L tend vers l’infini. De façon étonnante, ces asymptotiques ne dépendent de M que par sa dimension et son volume. On verra également que Z(L) s’équidistribue dans M asymptotiquement et en moyenne. Dans le cas du volume, cela généralise des résultats de Bérard et Zelditch. Si le temps le permet, j’évoquerai des résultats analogues pour un modèle de sous-variétés algébriques réelles aléatoires. |
| Déjeuner | ||
| 14h - 15h | Alain Joye (Grenoble) |
Chirality induced Interface Currents in the Chalker Coddington Model Abstract: Chalker & Coddington provided in 1988 a simplified description of the quantum dynamics of electrons in a plane, submitted to an electric potential and a strong perpendicular magnetic field, in a model that now bears their names. The one time step electronic motion is given by a unitary operator on l2(Z2) constructed in terms of scattering matrices attached to the sites of Z2 that contain the main physical characteristics of the potential and magnetic field at these sites. The transport properties of the electrons are then encoded in the spectral properties of the unitary operator, which is our main concern. We consider the situation where the model presents asymptotically pure anti-clockwise rotation on the left and clockwise rotation on the right and we investigate the presence of induced currents at the interface between these two different localised phases. The existence of interface currents is shown by proving that the absolutely continuous spectrum of the Chalker Coddington unitary operator covers the whole unit circle. The result is independent of the details of the model within the interface and possesses some topological features. This is joint work with J.Asch and O.Bourget |
| 15h15 - 16h15 | Yan Pautrat (Orsay) |
Landauer’s Principle in Repeated Interaction Systems Abstract: We study Landauer’s principle for repeated interaction systems consisting of a reference quantum system S in contact with an environment E made of a chain of independent quantum probes. The system S interacts with each probe sequentially, and the Landauer principle relates the energy variation of E and the decrease of entropy of S. We consider the adiabatic regime where the chain contains T probes and displays variations of order 1/T between the successive probes. We consider refinements of the Landauer bound at the level of the full statistics associated with a two-time measurement protocol. At the technical level, our results rely on a non-unitary adiabatic theorem and and an analysis of the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps. This is joint work with Eric Hanson, Alain Joye and Renaud Raquépas. |
| 11h15 - 12h15 | Vincent Bruneau (Bordeaux) |
Propriétés spectrales d'hamiltoniens magnétiques sur le demi-plan Résumé: Nous considérons l'opérateur de Schrödinger avec champ magnétique constant sur le demi-plan (avec condition de Dirichlet ou de Neumann). D'un point de vue classique la présence du bord génère un phénomène de propagation qui, au niveau quantique, engendre du spectre continu (alors que dans le plan entier le spectre est ponctuel). Nous décrirons le comportement du spectre de cet opérateur perturbé par un potentiel électrique décroissant à l'infini (via la fonction de comptage des valeurs propres et/ou la fonction de décalage spectrale). Nous verrons notamment comment pour des perturbations polynomialement décroissantes (de signe fixé) nous estimons les singularités près des niveaux de Landau alors que les perturbations à support compact laissent des questions ouvertes. Travail en collaboration avec Pablo Miranda (Santiago, Chili) |
| Déjeuner | ||
| 14h - 15h | Laurent Michel (Nice) |
Autour des petites valeurs propres du Laplacien de Witten Résumé: L’analyse des petites valeurs propres du Laplacien de Witten intervient de manière cruciale dans la description de dynamiques métastables. Dans cet exposé, on rappelera l’approche de Helffer-Klein-Nier, Hérau-Hitrik-Sjöstrand pour traiter ce problème, puis on donnera quelques généralisations à des situations dégénérées. |
| 15h15 - 16h15 | Stefano Olla (Dauphine) |
Hydrodynamic limits in chains of oscillators and Wigner distribution Abstract: Wigner distributions have been recently used as a tool for studying the separation of scales (macroscopic from microscopic) in order to obtain hydrodynamic limits in chains of oscillators in non-equilibrium. I will review the basic ideas and some results. |
| 11h15 - 12h15 | Adrien Hardy (Lille) |
Matrices aléatoires et électrostatique Résumé: Un résultat surprenant en théorie des matrices aléatoires est le suivant : la distribution des valeurs propres de matrices tirées suivant la mesure gaussienne sur l'espace des matrices Hermitienne de taille donnée est celle d'un gaz de Coulomb à température fixée. En exploitant cette correspondance on va pouvoir obtenir une description fine des espacements des valeurs propres quand la taille des matrices aléatoires tend vers l'infini. Plus précisément, on va obtenir une caractérisation de type équation DLR du processus limite des beta-ensembles qui fait intervenir l'énergie renormalisée introduite par Sandier & Serfaty. Ce premier résultat est un travail en cours avec D. Dereudre, T. Leblé et M. Maïda. Si le temps le permet, je discuterais un peu plus de la correspondance entre matrices aléatoires et gaz de Coulomb en étudiant la distribution jointe des valeurs propres de matrices tirées sur sous-variétés où les multiplicités sont prescrites, ce qui constitue un résultat indépendant. |
| Déjeuner | ||
| 14h - 15h | Diomba Sambou (Santiago du Chili) |
Sur une classe d'opérateurs de Pauli non auto-adjoints: étude des valeurs propres complexes Résumé: Récemment, l'analyse spectrale des modèles non auto-adjoints des opérateurs issus de la physique mathématique tels que Schrödinger et Dirac a suscité beaucoup d'intérêt. Cependant, la plupart des études concernent des modèles non magnétiques. Dans cet exposé, nous allons considérer une classe d'opérateurs de Pauli avec champs magnétiques non constants perturbés par des potentiels à valeur matricielle non auto-adjoints, et étudierons le comportement des valeurs propres complexes engendrées par ces perturbations près du bas du spectre essentiel. Plus précisément, nous allons présenter un critère simple permettant de créer des valeurs propres complexes puis de les localiser. |
| 15h15 - 16h15 | Wei-Xi Li (Wuhan) |
Compactness of the resolvent for the Witten Laplacian Abstract: In this talk we consider the Witten Laplacian on 0-forms and give sufficient conditions under which this operator admits a compact resolvent. These conditions are imposed on the potential itself, involving the control of high order derivatives by lower ones, as well as the control of the positive eigenvalues of the Hessian matrix. This compactness criterion for resolvent is inspired by the one for the Fokker-Planck operator. Our method relies on the nilpotent group techniques developed by Helffer-Nourrigat [Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, 1985]. |
| 11h15 - 12h15 | Mireille
Capitaine (Toulouse) |
Outliers de
modèles matriciels Hermitiens polynomiaux Résumé: Le but de cet exposé est de dégager une méthodologie générale pour localiser, en grande dimension, les éventuelles valeurs propres s’éloignant du reste du spectre (“outliers”) de polynômes auto-adjoints non commutatifs en des matrices aléatoires asymptotiquement libres. |
| Déjeuner | ||
| 14h - 15h | Peng Zhou
(IHES) |
Zero set of
eigenfunction for harmonic oscillators Abstract: Eigenfunctions of the Laplacian on a smooth Riemannian manifold are a classical object of study. One of Yau’s conjecture predicts that the `size’ of the nodal set of the N-th eigenfunction grows as N1/2 . In this talk, we present the analogous study for the Schrödinger equation, in the simplest example of the harmonic oscillator on Rn. For a fixed total energy E, the configuration space is partitioned into an `allowed region’ and a `forbidden region’, and roughly speaking, the eigenfunction oscillates in the allowed region and exponentially decays in the forbidden region. In the semiclassical limit, the nodal sets exhibit different behaviors in the two regions and across the interface. This is joint work with Steve Zelditch and Boris Hanin. (arXiv 1310.4532, arXiv 1602.06848) |
| 15h15 - 16h15 | Vladimir
Lotoreichik (Prague) |
A Faber-Krahn
inequality for the Robin Laplacian on exterior domains Abstract: We will discuss several generalizations of the Faber-Krahn inequality for the lowest eigenvalue of the Robin Laplacian with a negative boundary parameter on the exterior of a bounded, simply connected, smooth domain Ω⊂Rd. Some further generalizations for disconnected domains Ω will also be discussed. Our main motivation is to go beyond more traditional bounded domains in eigenvalue optimization. The ultimate goal is always to prove that the exterior of a ball maximizes the underlying lowest eigenvalue under a suitable constraint being imposed. In two dimensions, we constrain either the perimeter of Ω or its area. In higher dimensions, constraining the area of ∂Ω leads to an ill-posed optimization problem, as the large coupling asymptotics in Pankrashkin-Popoff-16 shows. Instead, for d≥3 we constrain the ratio between a Willmore-type energy of ∂Ω and its area, assuming, additionally, that Ω is convex. In the proofs we represent the lowest eigenvalue via the min-max principle on the level of quadratic forms, expressed in suitably chosen coordinates on Rd\Ω. We make use either of standard parallel coordinates, or of their modification worked out in Payne-Weinberger-61 and further refined in Savo-01. The trickiest part of the proof is to find a proper test function. These results are obtained in collaboration with David Krejcirík. |
| 11h15 - 12h15 | Rémy Rodiac (Louvain-la-Neuve) |
Condensats de Bose-Einstein à deux composantes avec couplage spin-orbite en 1D Résumé: On étudie la fonctionnelle de Gross-Pitaevski modélisant le comportement d’un condensat de Bose-Einstein à deux composantes, avec un terme de couplage spin-orbit en 1D. Grâce à un changement de fonctions on se ramène, dans certains régimes, à un problème de transition de phase de type Modica-Mortola (ou Allen-Cahn) avec une condition de Neumann non-homogène sur le bord, qui dépend du paramètre de couplage spin-orbit. Selon l’intensité de ce paramètre, deux régimes différents apparaissent. Dans le premier régime on a une seule transition incomplète sur le bord du domaine. La composante 1 occupe presque tout le domaine et la composante 2 l’entoure sur le bord du domaine. Dans le second régime on a de multiples transitions agencées de manière périodique, correspondant à des "bandes" alternées de composantes 1 et 2. Ceci est un travail en collaboration avec A.Aftalion (EHESS). |
| Déjeuner | ||
| 14h - 15h | Thomas Ourmières-Bonafos (Orsay) |
Dirac operators and delta interactions Abstract: In this talk, we will discuss different aspects of the Dirac operator in dimension three, coupled with a singular potential supported on a surface. After motivating the study of such objects, we will briefly be interested in the problem of self-adjointness for singular electrostatic or Lorentz-scalar potentials. For this last class of potentials, we will study the structure of the spectrum of such an operator and in particular, we will show that for an "attractive" potential, when the mass of the particle goes to infinity, the behavior of the eigenvalues is given by an effective operator on the surface. We will see that this effective operator is actually a Schrödinger operator with both a Yang-Mills potential and an electric potential, each one being of geometric nature. These are joint works with Markus Holzmann, Konstantin Pankrashkin and Luis Vega (arXiv :1612.07058, 1711.00746). |
| 15h15 - 16h15 | Laure Dumaz (Dauphine) |
Localisation de l’hamiltonien d’Anderson en dimension 1 Résumé: Dans cet exposé, nous étudierons la localisation d’un opérateur aléatoire de Schrödinger continu en dimension 1, appelé opérateur de Hill ou hamiltonien d’Anderson, où le potentiel est un bruit blanc sur le segment [0,L] avec conditions aux bords de Dirichlet ou de Neumann. Dans la limite où L tend vers l’infini, nous montrons la convergence des plus petites valeurs propres vers un processus de Poisson, ainsi que la localisation des vecteurs propres dans un sens précis (la forme du vecteur propre autour de son maximum est déterministe et ne dépend pas de la valeur propre). Travail en commun avec Cyril Labbé. |
| 11h15 - 12h15 | Alexander Watson
(Duke) |
Wave-packet
dynamics in locally periodic media Abstract: We study the dynamics of wave-packet solutions of Schrödinger’s equation and Maxwell’s equations in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. We derive systematically and rigorously the ‘anomalous velocity’ of wave-packets due to the Berry curvature of the Bloch band, and the dynamics of a wave-packet incident on a Bloch band degeneracy in one spatial dimension. Joint work with Michael Weinstein and Jianfeng Lu. |
| Déjeuner | ||
| 14h - 15h | Ivan Bardet
(IHES) |
Estimating the
decoherence time using quantum functional inequalities Abstract: Environment Induced Decoherence is a physical concept which provides a dynamical explanation to the disappearance of quantum phenomenon in the real world. Intuitively, it states that a quantum system is never perfectly isolated, so that quantum correlations disappear due to the action of the environment on the system. Focusing on finite dimensional quantum systems undergoing Markovian evolutions, we will discuss how we can adapt the theory of functional inequalities, namely hypercontractivity and log-Sobolev inequalities, to estimate the speed of decoherence. This study relies on the analysis of some new non-commutative norms called amalgamated norms. Surprisingly, we shall show several atypical behaviors compared to the known classical case. This is a joint work with Cambyse Rouzé (Cambridge). |
| 15h15 - 16h15 | Antti Knowles
(Genève) |
Local law and
eigenvector delocalization for supercritical Erdős-Rényi graphs Abstract: We consider the adjacency matrix of the Erdős-Renyi graph G(N;p) in the supercritical regime pN>C log N for some universal constant C. We show that the eigenvalue density is, with high probability, well approximated by the semicircle law on all spectral scales larger than the typical eigenvalue spacing. We also show that all eigenvectors are completely delocalized with high probability. Both results are optimal in the sense that they are known to be false for pN<log N. A key ingredient of the proof is a new family of large deviation estimates for multilinear forms of sparse vectors. Joint work with Yukun He and Matteo Marcozzi. |
| 11h15 - 12h15 | Rémy Rhodes (Paris-Est) |
Liouville quantum theory and the DOZZ formula Abstract: This talk will review some recent advances in the study of the Liouville quantum theory. This theory was introduced in physics by Polyakov in 1981 in the context of string theory. This is a conformal field theory based on a path integral à la Feynman and it can be seen as the natural extension of the theory of Riemann surfaces to the probabilistic framework. Solving this theory, namely computing the correlation functions, has been a great challenge in physics. In this direction and in the 90s, Dorn-Otto and the Zamolodchikov brothers conjectured that the 3-point correlation function satisfies a mysterious formula based on number theory, called the DOZZ formula. The main purpose of this talk is to explain this story and to present the construction of the path integral for the Liouville quantum theory. Then I will explain the DOZZ formula. Time depending, I will finally discuss how computing all other correlation functions reduces to DOZZ and to the spectral problem associated to the Hamiltonian of the Liouville theory. Based on works with F. David, A. Kupiainen and V. Vargas. |
| Déjeuner | ||
| 14h - 15h | Alexander Adam (Jussieu) |
Horocycle averages on closed manifolds Abstract: A well-known example of a contact Anosov flow is the geodesic flow on the unit tangent bundle SX of a closed Riemannian manifold X with variable negative sectional curvature. SX is foliated by the stable manifolds, on which one defines the horocycle flow. This flow is uniquely ergodic. What is the speed of convergence to the Birkhoff averages ? In 2003 Flaminio and Forni investigated this question in the case of constant negative curvature. They found that the speed of convergence is controlled by the existence of distributions which are invariant w.r.to the horocycle flow. These distributions are also eigendistributions of the geodesic flow. The speed of convergence is then determined by a (fractional) power spectrum, with exponents associated to those eigenvalues. I will report on this phenomenon for contact Anosov flows of sufficient regularity. |
| 15h15 - 16h15 | Maurizia Rossi (Paris 5) |
Asymptotic distribution of nodal intersections for arithmetic random waves Abstract: We focus on the nodal intersection number of random Gaussian toral Laplace eigenfunctions ("arithmetic random waves") against a fixed smooth reference curve. The expected intersection number is proportional to the the square root of the eigenvalue times the length of curve, independently of its geometry. The asymptotic behaviour of the variance was addressed by Rudnick-Wigman; they found a precise asymptotic law for "generic" curves with nowhere vanishing curvature, depending on both its geometry and the angular distribution of lattice points lying on the circles corresponding to the Laplace eigenvalues. They also discovered that there exist peculiar "static" curves, with variances of smaller order of magnitude, though did not describe the asymptotic behaviourn in this case. In this talk we investigate the finer aspects of the limit distribution of the nodal intersections number. For generic curves we prove a Central Limit Theorem (for most of the energies). For the aforementioned static curves, we establish a non-Gaussian limit theorem for the distribution of the nodal intersections, and obtain the asymptotic behaviour of their fluctuations, under a well-separatedness assumption on the corresponding lattice points, satisfied by most of the eigenvalues. This is a joint work with Igor Wigman. |