Apres un bref rappel des lois infiniment-divisibles, j’introduirai de nouvelles lois de probabilites construites a partir d’etats coherents qui sont des fonctions propres de laplaciens magnetiques a champ magnetique constant dans le plan complexe et dans le disque de Poincare. En dehors de l’etat fondammental, ces lois ne sont pas ID mais plutot quasi-ID et je donnerai l’analogue de leurs decomposition de Levy-Kintchine. J’expliquerai aussi le lien entre ces laplaciens et les sous-laplaciens provenant de la fibration de Heisenberg et d’Anti-de sitter, et dans le cas du disque le lien avec le laplacien de Maass. Si le temps le permets, je parlerai des processus determinantaux (fermioniques) associes aux projecteurs orthogonaux sur les classes de fonctions poly-analytiques.

Dans cet exposé, je discuterai d’une caractérisation variationnelle non-linéaire pour la valeur propre principale de l’opérateur de Dirac avec conditions au bord dites de masse infinie. Cette caractérisation variationnelle permet d’obtenir des résultats numériques et théoriques. En particulier, j’expliquerai comment nous l’utilisons pour comparer la valeur propre principale de l’opérateur dans de tels domaines avec celle du disque unité.

Il s’agit d’un travail en collaboration avec P.R.S. Antunes, R.D. Benguria et V. Lotoreichik.

Our best current model for describing gravitational waves, as for example measured in LIGO, is a family of solution to vacuum Einstein equations called asymptotically-flat space-times. Observers are then taken to be situated "at null-infinity" and, since the seminal work of Bondi’s group and Sachs, gravitational waves are related to the presence of a certain tensor, the "asymptotic shear", appearing in an asymptotic expansion away from null-infinity. Intuitively the presence of gravitational waves should correspond to the presence of some extra geometrical data at null-infinity but such a geometrisation however proved to be quite elusive. This is essentially because the geometry of null-infinity is of conformal nature which is notoriously tedious to work with. I will review these well-known facts and show how modern methods in conformal geometry (namely tractor calculus) can be adapted to the degenerate conformal geometry of null-infinity to encode the presence of gravitational waves in a completely geometrical way : the "asymptotic shear" is proved to be in 1-1 correspondence with choices of tractor connection, gravitational radiation is invariantly described by the tractor curvature and "soft-modes" of the vacuum correspond to the degeneracy of flat tractor connections.

Abstract in pdf on this link :

https://www.dropbox.com/s/piqvndmlb...

Many natural systems can be maintained in a stationary state through the exchange of matter, energy or information with their surroundings. These various currents break time-reversal invariance, generating a continuous increase of entropy in the universe. Such systems are out of equilibrium and can not be described by the Laws of Thermodynamics, or by using the classical principles of statistical physics, `a la Gibbs-Boltzmann. One may even fear that seeking an under- lying microscopic theory is a vain pursuit : equilibria are all alike, whereas every system can be far from equilibrium in its own way.
In the last two decades, however, important advances in our understanding of non-equilibrium processes have been achieved. Concepts of rare events, large deviations, fluctuations relations and macroscopic fluctuations provide a unified framework with the emergence of some universal features. The objective of this colloquium is to review these new ideas in non-equilibrium statistical physics and to illustrate them by examples inspired from soft-condensed matter.

We consider a slowly varying time dependent two-level system coupled to a Bose field in such a way that the instantaneous coupling is energy conserving and characterised by a regular radial form factor. We compute the transition probability from one eigenstate of the two-level system times the field vacuum to the other eigenstate at some later time as a function of the adiabatic parameter and coupling strength, and analyse the deviations from the adiabatic transition probability obtained in absence of field.
This is joint work with Marco Merkli and Dominique Spehner.