Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by Giulietti-­Liverani-­Pollicott. By combining the Faure Sjöstrand approach to Anosov flows and Melrose's microlocal radial estimates, Dyatlov and I gave a simple proof of that conjecture. The same radial estimates were used by Vasy to provide a microlocal explanation of the redshift effect and propagation estimates for Kerr de Sitter like spaces.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

Mot(s) clés libre(s) : mathématiques économiques, mathématiques financières, recherche opérationnelle, théorie des jeux, théorie économie

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

L’utilisation de méthodes de théorie des faisceaux (Kashiwara-Schapira)a été dévelopée ces dernières années par Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara et Schapira. Nous essaierons d’en donner un aperçu à la fois pour démontrer des résultats classiques, comme la conjecture d’Arnold, et pour des résultats nouveaux. The use of methods from the Sheaf Theory (Kashiwara-Schapira) was developped recently by Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara and Schapira. We will try to give an insight of that, in order to prove classical results, such as the Arnold conjecture, and to obtain new results.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

La philosophie du projet est d'utiliser l'Astronomie comme source d'exemples dans l'apprentissage des Mathématiques. Le plan suit le programme de Mathématiques de la licence afin que que ce soit plus accessible aux enseignants de mathématiques. A terme, son objectif est de couvrir tout le parcours de Mathématiques et de Physique des étudiants de L1, L2 et L3 comme c'est son objectif.
Mot(s) clés libre(s) : mathématiques, astronomie, pédagogie, illustration

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, number theory, summer school

In this talk I will describe recent work with Peter Hintz on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash Moser iteration scheme.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

Mathematical Explorations of Brain’s Activity The last century has been a fascinating period during which important experimental work have brought to light a vast body of findings characterizing brain’s activity in response of stimuli and their neuronal and molecular bases. These studies have revealed how millions of neurons interact together in order to process sensory information, analyze it and produce a rapid and adapted response. Recently, mathematics and computer science have taken an important role in advancing our knowledge of how the brain functions. The talk will (briefly) present some basic mechanisms at play in the brain, a few mathematical models representing brain’s activity at different scales, as well as a few new mathematical issues raised by their mathematical analysis. In particular, I will present a fascinating and somewhat mysterious synchronization phenomenon arising when the network heterogeneity or the randomness in each neurons’ activity increases. Explorations Mathématiques de l'activité du cerveau Le siècle dernier a été une période fascinante durant laquelle les recherches expérimentales ont fait des avancées majeures sur la caractérisation de l’activité du cerveau en réponse à des stimuli et leurs bases neuronales et moléculaires. Ces études ont révélé comment de gigantesques réseaux de millions de neurones s’activent afin de traiter des informations sensorielles, de l’analyser et d’y répondre de façon rapide et adaptée. Depuis peu, les mathématiques et l’informatique ont pris un rôle important dans les avancées sur la compréhension du fonctionnement du cerveau. L’exposé présentera (brièvement) les mécanismes de base du fonctionnement du cerveau, leurs modélisations mathématiques, ainsi que certains nouveaux problèmes mathématiques posés par l’analyse de ces équations. En particulier, nous présenterons un phénomène fascinant et encore mystérieux de synchronisation des neurones quand leur niveau de désordre ou l’aléas dans leur réponse augmente.
Mot(s) clés libre(s) : cerveau, modélisation mathématique, synchronisation des neurones

In the 1920's Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite "length". The basic question he addressed was whether the infinitesimal properties of the "length" of a set E in the plane yield geometric information on E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field known as Geometric Measure Theory (GMT). In this series of lectures we will present some of the main results in the area concerning the regularity of the support of a measure in terms of the behavior of its density or in terms of its tangent structure. We will discuss applications to PDEs, free boundary regularity problem and harmonic analysis. The aim is that the GMT component of the mini-course will be self contained.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation