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

Le récent article de McMullen « Dynamics with small entropy on projective K3 surfaces » éclaire d’un jour nouveau les nombres de Salem. Ces entiers algébriques gardent cependant tout leur mystère. On peut tous les obtenir grâce à la construction de Salem (Boyd (1977)) et cependant on ignore s’il en existe un inférieur à 1,1762... Après avoir rappelé la construction de Salem et le théorème de Boyd, on définira la mesure de Mahler logarithmique d’un polynôme de plusieurs variables. On prouvera que la mesure de Mahler d’un polynôme de deux variables est la limite d’une suite de mesures de Mahler de polynômes d’une variable (Boyd (1981)). On donnera des mesures explicites de mesures de Mahler de certaines classes de polynômes de 2 et 3 variables. En particulier dans le cas de 3 variables on présentera deux aspects de l’expression de cette mesure, un aspect arithmétique comme série L de Hecke d’un corps quadratique imaginaire et un aspect géométrique comme série L de la surface K3 définie par le polynôme qui s’exprime comme série L d’une forme modulaire de poids 3 à coefficients rationnels. Pour terminer, on évoquera des problèmes plus géométriques de fibrations elliptiques sur les surfaces K3 algébriques.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

Black holes play a central role in general relativity and astrophysics. The problem of proving the dynamical stability of the Kerr black hole spacetime, which is describes a rotating black hole in vacuum, is one of the most important open problems in general relativity. Following a brief introduction to the evolution problem for the Einstein equations, I will give some background on geometry of the Kerr spacetime. The analysis of fields on the exterior of the Kerr black hole serve as important model problems for the black hole stability problem. I will discuss some of the difficulties one encounters in analyzing waves in the Kerr exterior and how they can be overcome. A fundamentally important as pect of geometry and analysis in the Kerr spacetime is the fact that it is algebraically special, of Petrov type D, and therefore admits a Killing spinor of valence 2. I will introduce the 2 spinor and related formalisms which can be used to see how this structure leads to the Carter constant and the Teukolsky system. If there is time, I will discuss in this context some new conservation laws for fields of non zero spin.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural extensions, we apply these to the familiar expansions mentioned above in order to understand the structure and global behaviour of different number theoretic expansions, and to obtain new and old results in an elegant and straightforward manner.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

In order to control locally a space-­‐time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-­‐like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-­‐posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

In order to control locally a space-­‐time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-­‐like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-­‐posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

We will first introduce the basic concepts pertaining to Kobayashi pseudo-distances and hyperbolic complex spaces, including Brody’s theorem and the Ahlfors-Schwarz lemma. One of the main goals of the theory is to understand conditions under which a given algebraic variety is Kobayashi hyperbolic. This leads to the introduction of jet spaces and jet metrics, and provides a strong link between the existence of entire curves and the existence of global algebraic differential equations.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

We will first introduce the basic concepts pertaining to Kobayashi pseudo-distances and hyperbolic complex spaces, including Brody’s theorem and the Ahlfors-Schwarz lemma. One of the main goals of the theory is to understand conditions under which a given algebraic variety is Kobayashi hyperbolic. This leads to the introduction of jet spaces and jet metrics, and provides a strong link between the existence of entire curves and the existence of global algebraic differential equations.  
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

We will first introduce the basic concepts pertaining to Kobayashi pseudo-distances and hyperbolic complex spaces, including Brody’s theorem and the Ahlfors-Schwarz lemma. One of the main goals of the theory is to understand conditions under which a given algebraic variety is Kobayashi hyperbolic. This leads to the introduction of jet spaces and jet metrics, and provides a strong link between the existence of entire curves and the existence of global algebraic differential equations.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

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