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

indisponible
Mot(s) clés libre(s) : mathématiques, colloques, Grenoble (Isère), institut fourier

indisponible
Mot(s) clés libre(s) : colloques, mathématique, Grenoble (Isère), institut fourier, start

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

In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness, structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness, structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.  
Mot(s) clés libre(s) : Grenoble, école d'été, mathématique, institut fourier, summer school, geometric measure theory, calculus of variation