In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric analysis, metric geometry, topology

In these lectures, Gromov{Witten theory will be introduced, assuming only basic moduli theory covered in the rst week of the School. Then the Crepant Transformation Conjecture will be explained. Some examples, with emphasis on the projective/global cases, will be given. Note: The construction of virtual fundamental class, which forms the foundation of the GW theory, will be given in Jun Li's concurrent lectures and will not be explained here.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, courbes, institut fourier, summer school, Gromov-Witten

Le but de cet exposé est de présenter des généralisations multidimensionnelles des fractions continues et de l’algorithme d’Euclide d’un point de vue systèmes dynamiques, en nous concentrant sur les liens avec la numération et les substitutions. Nous allons considérer principalement deux types de généralisations, à savoir, les algorithmes définis par homographies, comme l’algorithme de Jacobi-Perron, et les fractions continues associées aux algorithmes de réduction dans les réseaux.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

Conformal compactification has long been recognised as an effective geometric framework for relating conformal geometry, and associated field theories ``at infinity'', to the asymptotic phenomena of an interior (pseudo-­‐)-­‐Riemannian geometry of one higher dimension. It provides an effective approach for analytic problems in GR, geometric scattering, conformal invariant theory, as well as the AdS/CFT correspondence of Physics. I will describe how the notion of conformal compactification can be linked to Cartan holonomy reduction. This leads to a conceptual way to define other notions of geometric compactification. The idea will be taken up, in particular, for the case of compactifying pseudo-­‐ Riemannian manifolds using projective geometry. A new characterisation of projectively compact metrics will be given, and some results on their asymptotics near the conformal infinity. This is joint work with Andreas Cap.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

Lecture I. I’ll give a complete elementary presentation of the essential features of the Perron Frobenius theory of nonnegative matrices for the central case of primitive matrices (the "Perron" part). (The "Frobenius" part, for irreducible matrices, and finally the case for general nonnegative matrices, will be described, with proofs left to accompanying notes.) For integer matrices we’ll relate "Perron numbers" to this and Mahler measures. Lecture II. I’ll describe how the Perron-Frobenius theory generalizes (and fails to generalize) to 1,2,... x 1,2,... nonnegative matrices. Lecture III. We’ll see the simple, potent formalism by which a certain zeta function can be associated to a nonnegative matrix, and its relation to the nonzero spectrum of the matrix, and how polynomial matrices can be used in this setting for constructions and conciseness. Lecture IV. We’ll describe a natural algebraic equivalence relation on finite square matrices over a semiring (such as Z, Z_+, R, ... ) which refines the nonzero spectrum and is related to K-theory.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, dynamics, institut fourier, summer school, number theory

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

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

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