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 the algebraic approach to quantum field theory on curved backgrounds, there exists a special class of quantum states for free fields, called of Hadamard form. These are of particular relevance since they yield finite quantum fluctuations of all observables and they can be used to implement interactions at a perturbative level. Although their existence is guaranteed on all globally hyperbolic spacetimes, for long time only few explicit examples were known. A way to bypass this problem exists on those manifolds which possess a null conformal boundary, such as all asymptotically flat spacetimes. In this talk we shall discuss this construction in particular for massless, conformally coupled scalar fields and for linearised gravity.
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, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation, minimal sets

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

A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

A celebrated theorem of Almgren shows that every integer rectifiable current which minimizes (locally) the area is a smooth submanifold except for a singular set of codimension at most 2. Almgren’s theorem is sharp in codimension higher than 1, because holomorphic subvarieties of Cn are area-minimizing. In fact the typical singularity of a 2-dimensional area-minimizing current is modelled by branch points of holomorphic curves. These singularities are rather difficult to analyze because they might be very high order phenomena.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation

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, COURBES PSEUDOHOLOMORPHES

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, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES

indisponible
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, feuilletages, COURBES PSEUDOHOLOMORPHES