Séminaire de Relativité Générale Mathématique

 

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (AdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an asymptotically AdS solution of the Einstein equations uniquely determined by its data on its conformal boundary at infinity? In this presentation, we establish a key step: we prove such a unique continuation result for wave equations on fixed asymptotically AdS spacetimes. In particular, we highlight the analytic and geometric features of AdS spacetime which enable this uniqueness result, as well as obstacles preventing such a result from holding in other cases. If time permits, we will also discuss some applications of this result toward symmetry extension and rigidity theorems.

Les foncteurs exponentiels (gradués commutatifs) apparaissent naturellement dans un certain nombre de calculs homologiques (homologie des groupes, des foncteurs…). Dans cet exposé, nous donnerons quelques résultats de structure des foncteurs exponentiels et des applications à des calculs concrets.

Soient V=(Z/2Z)n et X un V-CW complexe fini. On construit deux complexes dans la catégorie H*V-U, le premier "algébrique" basé sur la structure de H*V-module de la cohomologie équivariante H*_VX et le second "topologique" utilisant l'action du groupe V sur X. Sous certaines conditions, on montre que ces deux complexes sont acycliques et équivalents.

We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an algebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)–(4) provide many new questions to study.

I will make a quick review of old and new results concerning the Lorentz gas; discuss new directions in which I’d like to proceed (e.g. non periodic obstacles, interacting particles, …) and some (very) partial results toward such directions.

The two-dimensional one-component Coulomb plasma is an ensemble of identical charges in the plane repelling each other via the logarithmic Coulomb potential and confined by an external potential. I will discuss results that show that the fluctuations of linear statistics of many particles are of order 1 and given by a Gaussian free field, at any temperature. This is joint work with P. Bourgade, M. Nikula, and H.-T. Yau.

Totally positive matrices are characterized by having all their minors positive. They appear in various areas of physics and mathematics, including oscillations in mechanical systems, quantum groups, and algebraic geometry. It has been known since the work of Fomin that the two-point correlations functions of the two-dimensional Gaussian free field satisfy total positivity. I will present an analogous result for the correlations of the planar Ising model. The idea is to prove that determinants of such correlations have interpretations in terms of probabilities of events in the random current model. A natural open question is to identify all planar totally positive spin systems.

We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. These log-correlated non-Gaussian random fields arise in different branches of statistical mechanics. Existing results were mainly focused on the CLT for the linear functionals. In this talk I will describe some recent progress on the global maximum and local CLT for the field, thus confirming they are in the Gaussian universality class in a very strong sense. The proof uses a random walk representation (a la Helffer-Sjostrand) and an approximate harmonic coupling (by J. Miller).

Affine manifolds, i.e. manifolds which admit charts given by affine transformations, remain mysterious by the very few explicit examples and their famous open conjectures: the Auslander Conjecture, the Chern Conjecture and the Markus Conjecture. I will discuss an intermediate conjecture, somehow between the Auslander Conjecture and the Chern Conjecture, predicting the vanishing of the simplicial volume of affine manifolds. In a joint work with Chris Connell and Jean-François Lafont, we prove the latter conjecture under some hypothesis, thus providing further evidence for the veracity of the Auslander and Chern Conjectures. To do so, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups in their fundamental group to have vanishing simplicial volume. This answers a special case of a question due to Lück.
Joint work with Chris Connell and Jean-François Lafont.

Quasifuchsian manifolds are an important class of hyperbolic 3-manifolds, classically parametrized by two copies of Teichmüller space. Their volume is infinite, but they have a well-defined finite « renormalized volume » which has nice properties, both analytic and « coarse ». In particular, considered as a function over Teichmüller space, the renormalized volume provides a Kähler potential for the Weil-Petersson metric; moreover, it is within bounded additive constants of the volume of the convex core and is bounded from above by the Weil-Petersson distance between the conformal structures at infinity. After describing these properties, we will outline some recent applications (by Kojima, McShane, Brock, Bromberg, Bridgeman, and others) to the Weil-Petersson geometry of Teichmüller space or the geometry of hyperbolic 3-manifolds that fiber over the circle. We will then explain how properties of the renormalized volume suggest new questions and viewpoints on quasifuchsian manifolds.
The talk will be accessible to nonexperts.

On ne présente pas Grothendieck dans ce qui fut son enceinte, mais son héritage mathématique est si vaste qu’on risque toujours d’en négliger une partie : analyse fonctionnelle, algèbre homologique, géométrie algébrique, théorie des nombres …
Toutes ces méthodes sont encore d’actualité, et j’essayerai d’en donner une description vivante.