Groupes discrets, variétés riemanniennes, géométrie métrique
Responsable du projet Alain Valette
Bruno Colbois
Résumé This proposal consists, as usual, of two sub-projects.

Project A (Valette): Affine isometric actions on $L^p$-spaces, and metric embeddings.

The project will mainly deal with affine actions of groups on Hilbert and $L^p$-spaces. The main directions of research will be:

1) Equivariant $L^p$-compression, that quantifies equivariant embeddings of a group into $L^p$ (exact computations, behaviour under group constructions, invariance properties).

2) Unreduced and reduced 1-cohomology of unitary and uniformly bounded representations on Hilbert spaces; in particular study of the class of groups admitting a representation with non-zero 1-cohomology but vanishing reduced 1-cohomology.

3) Relations between coarse embeddings of box spaces of residually finite groups, and the Haagerup property for these groups.

4) Group properties of a discrete group, that can be defined through coefficients of representations and ideals in the algebra of bounded functions: new examples, and possible connection with exactness.

The first two directions are heavily related to the theses of the two PhD students, P.-N. Jolissaint and T. Pillon, hired on the project.

Project B (Colbois): Spectral theory on Riemannian manifolds and metric geometry.

The main topic of this proposal is spectral theory on Riemannian manifolds
and metric geometry, and more precisely the study of extremal metrics and of (upper) bounds for the spectrum of the Laplacian. A general objective is to choose a metric approach to the problem and work if possible in the context (or at least in the spirit) of metric measure spaces. The main direction of research concerns the Laplacian on weighted manifolds. It corresponds to the continuation of ongoing projects with A. El Soufi and A. Savo. The goal is to obtain some geometric upper bounds for the spectrum of weighted manifolds together with a study of the optimality of these bounds. The use of methods coming from mm-spaces will be developed with Z. Sinaei, for whom I am applying for a fellowship. Still in this direction, but with a more metric flavor, there is a project with P. Cerocchi, for whom I am apply for a fellowship, going around the control of the spectrum in relation with the Gromov-Hausdorff distance and a project around the control of the spectrum of submanifolds in relation with their distortion.

With Alexandre Girouard we will study the Steklov operator for compact hyperbolic surfaces with geodesic boundary.The last project is related the PhD thesis of A. Berger, for whom I am applying for a fellowship. It is the continuation of a general project about the use of numeric analysis in order to investigate the extremal domains for the spectrum of the Laplacian.
Mots-clés Coarse embeddings, Weighted manifolds
Type de projet Recherche fondamentale
Domaine de recherche Mathématiques
Source de financement FNS - Encouragement de projets (Div. I-III)
Etat Terminé
Début de projet 1-10-2013
Fin du projet 30-9-2015
Budget alloué 622'484.00
Contact Alain Valette