Geometric Analysis on groups and manifolds
Responsable du projet |
Alain Valette
Bruno Colbois |
Collaborateur |
Soyoung Moon
Olivier Isely Florent Baudier Marie Amélie Lawn Régis Straubhaar Jérémie Brieussel Erwan Hillion |
Résumé |
Project A: That project will deal with two closely related themes:
a)Affine actions on Hilbert (mainly) and Banach spaces: structural
properties of the Haagerup property (in particular for
permutational wreath products), equivariant compression,
bounded/proper alternative, structure of orbits. b)Exploring the
class of a-T-menable groups: finding new examples (one-relator
groups, automorphisms groups of rooted trees), existence of
amenable groups for a-T-menable groups, relations with the sofic
and hyperlinear properties. Project B: The main topic of this
proposal is spectral theory on riemannian manifolds, more precisely
the study of extremal metrics and bounds on the spectrum. A general
goal is to avoid, as much as possible, the use of constraints on
the curvature, but rather to impose metric and global conditions.
a) Large gap in the spectrum and concentration of the metric. We
plan to show that, under some conditions, the presence of a large
gap on the spectrum implies concentration of the metric. This is
true for a manifold with or without boundary (with the Neumann
condition in the former case). We will also exhibit concentration
phenomena for Laplace-like operators on a compact manifold. b)
Upper bound on the spectrum: complex submanifolds of CP^n. The goal
is to investigate the algebraic submanifolds of CP^n. We hope to get
upper bound on the spectrum of the submanifold in term of the degree
of the submanifold. This is known for the first eigenvalue, but
proved by methods seemingly not powerful enough for higher
eigenvalues. c) Critical and extremal metrics. The main goal of
this part of the project is to investigate critical or extremal
metrics which are not smooth. In a first time, we will look at very
special cases, like weighted graphs, orbifolds, and try to
understand examples in this context, thanks to numerical
investigations. d) Numerical investigations. A way to have a better
understanding of extremal metrics is to make numerical
investigations and this will be the subject of the thesis by Regis
Straubhaar, candoc on the project. The goal is first to investigate
the spectrum of surfaces and domains with Neumann boundary condition
under deformations, and apply this to investigate concrete
examples. |
Mots-clés |
Affine actions, Haagerup property, Amenable actions, Spectral theory on Riemannian manifolds, Upper bound on the spectrum, Extremal metrics, Laplace-type operators, Curvature, 1-cohomology, Wreath products, analysis on groups, geometric group theory, embeddings into Hilbert spaces and Banach spaces, proper isometric actions |
Type de projet | Recherche fondamentale |
Domaine de recherche | Mathématiques |
Source de financement | FNS - Encouragement de projets (Div. I-III) |
Etat | Terminé |
Début de projet | 1-10-2009 |
Fin du projet | 30-9-2011 |
Budget alloué | 584'949.00 |
Autre information |
http://p3.snf.ch/project-126689# |
Contact | Alain Valette |