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Analysis and geometry: groups, actions, manifolds, spectra
Titre du projet
Analysis and geometry: groups, actions, manifolds, spectra
Description
This proposal consists of two sub-projects.
Project A (Valette, Gournay): Metric and equivariant embeddings of groups into $L^p$-spaces.
The project will deal with two closely related themes: (a) Coarse embeddings of finitely generated groups in Hilbert spaces and $L^p$-spaces, and metrically proper, isometric actions of those groups on the same Banach spaces; quantitative aspects of those embeddings (metric and equivariant compression functions and exponents); explicit computations on concrete examples; behaviour of those invariants under various group constructions. (b) Obstructions to metric embeddings and to equivariant embeddings (variants of Kazhdan's property (T), e.g. property $(\tau)$; presence of expanders); link with Yu's property (A).
Project B (Colbois, Girouard): The main topic of this proposal is spectral theory on Riemannian manifolds, and more precisely the study of extremal metrics and of bounds on the spectrum. 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 space. The first two lines of research correspond to the continuation of ongoing projects and the next three are more speculative. The two last concern two Ph. D. theses. A new one about the Steklov problem and the contituation of a Ph.D thesis about a numerical approach to the problem.
Project A (Valette, Gournay): Metric and equivariant embeddings of groups into $L^p$-spaces.
The project will deal with two closely related themes: (a) Coarse embeddings of finitely generated groups in Hilbert spaces and $L^p$-spaces, and metrically proper, isometric actions of those groups on the same Banach spaces; quantitative aspects of those embeddings (metric and equivariant compression functions and exponents); explicit computations on concrete examples; behaviour of those invariants under various group constructions. (b) Obstructions to metric embeddings and to equivariant embeddings (variants of Kazhdan's property (T), e.g. property $(\tau)$; presence of expanders); link with Yu's property (A).
Project B (Colbois, Girouard): The main topic of this proposal is spectral theory on Riemannian manifolds, and more precisely the study of extremal metrics and of bounds on the spectrum. 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 space. The first two lines of research correspond to the continuation of ongoing projects and the next three are more speculative. The two last concern two Ph. D. theses. A new one about the Steklov problem and the contituation of a Ph.D thesis about a numerical approach to the problem.
Chercheur principal
Gournay, Antoine
Pillon, Thibault
Statut
Completed
Date de début
1 Octobre 2011
Date de fin
30 Septembre 2013
Chercheurs
Antolin-Pichel, Yago
Barthelme, Thomas
Organisations
Identifiant interne
14997
identifiant
Mots-clés
2 Résultats
Voici les éléments 1 - 2 sur 2
- PublicationMétadonnées seulementEigenvalue control for a Finsler--Laplace operator(2013-5-1)
- PublicationMétadonnées seulementNumerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces(2014-4-24)Let (M,g) be a smooth and complete surface, Ω⊂M be a domain in M, and Δg be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence 0<λ1(Ω)≤λ2(Ω)≤⋯↗∞. A classical question is to ask what is the domain Ω∗ which minimizes λm(Ω) among all domains of a given area, and what is the value of the corresponding λm(Ω∗m). The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for Ω∗m. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.