Projekt

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Analyse et géométrie: groupes, actions, variétés, spectres

Titel Englisch Analysis and geometry: groups, actions, manifolds, spectra
Gesuchsteller/in Valette Alain
Nummer 137696
Förderungsinstrument Projektförderung (Abt. I-III)
Forschungseinrichtung Institut de mathématiques Université de Neuchâtel
Hochschule Resource not found: '39a3a1f6-d571-4987-9696-9c1091518635'
Beginn/Ende 01.10.2011 - 30.09.2013
Bewilligter Betrag 574'902.00
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Keywords (1)

Coarse embeddings

Lay Summary (Englisch)

Lead
Analysis and geometry: groups, actions, manifolds, spectra.
Lay summary

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.



Direktlink auf Lay Summary Letzte Aktualisierung: 07.11.2012

Verantw. Gesuchsteller/in und weitere Gesuchstellende

Mitarbeitende

Publikationen

Publikation
Eigenvalue control for a Finsler-Laplace operator
T. Barthelm\'e B.Colbois (2013), Eigenvalue control for a Finsler-Laplace operator, in Ann. Global Anal. Geom., 44(1), 43-72.
Isoperimetric control of the spectrum of a compact hypersurface, 683 (2013) 49-66.
B. Colbois A. El Soufi A. Girouard (2013), Isoperimetric control of the spectrum of a compact hypersurface, 683 (2013) 49-66., in Journal f\"{u}r die reine und angewandte Mathematik, 683, 49-66.
Involutive isometries, eigenvalue bounds and a spectral property of Clifford tori
B. Colbois A. Savo (2012), Involutive isometries, eigenvalue bounds and a spectral property of Clifford tori, in Indiana University Mathematics Journal, 61(1), 337-357.
Proper actions of wreath products and generalizations
Cornulier Yves, Stalder Yves, Valette Alain (2012), Proper actions of wreath products and generalizations, in Trans. Amer. Math. Soc., 364(6), 3159-3184.
Reduced 1-cohomology and relative property (T)
Fernós Talia, Valette Alain, Martin Florian (2012), Reduced 1-cohomology and relative property (T), in Math. Z., 270(3-4), 613-626.
The Howe-Moore property for real and $p$-adic groups.
Cluckers Raf, Cornulier Yves, Louvet Nicolas, Tessera Romain, Valette Alain (2011), The Howe-Moore property for real and $p$-adic groups., in Math. Scand., 109(2), 201-224.

Zusammenarbeit

Gruppe / Person Land
Formen der Zusammenarbeit
Université de Chambéry (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
- Austausch von Mitarbeitern
Université de Paris-Sud (Orsay) (Europa)
- Publikation
Université de Tours (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
Ecole Polytechnique Fédérale de Lausanne (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
- Forschungsinfrastrukturen
Université de Montréal (Nordamerika)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
Université de Clermont-Ferrand (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
- Austausch von Mitarbeitern
Université de Nice (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
Université de Rome (La Sapienza) (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
Hebrew University (Asien)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
Katholieke Universiteit Leuven (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
Universität Wien (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Austausch von Mitarbeitern
Bristol University (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
Université de Fribourg (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
Université Catholique de Louvain-la-Neuve (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Publikation
- Austausch von Mitarbeitern

Wissenschaftliche Veranstaltungen



Selber organisiert

Titel Datum Ort
Geometry and Analysis on groups 30.05.2013 Neuchâtel,
CAT(0) spaces and groups 25.03.2012 Les Diablerets,

Veranstaltungen zum Wissenstransfer

Aktiver Beitrag

Titel Art des Beitrags Titel des Artikels oder Beitrages Datum Ort Beteiligte Personen


Verbundene Projekte

Nummer Titel Start Förderungsinstrument
126689 Analyse géométrique sur les groupes et les variétés 01.10.2009 Projektförderung (Abt. I-III)
149261 Groupes discrets, variétés riemanniennes, et géométrie métrique 01.10.2013 Projektförderung (Abt. I-III)
126689 Analyse géométrique sur les groupes et les variétés 01.10.2009 Projektförderung (Abt. I-III)

Abstract

This proposal consists, as usual, of two sub-projects.Project A (Valette, Gournay): Metric and equivariant embeddings of groups into $L^p$-spaces. Four persons to be hired on that project:1) Mr. Pierre-Nicolas JOLISSAINT, Phd Student, 1st and 2nd year of thesis, 2011-2013.2) M. Thibault PILLON, PhD student, 1st and 2nd year of thesis, 2011-2013.3) Dr. Ana KHUKHRO, post-doc, 2012-2013.4) Dr. Jean-Francois PLANCHAT, post-doc, 2012-2013.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.A projet between B. Colbois and A. Savo: Riemannian manifolds with an isometric involu- tion. We show that the presence of an isometric involution without fixed point for a compact manifold (M, g) has strong implication for the gap between the two first eigenvalues of Laplace-type operators on a vector bundle over M. In the case of submanifolds or of domains of the Euclidean space, some of these estimates are sharp, and we look at a characterization of equality and almost equality cases.A project between B. Colbois and A. El Soufi: qualitative information about extremal eigenvalues. We want to compare two consecutive extremal eigenvalues for a given problem: typically supremum of the k-th eigenvalue for the Neumann problem of domain of given volume in Euclidean or hyperbolic space.Laplacian with density. This is a collaboration with Th. Barthelm ´e, for whom we are asking for a fellowship. The question of studying the Laplacian with density is interesting in itself, but it appears also in a natural way in the Ph. D. thesis of Th. Barthelm ´e in relation with the Finsler geometry. Stability of the spectrum for domains with Dirichlet boundary conditions. This is a project with M. Iversen, for whom we are asking for a fellowship. We plan to investigate from a more geometric point of view the classical question ”if two domains are close, are their Dirichlet spectra also close ?” and to get uniform estimates.A geometric approach of the Steklov problem. This corresponds to the subject of a thesis for which we applying for a fellowship. This thesis will be codirected by A. Girouard and B. Colbois. There exist a lot of investigations of the Steklov problem from an analytic point of view, and we will propose a more geometric approach, comparable to what is done for the Laplacian.Numerical investigations. Second part of the thesis of R. Straubhaar, for whom we are applying for the continuation of a fellowship. A way to have a better understanding of extremal metrics is to make numerical investigations. In the second part of the thesis, we will mainly focus on domains in the hyperbolic plane or in the sphere (for Dirichlet or Neumann boundary conditions), using what was already done in the Euclidean plane.
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