Projekt

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Analyse géométrique sur les groupes et les variétés

Titel Englisch Geometric Analysis on groups and manifolds
Gesuchsteller/in Valette Alain
Nummer 126689
Förderungsinstrument Projektförderung (Abt. I-III)
Forschungseinrichtung Institut de mathématiques Université de Neuchâtel
Hochschule Universität Neuenburg - NE
Hauptdisziplin Mathematik
Beginn/Ende 01.10.2009 - 30.09.2011
Bewilligter Betrag 584'949.00
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Keywords (14)

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

Lay Summary (Englisch)

Lead
Lay summary
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.
Direktlink auf Lay Summary Letzte Aktualisierung: 21.02.2013

Verantw. Gesuchsteller/in und weitere Gesuchstellende

Mitarbeitende

Publikationen

Publikation
Free groups and reduced 1-cohomology of unitary representations
F. Martin and A. Valette (2012), Free groups and reduced 1-cohomology of unitary representations, in Clay Math. Proc. 11 (ed.), Amer. Math. Soc., Providence, 459-463.
Large eigenvalues and concentration
B. Colbois A. Savo , (2011), Large eigenvalues and concentration, in Pacific J. of Mathematics, 249(2), 271-290.
A new metric invariant for Banach spaces
F. Baudier N. J. Kalton and G. Lancien (2011), A new metric invariant for Banach spaces, in Studia Math., 199(1), 73-94.
Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property
Soyoung Moon (2011), Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property, in Ann. Math. Blaise Pascal, 18(2), 217-235.
Eigenvalue estimate for the rough Laplacian on differential forms
B. Colbois D. Maerten (2011), Eigenvalue estimate for the rough Laplacian on differential forms, in Manuscripta Mathematica, 132, 399-413.
Hilbert geometry for convex polygonal domains
B. Colbois C. Vernicos P. Verovic , (2011), Hilbert geometry for convex polygonal domains, in Journal of Geometry, 100(1-2), 37-64.
Isoperimetric control of the Steklov spectrum,
B. Colbois A. El Soufi and A. Girouard (2011), Isoperimetric control of the Steklov spectrum,, in J. Funct. Analysis, 261(5), 1384-1399.
Les Medailles Fields 2010
Alain Valette (2011), Les Medailles Fields 2010, in Losanges, 12, 35-41.
Permanence properties of amenable, transitive and faithful actions
Soyoung Moon (2011), Permanence properties of amenable, transitive and faithful actions, in Bull. Belgian Math. Soc. Simon Stevin, 18(2), 287-296.
Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms
M.-A. Lawn J. Roth (2011), Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms, in Journal Mathematical Physics, Analysis and Geometry, Vol. 14( Issue 3), 185-195.
Elon Lindenstrauss, medaille Fields 2010
A. Valette (2010), Elon Lindenstrauss, medaille Fields 2010, in Tangente, 137, 11-27.
Embeddings of proper metric spaces into Banach spaces
F. Baudier, Embeddings of proper metric spaces into Banach spaces, in Houston Journal of Mathematics..
Proper actions of wreath products and generalizations
Y. Cornulier Y. Stalder and A. Valette, Proper actions of wreath products and generalizations, in Trans. Amer. Math. Soc..
Reduced 1-cohomology and relative property (T)
T. Fernos and A. Valette, Reduced 1-cohomology and relative property (T), in Math. Zeitschrift.
The Howe- Moore property for real and p-adic groups
R. Cluckers Y. Cornulier N. Louvet R. Tessera and A. Valette, The Howe- Moore property for real and p-adic groups, in Math. Scandivadica.

Zusammenarbeit

Gruppe / Person Land
Formen der Zusammenarbeit
KU Leuven Belgien (Europa)
- vertiefter/weiterführender Austausch von Ansätzen, Methoden oder Resultaten
- Austausch von Mitarbeitern

Wissenschaftliche Veranstaltungen

Aktiver Beitrag

Titel Art des Beitrags Titel des Artikels oder Beitrages Datum Ort Beteiligte Personen
Non-Linear Geometry of Banach Spaces, Differentiability and Geometric Group Theory Vortrag im Rahmen einer Tagung Compression for metric embeddings 01.08.2011 College Station, Texas, USA, Vereinigte Staaten von Amerika Baudier Florent; Valette Alain;
Geometric and measured group theory Vortrag im Rahmen einer Tagung Haagerup property for wreath products 04.07.2011 Paris, Institut Henri Poincaré, Frankreich Valette Alain;
Geometric group theory Vortrag im Rahmen einer Tagung Proper isometric actions 20.06.2011 Bristol, UK, Grossbritannien und Nordirland Valette Alain;
Discrete groups and geometric stuctures Vortrag im Rahmen einer Tagung Compression and equivariant compression 30.05.2011 Ostende (Belgique), Belgien Valette Alain;
Metric embeddings, algorithms and hardness of approximation Vortrag im Rahmen einer Tagung Coarse geometry and C*-algebras 17.01.2011 Paris, Institut Henri Poincaré, Frankreich Valette Alain;
Embeddings workshop Vortrag im Rahmen einer Tagung Compression and equivariant compression 10.01.2011 Cambridge, UK, Grossbritannien und Nordirland Baudier Florent; Valette Alain;


Selber organisiert

Titel Datum Ort
Second Workshop on CR, pseudo-Hermitian and Sasakian Geometry 03.05.2011 Neuchâtel, Schweiz

Kommunikation mit der Öffentlichkeit

Kommunikation Titel Medien Ort Jahr
Referate/Veranstaltungen/Ausstellungen Graphs with large girth International 2011

Auszeichnungen

Titel Jahr
Chaire Francqui 2011, KU Leuven (Belgique) 2011

Verbundene Projekte

Nummer Titel Start Förderungsinstrument
137696 Analyse et géométrie: groupes, actions, variétés, spectres 01.10.2011 Projektförderung (Abt. I-III)
118014 Analyse géométrique sur les groupes et les variétés 01.10.2007 Projektförderung (Abt. I-III)
130435 Groupes sofiques: algèbre, analyse et dynamique 01.05.2010 Sinergia
149261 Groupes discrets, variétés riemanniennes, et géométrie métrique 01.10.2013 Projektförderung (Abt. I-III)
137696 Analyse et géométrie: groupes, actions, variétés, spectres 01.10.2011 Projektförderung (Abt. I-III)

Abstract

This is a joint research project, continuing the series of joint proposals by Profs. Colbois and Valette at the Mathematics Institute at Université Neuchâtel. It consists of two distinct projects:Project A (head: Alain VALETTE): Proper isometric actions on Hilbert spaces and cohomology.Project B (head: Bruno COLBOIS): Spectral theory on riemannian manifolds and metric geometry.
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