Coarse embeddings; $L^p$- equivariant compression; Box spaces; Uniformly bounded representations; Upper bound on the spectrum; Reduced 1-cohomology; Weighted manifolds; Laplace-type operators; Metric-measure spaces; Spectral theory on Riemannian manifolds; Affine isometric actions; Extremal metrics
B. Colbois A. El Soufi A. Savo (2015), Eigenvalues of the Laplacian on a compact manifold with density, in Communications in Analysis and Geometry
, 23, 639-670.
Alain Valette (2015), Le problème de Kadison-Singer (d'après A. Marcus, D. Spielman et N. Srivastava), in Astérisque
, 367-368, 451-476.
Cornulier Yves, Valette Alain (2015), On equivariant embeddings of generalized Baumslag-Solitar groups, in Geom. Dedicata
, 175, 385-401.
A. Berger (2015), The eigenvalues of the Laplacian with Dirichlet boundary condition in R2 are almost never minimized by disks, in Ann. Global Anal. Geom.
, 47, 285-304.
Petersen Henrik Densing, Valette Alain (2014), $L^2$-Betti numbers and Plancherel measure, in J. Funct. Anal.
, 266(5), 3156-3169.
Jolissaint Pierre-Nicolas, Valette Alain (2014), $L^p$-distortion and $p$-spectral gap of finite graphs., in Bull. Lond. Math. Soc.
, 46, 329-341.
Khukhro A. (2014), Embeddable box spaces of free groups, in Math. Ann.
, 360(1-2), 53-66.
B. Colbois A. El Soufi (2014), Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces,, in Math. Zeitschrift
, 278(1-2), 529-546.
T. Barthelm´e B. Colbois M. Crampon P. Verovic (2014), Laplacian and spectral gap in regular Hilbert geometries, in Tohoku Math. J
, 66, 377-407.
Franks Jordan, Valette Alain (2014), On 1-cocycles induced by a positive definite function on a locally compact abelian group., in Ann. Math. Blaise Pascal
, 21(1), 61-69.
B. Colbois A. Girouard (2014), The spectral gap of graphs and Steklov eigenvalues on surfaces, in Electronic Research Announcements in Mathematical Sciences
, 21, 19-27.
B. Colbois P. Verovic (2014), Two properties of volume growth entropy in Hilbert geometry, in Geometriae Dedicata
, 173, 163-175.
Jolissaint Pierre-Nicolas, Pillon Thibault (2013), $L^p$ compression of some HNN extensions, in J. Group Theory
, 16(6), 907-913.
T. Barthelm\'e B.Colbois (2013), Eigenvalue control for a Finsler-Laplace operator, in Ann. Global Anal. Geom
, 44(1), 43-72.
B. Colbois A. El Soufi A. Girouard (2013), Isoperimetric control of the spectrum of a compact hypersurface, in Journal f¨ur die reine und angewandte Mathematik
, 683, 49-66.
B. Colbois A. Girouard M. Iversen (2013), Uniform stability of the Dirichlet spectrum for rough outer perturbations, in Journal of Spectral Theory
, 3(4), 575-599.
Sylvain Arnt, Fibred coarse embeddability of box spaces and proper isometric affine actions on $L^p$- spaces, in Bull. Belgian Math. Soc.
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 manifoldsand 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.