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.