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.