The main topic of this proposal is spectral theory on Riemannian manifolds and 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. Still in this direction, but with a more metric flavour, there is a project with P. Cerocchi, 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. 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.