Interactions between analysis and geometry are of considerable importance in mathematics. The famous Yamabe problem and Perelman’s proof of Poincaré are remarkable illustrations of these interactions. The aim of this school is to provide an introduction to some of these topics, focusing on geometric problems that are expressed in terms of elliptic equations.
The first part of the school deals with two classical subjects : prescription of curvature and spectral geometry. These topics have a fairly long history, but are still subject of active research. We will provide participants with basic knowledge on these subjects before presenting some recent results and open problems. Thus, the school will begin with a general introduction to elliptic problems in Euclidean space and manifolds as well as the variatonal method to solve elliptic PDE. Regarding spectral geometry, we will concentrate on bounded domains of the Euclidean space and Riemannian surfaces, so that only a basic understanding of Riemannian geometry is necessary. As for the prescribed curvature problem, it will be discussed in the case of compact surfaces so as to minimize the needed geometric prerequisites.
The second part of the school is devoted to more specialized topics, such as quantitative isoperimetric and singular spectral pertubations.
There will be also of interactive sessions devoted to examples, exercises and numerical experiments.