Modes and quasimodes: concentration and non-concentration

In this course we will present basic tools for the study of high-frequency eigenfunctions or quasimodes of Schrödinger-type elliptic operators on a compact manifold. Our mail goal will be to give a quantitative description of concentration and oscillation effects that can be developed by a sequence of quasimodes, and in particular to show how this issue is related the global geometry of the manifold. We will give applications to unique continuation-type estimates and to the Calderón problem. We will mainly use tools of from the theory of pseudodifferential operators and semiclassical measures that will be reviewed during the course.