ERC Starting grant ANADEL (Feb. 2018-Jan.2024)

Analysis of Geometrical Effects on Dispersive Equations

We are concerned with localization properties of solutions to hyperbolic PDEs, especially problems with a geometric component : how do boundaries and heterogeneous media influence spreading and concentration of solutions. While our first focus is on the wave and the Schrödinger equations on manifolds with boundary, strong connections exist with phase space localization for (clusters of) eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models (in physically relevant settings), properties of eigenfunctions in quantum chaos (related to both physics of optic fiber design as well as number theoretic questions), or harmonic analysis on manifolds.

Microlocal analysis, nonlinear PDE, hyperbolic equations on manifolds with/without boundary, dispersion, long time behaviour of solutions, propagation and reflection of singularities, diffraction phenomenon.