The aim of the AURORA project CHARGE is to join the efforts of mathematicians in Norway and in France that have had a major impact in the last years in the applications of complex and harmonic analysis techniques to results on the time-frequency localizat ion. The problems CHARGE members address stem from applied analysis (PDEs, control theory, signal processing) as well as from recent developments in complex and harmonic analysis.
The main unifying theme of CHARGE is time-frequency localization. This is an important problem in mathematical analysis with applications to signal processing as well as theoretical physics. The aim here is to obtain good concentration properties both in the direct domain (usually tagged as time) and simultaneously in the Fouri er transform domain (frequency). Of course there are strong limitation to this which can have various mathematical formulations and are generically named uncertainty principles (UP). There has been an important rebirth of interest in the UP in recent year s as it appears in many parts of applied science. As a first example, let us mention the blooming subject of compressed sensing in which discrete versions of UPs play a central role.
Another example comes from unique continuation properties
of solutions of PDE's. Here one typically assumes that the difference between two solutions of a given PDE (for example the heat equation) is small at two distinct times and one concludes that the two solutions are almost the same at all times (or even coincide).
As a last example, let us mention the control theory of PDE's. The aim here is to obtain control of the solution of a PDE on a domain by fixing its values on a small part of that domain (or of its boundary). There are many approaches to such problems. It some of them a key property takes the form of the uncertainty principle (so called-annihilating pairs). The research started by some participants of CHARGE on UP gives interesting applications to control theory.