Autour du d-bar (partie 1)

"In certain problems in analysis one is interested in finding analytic functions with certain properties. The idea of the d-bar scheme is to first construct a smooth (not analytic) solution to the initial problem with the required properties - which in general is an easy task - and then to correct the solution maintaining the main properties of the problem : if $f$ is the smooth solution to the initial problem and if $u$ is a suitable solution to $\overliner{\partial}u=g$ where $g=\overliner{\partial}f$, then $F=f-u$ satisfies $\overliner{\partial}F=0$ so that $F$ is analytic. The challenge here is that the correction does not destroy the properties required by the initial problem (for instance values in given points, norms, etc.). The method will be illustrated on 3 examples : interpolation, corona theorem, separation of singularities. It should be mentioned that these problems are related with different applications such as for instance signal and control theory. The talk is aimed at an elementary level."