Spectral theory of nonlocal operators and infinite dimensional integrable systems , after P. Gérard, S. Grellier, T. Kappeler and P. Topalov

The so-called inverse spectral method was used in the years 1970 to give precise informations on the long time behaviour for several partial differential equations such as the KdV equation. In these works the spectral problem was a local one and reduced to fine analysis of ordinary differential equations. It was a longstanding open problem to perform the inverse spectral method for equations presenting non local spectral problems. This problem was recently solved in the context of the Benjamin-Ono equation. In this talk, we will present these developments together with the closely related analysis of the so-called cubic Szegö equation