Abstract : It is known that the minimal Sobolev regularity needed for the semi-linear local well-posedness of the non linear Schr?dinger equation, posed on a two dimensional spatial domain depends heavily on the geometry of this domain. In this talk we will observe a similar phenomenon in the study of the probabilistic well-posedness. We will show that in the study of solutions with low regularity gaussian random initial data the structure of the solutions on flat tori and on the standard sphere are totally different. As one may expect this phenomenon is related to the existence of stable closed geodesics but this time it manifests in a new way via the structure of the resonant manifold. Our methods are strong enough to deal with data distributed according to the Gibbs measure resolving thus a longstanding open problem. This is a joint work with Nicolas Burq, Nicolas Camps and Chenmin Sun.
