Nonlinear resonance for dispersive equations with a potential

In this talk we present a general framework for proving global
in time existence and asymptotic properties of solutions for nonlinear
dispersive equations with a potential. These problems arise naturally in
many contexts, e.g., quantum mechanics, nonlinear optics, general
relativity, and in the study of soliton stability. The main building block
is a theory of space-time resonance generalized to the spectral setting of
the corresponding Schrödinger operator.  As an example of the technique,
we provide the proof of small data global in time well-posedness and
scattering for a quadratic nonlinear Schrödinger equation in
three-dimensions.