Gross and Pitaevskii proposed to model the dynamics of the Bose-Einstein
condensate by a nonlinear Schrdinger equation, the Gross-Pitaevskii
equation. This equation plays a key role in the theory and experiments of
the Bose-Einstein condensation. The fundamental mathematical question is
to derive this equation from the first principle physics law, the
many-body Schrdinger equation. In the time-independent setting, this
problem was solved by Lieb-Seiringer-Yngvason. In this lecture, we shall
review the recent progress concerning the dynamical aspects of this
problem and the analytic methods developed for quantum dynamics of
many-body systems.