The time dependent Gross-Pitaevskii equation describes the
dynamics of initially trapped Bose-Einstein condensates. We present a
rigorous proof of this fact starting from a many-body bosonic Schroedinger
equation with a short scale repulsive interaction in the dilute limit. Our
proof shows the persistence of an explicit short scale correlation
structure in the condensate. This is a joint work with B. Schlein
and H.T. Yau.