  The initial value problem for the cubic defocusing nonlinear Schr\"odinger
equation $i \partial_t u + \Delta u = |u|^2 u$ on the plane is shown to be
globally well-posed for initial data in $H^s (\R^2)$ provided $s>1/2$. The
proof relies upon an almost conserved quantity constructed using multilinear
correction terms. The main new difficulty is to control the contribution of
resonant interactions to these correction terms. The resonant interactions are
significant due to the multidimensional setting of the problem and some
orthogonality issues which arise.
