Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2

The initial value problem for the cubic defocusing nonlinear Schr"odinger equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{2}u$ on the plane is shown to be globally well-posed for initial data in $H^s({\mathbb{R}}^2)$ provided $s>1\mathrm{/}2$. 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.