The cubic nonlinear Schrödinger equation in two dimensions with radial data

We establish global well-posedness and scattering for solutions to themass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$for large spherically symmetric L^2_x(\R^2) initial data; in the focusing casewe require, of course, that the mass is strictly less than that of the groundstate. As a consequence, we deduce that in the focusing case, any sphericallysymmetric blowup solution must concentrate at least the mass of the groundstate at the blowup time. We also establish some partial results towards the analogous claims in otherdimensions and without the assumption of spherical symmetry.