Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

In any dimension $n\ge 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr"odinger equation $i{u}_t+\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{n-2}}u$ in $\mathbb{R}\times {\mathbb{R}}^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n\ge 6$ some new technical difficulties arise because of the very low power of the non-linearity.