The defocusing energy-critical nonlinear Schr\"odinger equation in higher dimensions

We obtain global well-posedness, scattering, and global $L_{t,x}^{\frac{2(n+2)}{n-2}}$ spacetime bounds for energy-space solutions to the energy-critical nonlinear Schr"odinger equation in ${\mathbb{R}}_t\times {\mathbb{R}}_x^n$, $n\ge 5$.