Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d \geq3$

In this paper we prove that the defocusing, $d$-dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and solutions scatter for $u_{0} \in L^{2}(\mathbf {R}^{d})$, $d \geq 3$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made by Colliander, Keel, Staffilani, Takaoka, and Tao. Since we are considering an $L^{2}$-critical initial value problem we will localize to low frequencies. The main new ingredient in this proof is a long time Strichartz estimate for the solution to the first equation given in the paper at high frequencies. The long term Strichartz estimates allow us to estimate the error in the interaction Morawetz estimate caused by localizing to low frequencies.