Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr\"odinger equation when $d=1$

In this paper we prove that the defocusing, quintic nonlinear Schr"odinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}($\mathbf{R}$)$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in [11]. Since we are considering an $L^2$ - critical initial value problem we will localize to low frequencies.