Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 1

In this paper we prove global well-posedness and scattering for the defocusing, one dimensional mass-critical nonlinear Schr{\"o}dinger equation. We make use of a long-time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in earlier papers for dimensions $d\geq 3$ and $d = 2$.