Global well-posedness for the defocusing, quintic nonlinear Schr\"odinger equation in one dimension

In this paper, we prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schr"odinger equation. We show that a unique solution exists for $u_0\in H^s(\mathbf{R})$, $s>8\mathrm{/}29$. This improves the result in [13], which proved global well-posedness for $s > {1/3}$. The main new argument is that we obtain almost Morawetz estimates with improved error.