Global well--posedness for the mass--critical nonlinear Schr{ö}dinger equation on $\mathbb{T}$

We prove a global well–posedness result for the quintic NLS on $\mathbb{T}$ for initial data in $H^s(\mathbb{T})$, $s>1\mathrm{/}3$. This improves the previous best bound of $s>2\mathrm{/}5$.