Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$

The optimal $L^4$-Strichartz estimate for the Schr{\"o}dinger equation on the two-dimensional rational torus $\mathbb{T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schr\"odinger equation in $H^s(\mathbb{T}^2)$ for any $s>0$ and data which is small in the critical norm.