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

The optimal $L^4$-Strichartz estimate for the Schr{ö}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 Schrödinger equation in $H^s({\mathbb{T}}^2)$ for any $s>0$ and data which is small in the critical norm.