Global well-posedness of the cubic nonlinear Schrödinger equation on $\mathbb{T} ^{2}$

Abstract We prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> for initial data of arbitrary size in the defocusing case and data below the ground state threshold in the focusing case. The result is based on a new inverse Strichartz inequality, which is proved by using incidence geometry and additive combinatorics, in particular, the inverse theorems for Gowers uniformity norms by Green-Tao-Ziegler. This allows to transfer the analogous results of Dodson for the non-periodic mass-critical NLS to the periodic setting. In addition, we construct an approximate periodic solution that implies the sharpness of the results.