In 2014, Bourgain and Demeter proved almost sharp decoupling inequalities for the paraboloid and the light cone, leading to various applications to the Schrodinger and the wave equations. I will explain some subsequent developments, including important contributions by Guth, Maldague and Wang, my joint work with Shaoming Guo, Zane Li and Pavel Zorin-Kranich, and joint work with Andrew Hassell, Pierre Portal and Jan Rozendaal.

In this talk, we present the global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension $d=2$ for large initial data in $H^s,s>0$. 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. In addition, we construct an approximate periodic solution showing ill-behavior of the flow map at the $L^2$ regularity. This is based on joint works with Sebastian Herr.