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We consider the problem of large data scattering for the defocusing cubicnonlinear Schr\"odinger equation on $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$.This equation is critical both at the level of energy and mass. The keyingredients contain a global-in-time Stricharz estimate, resonant systemapproximation and profile decomposition. Assuming the large data scattering forthe 2d cubic resonant system, we can prove the large data scattering for thisproblem.

Global well-posedness and scattering for the defocusing cubic Schr\"odinger equation on waveguide $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$