Bounds on Sobolev norms for the defocusing nonlinearSchrödinger equation on general flat tori

We prove Strichartz estimates on general flat $d$-torus for arbitrary $d$. Using these estimates, we prove local wellposedness for the cubic nonlinear Schrodinger equations in appropriate Sobolev spaces. In dimensions $2$ and $3$, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.