I will present recent joint work with Beomjong Kwak on the Schrödinger equation with the cubic (focusing and defocusing) nonlinearity posed on the two-dimensional torus. First, the optimal L^4 Strichartz estimate will be discussed, leading to global well-posed for initial data which is small in L^2. This is based on a new counting argument which relies on incidence geometry, in particular the Szemeredi-Trotter Theorem. Second, the extension to large initial data will be discussed, which is based on an Inverse Strichartz Theorem. This is based on methods involving higher order Fourier analysis.