Herr and Kwak recently established sharp estimates for L^4 norms of solutions to the periodic Schrodinger equation in 2+1 dimensions by counting rectangles in the plane. Surprisingly, their proof relies heavily on the topology of $\mathbb{R}^2$. They made clever use of the Szemeredi-Trotter theorem (twice!) to prove their L^4 bound, which in turn allowed them to prove global well-posedness for the cubic nonlinear Schrödinger equation for periodic initial data in the mass-critical dimension d=2.  I will try to explain their work and highlight connections to Fourier analysis.