Abstract:

A few months ago, Herr and Kwak made a remarkable breakthrough where they proved an optimal bound for the periodic Schrodinger equation in 2+1 dimensions. The key was to rephrase the problem using number theory and geometry. For instance, it was relevant to know how many rectangles you can form, if one chooses vertices from a given finite set of points in the Euclidean plane. The resolution of such a question ultimately relies on the topology of $\mathbb{R}^2$. We will discuss these developments and survey some other related open questions.