One of the most successful lemmas of modern Harmonic Analysis is that "curvature induces decay of the Fourier transform". For example, if sources of electromagnetic waves are distributed on a sphere, the resulting waves will point in all possible directions and thus interact destructively; the same would not happen if those sources were distributed on a plane. In the 1970s, leading analysts such as E.M.Stein proposed to investigate and quantify this kind of phenomena. Since the main mathematical tool involved is the Fourier transform, this gave birth to a field of Harmonic Analysis known as "Fourier restriction theory".

This talk is aimed at non-specialists. We will study the following problem: how to optimally distribute wave sources on the sphere, maximizing the size of the corresponding superposition of waves? We will give a fully detailed solution of the 3-d case, then give a brief explanation of why this problem is still open in higher dimension.