Ward of the Waves: Fourier, Intertwiners, and Caustics

In the last lecture, the Fourier transform appeared not just as a computational device, but as part of the symmetry machinery of the Heisenberg group. In this talk I turn to a concrete example where that viewpoint becomes visible on the page: the Ward integral formula for solutions of the wave equation on a Rosen plane wave. I will begin in a largely self-contained way, writing down the formula and explaining why it solves the equation with very little prerequisite background. I will then reinterpret it using the Fourier machinery from the previous lecture: after Fourier transform in the null variable, the wave equation becomes a uu-dependent Schrödinger equation, and the Ward solution arises from the corresponding intertwiner picture. Time permitting, I will explain how changes of polarization lead naturally to caustics and to the Maslov index, which records the associated phase jumps. The aim is to show how an explicit formula on a curved spacetime fits into a broader symplectic and representation-theoretic framework.