Analysis Seminar

Classical billiard trajectories in the exterior of a convex polygon escape immediately to infinity, so one might reasonably conjecture that high-frequency waves should decay very rapidly. It turns out, however, that waves can diffract back and forth between vertices, and this diffractive effect is associated with families of resonances, slow decaying solutions to the wave equation. This effect of "diffractive trapping" shows up in a number of interesting geometric settings where singularities are present in coefficients or geometry. I will describe some of these settings, and discuss the trace formulae which are one of the essential tools for showing existence of resonances.