Seminar

When a solution to the wave equation encounters a conic singularity (or, for purposes of visualization, a corner in the plane), there is typically a diffractive effect, where singularities striking the cone tip produce a whole spherical wave of singularities emanating from it. This diffracted wave of course does not exist for the trivial cone (Euclidean space), or, by the method of images, by orbifold quotients of it. We conjecture that the only cones on which there are no diffracted singularities are in fact given by these quotients, i.e., they are cones over spherical space forms. We are able to prove this conjecture when the link of the cone is an analytic surface. In higher dimensions, we are able to prove that the link of an analytic, nondiffractive cone must be a Zoll manifold. This is joint work with Jeff Galkowski.