Blake Keeler (UNC Chapel Hill), GMA & Visions Seminar

Title: Fluid-Construction of the Hadamard Parametrix for the Wave Operator on a Riemannian Manifold

Abstract: A common object of interest in the study of PDEs is the fundamental solution of a differential operator. A fundamental solution allows one to invert the differential operator in question. In Euclidean space one can explicitly construct fundamental solutions for the wave operator, but on a general Riemannian manifold the wave operator has variable coefficients, and it is therefore much more difficult to construct a true fundamental solution. Thus, we look for the next best thing, called a parametrix, which is an approximate inverse to the wave operator in the sense that if we apply the wave operator to the parametrix, we obtain the identity map plus an operator which has smooth kernel. We will demonstrate a method for explicitly constructing this parametrix as a combination of increasingly smooth distributions associated with the flat wave operator which have useful recursion properties.

Location: Phillips Hall 381, 4:00 pm – 5:00 pm