- MSRI: Simons Auditorium
How do geometric features affect physics? In this talk I will start with a simple example in which we solve the initial value problem for the homogeneous heat equation on a half line, with the Dirichlet boundary condition at the origin. We will see how the Schwartz kernel of the fundamental solution, known as the heat kernel, ``feels'' the boundary at the origin, and how it also feels the boundary condition when we change between Dirichlet and Neumann boundary conditions. This is of course a very special example, because we can compute everything explicitly. However, even for a simple bounded domain in the plane, there is no analogous closed-form expression for the heat kernel. So, if we wish to understand how the geometric features of a domain in the plane affect the physical flow of heat, we need more tools: the tools of microlocal analysis! The main focus of this talk will be the microlocal construction of the heat kernel for curvilinear polygonal domains both in the plane and also in surfaces. This construction will introduce the concepts of ``blowing up'' and the incredible usefulness of ``blowing up.'' We will see how the microlocal construction can be applied to show that the heat kernel ``feels'' geometric features like curvature, boundary, and the presence (or lack thereof) of corners.