Blake Keeler – Spectral Asymptotics and the Heat Kernel – Advanced GMA Seminar

Title: Spectral Asymptotics and the Heat Kernel.

Abstract: In this lecture series, our goal will be to prove a result known as Weyl’s law, which tells us how the Laplace eigenvalues of a compact manifold are distributed. Since eigenvalues are inherently quite difficult to study, we will utilize a “back-door” approach via the heat equation. The heat kernel can be constructed using fairly classical techniques, and much of our time will be spent exploring its properties and using it to develop the spectral theory of the Laplacian. Lecture 1 will cover some preliminary concepts. We begin with a brief overview of the heat equation in Euclidean space, which will inform our intuition for what we expect on manifolds. We will then extend our notion of the Laplacian to Riemannian manifolds, which will allow us to write down an associated heat equation. Then, under the assumption that a fundamental solution to this heat equation exists, we will be able to show that the Laplacian has discrete spectrum and an associated orthonormal basis of eigenfunctions in L^2(M). In lecture 2, we will show that the heat kernel exists by actually constructing it. As a consequence of the construction, we will have an asymptotic expansion of the kernel for small timescales. In lecture 3, we will connect the heat trace to the distribution of eigenvalues using the Karamata Tauberian theorem, and Weyl’s law will follow as a straightforward corollary. Time permitting, we will also discuss some generalizations of Weyl’s law and improvements that can be made in the error term, as well as some relevant current research.