What does a random walk have to do with PDEs? We present the connection between the continuous limit of a random walk, Brownian motion, and the heat equation. This leads naturally to the study of the fundamental solution of the heat equation, known as the heat kernel. We discuss the heat kernel in gradually more general settings. Along the way we will learn what a Schrödinger operator is, and we will finish with some recent results on the heat kernels of Schrödinger operators, joint with Laurent Saloff-Coste.

Background for this talk will be kept to a minimum, although familiarity with the definitions of a probability distribution and of a Riemannian manifold will be helpful.