The Ubiquitous Heat Kernel

Like others, we came to the heat kernel via one direction of mathematics. However, as we progressed in that direction, we realized that the heat kernel plays a central role in almost all directions we can think of. That it bears a name related to physics only indicates that it was originally discovered in connection with heat. But even in physics, its significance goes way beyond giving a mathematical model for heat distribution. The name cannot be changed — it’s too late for that — but the impression some people may have that an occurrence of a certain kernel called the heat kernel means that one is necessarily doing physics is a false impression. Maybe one is and may be one isn’t. Also one may be using the language of physics and heat distribution simply as a convenient backdrop for some theorems giving bounds or asymptotic estimates for the heat kernel, as we do below (“diffusion”). For a recent example, cf. Norris [Nor 97], with a notable bibliography. The issue of Acta where this paper appears actually contains two articles on the heat kernel (the other is [BiJ 97]), constituting 3/5th of the issue, which is one way of being ubiquitous. Less anecdotically, there is a universal gadget which is a dominant factor practically everywhere in mathematics, also in physics, and has very simple and powerful properties. We have no a priori explanation (psychological, philosophical, mathematical) for the phenomenon of the existence of such a universal gadget.