Integrable systems, the heat equation and bispectrality

The fundamental solution of the heat equation on the real line has been linked to the theory of solitons from early days, by providing a tool for obtaining integrals of motion for the KdV equation. In this talk I will explain how one can go in the opposite direction and compute the heat kernel using Sato's theory. This approach allows to establish different properties of Hadamard's coefficients and it can be used to study the heat kernel where very little or even nothing is known (e.g. if we consider the heat equation on the integers). In particular, it allows to prove that finite heat kernel expansions characterize the rational solutions of the KdV/Toda lattice hierarchies, which arise within the context of the bispectral problem.