Time-dependent moments from the heat equation

We present a new connection between the classical theory of moments and the theory of partial differential equations (arXiv:2108.03505). For the classical heat equation we compute the moments of the unique solution. These moments are polynomials in the time variable, of degree comparable to the degree of the moment, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence. In the case of moment sequences, the polynomials trace a curve (the heat curve) which remains in the moment cone for positive time, but may wander outside for negative times. We also study how the determinacy of a moment sequence behaves along the heat curve. We show that for several other partial differential equations we have access to the time-dependent moments without calculating the solution of the partial differential equation. The talk is based on a joint work with R. Curto.