In 1950, de Bruijn studied the effect of evolving the Riemann zeta function (or more precisely, a closely related function known as the Riemann xi function) by the (backwards) heat equation. His analysis, together with later work by Newman, showed that there existed a finite constant Lambda, at most 1/2 in value, such that the Riemann hypothesis for this evolved function was true at times greater than or equal to Lambda, and false below that threshold. Thus the Riemann hypothesis for the zeta function is equivalent to Lambda being non-positive. Recently, in joint work with Brad Rodgers, I was able to establish the complementary estimate that Lambda is non-negative, confirming a conjecture of Newman; thus, the Riemann hypothesis for zeta, if true, is only “barely so”. The proof relies on an analysis of the dynamics of zeroes of entire functions under heat flow; it turns out that as one evolves forward in time, the zeroes “freeze” into approximate arithmetic progressions, while if one evolves backwards, the zeroes “vaporize” to leave the critical line. In followup work in an online collaborative “Polymath” project, the upper bound on Lambda has also been improved. We describe these results and their proofs in this talk.