it is an old result of polya and benz the backward heat flow preserves the set of polynomials with all real roots. recent results have shown a surprising connection between the evolution of real roots under the backward heat flow and the notion of “free convolution” in free probability. free convolution, in turn, is the operation that allows one to compute the eigenvalue distribution for sums of independent hermitian matrices in terms of the individual eigenvalue distributions.

the story gets even more interesting when one considers polynomials with complex roots. recent work of mine with ho indicates that under the heat flow, the complex roots of high-degree polynomials should evolve in straight lines with constant speed. this behavior also connects to random matrix theory and free probability. i will present some conjectures as well as recent rigorous results with ho, jalowy, and kabluchko.