It is an old result of Polya and Benz that applying the backward heat flow to a polynomial with all real zeros gives another polynomial with all real zeros. Much more recently, the limiting behavior of the real zeros as the degree goes to infinity has been worked out, with a surprising connection to random matrix theory. The situation is more complicated if we use the forward heat flow—in which case, the zeros will not remain real—or if we apply the heat flow to a polynomial with complex roots. Nevertheless, there is still a conjectural connection to random matrix theory. Consider, for example, the circular law in random matrix theory: If a random matrix Z has i.i.d. entries, its eigenvalues will be asymptotically uniform over a disk. The heat flow then conjecturally changes the circular law into the elliptical law: Applying the heat flow to the characteristic polynomial of Z should give a new polynomial whose zeros are asymptotically uniform over an ellipse. While the random matrix case remains a conjecture, we have rigorous results for random polynomials with independent coefficients. This is joint work with Ching Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The talk will be self-contained and have lots of pictures and animations.