The main goal of this project is to analyse the evolution of the point process of zeros of random polynomials as these undergo the heat flow. The problem belongs to an active line of research studying how differential operators acting on random functions affect their zeros. Even though the problem seems innocent at first glance, this project is rich of fascinating phenomena and features surprising links to other areas, like random matrix theory, free probability, (optimal) transport, differential equations and complex analysis that we shall explore and utilize. Initiated by the desire to connect the circular law, elliptic law and semicircle law from random matrix theory via a dynamic of the eigenvalues and as a push-forward under a transport map, we study the corresponding problem for the zeros of random polynomials with independent coefficients as they undergo the heat flow. Core questions are: - How does the geometry of the limiting (as the degree tends to infinity) zero distribution with the heat flow? - Is it possible to establish (global) universality for the limiting distribution? - What can we say about the dynamics of individual zeros or about the microscopic point process after the heat flow? - Does the theory extend to other differential operators and other models of (random) functions?