Abstract: Motivated by the desire to describe the evolution of zeros of (random) polynomials under differential operators, we study the zeros of a deterministic polynomial power P^n undergoing the (holomorphic) heat flow after time t. We determine the complex limiting empirical zero distribution of the heat-evolved polynomial power P_t^n as n tends to infinity. For small time, zeros turn out to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a semicircle law through the initial center of mass. These descriptions rely on geometric and potential theoretic tools, while we shall also discuss the dynamic from the perspective of PDE's, dynamical systems and (maybe) free probability. This talk is based on joint work with Antonia Höfert and Zakhar Kabluchko.
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