I will begin by reviewing results about the evolution of the roots of real-rooted polynomials under repeated differentiation. In this case, the limiting evolution of the (real) roots can be described in terms of the concept of fractional free convolution, which in turn is equivalent to the operation of taking corners of Hermitian random matrices. 

Then I will present new results about the evolution of the complex roots of random polynomials under repeated differentiation—and more generally under repeated applications of differential operators. In this case, the limiting evolution of the roots has an explicit form that is closely connected to free probability and random matrix theory. 

The talk will be self-contained and will have lots of pictures and animations.