Unsolved Problems in Polynomial Dynamics

Abstract:  Take a polynomial of degree n having all its roots on the real axis. Suppose n is really large and that those roots are distributed roughly like, say, a Gaussian. If you differentiate this polynomial n/2 times, the result is a polynomial of degree n/2 having n/2 roots on the real line – what can be said about the distribution of these n/2 roots? Do they still look like a Gaussian?  An old conjecture, dating back at least to Polya, is that they become more regular. Even though this is a question about polynomials, extremely little is known. I am describing the state of the art and a recently discovered dynamical system that seems to be able to predict the evolution. An old idea of Gauss plays a role; in the end, there are surprising connections to fluid dynamics.  Do roots of polynomials flow like a fluid?  The entire talk is elementary (we know, nothing, really – what we can prove requires only high school algebra and calculus) and there are many open research problems.