Complexity seminar

On the distribution of runners on a circle

I will discuss the following problem: consider runners on a circular track, running with constant speeds such that k of the speeds are distinct. Does it have to be the case that, at some point in time, their distribution on the circle is far from uniform? I will give an almost optimal solution, as a function of k. This has an interesting application to the distribution of complex arguments of roots of univariate polynomials, and the Real Tau Conjecture in arithmetic circuit complexity.