Combinatorics and the Geometry of Polynomials

This mini course will focus on the special structure of polynomial with positive coefficientes. These polynomial arise naturaly in combinatorial settings (say as generating polynomials) or in probabilistic settings (as probability generating functions) and, perhaps surprisingly, really do have special properties.

We will start with some results describing the structure of the set of zeros of polynomials with positive coefficients. This for example, will allow us to deduce the following gem: Let ƒ be a polynomial, if ƒm has all positive coefficients for some m, then ƒn has all positive coefficients for every large enough n. Fron here we will veer onto a more probabilistic track, where we think of positive polynomial as probability generating functions for some random variable on the integers and will consider questions motivated from the perspective of probability theory. For example, what is the conncetion between a random variable being (approximately) normally distributed and the roots of the probability generating functions? 
I will assume no prerequisities beyond a tiny bit of familiarly with complex analysis and develop any further techniques from scratch.