Title: Some recent applications of real-rooted polynomials

Abstract: In enumerative, geometric, algebraic and topological combinatorics the inequalities that hold amongst the coefficients of a combinatorial generating polynomial are frequently studied.  Typical questions ask whether or not the coefficient sequence is unimodal, log-concave, alternatingly increasing and/or gamma-nonnegative.  We will discuss some recent results that rely on the real zeros of polynomials to give answers to questions of this type.  The main applications will pertain to polytopal cell complexes and lattice polytopes.  Aside from giving answers, we will also pose some new problems motivated by these results.