Abstract
In the first lecture, I will prove the real rootedness of a large class of expected characteristic polynomials, completing Jan's proof of the existence of Ramanujan graphs. I will then explain two univariate techniques for bounding zeros of expected characteristic polynomials: the first is from algebraic combinatorics and the second is related to free probability theory.
In the second lecture, I will explain the proof of Weaver's conjecture, a question in discrepancy theory which is known to imply a solution of the Kadison-Singer problem. The key ingredient is a multivariate technique for controlling the roots of expected characteristic polynomials. I will conclude by describing connections to other research areas represented in this program, as well as some open questions.