Seminar

We survey the method of interlacing families of polynomials, with an emphasis on applications to functional analysis and combinatorics. This is a variant of the probabilistic method which relies on properties of real-rooted multivariate polynomials, and sometimes yields much sharper bounds than traditional probabilistic techniques. We will describe how it can be used to prove a sharp version of Bourgain and Tzafriri's Restricted Invertibility Theorem, as well as the existence of Ramanujan graphs of all degrees, and discuss connections to other areas such as operator theory, random matrix theory, and free probability.