Lecture 1 : Introduction , the Matching Polynomial

The Geometry of Polynomials, also known as the analytic theory of polynomials, refers the study of the zero loci of polynomials with complex coefficients (and their dynamics under various transformations of the polynomials) using methods of real and complex analysis. The course will focus on the fragment of this subject which deals with real-rooted polynomials and their multivariate generalizations, real stable and hyperbolic polynomials. We will explore this area via its interactions with questions in combinatorics, probability, and linear algebra, some of which will be algorithmically motivated. Specifically, we will be interested in the following kind of question: how are the properties of a graph/matrix/probability distribution reflected in the zeros of various generating polynomials associated with it? We begin by presenting two of the simplest examples of this interplay.