Lecture Notes for Interlacing Families

In these notes we follow the two papers [6, 7] where Marcus, Spielman and Srivastava developed a method that they coined the method of interlacing families. In [6] they proved the existence of infinite families of bipartite Ramanujan graphs for all degrees, and in [7] they provided a positive answer to the notorious Kadison– Singer problem. We take the opportunity to have a slightly more general and unified approach. We consistently use hyperbolic polynomials instead of determinants and barrier function arguments. One benefit of this is that the exposition is self-contained. We don’t for example use the Helton–Vinnikov theorem on determinantal representations, instead we use simple concavity properties of hyperbolic polynomials.