We will explain our recent solution of the Kadison-Singer Problem and the equivalent Bourgain-Tzafriri and Paving Conjectures. We will begin by introducing the method of interlacing families of polynomials and use of barrier function arguments to bound the roots of polynomials. As a simple demonstration of these techniques, we will present a very simple proof of the Restricted Invertibility Theorem of Bourgain and Tzafriri.

To prove the Paving Conjecture, we introduce the Mixed Characteristic Polynomial of a collection of matrices, and use the theory of Real Stable polynomials and multivariate generalizations of the barrier function arguments to bound their roots.

This is joint work with Adam Marcus and Nikhil Srivastava.