  We survey the techniques used in our recent resolution of the Kadison-Singer
problem and proof of existence of Ramanujan Graphs of every degree: mixed
characteristic polynomials and the method of interlacing families of
polynomials. To demonstrate the method of interlacing families of polynomials,
we give a simple proof of Bourgain and Tzafriri's restricted invertibility
principle in the isotropic case.
