  These lecture notes are meant to accompany two lectures given at the CDM 2016
conference, about the Kadison-Singer Problem. They are meant to complement the
survey by the same authors (along with Spielman) which appeared at the 2014
ICM. In the first part of this survey we will introduce the Kadison-Singer
problem from two perspectives ($C^*$ algebras and spectral graph theory) and
present some examples showing where the difficulties in solving it lie. In the
second part we will develop the framework of interlacing families of
polynomials, and show how it is used to solve the problem. None of the results
are new, but we have added annotations and examples which we hope are of
pedagogical value.
