The Solution of the Kadison-Singer Problem

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.