The Solution of the Kadison-Singer Problem

TL;DR

This paper explains the Kadison-Singer problem, introduces the interlacing families of polynomials framework, and discusses how it was used to solve the problem, with added pedagogical annotations.

Contribution

It provides a comprehensive, annotated explanation of the solution to the Kadison-Singer problem using interlacing polynomials, aimed at educational clarity.

Findings

The Kadison-Singer problem was solved using interlacing families of polynomials.

The paper offers pedagogical insights and examples on the problem and its solution.

It connects $C^*$ algebras and spectral graph theory in the context of the problem.

Abstract

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.