Ramanujan Graphs and the Solution of the Kadison-Singer Problem

TL;DR

This paper surveys advanced polynomial techniques used to resolve the Kadison-Singer problem and establish the existence of Ramanujan Graphs of all degrees, highlighting the interlacing families method and its applications.

Contribution

It introduces the interlacing families of polynomials method and demonstrates its effectiveness in solving longstanding problems in graph theory and operator algebras.

Findings

Resolution of the Kadison-Singer problem

Existence of Ramanujan Graphs of every degree

Simplified proof of Bourgain and Tzafriri's principle

Abstract

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.