Ramanujan Graphs and the Solution of the KadisonSinger Problem
Abstract
We survey the techniques used in our recent resolution of the KadisonSinger 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.4421
 Bibcode:
 2014arXiv1408.4421M
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Combinatorics;
 Mathematics  Operator Algebras;
 05C50;
 46L05;
 26C10
 EPrint:
 A version of this paper will appear in the proceedings of the 2014 International Congress of Mathematicians