This is an announcement for the paper "Ramanujan graphs and the solution of the Kadison-Singer problem" by Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava,.

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.

Archive classification: math.SP math.CO math.OA

Mathematics Subject Classification: 05C50, 46L05, 26C10

Remarks: A version of this paper will appear in the proceedings of the 2014

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1408.4421

or

http://arXiv.org/abs/1408.4421