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.
- Pub Date:
- August 2014
- Mathematics - Spectral Theory;
- Mathematics - Combinatorics;
- Mathematics - Operator Algebras;
- A version of this paper will appear in the proceedings of the 2014 International Congress of Mathematicians