Interlacing Families and the Kadison–Singer Problem Contents
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Ramanujan Graphs and the Solution of the Kadison–Singer Problem
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 isotr...
متن کامل2 6 Ju n 20 13 Interlacing Families II : Mixed Characteristic Polynomials and The Kadison - Singer Problem ∗
We use the method of interlacing families of polynomials introduced in [MSS13] to prove Weaver’s conjecture KS2 [Wea04], which is known to imply a positive solution to the Kadison–Singer problem via a projection paving conjecture of Akemann and Anderson [AA91]. Our proof goes through an analysis of the largest roots of a family of polynomials that we call the “mixed characteristic polynomials” ...
متن کاملA pr 2 01 4 Interlacing Families II : Mixed Characteristic Polynomials and The Kadison - Singer Problem ∗
We use the method of interlacing families of polynomials introduced in [MSS13] to prove two theorems known to imply a positive solution to the Kadison–Singer problem. The first is Weaver’s conjecture KS2 [Wea04], which is known to imply Kadison–Singer via a projection paving conjecture of Akemann and Anderson [AA91]. The second is a formulation due to Casazza, et al., [CEKP07] of Anderson’s ori...
متن کاملInterlacing families II:Mixed characteristic polynomials and the Kadison–Singer problem
We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison–Singer problem. The first is Weaver’s conjecture KS2, which is known to imply Kadison–Singer via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza et al. of Anderson’s original paving conjecture(s...
متن کاملLecture Notes for Interlacing Families
In these notes we follow the two papers [6, 7] where Marcus, Spielman and Srivastava developed a method that they coined the method of interlacing families. In [6] they proved the existence of infinite families of bipartite Ramanujan graphs for all degrees, and in [7] they provided a positive answer to the notorious Kadison– Singer problem. We take the opportunity to have a slightly more genera...
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