We will continue the proof of Paving Conjecture and finish the proof of Kadison-Singer problem using Marcus-Spielman-Srivastava theorem. Then, we will move to the proof Marcus-Spielman-Srivastava theorem. The proof breaks into two parts: The first part is about the control of mean characteristic polynomial. We start by examining the fact that characteristic functions of linear sum of positive semi-definite rank one fixed random Hermitian matrices are real stable random polynomials. Using properties of real stable polynomials, we will compare the maximal root of characteristic polynomial and that of the expected characteristic polynomial. The second part is about non-linear first-moment method. If time is permits, we will begin with the derivation of certain convexity property of real stable polynomials.
Speaker(s): Han Huang (University of Michigan)