Nikhil Srivastava: Interlacing Families, Mixed Characteristic Polynomials, and the Kadison-Singer Problem

The Kadison-Singer problem is a question in operator theory which arose while trying to make certain assertions in Dirac's formulation of quantum mechanics mathematically rigorous. Over several decades, this question was shown to be equivalent to several discrepancy-type conjectures about finite matrices, with applications in signal processing, harmonic analysis, and computer science. We prove a strong variant of the conjecture due to Nik Weaver, which says that every set of vectors satisfying some mild conditions can be divided into two sets each of which approximates the whole set spectrally. The proof is based on two significant ingredients: a new existence argument, which reduces the problem to bounding the roots of the expected characteristic polynomial of a certain random matrix derived from the vectors, and a systematic method to prove sharp bounds on the roots of such polynomials. The techniques are mostly elementary and based on tools from the theory of real stable polynomials.
Joint work with Adam Marcus and Dan Spielman.