Date & Time: Wednesday, March 26, 2014, 16:00-17:00.
Venue: Ramanujan Hall

Speaker: Nikhil Srivastava, Microsoft Research

Title: The Solution of the Kadison-Singer Problem

Abstract: The Kadison-Singer problem is a question in operator theory which arose in 1959 while trying to make Dirac's axioms for quantum mechanics mathematically rigorous in the context of von Neumann algebras. It asks whether every pure state on a discrete maximal abelian subalgebra of B(H) extends uniquely to a pure state on all of B(H), where H is a separable complex Hilbert space. In the 70's and 80's, it was realized that the linear-algebraic core of the problem lies in understanding when an arbitrary finite set of vectors in $\mathbb{C}^n$ can be partitioned into two disjoint subsets each of which approximate it spectrally.

We give a positive solution to the problem by proving essentially the strongest possible partitioning theorem of this type. The proof is based on two significant ingredients: a new existence argument, which reduces the problem to bounding the roots of the expected characteristic polynomials of certain random matrices, and a general method for proving upper bounds on the roots of such polynomials. The techniques are elementary, mostly based on tools from the theory of real stable polynomials, and the talk should be accessible to a broad audience.

Joint work with A. Marcus and D. Spielman.