Alladi Ramakrishnan Hall

The Solution of the Kadison-Singer Problem

Nikhil Srivastava

Microsoft Research, Bangalore

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.