# The solution of the Kadison-Singer Problem

# The solution of the Kadison-Singer Problem

In 1959, Kadison and Singer posed a problem in operator theory that has reappeared in many guises, including the Paving Conjecture, the Bourgain-Tzafriri Conjecture, the Feichtinger Conjecture, and Weaver's Conjecture. I will explain how we solve the Kadison-Singer Problem by proving Weaver's Conjecture in Discrepancy Theory. I will explain the "method of interlacing polynomials" that we introduced to solve this problem, and sketch the major steps in the proof. These are the introduction of "mixed characteristic polynomials"---the expected characteristic polynomials of a sum of random symmetric rank-1 matrices, the proof that these polynomials are real rooted, and the derivation of an upper bound on their largest roots. These techniques are elementary, and should be understandable by a broad mathematical audience. This is joint work with Adam Marcus and Nikhil Srivastava.