Beyond the Kadison-Singer problem

The aim of this talk is to give an overview of the solution of the Kadison-Singer problem (1959) by Marcus, Spielman, and Srivastava (2015). This problem was known to be equivalent to a large number of problems in analysis such as Anderson paving conjecture (1979), Bourgain-Tzafriri restricted invertibility conjecture (1991), Weaver’s conjecture (2004), and Feichtinger’s conjecture (2005). The amazing solution of this problem uses methods which are very far from analyst's toolbox such as real stable polynomials, interlacing families of polynomials, or multivariable barrier method. The solution involves a key concept of a mixed characteristic polynomial, which is an multilinear analogue of the usual characteristic polynomial. At the same time, the Kadison-Singer problem shows the unity of mathematics as it connects a large number of areas: operator algebras (pure states), set theory (ultrafilters), operator theory (paving), random matrix theory, linear and multilinear algebra, algebraic combinatorics (real stable polynomials), functional analysis (frame theory), and harmonic analysis (exponential frames). In the last part of the talk we discuss several developments beyond the Kadison-Singer problem.