The Kadison-Singer problem originally is a question about maximal abelian subalgebras of B(H), the bounded operators on a Hilbert space. It turned out to be equivalent to several famous questions throughout mathematics. A particularly innocent looking equivalent statement is the following: Does there exist an integer k such that every square matrix A of size n with zeros on the diagonal can be partitioned into k by k blocks (w.r.t. an arbitrary partition of 1,...,n into k subsets) such that all the diagonal blocks have norm at most half the norm of A?
In 2013, the Kadison-Singer problem was solved by Marcus, Spielman and Srivastava, using totally unexpected methods. I will explain the original problem, some of its reformulations, the solution of Marcus-Spielman-Srivastava, and a much more general conjecture for maximal abelian subalgebras of arbitrary von Neumann algebras, as proposed in a recent joint work with Sorin Popa.