Kadison–Singer problem

Let $\mathcal{D} \cong \ell^\infty(\mathbb{N})$ be the maximal abelian $C^*$-subalgebra of diagonal bounded linear operators inside the $C^*$-algebra $\mathcal{B}(\ell^2(\mathbb{N}))$ of all bounded linear operators on the separable Hilbert space $\ell^2(\mathbb{N})$. Does every pure state $\rho$ on $\mathcal{D}$ extend uniquely to a pure state on $\mathcal{B}(\ell^2(\mathbb{N}))$?

Solved in the affirmative in June 2013 by Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava (published in Annals of Mathematics in 2015). Rather than working directly with ultrafilters and $C^*$-algebras, they proved Nik Weaver's 2004 finite-dimensional discrepancy conjecture $\mathrm{KS}_2$: if $v_1, \dots, v_m \in \mathbb{C}^d$ satisfy $\sum_{i=1}^m v_i v_i^* = I_d$ and $\|v_i\|^2 \le \delta$ for all $i$, then $\{1, \dots, m\}$ can be partitioned into two sets $S_1, S_2$ such that $\left\|\sum_{i \in S_j} v_i v_i^*\right\| \le \frac{(1 + \sqrt{2\delta})^2}{2}$ for $j = 1, 2$. They proved this bound by analyzing the largest roots of "mixed characteristic polynomials" using their method of interlacing families and multivariate real-stable polynomials.