I will present a recent result showing that the direct product group $\Gamma = \mathbb F_2 \times \mathbb F_2$ is not Hilbert-Schmidt stable. Specifically, $\Gamma$ admits a sequence of asymptotic homomorphisms (with respect to the normalized Hilbert-Schmidt norm) which are not perturbations of genuine homomorphisms. As we will explain, while this result concerns unitary matrices, its proof relies on techniques and ideas from the theory of von Neumann algebras. We will also explain how this result can be used to settle in the negative a natural version of an old question of Rosenthal concerning almost commuting matrices. More precisely, we derive the existence of contraction matrices $A,B$ such that $A$ almost commutes with $B$ and $B^*$ (in the normalized Hilbert-Schmidt norm), but there are no matrices $A’,B’$ close to $A,B$ such that $A’$ commutes with $B’$ and $B’^*$.