  The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of
a free group factor is contained in a unique maximal amenable subalgebra. This
conjecture is motivated by related results in Popa's deformation/rigidity
theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an
approach to this conjecture in terms of so-called strong convergence of random
matrices by formulating a conjecture which is a natural generalization of the
Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom
conjecture. This random matrix conjecture is related to recent work of
Collins-Guionnet-Parraud.
