Connes' embedding problem asks whether every separable II_1-factor can be embedded in the ultrapower of the hyperfinite II_1-factor; this is equivalent to asking whether every finite set in every II_1-factor has microstates. We relate this to questions concerning the possible spectral distributions of a+b, where a and b are self-adjoint elements in a II_1-factor having given spectral distributions. The finite-dimensional version of the spectral distribution question was solved by Klyatchko, Totaro, Knudson and Tao, in terms of inequalities first formulated by Horn.

This is joint work with Benoit Collins.