The usual (or additive) Horn problem for matrices asks: given two Hermitian matrices C and D whose eigenvalues are known, what are the possible eigenvalues of their sum C+D? The additive problem was solved in the 1990's due to work of several authors including Klyachko, Knutson and Tao. The so-called mutliplicative Horn problem for matrices asks: given two n-by-n matrices A and B whose singular numbers are known, what are the possible singular numbers of their product AB? This problem was solved by Klyachko for invertible matrices, using the solution of the additive Horn problem. The answer is, curiously, essentially the logarithm of the solution for the additive Horn problem. The infinite dimensional version of the additive Horn problem (by which we mean, in finite von Neumann algebras), was solved a few years ago and the statement can phrased as a limit of the solutions in finite-dimensions. In more recent results, we show that also the multiplicative Horn problem (also for not-necessarily-invertible operators) has a solution in finite von Neumann algebras, which is a limit of the finite-dimensional solutions. Both of these results can be seen as modest evidence for a positive answer to Connes' embedding problem. (joint work with H. Bercovici, B. Collins, and W.S. Li)