Transport of measure refers to transforming one probability distribution $\mu$ into another probability distribution $\nu$ by pushing forward through some function $f$ such that $f_* \mu = \nu$, and a transport function $f$ is optimal if it minimizes $\int |f(x) - x|^2\,d\mu(x)$, and the minimum value of this quantity is the square of the Wasserstein distance. In the non-commutative setting, the probability distributions are replaced by traces on a universal free product $C[-R,R]^{*d}$, or equivalently functionals on a non-commutative polynomial algebra that represent the moments of tuples of operators from a non-commutative probability space, known as non-commutative laws. In the non-commutative setting, non-isomorphism of von Neumann algebras associated to $\mu$ and $\nu$ presents an obstacle to transport. The space of non-commutative laws with Biane and Voiculescu's non-commutative Wasserstein distance is furthermore much more wild than its commutative analog. However, for certain nice random matrix models (those associated to free Gibbs laws) one can obtain smooth non-commutative transport and hence isomorphism of the underlying von Neumann algebras. Moreover, the Monge-Kantorovich dual formulation of the Wasserstein distance adapts to the non-commutative setting.

This is based on joint work with Li and Shlyakhtenko, and with Gangbo, Nam, and Shlyakhtenko.