UCLA

We investigate the problem of finding explicitly a transport map from the $q$-Gaussians to the free group factors. To whit, we introduce a new graphical structure called semi-knots--diagrams which contain half the information of knots. With these we find semi-knot formulae for the conjugate system, and show that there is a formal semi-knot formula for the transport map. We further introduce free moment measures, which extend the classical notion, and show that a broad class of free Gibbs laws are free moment measures, in joint work with June Bahr.