**Abstract:** By solving a free analog of the Monge-Amp`ere equation, we prove a non-commutative analog of Brenier's monotone transport theorem: if an $n$-tuple of self-adjoint non-commutative random variables $Z_{1},...,Z_{n}$ satisfies a regularity condition (its conjugate variables $xi_{1},...,xi_{n}$ should be analytic in $Z_{1},...,Z_{n}$ and $xi_{j}$ should be close to $Z_{j}$ in a certain analytic norm), then there exist invertible non-commutative functions $F_{j}$ of an $n$-tuple of semicircular variables $S_{1},...,S_{n}$, so that $Z_{j}=F_{j}(S_{1},...,S_{n})$. Moreover, $F_{j}$ can be chosen to be monotone, in the sense that $F_{j}=mathscr{D}_{j}g$ and $g$ is a non-commutative function with a positive definite Hessian. In particular, we can deduce that $C^{*}(Z_{1},...,Z_{n})cong C^{*}(S_{1},...,S_{n})$ and $W^{*}(Z_{1},...,Z_{n})cong L(mathbb{F}(n))$. Thus our condition is a useful way to recognize when an $n$-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors $Gamma_{q}(mathbb{R}^{n})$ are isomorphic (for sufficiently small $q$, with bound depending on $n$) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport. |