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A central question in the field of optimalÌýtransportÌýstudies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimalÌýtransportÌýmaps on a Euclidean metric space areÌýLipschitz.ÌýLipschitzÌýregularity is a desirable property because it allows for theÌýtransferÌýof analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimalÌýtransport.

In this talk, we will further exploreÌýtransportÌýmaps withÌýlowÌýdistortion. The key point which we shall highlight is that, forÌýlowÌýdistortionÌýmappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction ofÌýtransportÌýmaps based onÌýprobabilisticÌýflows, and introduce a set of techniques to analyze theirÌýdistortion. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces.

Some applications, such as functional inequalities and normal approximations will also be discussed.

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