The optimal transport problem consists in finding the cheapest way to transport a distribution of mass from one place to another.
Apart from its natural applications in economics, optimal transport maps provide "efficient" changes of variables that have been used to
investigate the stability of minimizers to geometric/functional inequalities. However, in some cases, optimal maps may not alway been
the "right" choice and other changes of variables may be more suitable. For instance, this happens to be the case in the study of
universality in random matrix theory. In this talk, which is intended for a general audience, I'll give an overview of these results.