Abstract: A couple of years ago, with M. Goldman we devised a new approach to the regularity theory for Optimal Transportation that mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a nondimensional smallness condition, the displacement is close to the gradient of a harmonic function.

Probably the main advantage of this variational regularity theory over the one based on maximum principle — and attached to the name of Caffarelli — is that it does not require any regularity of the involved measures. Hence it can be applied to the popular matching problem, where it provides regularity on large scales

This lecture was part of the biannual Abel Symposium.

This year the title of the symposium was Partial Differential Equations waves, Nonlinearities and Nonlocalities.

The symposium was funded by
The Norwegian Academy of Sciences and Letters via the Abel board and The Norwegian Mathematical Society
NTNU Norwegian University of Science and Technology
Research Council of Norway via the grant Waves and Nonlinear Phenomena
Trond Mohn Foundation