GRK-Kolloquium, Prof. Dr. Felix Otto

Optimal matching, optimal transportation, and its regularity theory

The optimal matching of blue and red points is prima facie a combina-
torial problem. It turns out that when the position of the points is random,
namely distributed according to two independent Poisson point processes in
d-dimensional space, the problem depends crucially on dimension, with the
two-dimensional case being critical (Ajtai-Komlós-Tusnády).

Optimal matching is a discrete version of optimal transportation between
the two empirical measures. While the matching problem was first formu-
lated in its Monge version (p=1), the Wasserstein version (p=2) connects to
a powerful continuum theory. This connection to a partial differential equa-
tion, the Monge-Amp`ere equation as the Euler-Lagrange equation of optimal
transportation, enabled Parisi et. al. to give a finer characterization, made
rigorous by Ambrosio et. al..

The idea of Parisi et. al. was to (formally) linearize the Monge-Ampère
equation by the Poisson equation. I present an approach that quantifies this
linearization on the level of the optimization problem, locally approximating
the Wasserstein distance by an electrostatic energy. This approach (initiated
with M. Goldman) amounts to the approximation of the optimal displace-
ment by a harmonic gradient. Incidentally, such a harmonic approximation
is analogous to de Giorgi’s approach to the regularity theory for minimal
surfaces. Because this regularity theory is robust — measures don’t need to
have Lebesgue densities — it allows for sharper statements on the matching
problem (work with M. Huesmann and F. Mattesini).