COLLOQUIUM DI DIPARTIMENTO
Wednesday 1 March 2023, time: 15.00 - 16:00
Aula Dal Passo
Felix Otto (Max-Planck-Institut, Lipsia)
Title: Optimal matching, optimal transportation, and its regularity theory
Abstract: The optimal matching of blue and red points is prima facie a combinatorial 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-Koml\'os-Tusn\'ady].
Optimal matching is a discrete version of optimal transportation between the two empirical measures. While the matching problem was first formulated in its Monge version (p=1), the Wasserstein version (p=2) connects to a powerful continuum theory. This connection to a partial differential equation, the Monge-Ampere 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-Ampere 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 displacement 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).
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