Density Functional Theory (DFT) is the standard approach to quantum chemistry in simulations with more than a dozen electrons. The central object in this theory, the Hohenberg-Kohn (Levy-Lieb) functional, is in fact a regularized multi-marginal optimal transport problem in disguise.

In this talk, I will present an overview of the connection between Optimal Transport with Coulomb cost and DFT as well as discuss a conjecture posed by Seidl, Gori-Giorgi and Savin on the existence of Monge-type minizers for Optimal Transport problem with Coulomb costs, and the existence of fractal-like minimizers for a wide class of repulsive costs functions.