• Optimal Transportation
• Winter School

• Schedule
• Participants

Recent years have witnessed a lot of progress in the understanding of the two-dimensional Discrete Gaussian Free Field (DGFF). In my lectures I will discuss the asymptotic law of the extreme point process for the DGFF on lattice approximations of bounded open sets in the complex plane with zero boundary conditions outside. This limit turns out to be a a Poisson process with random intensity in the spatial coordinate and the Gumbel intensity in the field-value coordinate decorated by independent samples from a (dependent) cluster process. The random intensity measure obeys a canonical transformation rule under conformal maps of the domain and can be linked to the measure representing the volume form of the critical 2D Liouville Quantum Gravity. Based on recent joint work with Oren Louidor (Technion, Haifa).

I will begin by explaining the connection quantum mechanics (QM) - density functional theory (DFT) - optimal transport (OT), without assuming prior knowledge of either of the three. Roughly, DFT is a simplification of many-electron QM which makes it omputationally feasible; and OT is a simplification of DFT which arises by taking a semiclassical limit. As a result, one arrives at multi-marginal OT with Coulomb cost 1/|x-y|. Next, I will discuss the qualitative behaviour of the ensuing OT problem and in particular the question of whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (the intriguing answer, which differs from the case of positive power costs, is Yes for N=2 particles, maybe but not always for N>2 particles, and Never for infinitely many particles). After that, we will look at some recent applications of the OT theory to the simulation of real molecules. Finally I will derive the fact that in the large N limit, the highly correlated N-point optimizers weakly converge to independent states. In treating these topics I will contrast physicist's, OT theory and probabilistic approaches. For instance the Gangbo-McCann formula for the optimal map in terms of the Kantorovich potential is arrived at in an intriguingly simple way by physicists; whereas the large N limit can be nicely understood from the point of view of exchangeable sequences of random variables and de Finetti's theorem. Much of what I'll cover is based on joint work with Huajie Chen, Codina Cotar, Claudia Klueppelberg, Christian Mendl, and Brendan Pass over the past 3 years, e.g. Cotar, F., Klueppelberg, Comm. Pure Appl. Math 66, 548-599, 2013; Chen, F., Mendl, J. of Chem. Theory and Comp. 10, 4360, 2014; Cotar, F., Pass, to appear in Calc.Var.PDE, arXiv 1307.6540, 2013.

Optimal transport has become a powerful tool to attack non-smooth problems in analysis and geometry. A key role is played by the 2-Wasserstein metric, which induces a rich geometric structure on the space of probability measures. This structure allows to obtain gradient flow structures for diffusion equations and to exploit geodesic convexity of the entropy. However, in discrete settings the 2-Wasserstein metric degenerates and the theory seems to break down. In recent years a new class of transport metrics has emerged, which allows one to apply ideas from optimal transport to a number of different situations, which had been outside the scope of the existing theory. In these lectures we give an overview of these developments in the setting of Markov chains, chemical reaction networks, and quantum Markov processes.

In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently.

In this series of talks I will build an understanding of the modelling arguments that underlie the use of energies, entropies, and gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. I will explain all these concepts in detail in the lectures.