Variational Problems and Dynamics in Spaces of Large Dimensions

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This project investigates a broad array of challenging problems. One of them is on the theory of optimal transport, which dates back to Gaspard Monge in 1781. The theory studies, for instance, the optimal way to move a pile of sand to an excavation, where optimality is measured against a cost function which may be proportional to the distance traveled. Many works led to connections with partial differential equations, fluid mechanics, geometry, probability theory and functional analysis. Currently, optimal transport enjoys applications in signal and image representation, inverse problems, cancer detection, shape and image registration, and machine learning, to name a few. The project relies on the theory of optimal transport to make advances in games which consist of a large number of players, and keeps a focus on equations such as the so-called master equation pioneered by Lasry and Lions. It also features variational problems and dynamics of systems of finitely many and infinitely many particles and also deals with non-commutative optimal transport theory. The project offers training opportunities for undergraduate students, graduate students, and postdoctoral researchers, as well as learning seminars. The students are involved in a research program that aims to encourage interactions between mathematicians, computer scientists, and engineers. The project studies Hamilton-Jacobi equations with non-local terms, known to be challenging, especially when the non-locality appears in the gradient of the unknown function. In this context, the so-called Lasry-Lions monotonicity condition allowed to obtain the first well-posedness result on the master equation. In a previous research project, an alternative condition was proposed, which made it possible for the first time to handle non-separable Hamiltonians. This prior work relied heavily on the presence of the so-called individual noise. This project explores new ideas for handling non-separable Hamiltonians with only common noise in the master equations. It also initiates studies on inverse optimal transport and inverse Mean Field Games problems, with the goal of achieving desired outcomes by designing appropriate Lagrangians. Working on bounded domains makes it more difficult to predict the Lagrangians needed to produce desirable Nash equilibria in Mean Field Games. The project identifies minimal information on the boundary needed to deal with non-smooth augmented Lagrangians which appear in the resolution of these inverse problems. The project also considers inverse optimal transport problems in unbounded sets. Another part of this project is on partial differential equations on graphs and relies on a metric manifold of discrete densities. The project develops a well-posedness theory as well as a stability property for differential equations on graphs. While the role of optimal transport is well established in classical mechanics, establishing these ideas in the quantum setting, opens doors to several new avenues of research. For instance, one faces the challenge of identifying the non-commutative common noise operator, and studying its properties. To achieve these goals, there is a need to continue unearthing new properties of the Biane-Voiculescu's transport distance, and study non-commutative dynamical systems. Non-commutative optimal transport theory offers the appropriate setting for studying dynamics of random matrices, whose sizes will later tend to infinity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.