In optimal transportation theory, one looks for the best strategies to transport, say, a pile of dirt, to an excavation, where optimality is measured against a prescribed cost function. In his earlier research, the principal investigator (PI) created new tools for optimal transport theory, a theory which has experienced a mathematical renaissance over the last twenty-five years by weaving together threads from analysis, geometry, partial differential equations (PDEs) and dynamical systems. Motivated by his work on systems with large numbers of particles, the PI, with collaborators, has discovered some connections between optimal transportation, games with a large number of players, and Mean Field Games. These unexpected connections open doors to several new avenues of research, some of which being described in this proposal. The goal of this project is to advance knowledge in various other related topics. This includes the geometry of the space of closed differential forms with the non--linear factorization of differential forms. This proposal involves training students, mentoring postdocs and organizing learning seminars for graduate students. The PI will remain involved in national as well as international activities. These, constitute the broader impacts of the proposed activities.
The PI studies the mathematical aspects of Density Functional Theory (DFT), as well as certain geometric clustering problems originating in combinatorial optimization. What many of these problems have in common, and what is sometimes novel in the approach for studying them, is their formulation on infinite dimensional ``manifolds'' equipped with metric, differential, and symplectic pseudo-structures. This infinite dimensional setting often lead to a new outlook and --at least on the formal level-- to novel and simpler approaches to otherwise complex problems. A typical example is how Otto's calculus turns many nonlinear PDEs into gradient flows of geodesically convex functionals. In a similar fashion, our searching for geodesic of minimal lengths on the set of differential forms, leads to a new concept of ``quasiconvexity," that looks like an extension of the concept originally introduced by C. Morrey and used extensively in non--linear elasticity theory. Part of the challenge, however, includes turning formal arguments into rigorous ones to yield new results. The intellectual merit of the research is that the problems to be studied have the potential of revealing principles which can be extended to a more general context.
