Infinite dimensional variational problems and their dynamics

在最优运输理论中，人们寻找最佳的运输策略，比如说，一堆泥土，挖掘，在那里最优性是根据规定的成本函数来衡量的。在他早期的研究中，首席研究员（PI）为最优运输理论创造了新的工具，该理论在过去的二十五年中经历了数学复兴，将分析，几何，偏微分方程（PDE）和动力系统编织在一起。在大量粒子系统研究的激励下，PI与合作者发现了最优运输、大量参与者博弈和平均场博弈之间的一些联系。这些意想不到的联系为几种新的研究途径打开了大门，其中一些在本提案中有所描述。 该项目的目标是推进各种其他相关主题的知识。这包括几何空间的封闭微分形式与非线性因式分解的微分形式。 这项建议涉及培训学生，指导博士后和组织研究生学习研讨会。PI将继续参与国家和国际活动。这些构成了拟议活动的更广泛影响。PI研究密度泛函理论（DFT）的数学方面，以及源自组合优化的某些几何聚类问题。许多这些问题有什么共同之处，什么是有时新颖的方法来研究它们，是他们制定的无限维“流形”配备了度量，微分和辛伪结构。 这种无限的维度设置往往导致一个新的前景和-至少在正式的水平-以新颖和简单的方法，否则复杂的问题。一个典型的例子是奥托的演算如何将许多非线性偏微分方程转化为测地凸泛函的梯度流。以类似的方式，我们在微分形式集上寻找最小长度的测地线，导致了一个新的“拟凸性”概念，它看起来像是最初由C引入的概念的扩展。Morrey的理论，并在非线性弹性理论中得到了广泛的应用.然而，挑战的一部分包括将正式的论点转化为严格的论点，以产生新的结果。 这项研究的学术价值在于，所研究的问题有可能揭示出一些原则，这些原则可以推广到更普遍的情况。

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.