Infinite dimensional variational problems and their dynamics

在最优运输理论中，人们寻找运输的最佳策略，比如说，一堆土，挖掘，其中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）为最优输运理论创造了新的工具，这一理论在过去的25年里通过将分析、几何、偏微分方程（PDEs）和动力系统的思想编织在一起，经历了数学的复兴。受大量粒子系统工作的启发，PI和合作者发现了最佳运输、大量玩家的游戏和Mean Field游戏之间的一些联系。这些意想不到的联系为一些新的研究途径打开了大门，其中一些在本提案中进行了描述。这个项目的目标是促进其他相关主题的知识。这包括封闭微分形式空间的几何与微分形式的非线性分解。该提案涉及培训学生，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。这些是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，以及起源于组合优化的某些几何聚类问题。这些问题的共同点和研究方法的新颖之处在于，它们在具有度量、微分和辛伪结构的无限维“流形”上的表述。这种无限维度的设置通常会导致一种新的观点，并且-至少在正式层面上-为其他复杂问题提供新颖和更简单的方法。一个典型的例子是Otto的微积分如何将许多非线性偏微分方程转化为测地线凸泛函的梯度流。以类似的方式，我们在微分形式集合上寻找最小长度的测地线，导致了“准自凸性”的新概念，它看起来像是C. Morrey最初引入的概念的扩展，并在非线性弹性理论中广泛使用。然而，部分挑战包括将正式的论证转化为严谨的论证，以产生新的结果。这项研究的智力价值在于，所研究的问题具有揭示原则的潜力，这些原则可以扩展到更普遍的背景下。

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.