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，例如，将一堆泥土运输到挖掘中，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)为最优运输理论创造了新的工具，这一理论在过去25年里经历了数学复兴，将分析、几何、偏微分方程(PDE)和动力学系统的线索编织在一起。在他对含有大量粒子的系统的工作的激励下，PI与合作者一起，发现了最佳运输、有大量玩家的游戏和平均场游戏之间的一些联系。这些意想不到的联系为几个新的研究途径打开了大门，其中一些在本提案中得到了描述。这个项目的目标是增进各种其他相关主题的知识。这包括具有微分形式的非线性因式分解的闭微分形式空间的几何。这项建议包括培训学生、指导博士后和为研究生组织学习研讨会。国际和平协会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集问题。这些问题中的许多共同之处，以及在研究它们的方法上有时是新奇的，是它们在配备了度量、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带来一种新的观点，并--至少在形式层面上--对原本复杂的问题采取新的、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组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C.Morrey最初引入的、在非线性弹性理论中广泛使用的概念的扩展。然而，部分挑战包括将形式论证转化为严格的论证以产生新的结果。研究的智力价值在于，所研究的问题有可能揭示原理，这些原理可以扩展到更一般的背景。

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.