Variational Problems and Dynamics in Spaces of Large Dimensions

这个项目调查了一系列具有挑战性的问题。其中之一是最优运输理论，这可以追溯到1781年的Gaspard Monge。例如，该理论研究将一堆沙子移动到挖掘处的最佳方式，其中最优性是根据可能与行进距离成比例的成本函数来衡量的。许多工程导致连接与偏微分方程，流体力学，几何，概率论和功能分析。目前，最优传输在信号和图像表示、逆问题、癌症检测、形状和图像配准以及机器学习等方面都有应用。该项目依赖于最优运输理论，在由大量玩家组成的游戏中取得进展，并专注于方程，如Lasry和Lions开创的所谓主方程。它还具有变分问题和动力学系统的多个和无限多个粒子，也涉及非交换的最佳运输理论。该项目为本科生、研究生和博士后研究人员提供培训机会，并举办学习研讨会。学生们参与了一项旨在鼓励数学家，计算机科学家和工程师之间互动的研究计划。该项目研究具有非局部项的Hamilton-Jacobi方程，已知具有挑战性，特别是当非局部性出现在未知函数的梯度中时。在这种情况下，所谓的Lasry-Lions单调性条件允许获得主方程的第一个适定性结果。在以前的一个研究项目中，提出了一个替代条件，这使得第一次处理不可分的哈密顿成为可能。这项先前的工作在很大程度上依赖于所谓的个体噪声的存在。这个项目探索了新的想法来处理不可分离的哈密顿量，只有共同的噪音在主方程。它还启动了逆最优运输和逆平均场博弈问题的研究，其目标是通过设计适当的拉格朗日函数来实现预期的结果。在有界域上工作使得预测平均场博弈中产生理想纳什均衡所需的拉格朗日函数变得更加困难。该项目确定了最小的边界信息需要处理非光滑增广拉格朗日出现在这些逆问题的解决方案。该项目还考虑了无界集合中的逆最优运输问题。该项目的另一部分是图上的偏微分方程，并依赖于离散密度的度量流形。该项目开发了一个适定性理论以及图上微分方程的稳定性。虽然最佳传输的作用在经典力学中已经确立，但在量子环境中建立这些想法，为几个新的研究途径打开了大门。例如，人们面临着识别非交换公共噪声算子并研究其性质的挑战。为了实现这些目标，需要继续发掘Biane-Voiculescu输运距离的新性质，并研究非对易动力系统。非交换最优传输理论为研究随机矩阵的动力学提供了合适的设置，其大小后来趋于无穷大。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.