Optimality in analysis and geometry of probability measures

拟议研究的主要目标是就概率度量与优化相结合时出现的基本问题建立通用的理论。这种情况在科学和工程中是普遍存在的，如概率度量模型随机性、数据集和质量分布。我们通过最优输运理论的透镜来看待这些问题，最优输运理论考虑了这样一种现象，即当质量分布以这样一种方式匹配时，将质量从一个位置移动到另一个位置的特定传输成本最小化。这是一个快速发展的领域，对几何、概率和偏微分方程问题做出了基本贡献，过去10年里有两个菲尔兹奖章来庆祝这一点。 我在建议中的目标是： (1)发展数学方法，当存在附加的概率约束时，分析最优传输的结构，这些约束要求质量通过特定的随机过程移动，例如，令或布朗运动。这些附加的约束为最优运输提供了更丰富但不平凡的结构，一个成功的理论将在概率、偏微分方程式和几何之间建立新的联系。它还将应用于金融学中的定价理论，以及理解人群或群体中大量粒子的运动。 (2)发展理解和利用概率度量之间的几何平均的数学方法，称为沃瑟斯坦重心。我将集中揭示沃瑟斯坦重心的分析特征，这将涉及该领域的一些悬而未决的问题，特别是关于随机形状统计的基本问题。 计算方法的最新进展使最优传输理论能够有效地处理各种应用，包括流体力学、经济学、计算机图形学，甚至机器学习。这些发展通过提供大量的例子和相关问题，加强了我在本提案中强调的理论研究。另一方面，拟议研究计划中的理论进展将有助于创造出更广泛应用的创新方法。例如，沃瑟斯坦GAN最近在机器学习方面的显著发展源于概率度量空间上的沃瑟斯坦距离理论。这项建议还包含对与植物根有关的生物学问题的应用，了解植物根的形状可能会支持农业研究，以管理和改进粮食生产。

The main goal of the proposed research is to establish versatile theories on fundamental questions that arise when probability measures are coupled with optimization. Such situations are ubiquitous in science and engineering as probability measures model randomness, datasets, and mass distributions. We view these through the lens of optimal transport theory, which considers the phenomena when mass distributions are matched in such a way to minimize a certain transport cost of moving mass from one location to another. This is a rapidly growing area with fundamental contributions to problems in geometry, probability, and partial differential equations, which is celebrated by two Fields medals in the last 10 years. My objectives in the proposal are: (1) To develop mathematical methods for analyzing the structure of optimal transport when there are additional probabilistic constraints which require the mass to be moved by a specified stochastic process, such as a martingale or Brownian motion. These additional constraints give more fruitful but nontrivial structures to optimal transport, and a successful theory will make new connections between probability, partial differential equations, and geometry. It will also have applications to pricing theory in finance as well as to understanding the motion of large numbers of particles in crowds or swarms. (2) To develop mathematical methods for understanding and utilizing the geometric average between probability measures, called the Wasserstein barycentre. I will focus on revealing analytical features of Wasserstein barycentres, and this will involve some of the outstanding open problems in the area, in particular, fundamental questions on the statistics of random shapes. Recent progress in computational methods is enabling optimal transport theory to effectively handle a variety of applications, to areas including fluid mechanics, economics, computer graphics, and even to machine learning. These developments enhance the theoretical investigations that I emphasize in this proposal, by providing an abundance of examples and related questions. On the other hand, theoretical progress in the proposed research program will contribute to making innovative methods for wider applications. For example, the remarkable recent development of the Wasserstein GAN in machine learning originated from the theory of Wasserstein distances on the space of probability measures. This proposal also contains applications to biological problems concerning plant roots, where understanding their shapes may support agricultural research into managing and improving food production.