Optimality in analysis and geometry of probability measures

提出的研究的主要目标是建立通用的理论，当概率措施与优化相结合时出现的基本问题。这种情况在科学和工程中普遍存在，因为概率度量为随机性、数据集和质量分布建模。我们通过最优传输理论的视角来看待这些问题，该理论考虑了质量分布匹配的现象，以使质量从一个位置移动到另一个位置的某种传输成本最小化。这是一个快速发展的领域，对几何、概率和偏微分方程的问题做出了根本性的贡献，在过去十年中获得了两次菲尔兹奖。我在提案中的目标是：(1)当有额外的概率约束要求质量通过特定的随机过程（如鞅或布朗运动）移动时，开发用于分析最佳传输结构的数学方法。这些额外的约束为最优输运提供了更富有成效但不平凡的结构，一个成功的理论将在概率、偏微分方程和几何之间建立新的联系。它还将应用于金融定价理论，以及理解群体或群体中大量粒子的运动。(2)发展理解和利用概率测度之间的几何平均值的数学方法，称为瓦瑟斯坦重心。我将专注于揭示Wasserstein质心的分析特征，这将涉及该领域的一些突出的开放问题，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输运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等领域。这些发展通过提供大量的例子和相关问题，加强了我在本建议中强调的理论研究。另一方面，本研究计划的理论进展将有助于创造更广泛应用的创新方法。例如，Wasserstein GAN最近在机器学习中的显著发展源于概率度量空间上的Wasserstein距离理论。这一建议也包含了对植物根系的生物学问题的应用，了解它们的形状可能会支持农业研究，以管理和改善粮食生产。

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.