Optimality in analysis and geometry of probability measures

拟议的研究的主要目标是建立通用的理论时出现的基本问题，概率措施与优化相结合。这种情况在科学和工程中无处不在，因为概率度量模型随机性，数据集和质量分布。我们认为这些通过最佳传输理论的透镜，它认为的现象时，质量分布相匹配的方式，以尽量减少一定的运输成本移动质量从一个位置到另一个。这是一个快速增长的领域，对几何，概率和偏微分方程问题做出了根本性的贡献，在过去的10年里，这一领域获得了两枚菲尔兹奖章。我在该提案中的目标是：**（1）当存在额外的概率约束时，开发分析最佳运输结构的数学方法，这些约束要求质量按照指定的随机过程（例如鞅或布朗运动）移动。 这些额外的约束为最优运输提供了更富有成效但非平凡的结构，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中大量粒子的运动。**（2）发展数学方法来理解和利用概率测度之间的几何平均，称为瓦瑟斯坦重心。我将专注于揭示分析功能的Wasserstein重心，这将涉及一些悬而未决的问题，在该地区，特别是基本问题的统计随机形状。** 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。这些发展通过提供大量的例子和相关的问题，加强了我在本建议中强调的理论研究。另一方面，在拟议的研究计划的理论进展将有助于使创新的方法，更广泛的应用。例如，Wasserstein GAN在机器学习中的显着最新发展起源于概率测度空间上的Wasserstein距离理论。该提案还包含了有关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管理和提高粮食生产。

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.