Measure Transportation And Notions Of Dimensionality In High Dimensional Probability

高维概率是一种数学理论，旨在解释具有大量自由度的系统的行为。这些系统在物理科学以及社交媒体和通信网络不断生成大量数据集的世界中至关重要。该项目旨在通过两种方法开发高维概率的理论工具。第一种方法将利用直觉，即在许多实际情况下，尽管系统包含许多自由度，但相关信息包含在较小的变量集中。该项目旨在通过发展内在维度的概念来量化这种直觉。第二种方法旨在将给定的复杂系统表示为一个简单得多的系统的转换。该项目还包括教育工作，包括指导本科生和研究生，以及在专业会议和公众中传播工作成果。长期目标是通过两种方法提高对高维概率度量的理解。 第一种方法是在函数不等式的背景下发展内在维度的原始概念。维数概念通过曲率维数条件在函数不等式中发挥着至关重要的作用，但它忽略了感兴趣的度量可以存在于低维空间中的事实。这种遗漏可能会导致低效的功能不平等；该项目希望弥补这一差距。第二种方法是开发基于随机过程和概率流的新测量传输方法。例如，Lipschitz 传输地图提供了一种将功能不平等从简单的源测量转移到复杂的目标测量的强大方法。目前已知只有少数这样的利普希茨输运图存在，这限制了输运方法的适用性。该项目将利用随机分析和重整化群方法的工具来构建具有理想规律性的交通地图。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

High dimensional probability is a mathematical theory which aims to explain the behavior of systems with very large number of degrees of freedom. Such systems are of utmost importance in the physical sciences and in a world where social media and communication networks constantly generate huge data sets. This project aims to develop theoretical tools in high dimensional probability via two methods. The first method will exploit the intuition that in many practical situations, although the system contains numerous degrees of freedom, the relevant information is contained in a much smaller set of variables. The project will aim to quantify this intuition by developing the notion of intrinsic dimensionality. The second method aims to represent a given complex system as the transformation of a much simpler system. The project also includes educational efforts, including mentoring undergraduate and graduate students, and the dissemination of the work in professional meetings and to the public. The long-term goal is to improve understanding of high-dimensional probability measures via two approaches. The first approach is to develop an original concept of intrinsic dimensionality in the context of functional inequalities. The notion of dimension plays a crucial role in functional inequalities via the curvature-dimension condition, but it ignores the fact that the measures of interest can live on lower-dimensional spaces. This omission can lead to inefficient functional inequalities; this project hopes to bridge this gap. The second approach is to develop new measure transportation methods based on stochastic processes and probabilistic flows. For example, Lipschitz transport maps provide a powerful way to transfer functional inequalities from simple source measures to complicated target measures. Only a few such Lipschitz transport maps are known to exist, which limits the applicability of the transport method. The project will utilize the tools of stochastic analysis and renormalization group methods to build transport maps with desirable regularity.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.