New High Dimensional Phenomena and Related Questions

该项目涵盖了许多数学主题，重点是研究新的高维现象，在概率，几何和信息论方面。本研究的核心部分涉及多维欧几里得空间上概率分布的测度现象的集中，包括它们与越来越多的相关随机变量的光滑泛函的渐近行为的关系。作为过去三十年来许多令人兴奋的发展的主题，集中工具有助于探索一般复杂系统的最基本性质，其中大量小部分的随机性导致稳定的极限行为。这一研究领域与许多具有挑战性的数学问题密切相关，并在统计学、信息论、计算机科学、机器学习等其他领域有许多应用。作为一个具有深远意义的总体目标，该项目旨在探索增长维度在高维模型中作为统一源的作用及其对随机过程的整个时间演化的影响。该项目将通过在不同的数学领域之间建立新的有前途的联系，并为他们提供强大的跨学科工具，从而产生更广泛的影响。该项目还将对数学科学教育产生重要影响。更具体地说，该项目涉及先进的集中技术，这些技术将在具有许多对称性的空间框架中开发，例如格拉斯曼流形。利用新的工具，本研究的目标之一是在相关型条件下关于加权和分布的集中和相关随机变量的二次型的问题。PI计划探索凸测度的高维投影的精细浓度特性，包括负责光谱间隙和等周常数的新的积分几何特征。该项目的一部分致力于在Renyi散度和相对Fisher信息方面的中心极限定理，以及泊松近似区域的edgeworth型展开式。该项目还涉及运输不平等及其在匹配问题中的应用。该项目的主题涉及长期悬而未决的问题或与最近发展有关的挑战性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project covers a number of topics in mathematics with focus on the study of new high dimensional phenomena, in probabilistic, geometric, and information-theoretic aspects. The core part of this investigation deals with the concentration of measure phenomena for probability distributions on multidimensional Euclidean spaces, including their relationship with the asymptotic behavior of smooth functionals of a growing number of dependent random variables. Being the subject of many exciting developments over the last three decades, the concentration tools help explore most essential properties of general complex systems where the randomness of a large number of small parts results in a stable limit behavior. This research area is deeply connected to many challenging mathematical problems and has numerous applications in other fields such as statistics, information theory, computer science, machine learning.As a general far-reaching goal, the project aims to explore the role of the growing dimension as a unifying source in high-dimensional models and its influence on the entire time evolution of random processes. The project will have a broader impact by creating new promising connections between different mathematical fields, and providing them with powerful interdisciplinary tools. The project will also have an important impact on education in mathematical sciences.More specifically, the project deals with advanced concentration techniques that will be developed in the framework of spaces with many symmetries such as Grassmanian manifolds. With new tools, one of the targets of this investigation is the circle of problems about the concentration of distributions of weighted sums and quadratic forms of dependent random variables under correlation-type conditions. The PI intends to explore refined concentration properties of high dimensional projections of convex measures, including new integral geometric characteristics that are responsible for spectral gap and isoperimetric constants. Part of the project is devoted to the central limit theorem in terms of the Renyi divergence and relative Fisher information, and to the Edgeworth-type expansions in the area of Poisson approximation. The project also deals with transport inequalities and their applications to matching problems. The themes of this project concern either long-standing open problems or challenging questions related to recent developments.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.