Unusual Concentration Phenomena in Probability, Analysis, and Geometry

许多复杂系统在最小的尺度上以随机的方式运行。那么，为什么我们周围的世界似乎如此可预测呢？在一个非常普遍的背景下，这可以通过一个被称为测度集中的原理来解释：许多独立随机变量的平滑函数的行为基本上是可预测的。这种现象的精确数学公式为研究复杂的随机结构提供了有力的工具。该项目旨在研究与几个不同数学领域的意外联系所产生的新的集中现象：从嵌入的研究（数据在特定空间中的表现有多好？）对奇异形状的研究可以追溯到一个多世纪前的几何老问题；并从非齐次随机矩阵的研究出发，在现代数据科学中得到了广泛的应用。这些问题的不同寻常的特征激发了理论在新方向上的发展，也引入了新的工具，这些工具可以应用于在纯数学和应用数学中出现的广泛的随机结构。该项目包括教育、指导和外联活动，旨在吸引和培训下一代数学家，提高数学科学的参与度和多样性。它还为研究生提供研究培训机会。这个项目的目的是系统地发展新的集中现象，这些现象产生于概率、泛函分析、度量几何和凸几何中的问题。该项目围绕三个主题进行组织。第一个主题的目的是发展在赋范空间中取值的函数的集中不等式的一般理论。第二个主题旨在理解凸几何中某些长期存在的问题，这些问题可能被视为浓度现象的不寻常类似物。第三个主题是关于非齐次随机矩阵的范数的集中不等式。对非标准环境中集中现象的研究激发了新问题和新工具的发展，这些问题和工具既具有直接的概率意义，又为其他数学领域的问题提供了新的视角。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。

Many complex systems behave in a random fashion at the smallest scales. Why then does the world around us appear to be so predictable? This is explained in a very general context by a principle known as concentration of measure: smooth functions of many independent random variables behave in an essentially predictable manner. Precise mathematical formulations of this phenomenon provide powerful tools for studying complex random structures. This project aims to study new concentration phenomena that arise from unexpected connections with several different areas of mathematics: from the study of embeddings (how well can data be represented in a particular space?); from the study of exotic shapes which date back to old problems in geometry from over a century ago; and from the study of non homogeneous random matrices, which are widely used in modern data science. The unusual features of these problems motivate the development of the theory in new directions, as well as the introduction of new tools that may be applied to a wide range of random structures that arise in both pure and applied mathematics. The project includes educational, mentoring and outreach activities that are aimed at attracting and training the next generation of mathematicians, and at increasing participation and diversity in the mathematical sciences. It also provides research training opportunities for graduate students.The aim of this project is to systematically develop novel concentration phenomena that arise from problems in probability, functional analysis, metric geometry, and convex geometry. The project is organized around three topics. The first topic aims to develop a general theory of concentration inequalities for functions taking values in normed spaces. The second topic aims to understand certain long-standing questions in convex geometry that may be viewed as unusual analogues of the concentration phenomenon. The third topic is concerned with concentration inequalities for the norms of non-homogeneous random matrices. The investigation of concentration phenomena in nonstandard settings motivates new questions and the development of new tools that are both of direct probabilistic significance, and that provide new perspectives on problems in other areas of mathematics.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.