Unusual Concentration Phenomena in Probability, Analysis, and Geometry

概率、分析和几何中的异常集中现象

• 批准号: 2054565
• 负责人: Ramon Van Handel
• 金额: $ 36.45万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054565&HistoricalAwards=false
• 关键词: Unusual; Concentration; Phenomena; Probability; Analysis

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.

许多复杂的系统在最小尺度上以随机的方式运行。那么，为什么我们周围的世界看起来如此可预测？这在一个非常一般的上下文中被称为测量集中的原理解释：许多独立随机变量的光滑函数以基本上可预测的方式表现。这种现象的精确数学公式为研究复杂的随机结构提供了强有力的工具。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如何？）;从奇异形状的研究可以追溯到世纪前的几何老问题;从非齐次随机矩阵的研究，这是广泛使用的现代数据科学。这些问题的不寻常的特点促使理论向新的方向发展，以及引入新的工具，这些工具可以应用于纯数学和应用数学中出现的各种随机结构。该项目包括教育、辅导和外联活动，旨在吸引和培训下一代数学家，并增加数学科学的参与和多样性。本项目的目的是系统地开发从概率、泛函分析、度量几何和凸几何问题中产生的新的浓度现象。该项目围绕三个主题展开。第一个主题的目的是发展一个一般理论的浓度不等式的功能采取的价值在赋范空间。第二个主题旨在了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范数的集中不等式。在非标准设置的浓度现象的调查激发了新的问题和新的工具，都是直接的概率意义的发展，并提供了新的角度对问题的其他领域的mathematics.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。