Analytic, Geometric, and Probabilistic Aspects of High-Dimensional Phenomena

高维现象的分析、几何和概率方面

基本信息

批准号：
1955175
负责人：
Tomasz Tkocz
金额：
$ 20.52万
依托单位：
Carnegie-Mellon University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1955175&HistoricalAwards=false
关键词：
Analytic Geometric Probabilistic Aspects Dimensional

项目摘要

The complexity of mathematical objects arising in geometry and probability increases as the dimension of the object increases. This is a result of a growing number of possible configurations as well as a lack of intuition, which is primarily built on low-dimensional examples. Sometimes, due to certain underlying fundamental properties such as symmetry or independence of these objects, we witness an order and universality present in high dimensions. This project aims to deepen our mathematical understanding of such phenomena in several contexts, such as volumetric aspects of high-dimensional random polytopes (geometric objects with "flat" sides), or the sums of many random quantities in which each quantity comes with a deterministic weight. In addition to their fundamental interest, such problems are motivated by, and often find applications in, related areas of statistics, computer science, big data and machine learning. A vital part of this project is the student training and educational activities that will result. More specifically, this project is devoted to three topics related to analytic, geometric and probabilistic aspects of high-dimensional phenomena: estimates for moments and tails of sums of random variables, thresholds for the volume of random polytopes, and efficient coverings of convex sets with its homothetic copies (the Hadwiger covering/illumination problem). Our work on probabilistic comparison inequalities, involving analytic and probabilistic techniques such as chaining, will help us understand the concentration of measure phenomena for random sums, with applications to the geometry of Banach spaces. Volume threshold phenomena of random polytopes in high dimensions have been established and satisfactorily understood only in the presence of a product structure or rotational symmetry. The lack of these two in our problems creates a need for new, more robust techniques and approaches. The illumination conjecture touches upon very basic concepts: coverings and intersections of convex sets. This project will exploit recent developments in geometric functional analysis to open up perspectives on improving best asymptotic bounds.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.

几何和概率中出现的数学对象的复杂性随着对象维度的增加而增加。这是由于可能的配置数量不断增加以及缺乏直觉的结果，直觉主要建立在低维示例的基础上。有时，由于某些潜在的基本属性，例如这些物体的对称性或独立性，我们见证了高维度中存在的秩序和普遍性。该项目旨在加深我们对多种背景下此类现象的数学理解，例如高维随机多胞体（具有“平坦”边的几何对象）的体积方面，或许多随机量的总和，其中每个量都具有确定性重量。除了它们的基本兴趣之外，这些问题还受到统计、计算机科学、大数据和机器学习相关领域的推动，并且经常在这些领域找到应用。该项目的一个重要部分是由此产生的学生培训和教育活动。更具体地说，该项目致力于与高维现象的分析、几何和概率方面相关的三个主题：随机变量之和的矩和尾部的估计、随机多胞体体积的阈值以及凸集的有效覆盖它的拟似副本（Hadwiger 覆盖/照明问题）。我们在概率比较不等式方面的工作，涉及分析和概率技术（例如链接），将帮助我们理解随机和的测度现象的集中性，并将其应用于巴纳赫空间的几何。高维随机多胞体的体积阈值现象仅在存在乘积结构或旋转对称性的情况下才被建立并令人满意地理解。我们的问题中缺少这两个因素，因此需要新的、更强大的技术和方法。光照猜想涉及非常基本的概念：凸集的覆盖和交集。该项目将利用几何泛函分析的最新进展，开辟改进最佳渐近界的视角。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。