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.

在几何形状和概率中产生的数学对象的复杂性随着对象的尺寸的增加而增加。这是由于越来越多的可能配置以及缺乏直觉的结果，这主要建立在低维示例上。有时，由于某些潜在的基本属性，例如对称性或这些对象的独立性，因此我们目睹了在高维度中存在的秩序和普遍性。该项目的目的是在几种情况下加深我们对这种现象的数学理解，例如高维随机多型的体积方面（具有“平坦”侧面的几何对象），或者每个数量的许多随机数量的总和都带有确定性的重量。除了基本的兴趣外，此类问题还由统计，计算机科学，大数据和机器学习的相关领域的应用，并经常在相关领域中找到应用。该项目的重要组成部分是将导致的学生培训和教育活动。 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).我们在概率比较不平等方面的工作，涉及分析性和概率技术（例如链接），将有助于我们了解随机总和的测量现象集中度，并应用于Banach空间的几何形状。仅在产物结构或旋转对称性的存在下，已经建立了高维的随机多型的体积阈值现象。在我们的问题中缺乏这两个，因此需要新的，更强大的技术和方法。照明猜想涉及非常基本的概念：凸组的覆盖物和交叉点。该项目将利用几何功能分析的最新发展，以开放有关改善最佳渐近界限的观点。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。