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

几何和概率中数学对象的复杂性随着对象维数的增加而增加。这是由于越来越多的可能的配置以及缺乏直觉，这主要是建立在低维的例子。有时，由于某些潜在的基本属性，如对称性或这些对象的独立性，我们见证了一个秩序和普遍性存在于高维度。该项目旨在加深我们在几种情况下对此类现象的数学理解，例如高维随机多面体（具有“平坦”侧面的几何对象）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权重。除了其根本利益之外，这些问题的动机还在于统计学、计算机科学、大数据和机器学习等相关领域，并经常在这些领域得到应用。这个项目的一个重要组成部分是学生培训和教育活动，将导致。更具体地说，这个项目致力于三个主题相关的分析，几何和概率方面的高维现象：估计的时刻和尾部的总和随机变量，阈值的体积随机多面体，和有效覆盖的凸集与其位似副本（Hadwiger覆盖/照明问题）。我们的工作概率比较不等式，涉及分析和概率技术，如链接，将帮助我们了解浓度的措施随机和现象，与应用几何的Banach空间。在高维随机多面体的体积阈值现象已经建立和令人满意的理解，只有在存在的产品结构或旋转对称性。在我们的问题中缺少这两个，就需要新的、更健壮的技术和方法。光照猜想涉及到非常基本的概念：凸集的覆盖和交集。该项目将利用几何功能分析的最新发展，开辟改善最佳渐近边界的前景。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.