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

数学对象在几何和概率方面的复杂性随着对象的维度的增加而增加。这是由于可能的配置越来越多，以及缺乏直觉的结果，直觉主要建立在低维的例子上。有时，由于某些潜在的基本性质，如这些对象的对称性或独立性，我们见证了高维中存在的秩序和普遍性。这个项目旨在加深我们在几种情况下对这种现象的数学理解，例如高维随机多面体的体积方面(具有“平”边的几何对象)，或者许多随机量的总和，其中每个量都带有确定性的权重。除了它们的根本利益之外，这些问题还受到统计学、计算机科学、大数据和机器学习等相关领域的推动，并经常在这些领域得到应用。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，这个项目致力于与高维现象的解析、几何和概率方面有关的三个主题：随机变量和的矩和尾部的估计，随机多面体体积的阈值，以及凸集及其同质副本的有效覆盖(Hadwiger覆盖/照明问题)。我们在概率比较不等式方面所做的工作，涉及分析和概率技术，如链，将帮助我们理解随机和的测度现象的集中度，并将其应用到Banach空间的几何中。高维随机多面体的体积阈值现象已经被建立，并且只有在存在乘积结构或旋转对称性的情况下才能令人满意地被理解。在我们的问题中，这两个方面的缺失创造了对新的、更强大的技术和方法的需求。照明猜想涉及到非常基本的概念：凸集的覆盖和交集。这个项目将利用几何泛函分析的最新发展来开辟改善最佳渐近边界的视角。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.