NSF-BSF: convexity and symmetry in high dimensions, with applications

几何中最古老的结果之一是等周不等式。它指出，在所有具有给定体积的物体中，球的表面积可能最小。这就是为什么肥皂泡是圆形的原因——它们使表面压力最小化，从而呈现出表面积最小的形状。近年来，很明显，在额外的对称性和凸性假设下，许多等周型结果变得更强。在这个项目中，对称性和凸性在高维等周型问题中的作用以及该领域的其他几个问题将被研究。除了这个方向，关于高维空间几何的渐近问题也将被研究。当参数数量增加时，问题似乎会变得更加复杂，因此，在高维空间中研究一些精确的问题似乎非常困难。然而，事实证明，许多参数带来了美丽和简单，并且事情开始以某种可预测的方式运行。这种现象的一个简单例子是，民意调查通常表现出接近正态分布的行为。这方面的几个问题将在学习理论中有潜在的应用。研究具有所谓非均匀轮廓的大型随机矩阵的行为。这是一个在统计学、计算机科学、物理学等领域具有潜在应用的方向。这个项目的其他部分将包括指导博士后、研究生和本科生，以及组织研讨会和会议。等周不等式是著名的布伦-闵可夫斯基不等式的结果，这是凸几何中普遍使用的一个重要结果。在过去的20年里，很明显，布伦-闵可夫斯基不平等并不是故事的结局。如果只处理具有一定对称性的物体，那么更强的不平等也应该成立。这产生了许多猜想，如对数布伦-闵可夫斯基猜想、(B)-猜想和维布伦-闵可夫斯基猜想。通过等周型不等式的局部版本，我们将探讨这些基本的开放性问题。此外，渐近几何分析中的几个方向也是该项目的一部分，例如创建一个快速算法，该算法可以根据一般的对数凹度量来学习凸体。本文将使用先前开发的方法研究非齐次随机矩阵的系综，该方法涉及一种新的有效的单位球离散化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the oldest results in geometry is the isoperimetric inequality. It states that among all the bodies with a given volume, the ball has the smallest possible surface area. This is the reason why soap bubbles are round - they minimize their surface pressure and thus assume the shape with the smallest surface area. In the recent years, it has become clear that under additional symmetry and convexity assumptions, many isoperimetric-type results become stronger. In this project, the role that symmetry and convexity play in isoperimetric-type questions in high dimensions and several other questions in this field will be researched. In addition to this direction, asymptotic questions about the geometry of high-dimensional spaces will be investigated. When the number of parameters increases, it seems that the question should become more complicated, and therefore, studying some precise questions in a high-dimensional space seems hopelessly difficult. However, it turns out that with many parameters comes beauty and simplicity, and things start behaving in some predictable way. A simple example of this phenomenon is the fact that polls usually exhibit behavior close to the normal distribution. Several questions in that vein will be approached, with potential applications in learning theory. A study of the behavior of large random matrices with so-called inhomogeneous profiles will be performed. This is a direction with potential applications in areas such as Statistics, Computer Science, Physics, and more. Other parts of this project will involve supervising postdocs, graduate and undergraduate students, and organizing seminars and conferences.The isoperimetric inequality is a consequence of the celebrated Brunn-Minkowski inequality, an important result which is used ubiquitously in convex geometry. Over the last 20 years it has become clear that the Brunn-Minkowski inequality is not the end of the story. If one deals only with bodies that have certain symmetries, much stronger inequalities should also be true. This yielded many conjectures such as the log-Brunn-Minkowski conjecture, the (B)-conjecture, and the Dimensional Brunn-Minkowski conjecture. Using isoperimetric-type inequalities via their local versions these fundamental open questions will be approached. In addition, several directions in Asymptotic Geometric Analysis are part of this project, such as creating a fast algorithm which learns a convex body with respect to a general log-concave measure. The ensemble of inhomogeneous random matrices will be studied, using the previously developed methods which involve a new efficient discretization of the unit sphere.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.