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

几何学中最古老的结果之一是等周不等式。它指出，在具有给定体积的所有物体中，球的表面积可能最小。这就是为什么肥皂泡是圆形的--它们的表面压力最小，因此呈现出表面积最小的形状。近年来，很明显，在附加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，将研究对称性和凸性在高维等周型问题中的作用以及该领域中的其他几个问题。除了这个方向，还将研究有关高维空间几何的渐近问题。当参数的数量增加时，问题似乎应该变得更加复杂，因此，在高维空间中研究一些精确的问题似乎困难得无可救药。然而，事实证明，伴随着许多参数而来的是美丽和简单，事情开始以某种可预测的方式发生。这种现象的一个简单例子是，民意调查通常显示出接近正态分布的行为。本课程将探讨这方面的几个问题，并探讨其在学习理论中的潜在应用。将对具有所谓非均匀分布的大型随机矩阵的行为进行研究。这是一个在统计、计算机科学、物理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Brunn-Minkowski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，布鲁恩-明科夫斯基不平等并不是故事的结束，这一点已经变得很明显。如果一个人只处理具有一定对称性的物体，那么更严重的不平等也应该是正确的。这产生了许多猜想，如LOG-Brunn-Minkowski猜想、(B)-猜想和维Brunn-Minkowski猜想。使用等周型不等式，通过它们的本地版本，这些基本的开放问题将被解决。此外，渐近几何分析中的几个方向也是该项目的一部分，例如创建一个关于一般对数凹度学习凸体的快速算法。将使用先前开发的方法来研究非齐次随机矩阵的集合，该方法涉及单位球体的新的有效离散化。该奖项反映了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。

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.