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

NSF-BSF：高维凸性和对称性及其应用

基本信息

批准号：
2247834
负责人：
Galyna Livshyts
金额：
$ 47.37万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-04-15 至 2026-03-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247834&HistoricalAwards=false
关键词：
NSF BSF convexity symmetry dimensions

项目摘要

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.

几何结果最古老的结果之一是等等不等式。它指出，在给定体积的所有身体中，球具有最小的表面积。这就是为什么肥皂气泡圆形的原因 - 它们最小化表面压力，从而假定表面积最小的形状。近年来，很明显，在额外的对称性和凸度假设下，许多等速度型结果变得更强大。在这个项目中，将研究对称性和凸性在高维度和该领域的其他几个问题中的等级类型问题中发挥的作用。除了这个方向外，还将研究有关高维空间几何形状的渐近问题。当参数数量增加时，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是无望的。但是，事实证明，随着许多参数的来源，美丽而简单，事情开始以某种可预测的方式行事。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。将在学习理论中使用潜在的应用，以解决这个问题的几个问题。将对具有所谓不均匀概况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，物理等领域中具有潜在应用的方向。该项目的其他部分将涉及监督博士后，研究生和本科生，并组织研讨会和会议。等等不平等是著名的Brunn-Minkowski不平等现象的结果，这是一个重要的结果，这是在凸线上无处理的。在过去的20年中，很明显，布鲁恩·米科夫斯基（Brunn-Minkowski）的不平等并不是故事的终结。如果仅处理具有某些对称性的身体，那么不平等现象也应该是正确的。这产生了许多猜想，例如log-brunn-minkowski猜想，（b） - 注射器和尺寸Brunn-Minkowski猜想。通过其本地版本，将解决这些基本的开放问题。此外，渐近几何分析中的几个方向是该项目的一部分，例如创建一种快速算法，该算法了解有关一般对数符号测量的凸体。将使用先前开发的方法来研究不均匀的随机矩阵的合奏，该方法涉及单位领域的新有效离散化。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优点和更广泛影响的评估标准来通过评估来进行评估的。