Analytic and Probabilistic Methods in Geometric Functional Analysis

几何泛函分析中的解析和概率方法

• 批准号: 2246484
• 负责人: Tomasz Tkocz
• 金额: $ 24.24万
• 依托单位: Carnegie-Mellon University
• 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: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246484&HistoricalAwards=false
• 关键词: Analytic; Probabilistic; Methods; Geometric; Functional

Geometric functional analysis is concerned with various geometric properties of infinite dimensional linear spaces through their finite dimensional subspaces, a good analogy of which is a CT scan in medical imaging. This project aims to deepen our understanding of existing, as well as to develop new methods, heavily involving probabilistic and analytic ideas, in order to study quantitatively the said geometric properties. A vital part will be student training at both graduate and undergraduate levels and educational activities that will result.More specifically, there are three fundamental questions this project aims to tackle: sharp bounds on Gaussian measures of dilates of symmetric convex sets, volumetric bounds on sections of convex bodies, and quantitative aspects of concentration with concrete applications to convex geometry and geometry of numbers. The focus is on new methods, particularly, the Fourier approach to geometric tomography, blended with the novel reverse Hölder inequalities for negative moments, as well as applications of entropy to quantify concentration.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.

几何泛函分析涉及无限维线性空间通过其有限维子空间的各种几何特性，其中一个很好的类比是医学成像中的 CT 扫描，该项目旨在加深我们对现有方法的理解，并开发新方法。 ，大量涉及概率和分析思想，为了定量研究上述几何性质，一个重要的部分将是研究生和本科生的学生培训以及由此产生的教育活动。更具体地说，该项目旨在解决三个基本问题。铲球：锋利对称凸集扩张的高斯测度的界限、凸体截面的体积界限、定量和集中方面以及凸几何和数字几何的具体应用重点是新方法，特别是几何层析成像的傅里叶方法。 ，与负矩的新颖的逆霍尔德不等式相结合，以及熵的应用来量化浓度。该奖项反映了 NSF 的法定使命，并通过使用评估来认为值得支持基金会的智力价值和更广泛的影响审查标准。