Analytic and Probabilistic Methods in Geometric Functional Analysis

几何泛函分析是通过无限维线性空间的有限维子空间来研究无限维线性空间的各种几何性质，医学成像中的CT扫描就是一个很好的类比。该项目旨在加深我们对现有的理解，以及开发新的方法，大量涉及概率和分析思想，以定量研究上述几何性质。研究生和本科阶段的学生培训以及由此产生的教育活动将是至关重要的一部分。更具体地说，本项目旨在解决三个基本问题：对称凸集膨胀的高斯测度的尖锐界限，凸体部分的体积界限，以及凸几何和数字几何具体应用的集中的定量方面。重点是新方法，特别是几何层析的傅里叶方法，与负矩的新颖反向Hölder不等式混合，以及熵的应用来量化浓度。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。

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.