Analytic and Probabilistic Methods in Geometric Functional Analysis

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性质，其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，以定量研究所述几何性质。一个至关重要的部分将是学生培训在研究生和本科生水平和教育活动，将导致。更具体地说，有三个基本问题，这个项目的目的是解决：尖锐的界限高斯措施的膨胀对称凸集，体积界限的部分凸体，和定量方面的浓度与具体应用凸几何和几何的数字。重点是新的方法，特别是傅立叶方法的几何层析成像，混合了新的反向Hölder不等式的负矩，以及应用熵来量化浓度。该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.