Analytic and Probabilistic Methods in Geometric Functional Analysis

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.