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

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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.