Analytic, Geometric, and Probabilistic Aspects of High-Dimensional Phenomena

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

The complexity of mathematical objects arising in geometry and probability increases as the dimension of the object increases. This is a result of a growing number of possible configurations as well as a lack of intuition, which is primarily built on low-dimensional examples. Sometimes, due to certain underlying fundamental properties such as symmetry or independence of these objects, we witness an order and universality present in high dimensions. This project aims to deepen our mathematical understanding of such phenomena in several contexts, such as volumetric aspects of high-dimensional random polytopes (geometric objects with "flat" sides), or the sums of many random quantities in which each quantity comes with a deterministic weight. In addition to their fundamental interest, such problems are motivated by, and often find applications in, related areas of statistics, computer science, big data and machine learning. A vital part of this project is the student training and educational activities that will result. More specifically, this project is devoted to three topics related to analytic, geometric and probabilistic aspects of high-dimensional phenomena: estimates for moments and tails of sums of random variables, thresholds for the volume of random polytopes, and efficient coverings of convex sets with its homothetic copies (the Hadwiger covering/illumination problem). Our work on probabilistic comparison inequalities, involving analytic and probabilistic techniques such as chaining, will help us understand the concentration of measure phenomena for random sums, with applications to the geometry of Banach spaces. Volume threshold phenomena of random polytopes in high dimensions have been established and satisfactorily understood only in the presence of a product structure or rotational symmetry. The lack of these two in our problems creates a need for new, more robust techniques and approaches. The illumination conjecture touches upon very basic concepts: coverings and intersections of convex sets. This project will exploit recent developments in geometric functional analysis to open up perspectives on improving best asymptotic bounds. 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.