Convexity and stochastic isoperimetry

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

This project focuses on two distinct but related fields of mathematics, namely, Convex Geometry and Probability. Mathematical relationships known as isoperimetric inequalities govern fundamental principles in geometry and analysis; they determine the formation of structures, like soap bubbles, honeycombs, and crystals, among many others. Recently, fundamental isoperimetric inequalities, especially for convex shapes, have admitted stronger probabilistic versions that apply to an object's typical random substructures. This probabilistic shift coincides with a demand for new tools to quantify regularity in high-dimensional random objects. This, in turn, is reshaping connections between isoperimetric inequalities and motivating new principles that can be applied outside of convex geometric analysis. This research aims to re-examine fundamental relationships between convexity and isoperimetry from a stochastic, that is random, viewpoint. Developments in Euclidean space provide a foundation to see how far such principles extend - from sets to functions, beyond Euclidean spaces, and to more abstract mathematical entities, such as functionals of matrices and related more general notions of convexity. Progress in these directions will have direct applications in high-dimensional probability, including problems on the behavior of products of large random matrices. The project includes topics tailored to undergraduate, graduate, as well as postdoctoral research. The Principal Investigator will continue to mentor these early career researchers. He will also develop a special graduate course and an expository monograph on stochastic isoperimetry and its applications. The results of the research will be disseminated through talks given at national and international research meetings. Convexity and randomness provide a natural bridge between geometric notions and probabilistic behavior, for example, diameters of random sets translate naturally to largest singular values of random matrices. In this way, stochastic isoperimetric principles become distributional inequalities for high-dimensional random objects. The Principal Investigator will develop a comprehensive theory of stochastic isoperimetric inequalities for random functions, especially related to centroid bodies and duality. In large part, stochastic isoperimetry has relied on Euclidean symmetrization methods in product spaces. Other forms of symmetrization also lend themselves to the stochastic approach. Even for the sphere, core isoperimetric principles remain at the stage of conjectures. Starting with a stochastic point of view will provide a basis to develop geometric, analytic and probabilistic aspects simultaneously. Moreover, important functionals of random matrices follow the same set of principles. This motivates isoperimetry for geometric functionals of random matrices, especially non-spectral quantities and operators acting in spaces equipped with non-Euclidean norms. The appeal of new links between geometric analysis, probability and random matrix theory is a major driving force behind this project. 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.