Convexity and stochastic isoperimetry

这个项目专注于两个截然不同但又相互关联的数学领域，即凸几何和概率。被称为等周不等式的数学关系支配着几何和分析中的基本原理；它们决定了结构的形成，如肥皂泡、蜂窝和晶体等。最近，基本的等周不等式，特别是对于凸形，已经承认了适用于对象的典型随机子结构的更强的概率形式。这种概率变化与对量化高维随机对象中的规律性的新工具的需求不谋而合。这反过来又重塑了等周不平等之间的联系，并激发了可以应用于凸几何分析之外的新原则。本研究旨在从随机性即随机性的角度重新审视凸性与等周度之间的基本关系。欧几里德空间的发展提供了一个基础，以了解这些原理的扩展程度-从集合到函数，超越欧几里得空间，以及更抽象的数学实体，如矩阵泛函和相关的更一般的凸性概念。这些方向的进展将直接应用于高维概率，包括关于大型随机矩阵乘积的行为的问题。该项目包括为本科生、研究生以及博士后研究量身定做的主题。首席调查员将继续指导这些早期职业研究人员。他还将开发一门专门的研究生课程和一本关于随机等周学及其应用的说明性专著。研究结果将通过在国内和国际研究会议上的演讲来传播。凸性和随机性在几何概念和概率行为之间提供了一座天然的桥梁，例如，随机集的直径自然地转化为随机矩阵的最大奇异值。这样，随机等周原理就变成了高维随机对象的分布不等式。首席调查员将为随机函数，特别是与形心体和对偶有关的随机等周不等式发展一个全面的理论。随机等周法在很大程度上依赖于乘积空间中的欧几里德对称化方法。其他形式的对称化也适用于随机方法。即使对于球体，核心等周原理仍然停留在猜想阶段。从随机的角度出发，将提供同时发展几何、解析和概率方面的基础。此外，随机矩阵的重要泛函遵循相同的一组原理。这激发了随机矩阵的几何泛函的等周测定，特别是在配备非欧几里德范数的空间中作用的非谱量和算子。几何分析、概率和随机矩阵理论之间的新联系的吸引力是该项目背后的主要推动力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.