High-dimensional random matrices and convex bodies; applications

高维随机矩阵和凸体；

• 批准号: RGPIN-2018-04722
• 负责人: TomczakJaegermann, Nicole
• 金额: $ 2.55万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758363
• 关键词: dimensional; random; matrices; convex; bodies

The project concentrates on several related directions of Asymptotic Geometric Analysis (AGA). This field is concerned with geometric and linear properties of finite dimensional objects, such as convex sets and normed spaces, especially with the characteristic behavior that emerges when the dimension, or a number of other relevant free parameters, is suitably large or tends to infinity. High-dimensional systems are very frequent in mathematics and applied sciences, hence, understanding high--dimensional phenomena is becoming increasingly important. The last decade has seen a tremendous growth of AGA, with the development of new powerful techniques, mainly of probabilistic nature. By virtue of AGA's general framework, methods, and its impact on related fields, AGA can be situated at the "crossroads" of many branches of mathematics: functional analysis, convex and discrete geometry, and several areas of probability. Many phenomena in AGA are closely related to the behavior of singular values of random matrices. Questions on distributions of singular values of random matrices are of major importance due to many applications in pure and applied mathematics, statistics, computer sciences, electrical engineering, among others. Classical random matrix theory extensively studied corresponding limiting distributions already for a long time. In sharp contrast, our interest concentrates on the non--limiting regime. We consider a high dimensional random matrix and seek asymptotically sharp bounds for the largest and smallest singular values which hold with an overwhelming probability. This project will bring significant contributions to several directions of AGA. It will lead to development of new understanding, new techniques, and new results in the fast growing cutting edge asymptotic non--limiting theory of random matrices. It will also lead to the development of the theory of adjacency matrices of random graphs as well as to solving open problems in other directions of AGA. Finally the new approach via finite dimensional random matrices may lead to solutions of some old Banach space questions left open from eighties. The project will also serve to train graduate students and postdoctoral fellows.

本项目主要研究渐近几何分析（AGA）的几个相关方向。该领域关注有限维对象的几何和线性性质，如凸集和赋范空间，特别是当维度或许多其他相关自由参数适当大或趋于无穷大时出现的特征行为。高维系统在数学和应用科学中非常常见，因此，理解高维现象变得越来越重要。在过去的十年里，随着主要是概率性质的新型强大技术的发展，AGA取得了巨大的发展。由于AGA的总体框架、方法及其对相关领域的影响，AGA可以位于许多数学分支的“十字路口”：泛函分析、凸和离散几何以及几个概率论领域。遗传算法中的许多现象都与随机矩阵的奇异值行为密切相关。随机矩阵的奇异值分布问题在纯数学和应用数学、统计学、计算机科学、电子工程等领域有着重要的应用。经典随机矩阵理论长期以来对相应的极限分布进行了广泛的研究。与之形成鲜明对比的是，我们的兴趣集中在非极限体制上。我们考虑一个高维随机矩阵，并寻求最大和最小奇异值的渐近锐界，这些奇异值具有压倒性的概率。该项目将为AGA的几个方向带来重大贡献。它将导致发展的新认识，新技术，并在快速发展的前沿的随机矩阵的渐近非极限理论的新结果。它还将导致随机图邻接矩阵理论的发展，以及解决AGA其他方向的开放性问题。最后，利用有限维随机矩阵的新方法可以解决一些80年代遗留下来的老巴拿赫空间问题。该项目还将用于培养研究生和博士后。