Asymptotic Geometric Analysis, Random Matrices, and Applications

渐近几何分析、随机矩阵及其应用

• 批准号: RGPIN-2022-03483
• 负责人: Litvak, Alexander
• 金额: $ 2.26万
• 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=758355
• 关键词: Asymptotic; Geometric; Analysis; Random; Matrices; Asymptotic; Geometric; Analysis; Random; Matrices

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 solving open problems in other directions of AGA. The project will also serve to train graduate students and postdoctoral fellows.

在不对称几何分析（AGA）的几个相关方向上的项目浓度。该字段与有限维对象的几何和线性特性有关，例如凸组和规范空间，尤其是当尺寸或许多其他相关的自由参数时出现的特征行为非常大，或者是无限的。高维系统在数学和应用科学中很常见，因此，了解高维现象变得越来越重要。在过去的十年中，AGA的增长巨大，随着新的强大技术的发展，主要是概率性质。借助AGA的一般框架，方法及其对相关领域的影响，AGA可以位于数学许多分支的“十字路口”：功能分析，凸和离散的几何形状以及几个领域的概率。 AGA中的许多现象与随机物质的奇异值的行为密切相关。由于在纯数学和应用数学，统计学，计算机科学，电气工程等中的许多应用中，关于随机物品的奇异价值的分布问题至关重要。经典的随机矩阵理论已经长时间广泛研究了相应的限制分布。相比之下，我们的利益集中在非限制性方案上。我们考虑一个高维随机矩阵，并为最大，最小的奇异值寻求不对称的锋利边界，这些值具有压倒性的概率。该项目将为AGA的多个方向带来重大贡献。这将导致新的理解，新技术和新的结果的发展，以快速发展的最不对称的非对称非对称矩阵理论。这也将导致在AGA的其他方向上解决开放问题。该项目还将用于培训研究生和博士后研究员。