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

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.

该项目集中于渐近几何分析（AGA）的几个相关方向，该领域涉及有限维对象的几何和线性属性，例如凸集和赋范空间，特别是当维度或维度时出现的特征行为。其他相关自由参数的数量适当大或趋于无穷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。随着新的强大技术（主要是概率性质）的发展，AGA 凭借 AGA 的一般框架、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、凸函数。 AGA 中的许多现象与随机矩阵的奇异值的行为密切相关，由于纯数学中的许多应用，随机矩阵的奇异值分布问题非常重要。并应用了数学、统计学、计算机科学、电气工程等领域已经研究了相应的极限分布很长时间，与之形成鲜明对比的是，我们关注的是高维随机矩阵。寻找以压倒性概率成立的最大和最小奇异值的渐近尖锐界限，该项目将为 AGA 的多个方向带来重大贡献，它将导致新理解、新技术和新结果的快速发展。渐近增长尖端该项目还将帮助解决 AGA 其他方向的开放问题。