Asymptotic geometric analysis, random matrices and related topics

渐近几何分析、随机矩阵及相关主题

• 批准号: 251088-2011
• 负责人: Litvak, Alexander
• 金额: $ 1.31万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570941
• 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 the development of the theory of entropy extensions and factorizations as well as to solving open problems in other directions of AGA. 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。