High-dimensional Phenomena in Asymptotic Geometric Analysis and Finite-dimensional Random Matrices

渐近几何分析和有限维随机矩阵中的高维现象

• 批准号: 8854-2013
• 负责人: TomczakJaegermann, Nicole
• 金额: $ 2.48万
• 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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=592941
• 关键词: dimensional; Phenomena; Asymptotic; Geometric; Analysis

The proposed research program is in the area of Pure Mathematics (Modern Analysis) called Asymptotic Geometric Analysis (AGA, in short). This is a fast growing interdisciplinary area at the "crossroads" of analysis, convex and discrete geometry, several areas of probability. While AGA takes roots in classical functional analysis, it is based on a new asymptotic point of view: properties of interest appear in quantitative forms that depend on the underlying finite dimension or a finite number of other relevant parameters. Combining them with structural or geometric hypotheses specific for the problems on hand shows the tendency of high dimensional systems to gather around a "typical" behaviour, and expresses this in a quantitative manner. High-dimensional systems are very frequent in mathematics and applied sciences hence understanding high-dimensional phenomena is becoming increasingly important.

提出的研究计划是在纯数学（现代分析）领域称为渐近几何分析（AGA，简称）。这是一个快速发展的跨学科领域，处于分析、凸几何和离散几何、概率的几个领域的“十字路口”。虽然AGA植根于经典泛函分析，但它基于一种新的渐近观点：感兴趣的性质以依赖于潜在有限维或有限数量的其他相关参数的定量形式出现。将它们与特定于手头问题的结构或几何假设相结合，显示了高维系统围绕“典型”行为聚集的趋势，并以定量的方式表达了这一点。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。