Asymptotic Geometric Analysis: Matrices, Operators and Noncommutative Phenomena

渐近几何分析：矩阵、运算符和非交换现象

• 批准号: 0109362
• 负责人: Stanislaw Szarek
• 金额: $ 24.15万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0109362&HistoricalAwards=false
• 关键词: Asymptotic; Geometric; Analysis; Matrices; Operators

This project involves continued research on the geometric,probabilistic and combinatorial aspects of functional analysis andconvexity theory, with particular attention to noncommutative objects andphenomena. While most of the problems considered in the project havemotivation in other fields of mathematical sciences, they are typicallyexpressed in the language of local or finite dimensional geometry of Banachspaces and/or analyzed using the methods prevalent in that area. Sampleresearch topics include: entropy of linear operators and duality of suchentropy, structural properties of high-dimensional convex bodies, variousanalytic and geometric questions related to random matrices, some problemsin the "local" operator theory, Gaussian correlation of convex sets,concentration phenomenon in the noncommutative context and some geometricquestions related to free and quantum information theories. On an elementary level, Analysis is a study of functions, orrelationships between quantities and the parameters on which they depend.As it happens, very many naturally appearing relationships are linear orat least convex. Thus, a good understanding of convex functions and,consequently, of convex sets is a prerequisite for understanding thoserelationships. The number of free parameters in the underlying problem canoften be related to the dimension of sets in the corresponding mathematicalmodel. Since real-life problems usually do depend on very many parameters,the high-dimensional setting is of particular interest. This is exactlythe domain of Asymptotic Geometric Analysis, which studies quantitativeproperties of convex sets as the dimension goes to infinity. Itconstitutes a fertile middle ground between the classical FunctionalAnalysis and the classical Geometry. Functional Analysis is usuallyconcerned with the infinite-dimensional setting (which frequently is anidealization of a very large dimension), but it often provides onlyqualitative information. On the other hand, Geometry typically yieldsvery precise information for a specific not-too-large dimension. For thelast two decades or so the asymptotic theory has been quite successful inidentifying and exploiting "approximate" symmetries of various problemsthat escaped the earlier "too qualitative" or "too rigid" methods. (Thisled, among others, to the discovery of many links to Computer Science.)Finally, to explain our interest in noncommutativity we point out that itsimply reflects the fact that the final outcome of a process may depend onthe order of operations involved; the best known, but by far not the onlymanifestation of that principle is quantum mechanics.

该项目涉及泛函分析和凸性理论的几何、概率和组合方面的持续研究，特别关注非交换对象和现象。 虽然该项目中考虑的大多数问题都有数学科学其他领域的动机，但它们通常用巴纳赫空间的局部或有限维几何语言来表达和/或使用该领域流行的方法进行分析。 研究主题包括：线性算子的熵和熵的对偶性、高维凸体的结构性质、与随机矩阵相关的各种分析和几何问题、“局部”算子理论中的一些问题、凸集的高斯相关性、非交换环境中的集中现象以及与自由和量子信息论相关的一些几何问题。 在基本层面上，分析是对函数、数量及其所依赖的参数之间的关系的研究。碰巧，许多自然出现的关系是线性的或至少是凸的。 因此，对凸函数以及凸集的良好理解是理解这些关系的先决条件。基础问题中自由参数的数量通常与相应数学模型中集合的维数相关。 由于现实生活中的问题通常取决于很多参数，因此高维设置特别令人感兴趣。 这正是渐近几何分析的领域，它研究维度趋向无穷大时凸集的定量性质。 它构成了经典泛函分析和经典几何之间肥沃的中间地带。 泛函分析通常关注无限维设置（通常是非常大维度的理想化），但它通常仅提供定性信息。 另一方面，几何通常会针对特定的不太大的维度产生非常精确的信息。 在过去的二十年左右的时间里，渐近理论在识别和利用各种问题的“近似”对称性方面非常成功，这些问题逃脱了早期“过于定性”或“过于严格”的方法。 （这导致了与计算机科学的许多联系的发现。）最后，为了解释我们对非交换性的兴趣，我们指出它简单地反映了一个事实，即一个过程的最终结果可能取决于所涉及的操作的顺序； 最著名但迄今为止并非该原理的唯一体现是量子力学。