Mathematical Sciences: Geometric & Probabilistic Aspects of Convexity and Functional Analysis

数学科学：几何

• 批准号: 9623984
• 负责人: Stanislaw Szarek
• 金额: $ 7.05万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623984&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometric; & Probabilistic

NSF PROPOSAL # 9623984 Dr. Stanislaw J. Szarek Case Western Reserve University ABSTRACT The project involves continued research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory. Many of the problems considered are expressed in the language of the asymptotic theory of finite dimensional normed spaces or, what is essentially equivalent, as "isomorphic" statements about high dimensional convex bodies. Some questions have motivation in the theory of operator algebras, mathematical physics, operations research, computer science or probability and statistics. Typical research topics include: entropy of linear operators, existence of nontrivial linear operators on general Banach spaces, norms of random matrices, some problems in the "local" operator theory, Gaussian correlation of convex sets, balancing vectors in normed spaces and other questions on the border of probability, combinatorics and convexity. On an elementary level, analysis is a study of "functions", or relationships, between quantities and the parameters on which they depend. As it happens, very many naturally appearing relationships are linear or at least convex. Thus a good understanding of convex functions (and consequently of convex sets) is a prerequisite for understanding those relationships. Depending on whether the set of parameters is finite or infinite (the latter may also be an "idealization" of "very large"; real life problems usually do depend on very many parameters), we are dealing with either finite dimensional convexity or infinite dimensional functional analysis. The "isomorphic" asymptotic theory was, over the last twenty years or so, quite successful in identifying and exploiting "approximate" symmetries of various problems (involving high dimensional convex sets) that could not be analyzed with earlier more "rigid" methods. One example of a specific link to applications is a recent advance (in computer science) in the area of efficient computation of volume of high dimensional convex sets, which utilized results and ideas from the asymptotic theory.

NSF提案编号9623984 Stanislaw J. Szarek博士凯斯西储大学 摘要 该项目涉及对泛函分析和凸性理论的几何、概率和组合方面的持续研究。 考虑的许多问题都表示在语言的渐近理论的有限维赋范空间，或者，什么是基本上等同的，作为“同构”的声明高维凸体。有些问题的动机在算子代数理论，数学物理，运筹学，计算机科学或概率统计。典型的研究课题包括：线性算子的熵，一般Banach空间上非平凡线性算子的存在性，随机矩阵的范数，“局部”算子理论中的一些问题，凸集的高斯相关，赋范空间中的平衡向量以及概率、组合学和凸性边界上的其他问题。 在一个基本的水平上，分析是一个研究“功能”，或关系，数量和参数之间的依赖。 事实上，很多自然出现的关系都是线性的，或者至少是凸的。 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。 根据参数集是有限的还是无限的（后者也可能是“非常大”的“理想化”;真实的生活问题通常依赖于非常多的参数），我们处理的是有限维凸性或无限维泛函分析。 “同构”渐近理论是，在过去的二十年左右，相当成功地确定和利用“近似”对称性的各种问题（涉及高维凸集），不能分析与早期更“刚性”的方法。 应用程序的特定链接的一个示例是 最近的进展（在计算机科学）在该地区的有效计算的体积的高维凸集，其中利用的结果和思想，从渐近理论。