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 该项目涉及泛函分析和凸性理论的几何、概率和组合方面的持续研究。 所考虑的许多问题都用有限维赋范空间渐近理论的语言来表达，或者本质上等效的，作为关于高维凸体的“同构”陈述。有些问题的动机来自算子代数、数学物理、运筹学、计算机科学或概率与统计理论。典型的研究主题包括：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、凸集的高斯相关性、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。 在基本层面上，分析是对数量及其所依赖的参数之间的“函数”或关系的研究。 事实上，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。 根据参数集是有限还是无限（后者也可能是“非常大”的“理想化”；现实生活中的问题通常确实取决于非常多的参数），我们正在处理有限维凸性或无限维泛函分析。 在过去二十年左右的时间里，“同构”渐近理论在识别和利用各种问题（涉及高维凸集）的“近似”对称性方面非常成功，而这些问题无法用早期更“严格”的方法进行分析。 与应用程序的特定链接的一个例子是高维凸集体积的有效计算领域的最新进展（计算机科学），它利用了渐近理论的结果和思想。