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＃9623984 Stanislaw J. Szarek案例西部储备大学摘要该项目涉及对功能分析和凸理论的几何，概率和组合方面的持续研究。 所考虑的许多问题都是用有限维度规范空间的渐近理论的语言表达的，或者本质上等效的是关于高维凸体的“同构”陈述。一些问题在运营商代数，数学物理学，操作研究，计算机科学或概率和统计的理论中具有动力。典型的研究主题包括：线性操作员的熵，在一般Banach空间上的非平凡线性操作员的存在，随机矩阵的规范，“本地”操作员理论中的某些问题，凸组的高斯相关性，在规范空间中平衡矢量以及在可能性的问题上以及其他问题上的概率，结合性，结合性，合并器和侧侧的问题。 在基础上，分析是对数量和依赖参数之间的“函数”或关系的研究。 碰巧的是，许多自然出现的关系是线性或至少凸的。 因此，对凸功能的良好理解（以及凸集）是理解这些关系的先决条件。 取决于一组参数是有限的还是无限的（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参数），我们正在处理有限的尺寸凸度或无限尺寸函数分析。 在过去的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的“近似”对称性（涉及高维凸集）方面非常成功，这些对称性无法使用早期的“刚性”方法来分析。 与应用程序的特定链接的一个示例是在有效计算高维凸集体积的领域的最新进步（计算机科学），该集合的量有效地计算，该集合利用了渐近理论的结果和思想。