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

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。所考虑的许多问题都是用有限维赋范空间的渐近理论的语言表达的，或者本质上等价的，作为关于高维凸体的“同构”陈述。有些问题在算子代数理论、数学物理、运筹学、计算机科学或概率论和统计学中具有动机。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、赋范空间中向量的平衡以及概率、组合学和凸性边界上的其他问题。在基本层面上，分析是对数量和它们所依赖的参数之间的“函数”或关系的研究。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，很好地理解凸函数（以及凸集）是理解这些关系的先决条件。取决于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”；现实生活中的问题通常依赖于非常多的参数），我们处理的要么是有限维的凸性，要么是无限维的泛函分析。“同构”渐近理论在过去二十年左右的时间里，在识别和利用各种问题（涉及高维凸集）的“近似”对称性方面取得了相当大的成功，这些问题无法用早期更“严格”的方法来分析。一个与应用相关的例子是最近（在计算机科学中）在高维凸集体积的有效计算领域取得的进展，它利用了渐近理论的结果和思想。

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.