Asymptotic Geometric Analysis: Matrices, Operators and Noncommutative Phenomena

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，特别关注非交换对象和现象。虽然项目中考虑的大多数问题都有其他数学科学领域的动机，但它们通常用巴纳赫空间的局部或有限维几何语言表达，并且/或使用该领域流行的方法进行分析。样本研究课题包括：线性算子的熵及其对偶性、高维凸体的结构性质、随机矩阵的各种解析和几何问题、局部算子理论中的一些问题、凸集的高斯相关、非交换环境中的集中现象以及与自由信息理论和量子信息理论有关的一些几何问题。在初级层面上，分析是对函数的研究，或者是对数量和它们所依赖的参数之间关系的研究。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，很好地理解凸函数和凸集是理解这些关系的先决条件。潜在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有关。由于现实生活中的问题通常依赖于非常多的参数，因此高维设置特别有趣。这正是渐近几何分析（Asymptotic Geometric Analysis）的研究领域，它研究的是凸集在维数趋于无穷时的定量性质。它构成了古典功能分析和古典几何之间肥沃的中间地带。功能分析通常关注无限维的设置（这通常是一个非常大的维度的理想化），但它通常只提供定性信息。另一方面，几何通常为特定的不太大的维度提供非常精确的信息。近二十年来，渐近理论在识别和利用各种问题的“近似”对称性方面取得了相当大的成功，这些问题逃脱了早期“过于定性”或“过于僵化”的方法。（这导致了许多与计算机科学相关的发现。）最后，为了解释我们对非交换性的兴趣，我们指出它只是反映了一个事实，即过程的最终结果可能取决于所涉及的操作的顺序；这一原理最著名的，但迄今为止不是唯一的表现是量子力学。

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.