Asymptotic Geometric Analysis: Matrices, Operators and Noncommutative Phenomena

这个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，特别关注非对易对象和现象。虽然项目中考虑的大多数问题在数学科学的其他领域都有影响，但它们通常用Banach空间的局部或有限维几何的语言来表示和/或使用该领域流行的方法进行分析。研究内容包括：线性算子的熵及其对偶，高维凸体的结构性质，与随机矩阵有关的各种解析和几何问题，“局部”算子理论中的一些问题，凸集的高斯相关性，非对易背景下的集中现象，以及与自由和量子信息论有关的一些几何问题。在初级水平上，分析是研究函数或量与它们所依赖的参数之间的关系。碰巧的是，许多自然出现的关系是线性的，或者至少是凸的。因此，很好地理解凸函数，进而理解凸集是理解这些关系的先决条件。基本问题中的自由参数的数量通常与相应数学模型中的集合的维度有关。由于现实生活中的问题通常依赖于许多参数，因此高维设置尤其令人感兴趣。这正是渐近几何分析的领域，它研究当维度达到无穷大时凸集的定量性质。它构成了古典泛函分析和古典几何之间的一块肥沃的中间地带。泛函分析通常涉及无限维环境(通常是非常大维度的理想化)，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。在过去二十年左右的时间里，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成功，这些问题逃脱了早期的“过于定性”或“过于僵化”的方法。最后，为了解释我们对非对易的兴趣，我们指出，它只是反映了这样一个事实，即一个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，但到目前为止并不是唯一的表现就是量子力学。

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.