Topics in Asymptotic Geometric Analysis and its Applications

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，这个松散定义的领域最近被称为“渐近几何分析”。一方面将特别注意不可交换的对象和现象，另一方面将特别注意与数学和其他数学科学的其他领域的联系，这些领域激发了正在审议的大多数问题。拟研究的研究课题包括：高维凸体和高维赋范空间的结构性质、度量熵（凸集或线性算子的度量熵）和度量熵的对偶性，以及与量子信息理论和量子相关的一些几何问题。这些问题通常以巴拿赫空间的局部或有限维几何语言表达，并主要使用在该背景下产生或发展的各种方法进行分析。在基本层面上，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。由于很多自然出现的关系都是线性的，或者至少是凸的，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。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的领域，它研究凸集（或其他几何结构）在维数趋于无穷时的定量性质。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大的成功，这些问题逃脱了早期经典泛函分析和经典几何的“过于定性”或“过于严格”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强调，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。

AbstractSzarekThis project involves continued research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory, the loosely defined area that has been lately referred to as "asymptotic geometric analysis." Particular attention will be paid to non-commutative objects and phenomena on one hand and, on the other hand, to links with other areas of mathematics and other mathematical sciences, which motivate most of the problems that are being considered. Sample research topics that are proposed to be studied include: structural properties of high-dimensional convex bodies and of high-dimensional normed spaces, metric entropy (of convex sets or of linear operators) and duality of such entropy and some geometric questions related to quantum information theory and quantum. The questions are typically expressed in the language of local or finite dimensional geometry of Banach spaces and are to be analyzed using mainly the diverse methods that originated or were developed in that context.On an elementary level, Analysis is a study of functions, or relationships between quantities and the parameters on which they depend. Since very many naturally appearing relationships are linear or at least convex, a good understanding of convex functions and, consequently, of convex sets is a prerequisite for understanding those relationships. The number of free parameters in the underlying problem can often be related to the dimension of sets in the corresponding mathematical model. Since real-life problems usually depend on very many parameters, the high-dimensional setting is of particular interest. This is exactly the domain of asymptotic geometric analysis, which studies quantitative properties of convex sets (or other geometric structures) as the dimension goes to infinity. For the last two decades or so the asymptotic theory has been quite successful in identifying and exploiting "approximate" symmetries of various problems that escaped the earlier "too qualitative" or "too rigid" methods of classical functional analysis and classical geometry. (This led, among others, to the discovery of many links to computer science.) Finally, to explain our emphasis on non-commutativity we point out that it simply reflects the fact that the final outcome of a process may depend on the order of operations involved; the best known, but by far not the only manifestation of that principle is quantum mechanics.