Topics in Asymptotic Geometric Analysis and its Applications

AbstractSzarek这个项目涉及的几何，概率和组合方面的功能分析和凸性理论，松散定义的区域，最近被称为“渐近几何分析的继续研究。“一方面，将特别注意非交换对象和现象，另一方面，与数学和其他数学科学的其他领域的联系，这激发了正在考虑的大多数问题。建议研究的样本研究课题包括：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）和这种熵的对偶性以及与量子信息理论和量子有关的一些几何问题。 这些问题通常用Banach空间的局部或有限维几何的语言来表达，并主要使用起源于或在此背景下开发的各种方法进行分析。在初级水平上，分析是对函数或量与它们所依赖的参数之间的关系的研究。由于许多自然出现的关系是线性的或至少是凸的，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。基本问题中的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取决于很多参数，高维设置是特别感兴趣的。这正是渐近几何分析的领域，它研究凸集（或其他几何结构）在维数达到无穷大时的定量性质。在过去的二十年左右的渐近理论已经相当成功地确定和利用“近似”的对称性的各种问题，逃脱了早期的“太定性”或“太严格”的方法，经典的功能分析和经典几何。(This导致，除其他外，发现了许多与计算机科学的联系。最后，为了解释我们对非对易性的强调，我们指出，它仅仅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远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.