Nonlinear and noncommutative perspectives on Banach space theory

几个世纪以来，基础微积分和代数等经典数学工具被广泛用于研究许多现实世界的现象。然而，有两个重要的假设对它们的有用性起着至关重要的作用，但它们并不总是成立的。首先，微积分只有在像地球表面一样，从近距离观察时才能被使用。此外，代数中的许多性质利用了这样一个事实，即当乘数时，改变因子的顺序不会影响它们的乘积。不幸的是，在这些假设下，许多真实世界的情况都不能得到充分的模拟。例如，当在线商家根据以前的购买推荐某一特定产品时，相关信息由数量组成，其变化可以更准确地描述为“跳跃”：我们购买某一特定产品是否会导致销售量的突然变化。在量子物理学或纳米尺度的物理学中，因素相乘的顺序可能会影响乘积。PI在这个项目中提议，在没有那些看似简单的假设的情况下，开发现有数学工具的现代版本。将考虑的特定工具与计算机科学和量子物理相关。PI将进行一项计划，在度量空间和算子空间的背景下研究Banach空间理论的各个方面的对应关系(主要涉及算子理想和逼近性质的研究)。这将通过将Banach空间局部理论的方法和技术与现代非线性和非对易方法相结合来实现。非线性方面包括利用无Lipschitz空间来研究Banach空间的某些逼近性质在Lipschitz同构下是否不变；以及回答关于将有限度量空间嵌入到Hilbert空间的定量问题。在非对易方面，该项目旨在证明作用于算子空间之间的某些映射类的合成定理，构造有限维空间的迹类算子的几乎欧几里德子空间，并定义算子空间的Radon-Nikodym性质。

For many centuries the classical mathematical tools of basic calculus and algebra have been widely used in the study of many real-world phenomena. However there are two important assumptions that play a crucial role in their usefulness that do not always hold. First, calculus can only be used when, like the surface of the earth, the quantities of interest "appear to be flat" from up close. In addition, many properties in algebra use the fact that when multiplying numbers, changing the order of the factors does not affect their product. Unfortunately there many real-world situations that cannot be modeled adequately under these assumptions. For example, when an online merchant recommends a particular product based on previous purchases, the information that is relevant consists of quantities whose variations can be more accurately described as "jumps": whether or not we buy a particular product causes a sudden change in quantity sold. In quantum physics, or physics at nanoscopic scales, the order in which factors are multiplied can affect the product. The PI proposes in this project to develop modern versions of existing mathematical tools in the absence of those seemingly simple assumptions. The particular tools that will be considered are relevant for computer science and quantum physics.The PI will carry on a program that studies the counterparts of various aspects of Banach space theory (mainly related to study of ideals of operators and approximation properties) in the context of metric spaces and operator spaces. This will be done by combining methods and techniques from the local theory of Banach spaces with modern nonlinear and noncommutative approaches. The nonlinear aspects include using Lipschitz-free spaces to investigate whether or not certain approximation properties for Banach spaces are invariant under Lipschitz isomorphisms; and also answering quantitative questions about embedding finite metric spaces into Hilbert spaces. On the noncommutative side, the project aims at proving composition theorems for certain classes of mappings acting between operator spaces, constructing almost Euclidean subspaces of finite-dimensional spaces of trace-class operators, and defining a Radon-Nikodym property for operator spaces.