Applications of asymptotic structures in Banach spaces

Banach空间是抽象的数学对象，近一个世纪以来一直被用作科学工具箱。它们是具有某些几何性质的有限或无限维矢量集合(抽象意义上)。传统上，它们被用来求解微分方程，即一种直接应用于物理和工程的方程。Banach空间理论的领域围绕着工具箱本身的研究和发展。在过去的几十年里，人们发现了与其他数学领域(例如，组合学、描述集合论和概率论)的紧密联系。这些联系中的一些已经导致在诸如计算机科学等现代领域中的应用，例如，通过在Banach空间中实现图来设计有效的算法。因此，研究Banach空间及与其相互作用的其他对象，如有界线性算子的抽象性质具有很强的潜在价值。该程序的核心是研究Banach空间的几何性质。我们广泛地将它们分为局部性质、渐近性质和全局性质。所有这些都发挥了重要作用，但对第二种类型给予了特别关注。这一深奥定义的房产家族在该领域的专家中引起了大量关注。在过去的二十年里，人们已经确定它们可以用来研究开放定义的概念，如有界线性算子和Banach空间中的图的表示。有界线性算子出现在各种应用中，例如在求解复杂方程组或数据压缩中。Banach空间中的图的表示可以用来研究由该图使用环境Banach空间的结构来建模的问题。在该程序中，将研究Banach空间几何中的某些问题，例如，不同类型的渐近结构如何相互作用，以及Banach空间表现出极端非齐次的例子。本研究的结论将被用来更好地控制某些Banach空间上有界线性算子的行为。此外，对Banach空间的度量性质(例如，图的度量表示)与其渐近性质之间的关系的理解也将得到改善。这构成了研究人员在研究问题的某些方面的研究计划的继续，例如著名的不变子空间问题、标量加紧问题和自反性的度量特征。其中一些问题的部分或全部解决将是对该理论的重大贡献。在这个过程中开发的一些工具有望解开Banach空间和其他数学领域之间的隐藏联系，并导致更多的应用。

Banach spaces are abstract mathematical objects that have been used as a scientific toolbox for almost a century. They are finite or infinite dimensional collections of vectors (in an abstract sense) that are imbued with certain geometric properties. Traditionally, they have been used to find solutions to differential equations, i.e., a kind of equations with direct applications to physics and engineering. The field of Banach space theory revolves around the study and development of the toolbox itself. Over the decades, strong connections to other fields of mathematics (e.g., combinatorics, descriptive set theory, and probability) have been discovered. Some of these connections have lead to applications in such modern fields as computer science, e.g., in the design of efficient algorithms via the realization of graphs inside Banach spaces. Therefore, there exists strong potential value in the study of abstract properties of Banach spaces and other objects that interact with them, such as bounded linear operators. At the heart of the program lies the study of geometric properties of Banach spaces. We broadly classify them into local, asymptotic, and global properties. All of them play an important role but particular focus is given to the second type. This esoterically defined family of properties has gathered a large amount of attention among experts in the field. In the past two decades it has been established that they can be used to study exoterically defined notions such as bounded linear operators and representations of graphs inside Banach spaces. Bounded linear operators appear in all sorts of applications, e.g., in the solution of complicated systems of equations or in the compression of data. A representations of a graph inside a Banach space can be used to study a problem modeled by this graph using the structure of the ambient Banach space. During the program, certain problems in the geometry of Banach spaces with be studied, e.g., how different types of asymptotic structures interact with one another and examples of Banach spaces exhibiting extreme non-homogeneity will be designed. The conclusions of this study will be used to better control the behavior of bounded linear operators on certain Banach spaces. Also, the understanding of relations between metric properties (e.g., metric representations of graphs) of a Banach space and its asymptotic properties will be improved. This constitutes a continuation of the investigator's research program in the study of certain aspects of problems such as the famous invariant subspace problem, the scalar-plus-compact problem, and the metric characterization of reflexivity. The partial, or full, solution of some of these problems will be a major contribution to the theory. Some of the tools developed during the process are expected to unravel hidden connections between Banach spaces and other mathematical areas and lead to more applications.