Applications of asymptotic structures in Banach spaces

Banach空间是抽象的数学对象，几乎世纪以来一直被用作科学工具箱。它们是有限维或无限维的向量集合（在抽象意义上），具有某些几何性质。传统上，它们被用来寻找微分方程的解，即，一类直接应用于物理和工程的方程。巴拿赫空间理论的领域围绕着工具箱本身的研究和发展。几十年来，与其他数学领域的密切联系（例如，组合学、描述性集合论和概率）。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，在设计有效的算法，通过实现图内的Banach空间。因此，对Banach空间的抽象性质以及与之相互作用的其他对象，如有界线性算子的研究具有很强的潜在价值。在该计划的核心在于Banach空间的几何性质的研究。我们广泛地将它们分为局部，渐近和全局属性。所有这些都发挥着重要作用，但特别注重第二类。这个深奥定义的属性家族在该领域的专家中引起了大量的关注。在过去的二十年中，人们已经确定，它们可以用来研究exoterically定义的概念，如有界线性算子和表示图内的Banach空间。有界线性算子出现在各种应用中，例如，在复杂方程组的解或数据压缩中。在Banach空间中的一个图的表示可以用来研究一个问题，该问题由这个图使用周围的Banach空间的结构来建模。在该计划中，某些问题的几何Banach空间进行了研究，例如，不同类型的渐近结构如何相互作用，并将设计表现出极端非齐性的Banach空间的例子。本文的研究结论将用于更好地控制某些Banach空间上有界线性算子的行为。此外，对度量属性之间关系的理解（例如，图的度量表示）及其渐近性质。这构成了调查员的研究计划在某些方面的研究问题，如著名的不变子空间问题，标量加紧问题，和度量表征的自反性。部分或全部解决这些问题将是对理论的重大贡献。在此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.