Applications of asymptotic structures in Banach spaces

点击翻译按钮获取中文摘要

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.