Banach spaces together with their geometry provide an important framework for studying problems in physics, signal processing, and the analysis of large data sets. This research project on Banach spaces will have two main directions. The first one is the study of coordinate systems of Banach spaces. If we model a given problem in physics or signal analysis using a certain Banach space we will also need an "appropriate coordinate system" for that space, i.e. we want to represent the elements of this space by a sequence of numbers. What constitutes an "appropriate coordinate system" will of course depend on the specific problem, but generally the goal is to approximate a given element of a Banach space as well as possible with the least amount of coordinates, and to reconstruct the element from the given sequence of coordinates with the smallest possible error and the least amount of effort. Not every Banach space admits coordinate systems which have the minimality properties; one would like a coordinate system to satisfy. Therefore one also needs to find for a given Banach space criteria to embed it into a space with coordinate systems, without losing the topological and geometrical properties of the original space. Metric spaces are often used in Computer Science to model large data sets, and our second objective is to investigate embeddings of metric spaces into Banach spaces, and to obtain on the one hand information about the structure of the metric space, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain metrics embed or do not embed in it.
Many of the problems under study in this project either originate from, or are related to, other areas of mathematics such as descriptive set theory, harmonic analysis, metric geometry, and approximation theory. The techniques to be employed will involve a combination of analysis, geometry, infinite combinatorics, and logic. One of the problems considered is an old one from harmonic analysis, which asks whether the space of p-integrable functions and other function spaces have a Schauder basis formed by translates of only one element. Another prominent problem is the embedding of uniformly convex Banach spaces into such spaces with a basis or a finite dimensional decomposition. Together with his colleague Sivakumar and their joint student, Keaton Hamm, the investigator intends to study the representation and approximation of elements of certain function classes using redundant coordinate systems. The research also pursues a new direction and investigates problems in metric geometry. This work aims to characterize geometric and topological properties of Banach space like reflexivity by the embeddability of certain metric spaces.
