This project specifically addresses problems in the geometry of Banach spaces with a focus on the analysis of coordinate systems such as bases and frames. Coordinate systems are widely used in both application and theory, strongly connecting these problems to other areas of mathematics such as approximation theory, descriptive set theory, and differential topology. For example, new Banach spaces are often constructed by explicitly building a basis for the space. In this spirit, this project will address the construction of new Banach spaces with the property that every bounded operator on the space is a multiple of the identity plus a compact operator. Moreover, the project will also study the greedy approximation properties of coordinate systems in Banach spaces. Greedy approximation is based on the idea of always taking the "biggest piece" in each step of an iterative algorithm. This project will consider the existence of greedy bases and the convergence of greedy algorithms in particular Banach spaces. Beyond greedy approximation, this project intends to extend the descriptive set theory approach to bases, which has given remarkable insight into the structural theory of Banach spaces, to that of frames. Furthermore, this project will work on adapting the techniques and structure of Hilbert and Banach frames to the continuously varying setting of vector bundles.
The structural attributes of Banach spaces and Hilbert spaces make them ideal settings for analyzing many problems in mathematics and engineering. A common example is encoding and transmitting signals. Bases in a Hilbert space or Banach space give a unique representation for the vectors in the space while the representation given by a frame is redundant. Signal encoding and transmission is often accomplished by sending coefficients with respect to some basis. This strategy, however, is not robust in the face of error, as any loss or corruption of basis coefficients results in the loss of entire dimensions of the signal. This is where frames come in as their redundancy distributes error loss over the whole space instead of concentrating it in isolated dimensions. Frames now play an important role in signal processing, and the study of their geometry in both Hilbert and Banach spaces is a growing area of research. Additionally, sometimes it is important to consider not just a single vector space, but some related collection of spaces. For example, the tangent bundle of a surface is the collection of tangent planes to the surface. In this case we want a basis for the tangent space at each point which moves smoothly over the surface. It is impossible to find such a basis for many surfaces. On the contrary, it is always possible to find a redundant frame for the tangent space which moves smoothly. Given this, it is naturally of interest to study such frames.
