ABSTRACT

Banach spaces are a useful and powerful abstract framework to understand real-world data such as images, sound, or experimental results, and they do so in at least two different levels. First, for a particular instance of data, a Banach space provides a way to rigorously quantify various characteristics of that data. At a second level of abstraction, studying the structure of the whole space consisting of all possible instances of data has been crucial to the solution of certain problems, such as the prediction of the future behavior of the system being modeled. For this project, the spaces under consideration come from both signal processing (which deals with the problem of storing information about an object by considering it as a sum of simpler ones) and Quantum Information Science (which studies a mathematical framework for communications where one can encode information not as a string of 0s and 1s as today's computers do, but rather in the state of a quantum-mechanical system). The PI seeks to advance the knowledge of some spaces coming from the two aforementioned practical settings -- as well as related analytical considerations -- via fundamental research, including research conducted by undergraduate students under the PI's supervision. In addition, through his outreach and mentoring of postdocs, the PI will contribute to growing and diversifying the group of students and researchers in STEM fields.

The project is divided into three parts. The first is inspired by an uncertainty principle in time-frequency analysis, which is related to Sobolev-style inequalities and spaces associated to finite graphs endowed with an extra "magnetic" structure. The name comes from the fact that the presence of a magnetic potential in some quantum-mechanical models of bonds between atoms is modeled not just with a graph, but also with an additional assignment of a complex number of modulus one to each edge of the graph. The second part of the project seeks to generalize the theory of frames, i.e. overcomplete bases, from the Hilbert space setting to the general Banach space one, where we no longer enjoy the advantages of having a large group of symmetries. Significant work has already been done in this direction, but mostly in the infinite-dimensional setting, and the PI will continue developing the nascent theory of frames on finite-dimensional Banach spaces. The third part is focused on quantum graphs, which are linear spaces of complex-valued matrices that come from Quantum Information Theory and can be considered as generalizations of classical combinatorial graphs. The PI will investigate quantum versions of a variety of classical results in graph theory, particularly those related to the aforementioned Sobolev-style inequalities.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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