Banach Spaces with a Focus on Sobolev-Style Spaces, Frame Theory, and Quantum Graphs

巴拿赫空间是一种有用且强大的抽象框架，用于理解真实世界的数据，如图像、声音或实验结果，它们至少在两个不同的层次上这样做。首先，对于特定的数据实例，Banach空间提供了一种严格量化该数据的各种特征的方法。在第二个抽象层次上，研究由所有可能的数据实例组成的整个空间的结构对于解决某些问题至关重要，例如预测被建模系统的未来行为。对于这个项目，考虑的空间来自信号处理（通过将对象视为简单对象的总和来处理存储有关对象的信息的问题）和量子信息科学（研究通信的数学框架，其中人们可以不像今天的计算机那样将信息编码为一串0和1，而是在量子力学系统的状态下）。PI旨在通过基础研究，包括由PI指导下的本科生进行的研究，来提高对上述两种实际环境以及相关分析考虑的一些空间的认识。此外，通过他对博士后的拓展和指导，PI将为STEM领域的学生和研究人员群体的增长和多样化做出贡献。本项目分为三个部分。第一个是受时频分析中的不确定性原理的启发，它与sobolev式不等式和与赋予额外“磁性”结构的有限图相关的空间有关。这个名字来源于这样一个事实：在一些原子间键的量子力学模型中，磁势的存在不仅是用图来建模的，而且还用图的每条边的复数模数1的附加赋值来建模。项目的第二部分试图将框架理论，即过完备基，从希尔伯特空间推广到一般的巴拿赫空间，在那里我们不再享有拥有大量对称的优势。在这个方向上已经做了重要的工作，但主要是在无限维的环境中，PI将继续发展有限维巴拿赫空间上的新生框架理论。第三部分是量子图，量子图是由复值矩阵构成的线性空间，它来源于量子信息论，可以看作是经典组合图的推广。PI将研究图论中各种经典结果的量子版本，特别是那些与前面提到的sobolev式不等式相关的结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.