RUI: Opportunities in Operator Theory

这个项目涉及几个相互关联的线程，都在近年来结出了果实。该项目最新颖和动态的方面是某些离散对象的几何性质之间令人兴奋的相互作用，以及对形成量子力学自然环境的无限维空间类型之间连续变换的研究。该项目还将研究某些具有特殊对称性的变换的性质和分支。潜在的应用是信号处理，控制工程和量子力学。拟议的研究将在数学科学的不同领域之间建立桥梁。为此，首席研究员将与新老同事合作，并赞助本科生研究。事实上，这个项目产生的许多问题都适合本科生研究，首席研究员将招募各种各样的学生来研究它们。这些学生将配备必要的技能，专业知识，信心和激情，以追求职业生涯中的数学科学。该项目不仅将创造新的数学，而且还将创造新的数学家，以加强和丰富国家的研究和教育基础设施。最近研究的一个有前途的方面涉及离散几何（格的性质）和结构矩阵（如Toeplitz矩阵）的渐近性质之间的新联系。在另一个方向，主要研究者和他的合作者已经开发了框架理论的结构和运营商相关的方法来证明新的结果格。 与合作者一起，首席研究员还将研究复对称算子和截断Toeplitz算子。复对称算子是一类广泛的算子，直到最近才得到充分的研究。截断Toeplitz算子的研究是函数论算子理论的一个快速发展的分支，从2007年Sarason的一篇开创性论文开始，已经经历了蓬勃的发展。首席研究员和他越来越多的合作者最近发表的一系列文章揭示了这两个类别之间令人惊讶的联系。关于这一主题的研究有可能产生变革，与若干领域有关。例如，与函数理论和矩阵分析的联系已经吸引了来自大型机构和小型学院的研究人员。此外，复对称和截断Toeplitz算子的研究已被证明是本科生研究的沃土。该奖项反映了NSF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。

This project involves several interrelated threads that have all born fruit in recent years. The most novel and dynamic aspect of the project is the exciting interplay between the geometric properties of certain discrete objects and the study of continuous transformations between the type of infinite-dimensional spaces that form the natural setting for quantum mechanics. The project will also study the properties and ramifications of certain transformations that enjoy special symmetries. Potential applications are to signal processing, control engineering, and quantum mechanics. The proposed research will build bridges between different areas of the mathematical sciences. To do this, the principal investigator will collaborate with colleagues old and new, as well as sponsor undergraduate research. Indeed, many questions stemming from this project are suitable for undergraduate research and the principal investigator will recruit a diverse array of students to work on them. These students will be equipped with the skills, expertise, confidence, and passion necessary to pursue careers in the mathematical sciences. This project will create not only new mathematics, but also the new mathematicians necessary to enhance and enrich the nation's infrastructure for research and education.A promising aspect of recent research concerns new links between discrete geometry (properties of lattices) and the asymptotic properties of structured matrices (such as Toeplitz matrices). In the other direction, the principal investigator and his collaborators have developed frame-theoretic constructions and operator-related methods to prove new results about lattices. Together with collaborators, the principal investigator will also study complex symmetric operators and truncated Toeplitz operators. Complex symmetric operators are a broad class of operators that has not been adequately studied in generality until recently. The study of truncated Toeplitz operators, a rapidly growing branch of function-theoretic operator theory, has undergone spirited development stemming from a seminal 2007 paper of Sarason. A recent series of articles by the principal investigator and his growing list of collaborators have unearthed surprising links between these two classes. Research on this subject has the potential to be transformative, having relevance to a number of fields. For instance, connections to function theory and matrix analysis have already engaged researchers from both large institutions and small colleges. Moreover, the study of complex symmetric and truncated Toeplitz operators has already proven to be fertile ground for undergraduate research.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.