Modern Aspects of Multivariable Operator Theory and Matrix Analysis

算子论和矩阵分析都是数学的基本领域。在很大程度上，它们最初的发展是为了为量子力学和其他物理现象提供理论基础。随着算子理论和矩阵分析的突破，出现了越来越多的应用领域，这证明了它们对科学和工程的潜在重要性。最接近该项目研究的应用领域是控制系统工程、电气工程、信号处理、图像处理和量子计算。首席研究人员将在过去工作的基础上，进一步发展自由函数理论、算子完成、统计信号处理、矩阵不等式和最优化等子学科之间的相互作用。为了实现这一目标，首席研究员将继续现有的合作以及开发新的合作，并积极参与新兴研究的研究生和本科生。首席研究员将继续保持一个促进学生参与和发展的智力环境，为学生提供在数学科学领域成功追求职业生涯的技能、经验和信心。因此，该项目将产生新的、有影响力的数学结果，以及准备加入对这个国家至关重要的科学和教育大军的训练有素的数学家。系统和控制理论、滤波器设计、信号和图像处理中的许多问题都归结为函数论问题。多变量情形是一个非常活跃的研究领域，其中多变量算子理论的技巧是非常有效的。本项目的具体主题包括(I)矩阵补全，(Ii)矩问题，(Iii)自由函数理论，(Iv)实现，(V)行列式表示，(Vi)数值值域和半径及其推广，(Vii)超几何函数，和(Viii)逆特征值问题。这一领域的结合将导致不同研究小组感兴趣的新的研究途径。所有项目也都有计算部分，使研究人员能够在科学和工程的所有领域利用成果和潜力。首席研究员将继续在德雷克塞尔大学举办分析研讨会，有当地和国际研究人员以及德雷克塞尔的学生参加。首席研究员和他的学生将通过会议报告和各种领先的数学期刊和预印本服务器上的出版物来传播结果。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Operator theory and matrix analysis are both fundamental areas of mathematics. In large part, they were originally developed to provide a theoretical basis for quantum mechanics and other physical phenomena. An increasing number of application areas have emerged as a result of breakthroughs in operator theory and matrix analysis, which testify to their underlying importance to science and engineering. The areas of application closest to the research in this project are control systems engineering, electrical engineering, signal processing, image processing, and quantum computation. The principal investigator will build on his past work to further develop the interplay between the subdisciplines of free function theory, operator completions, statistical signal processing, matrix inequalities, and optimization. To accomplish this goal the principal investigator will continue existing collaborations as well as develop new ones, and engage actively with both graduate students and undergraduate students in the emerging research. The principal investigator will continue to maintain an intellectual environment fostering student involvement and development, providing the students with the skills, experience, and confidence to successfully pursue a career in the mathematical sciences. Thus, the project will yield both new, impactful mathematical results, as well as highly trained mathematicians prepared to join the scientific and educational workforce crucial to this nation. Many questions in system and control theory, filter design, signal and image processing come down to function theoretic questions. The case of several variables is a highly active research area where the techniques of multivariable operator theory are highly effective. The specific themes of the current project include (i) Matrix completions, (ii) Moment problems, (iii) Free function theory, (iv) Realizations, (v) Determinantal representations, (vi) Numerical range and radius, and their generalizations, (vii) Hypergeometric functions, and (viii) Inverse eigenvalue problems. This combination of areas will lead to new avenues of research that are of interest to different research groups. All projects also have a computational component, allowing for the implementation of the results and the potential to be used by researchers in all areas of science and engineering. The principal investigator will continue running an Analysis Seminar at Drexel University featuring local and international researchers, as well as Drexel students. The principal investigator and his students will disseminate the results via conference presentations and publications in a variety of leading mathematical journals and preprint servers.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.