Free Analysis: Exploring the Interactions between Operator Theory and Noncommutative Function Theory

这个项目属于数学分析的分支，即算子论和泛函分析。这些学科最初是在20世纪初作为量子力学数学基础发展的一部分而发展起来的。这些想法后来在许多意想不到的方向上发展起来，远离了最初的来源，不仅应用于物理学(例如目前非常活跃的量子计算和量子信息理论领域)，还应用于电气和机械工程(这些想法应用于自动控制系统的设计、信号和图像处理)，甚至人工智能和机器学习。这个领域的一个特别新的和令人兴奋的分支被称为“非对易函数理论”，它起源于对工程中某些类型的优化问题的研究，但已经成长为自己的生命力。“多变量算符理论”的邻近领域是多方面的，但与这些应用中出现的许多问题(如量子通道的研究和最优化中的“线性矩阵不等式”理论)密切相关。该项目的目的是扩大可用于研究这些问题的数学工具的范围，并加深我们对这些不同数学思想之间相互作用的理解。该项目将进一步整合研究和教育，以及初级研究人员的专业发展。将在当地一所学校开展“数学圈”项目。该项目将综合运用算子论(特别是多变量)、泛函分析和复分析的技术，研究算子论与迅速发展的“自由分析”或“非对易”函数论领域的相互关系。这些最新的快速发展为研究数学分析中的一些基本问题提供了新的工具，并已在数学的各个领域找到了应用，包括最优化理论、凸分析和量子信息理论。与研究生共同工作的目的是将这些新方法应用于泛函分析和复函数论的交叉问题，特别是“非对易”区域中的函数论算子理论的研究，包括非对易有理函数的因式分解、零集和实现理论的问题。该项目将借鉴几个目前活跃的数学分析领域的技术，目的是扩大和加深我们对函数论、算子理论和复杂分析之间相互作用的理解。该奖项反映了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。

This project belongs to the branches of mathematical analysis known as Operator Theory and Functional Analysis. These subjects were developed initially in the early part of the 20th century as part of the development of the mathematical foundations of quantum mechanics. These ideas have subsequently evolved in many unexpected directions, far from their original source, with applications not only in physics (such as the currently very active areas of quantum computing and quantum information theory), but also in electrical and mechanical engineering (where these ideas are applied in the design of automatic control systems, in signal and image processing), and even artificial intelligence and machine learning. A particularly new and exciting branch of this field is known as “noncommutative function theory” which has its origins in the study of certain kinds of optimization questions in engineering, but has grown to take on a life of its own. The adjacent area of “multivariable operator theory” is multi-faceted but is closely connected with many questions arising in these applications (such as the study of quantum channels, and the theory of “linear matrix inequalities” in optimization). The project is aimed at expanding the array of mathematical tools available for the study of these questions, and at deepening our understanding of the interplay between these diverse mathematical ideas. The project further will integrate research and education, and professional development of junior researchers. A “Math Circle” program at a local school is to be carried out.The project will employ a blend of techniques from operator theory (especially in several variables), functional analysis, and complex analysis to study the interrelation between operator theory and the rapidly developing field of "free analysis" or "noncommutative" function theory. These recent, rapid developments have made new tools available to the study of some basic questions in mathematical analysis and have already found applications to diverse areas of mathematics, including optimization theory, convex analysis, and quantum information theory. The goal of joint work with graduate students is to apply these new methods to questions at the intersection of functional analysis and complex function theory, particularly the study of function-theoretic operator theory in "noncommutative" domains, including questions about factorizations, zero sets, and realization theory for noncommutative rational functions. The project will draw on techniques from several currently active areas of mathematical analysis, with the aim of broadening and deepening our understanding of the interplay between function theory, operator theory, and complex analysis.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.