Operator Analysis and Applications

算子理论最初是作为一种描述量子力学的数学语言发展起来的。它已经扩展到纯数学和应用数学的许多其他领域，例如控制论。控制理论是设计像自动驾驶和自动驾驶汽车这样的系统，它们必须对环境做出反应，但不能过度反应。随着系统变得越来越复杂，并且包含了更多的特性，新的问题就出现了。PI将致力于开发新的数学工具来处理这些问题。该项目还通过培养研究生为美国的劳动力发展做出贡献。PI将研究算符理论、函数理论以及它们之间的相互作用。非交换函数是输入是两个（或更多）运算符，输出是一个运算符的函数。这是一个相对较新的领域，但已经在控制理论、代数几何和半定规划中得到了应用。PI将研究非交换函数，这通常可以解释交换情况，并且最近在控制理论中变得重要。这些方法将结合多变量算子理论、泛函分析和几个复杂变量的技术。此外，PI还将利用数学模型研究阿尔茨海默病的发展、声学信号分析、COVID-19风险测量模型等。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。

Operator Theory was originally developed as a mathematical language to describe Quantum Mechanics. It has spread to many other areas in pure and applied mathematics, such as Control Theory. Control Theory is the design of systems like automatic pilots and self-driving cars that must react, but not over-react, to their environment. As the systems get more complex, and more features are incorporated, new problems arise. The PI will work on developing new mathematical tools to deal with these problems. This project also contributes to US workforce development through the training of graduate students.The PI will study problems in Operator theory, function theory, and the interaction between them. Non-commutative functions are functions whose input is two (or more) operators, and whose output is an operator. This is a relatively new field, but already has applications in control theory, algebraic geometry, and semi-definite programming. The PI will study non-commutative functions, which can often shed light on the commutative case, and have recently become important in Control Theory. The methods will be a combination of techniques from multivariable operator theory, functional analysis and several complex variables. In addition, the PI will work on using mathematical models to study the development of Alzheimer's disease, the analysis of acoustic signals, and risk measurement models for COVID-19.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.