Operator Theory and Applications

算子理论与应用

• 批准号: 1565243
• 负责人: John McCarthy
• 金额: $ 62.5万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565243&HistoricalAwards=false
• 关键词: Operator; Theory; Applications; Operator; Theory; Applications

The Heisenberg uncertainty principle asserts that the order in which measurements are made matters; measuring the position of a particle before or after measuring its momentum affects the result. John von Neumann realized that the best way to capture this feature in a mathematical formulation was in terms of mathematical objects called operators. Studying operator theory is still fundamental not only in quantum mechanics, but in many areas of both pure and applied mathematics. Control theory, which is the design of things like automatic pilots and self-driving cars, depends critically on operator theory, and as these systems get more complex, new mathematical questions arise. The principal investigator will work on answering such questions.This project will study problems in operator theory, in function theory, and in the theory of noncommutative functions. Noncommutative functions are functions whose input consists of two (or more) matrices and whose output is a matrix. Roughly speaking they are generalized noncommutative polynomials in the same way that an analytic function is a generalized commutative polynomial. The theory of noncommutative functions is very new, but it has been successfully applied in diverse areas, including control theory, realization formulas, noncommutative algebraic geometry, and semi-definite programming. The principal investigator will use noncommutative function theory to study spectral theory, and operator monotonicity of functions, shedding light on the commutative theory also. In addition, he will work on using mathematical models to help understand the development of Alzheimer's disease.

海森伯格不确定性原则断言，测量的顺序是重要的。测量粒子在测量其动量之前或之后的位置会影响结果。约翰·冯·诺伊曼（John von Neumann）意识到，在数学公式中捕获此功能的最佳方法是在称为算子的数学对象方面。研究操作者理论不仅在量子力学中仍然是基本的，而且在纯数学和应用数学的许多领域中。控制理论是自动飞行员和自动驾驶汽车等事物的设计，这取决于操作者的理论，随着这些系统变得更加复杂，新的数学问题就会出现。首席研究者将致力于回答此类问题。该项目将研究操作者理论，功能理论和非交流功能理论中的问题。 非共同函数是函数，其输入由两个（或更多）矩阵组成，其输出为矩阵。大概的话，它们是普遍的非交换多项式，就像分析函数是广义的交换多项式一样。非共同功能的理论非常新，但是它已成功地应用于不同领域，包括控制理论，实现公式，非交通性代数几何形状和半明确编程。首席研究者将使用非交通函数理论来研究光谱理论和功能的运算符单调性，从而阐明了交换理论。此外，他将努力使用数学模型来帮助了解阿尔茨海默氏病的发展。