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

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.

海森堡不确定性原理断言测量的顺序很重要。在测量粒子动量之前或之后测量粒子的位置会影响结果。约翰·冯·诺依曼意识到，在数学公式中捕捉这一特征的最佳方法是使用称为运算符的数学对象。研究算子理论不仅在量子力学中仍然是基础，而且在纯数学和应用数学的许多领域也是如此。控制理论是自动驾驶仪和自动驾驶汽车等事物的设计，它很大程度上取决于算子理论，随着这些系统变得越来越复杂，新的数学问题就会出现。首席研究员将致力于回答这些问题。该项目将研究算子理论、函数论和非交换函数理论中的问题。 非交换函数是输入由两个（或多个）矩阵组成且输出是矩阵的函数。粗略地说，它们是广义非交换多项式，就像解析函数是广义交换多项式一样。非交换函数理论非常新，但它已成功应用于各个领域，包括控制理论、实现公式、非交换代数几何和半定规划。首席研究员将使用非交换函数理论来研究谱理论和函数的算子单调性，并为交换理论提供线索。此外，他还将致力于使用数学模型来帮助了解阿尔茨海默病的发展。

项目成果

The implicit function theorem and free algebraic sets

隐函数定理和自由​​代数集

• DOI: 10.1090/tran/6546
• 发表时间: 2016
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Agler, Jim