Operator Theory and Applications

海森堡测不准原理断言，测量的顺序很重要；在测量一个粒子的动量之前或之后测量它的位置都会影响结果。约翰·冯·诺伊曼意识到，在数学公式中捕捉这一特征的最好方法是用称为算子的数学对象。研究算符理论仍然是基础，不仅在量子力学，但在许多领域的纯数学和应用数学。控制理论，也就是自动驾驶仪和自动驾驶汽车的设计，主要依赖于算子理论，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。这个项目将研究算子理论、函数理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，输出为一个矩阵的函数。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。非交换函数理论是一个非常新的理论，但它已经成功地应用于许多领域，包括控制理论、实现公式、非交换代数几何和半定规划。主要研究者将利用非交换函数理论来研究谱理论，以及函数的算子单调性，从而对交换理论有所启发。此外，他将使用数学模型来帮助理解阿尔茨海默病的发展。

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.