CAREER: Noncommutative Analysis

本研究项目主要研究各种分析问题中的非交换性及其在数学物理中的应用。如果对于集合中的所有元素x和y， x*y等于y*x，则对集合中元素的操作*是可交换的。例如，实数集上常见的加法和乘法运算是可交换的，而除法运算则不是可交换的，因为，例如，2除以3不等于3除以2。不可交换性是现代数学及其应用中许多领域的固有现象。自从冯·诺伊曼（J. von Neumann）提出量子力学公式以来，它既是复杂障碍的源泉，也是数学的深度和美感的源泉。本项目的综合研究和教育计划旨在处理在各种分析问题中出现的非交换性效应及其在非交换几何、数学物理和算子代数中的应用，以及加强这些数学领域之间的联系。针对研究生和初级研究人员的讲习班将与为更广泛的数学界举办的研究会议一起组织，以促进学生的研究和职业发展。大学生将参加一项扩展计划，以提高高中学生和教师的数学技能和知识，并提高一般社区的数学鉴赏力。研究项目垂直整合，为学生提供不同层次的研究机会。本研究项目包括建立与各自标量函数的经典性质相似的单算子和多算子函数的性质，推导相关舒尔乘子和类似变换的界，描述微分算子微扰的谱特征，理解高阶微扰几何及其在非交换几何中的意义，描述算子代数中元素的结构。这些问题的解决取决于创新算子理论方法的发展，这些方法有望从分析和算子理论的各种技术的综合中出现。该项目旨在最终使数学领域受益，而不仅仅是在项目中调查的主题。

This research project centers on noncommutativity in various problems of analysis and its applications to mathematical physics. An operation * on the elements of a set is said to be commutative if x*y is equal to y*x for all elements x and y in the set. For example, the usual operations of addition and multiplication on the set of real numbers are commutative, whereas the operation of division is not commutative since, for instance, 2 divided by 3 is not equal to 3 divided by 2. Noncommutativity is an inherent phenomenon in many areas of modern mathematics and its applications. It has been a source of intricate obstacles as well as mathematical depth and beauty since J. von Neumann's formulation of quantum mechanics. The integrated program of research and education in this project is aimed at treating noncommutativity effects that arise in various problems of analysis and its applications to noncommutative geometry, mathematical physics, and operator algebras, as well as to strengthening connections between these areas of mathematics. Research and career development of students will be facilitated by workshops targeted at graduate students and junior researchers, organized in conjunction with research conferences for the broader mathematical community. University students will participate in an outreach program to enhance mathematical skills and knowledge of high school students and teachers and raise appreciation of mathematics by a general community. The research projects are vertically integrated to provide ample opportunity for student research at different levels.This research project includes establishing properties of single and multivariate operator functions that are similar to classical properties of the respective scalar functions, deriving bounds for related Schur multipliers and similar transformations, describing spectral characteristics of perturbations of differential operators, understanding geometry of higher order perturbations and finding its meaning in noncommutative geometry, and describing structure of elements in operator algebras. Resolution of these problems depends on development of innovative operator theoretic methods that are expected to emerge from a synthesis of various techniques in analysis and operator theory. The project aims to ultimately benefit areas of mathematics beyond the topics investigated in the project.