CAREER: Noncommutative Analysis

本研究计画的中心是各种分析问题中的非对易性及其在数学物理上的应用。 对一个集合的元素的运算 * 称为交换的，如果对于集合中的所有元素x和y，x*y等于y*x。 例如，在真实的数集合上的加法和乘法运算是可交换的，而除法运算不是可交换的，因为2除以3不等于3除以2。 非对易性是现代数学及其应用的许多领域中的固有现象。自从J·冯·诺依曼的量子力学公式化以来，它一直是错综复杂的障碍以及数学深度和美感的来源。该项目的研究和教育的综合计划旨在处理各种分析问题中出现的非对易性效应及其在非对易几何，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。学生的研究和职业发展将通过针对研究生和初级研究人员的研讨会来促进，这些研讨会将与更广泛的数学界的研究会议一起组织。大学生将参加一个推广计划，以提高高中学生和教师的数学技能和知识，并提高数学的一般社区的赞赏。本研究计划的内容包括：建立单变量和多变量算子函数与各自标量函数的经典性质相似的性质，推导相关Schur乘子和类似变换的界，描述微分算子扰动的谱特征，理解高阶扰动的几何学，并发现它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的运营商理论方法的发展，预计将出现在分析和运营商理论的各种技术的综合。 该项目的目的是最终受益的数学领域以外的项目中调查的主题。

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.