Mathematical Sciences: Cyclic Cohomology, Symbols, Local Index, Toeplitz-type Operators and Hyponormal Operators

数学科学：循环上同调、符号、局部指数、Toeplitz 型算子和次正规算子

• 批准号: 8700048
• 负责人: Daoxing Xia
• 金额: $ 4.17万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8700048&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cyclic; Cohomology; Symbols

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum mechanics, etc. could be studied productively via the integral equations that model the phenomena. Currently, operator theory has a strong interaction not only with physics, but also engineering and many branches of mathematics. Notable in the latter regard are K-theory, non-commutative differential geometry, index theory, and cohomology. Professor Xia is an expert in all of these aspects of operator theory, with deep and significant contributions to his credit. For many years, before taking residency in the United States, he was an active, prolific leader in Chinese mathematics. He has done outstanding work in function theory, quasi-invariant measures on infinite-dimensional spaces, and topological algebras. He has worked on scattering in quantum field theory, and in harmonic analysis he has collaborated with the great Soviet mathematician Gelfand. In recent years, he has concentrated on operator theory. This work has involved deep connections among hyponormal operators, contractions, and K-theory; cyclic cohomology and almost Lie algebras of operators; almost Lie groups of operators; and difference operators and mathematical physics. The proposed research focuses on three major areas of research. The first involves determining cyclic cohomology and relations among cocycles, symbols, and index. This relates to K- theory, non-commutative differential geometry, and quantum field theory. The second is the study of analytic models of hyponormal operators and subnormal operators, with relationship to invariant subspaces, cohomology, and algebraic geometry. Finally, Professor Xia will study Toeplitz operators and applications to differential equations and pseudo-differential operators.

算子理论是现代数学的一门中心学科， 分析. 它的起源在于数学物理学的研究 和偏微分方程在20世纪初 世纪。 当时，人们看到，许多物理 平衡论、振动论、量子论中的问题 力学等可以通过积分进行富有成效的研究 模拟这些现象的方程。 目前，算子理论 不仅与物理学有很强的相互作用， 工程学和数学的许多分支。 值得注意的是 后者是K-理论、非交换微分 几何学、指数理论和上同调。 夏教授在所有这些方面都是专家， 算子理论，对他的理论有着深刻而重大的贡献。 信用 多年来，在进入美国之前， 他是中国数学界一位活跃、多产的领导者。 他在函数论、准不变式 无限维空间上的测度和拓扑 代数 他致力于量子场论中的散射， 在和声分析中，他与伟大的 苏联数学家盖尔芬德。 近年来 专注于运营商理论。 这项工作涉及很深 亚正规算子、收缩算子和 算子的循环上同调与殆李代数; 几乎李群的运营商;和差异运营商和 数学物理 拟议的研究集中在三个主要领域， research. 第一个涉及确定循环上同调， 上循环、符号和索引之间的关系。 这与K- 非对易微分几何与量子场 理论 二是亚常态分析模型的研究 运算符和次正规运算符，与不变量的关系 子空间、上同调和代数几何。 最后， 夏教授将研究Toeplitz算子及其应用， 微分方程和伪微分算子。