Mathematical Science: Cohomology, Multivariable Operator Theory and Perturbation of Differential Operator

数学科学：上同调、多变量算子理论和微分算子的摄动

• 批准号: 9400766
• 负责人: Daoxing Xia
• 金额: $ 6.6万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1997-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400766&HistoricalAwards=false
• 关键词: Mathematical; Science; Cohomology; Multivariable; Operator

9400766 Xia The project involves work in three directions. First work is proposed on finding complete unitary invariants for operator tuples. The operators in these tuples commute and are chosen from classes of non-normal operators with trace class self-commutators. The goal is to derive trace formulae for products of commutators and to establish some K-theory related to the cyclic cohomology associated with almost commuting operators. The second problem is to study the operators of the form P+Q where P is a partial differential operator. The project here is to derive results on the cyclic cohomology associated with a product of twisted commutators for this type of perturbation and to study the related spectral analysis. The final project is to study the Gelfand-Fuks cohomology of a Poisson algebra and its quantized version. This is a project on operator theory. The roots of this project can be traced back to mathematical physics and the study of the electron. The behavior of the electron was modeled by a pair of mathematical objects called operators. The states of the electron are determined by the "spectra" of the operators. Here the project involves a study of tuples of operators that are similar to the pair studied in quantum mechanics. The results will have applications to quantum mechanics and the mathematical areas of pseudo-differential operators and algebraic geometry. ***

9400766 Xia该项目涉及三个方向的工作。提出了第一项工作，以寻找操作员元组的完全统一不变。这些元组中的运营商通勤，并从具有跟踪类自我交流者的非正常运算符类中选择。目的是为换向器的产品得出痕量公式，并建立与几乎与通勤运营商相关的环状共同体相关的一些K理论。 第二个问题是研究P+Q表格的操作员，其中P是部分差分运算符。此处的项目是为了获得与扭曲换向器的产物相关的循环共同体的结果，并研究了相关的光谱分析。最终项目是研究泊松代数及其量化版本的Gelfand-Fuks共同体。 这是一个关于操作者理论的项目。该项目的根源可以追溯到数学物理学和电子研究。电子的行为是由称为算子的一对数学对象建模的。电子的状态由操作员的“光谱”确定。在这里，该项目涉及一项与量子力学中研究的二对相似的操作员的研究。结果将在量子力学和伪差异操作员和代数几何形状的数学领域中应用。 ***