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. ***

小行星9400766 该项目涉及三个方向的工作。第一个工作是提出寻找完整的酉不变量的操作元组。这些元组中的算子是可交换的，并且是从具有迹类自同构算子的非正规算子类中选择的。本文的目的是导出几乎可换算子乘积的迹公式，并建立与几乎可换算子相关的循环上同调的K-理论。 第二个问题是研究P+Q型算子，其中P是偏微分算子。这里的项目是得到结果的循环上同调与产品的扭曲？最后一个项目是研究Poisson代数及其量子化版本的Gelfand-Fuks上同调。 这是一个关于算子理论的项目。这个项目的根源可以追溯到数学物理学和电子的研究。电子的行为是由一对称为算子的数学对象建模的。电子的状态由算符的“谱”决定。在这里，该项目涉及到对类似于量子力学中研究的算子对的元组的研究。结果将应用于量子力学和数学领域的伪微分算子和代数几何。 ***