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

9400766夏这个项目涉及三个方向的工作。第一个工作是寻找算子元组的完全酉不变量。这些元组中的操作符可交换，并且是从具有跟踪类自交换子的非正规操作符类中选择的。目的是推导换向子乘积的迹公式，并建立与几乎交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形式的算子，其中P是一个偏微分算子。本文的目的是推导与这类扰动的扭向子乘积相关的循环上同调的结果，并研究相关的谱分析。最后的课题是研究泊松代数的Gelfand-Fuks上同调及其量子化版本。这是一个关于算子理论的课题。这个项目的根源可以追溯到数学物理和对电子的研究。电子的行为是由一对称为算符的数学对象来模拟的。电子的状态是由算符的“谱”决定的。在这里，这个项目涉及对元算子的研究，类似于量子力学中对算子的研究。研究结果将应用于量子力学、伪微分算子和代数几何等数学领域。＊＊＊

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. ***