Mathematical Sciences: Cyclic Cohomology, Local Index and N-Tuple of Operators

数学科学：循环上同调、局部索引和算子 N 元组

• 批准号: 8901506
• 负责人: Daoxing Xia
• 金额: $ 5.85万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8901506&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cyclic; Cohomology; Local

Professor Xia's project will focus on situations in which operators on Hilbert space almost satisfy various algebraic relationships, where almost means up to perturbation by compact, trace class, or other suitably small operators. He is particularly interested in almost Lie algebras of operators, and in n-tuples of operators almost satisfying identities of Schrodinger type. Formulas explicitly giving the trace of operators known in such contexts to be trace class will be sought, and powerful generalizations of trace formulas involving the pairing of cyclic cohomology with K-theory will be explored. This is mathematical research in the theory of operators on Hilbert space. Operators may be thought of as enriched numbers, obeying the same laws of arithmetic as ordinary numbers with two notable exceptions: multiplication of operators depends upon the order in which the factors are taken, and not every non-zero operator has an inverse. For finite-dimensional operators, i.e. square matrices of numbers, these phenomena account for the richness of the subject of linear algebra. In the infinite- dimensional setting of Hilbert space, there is furthermore the phenomenon of operators obeying the laws of arithmetic for numbers except for error terms that are small in an appropriate sense and that contain important information about whatever situation gave rise to the operators. In a broad sense, Professor Xia's project is concerned with how one can extract this information efficiently.

夏教授的项目将集中在以下情况下， Hilbert空间上的算子几乎满足各种代数 关系，其中几乎意味着由紧凑的扰动， 跟踪类或其他适当的小操作符。他是 特别感兴趣的几乎李代数的运营商，和 在几乎满足恒等式的算子的n元组中， 薛定谔类型。公式明确给出的痕迹 在这种上下文中已知为跟踪类的运算符将是 寻求，和强大的推广迹公式，涉及 将探讨循环上同调与K-理论的配对。 这是数学研究中的算子理论 Hilbert空间运算符可以被认为是丰富的数， 遵守与普通数相同的算术定律， 值得注意的例外：运算符的乘法取决于 取因子的顺序，而不是每个非零 运算符有一个逆。对于有限维运算符，即 数字的方阵，这些现象解释了 线性代数学科的丰富性。在无限的- 希尔伯特空间的维数设置，还有 运算符遵守算术定律的现象， 除了在适当的情况下很小的误差项之外， 它包含了重要的信息， 这种情况引起了运营商。从广义上讲，教授 Xia的项目关注的是如何提取这种 信息高效。