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.

Xia教授的项目将集中在希尔伯特领域的运营商几乎满足各种代数关系的情况下，几乎意味着要通过紧凑型，跟踪阶级或其他适当的小型运营商进行扰动。他对几乎是运营商的代数以及几乎满足Schrodinger类型身份的运营商的代数特别感兴趣。将寻求明确的公式，从而探索将要探索涉及循环共同体与K理论配对的痕量公式的强大概括。 这是希尔伯特空间运营商理论中的数学研究。可以将操作员视为富集的数字，遵守与普通数字相同的算法定律，有两个值得注意的例外：操作员的乘法取决于接收因素的顺序，而不是每个非零操作员都具有反向。对于有限维操作员，即数字的平方矩阵，这些现象解释了线性代数的丰富性。在希尔伯特空间的无限环境中，还有操作员的现象，遵守算术定律的数字，除了适当意义上的错误术语，并且包含有关任何情况的重要信息。从广义上讲，Xia教授的项目与如何有效提取此信息有关。