Mathematical Sciences: Non-Commutative Differential Geometry

数学科学：非交换微分几何

• 批准号: 8910928
• 负责人: Mariusz Wodzicki
• 金额: $ 6.93万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-05-15 至 1992-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8910928&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Commutative; Differential

At a technical level, Professor Wodzicki's project is largely concerned with computing the cyclic homology of algebras that come, variously, from differential or algebraic geometry or from functional analysis. Algebras of differential operators, especially on smooth analytic varieties, play a major role in this research, but there is also an important aspect of it that deals with C*-algebras (certain algebras of operators on Hilbert space) and their K-theory and cyclic homology. The mathematical research here envisioned is a far-reaching outgrowth of calculus on manifolds, roughly surfaces and their higher-dimensional analogues. Differential operators, which come from calculus, see the manifold only in small pieces, but nevertheless contain information about its overall shape and conformation. One strategy, relevant to the work of Professor Wodzicki, for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. The further development of this machinery, two important and intimately related parts of which are called K-theory and cyclic homology, and its use in elucidating geometric and topological structure constitute the agenda of the research supported by this award.

在技术层面上，沃齐基教授的项目 主要涉及计算代数的循环同调 它们来自微分几何、代数几何， 从功能分析。微分算子代数， 特别是在光滑解析簇上， 这项研究，但也有一个重要的方面， 本文讨论了C ~*-代数（Hilbert空间上的某些算子代数 空间）及其K-理论和循环同调。 这里设想的数学研究是一个意义深远的 流形上的微积分的衍生，粗糙曲面及其 更高维度的类似物微分算子， 从微积分中，只能看到流形的一小部分， 然而包含关于其整体形状的信息， 构象一项与教授的工作有关的战略 Wodzicki，为了从一个 微分算子涉及将算子组装成 代数结构，然后使用代数机器。的 为了进一步发展这一机制， 其中密切相关的部分被称为K理论和循环 同调及其在阐明几何和拓扑中应用 结构构成的研究议程，由这一支持 奖