Mathematical Sciences: Non-Commutative Differential Geometryof the Deformations of Commutative Rings

数学科学：交换环变形的非交换微分几何

• 批准号: 9101817
• 负责人: Boris Tsygan
• 金额: $ 5.23万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1993-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101817&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Commutative; Differential

This research is concerned with the noncommutative differential geometry of the deformations of commutative algebras; algebras of differential operators; symbols of pseudo-differential operators; coordinate algebras of quantum groups and spaces; and differential operators on quantum manifolds. The principal investigator will develop a Riemann-Roch theorem for quantum manifolds through his work on the cyclic homology theory of quantum groups and spaces. Applications of these results to arithmetic and conformal quantum field theory will also be explored. The research in this project is an outgrowth of calculus on manifolds. Roughly speaking, manifolds are 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 for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. This project is concerned with the development of this machinery.

这项研究涉及交换代数变形的非交通差异几何。差分运营商的代数；伪差异操作员的符号；坐标量子组和空间的代数；以及量子歧管上的差分运算符。 首席研究者将通过他在量子组和空间的环状同源性理论方面的工作来开发一种Riemann-Roch定理，以供量子歧管。 这些结果在算术和保形量子场理论中的应用也将被探讨。 该项目中的研究是歧管上的微积分的产物。 粗略地说，歧管是表面及其较高维度的类似物。 来自微积分的差分操作员仅在小块中看到歧管，但仍然包含有关其整体形状和构象的信息。从差分运算符中提取此类信息的一种策略涉及将操作员组装成代数结构，然后使用代数机械。 该项目与该机械的开发有关。