Mathematical Sciences: Non-Commutative Differential Geometry of Deformations of Commutative Rings: Operations Index Theorems and Characteristic Classes

数学科学：交换环变形的非交换微分几何：运算指数定理和特征类

• 批准号: 9623051
• 负责人: Boris Tsygan
• 金额: $ 6万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623051&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Commutative; Differential

9623051 Tsygan This grant supports the research of Professor B. Tsygan to work on various problems in the deformation theory of Poisson manifolds. In particular he will work on algebraic index theorems, the homological algebra of characteristic classes and quantum cohomology and applications of structure theorems for deformations to both Riemann-Roch and the Verlinde formulas. This is research that starts out in the field of algebra but moves beyond that to touch both geometry and theoretical physics. The reasons for these connections are that algebra can be though of as the study of symmetry in the abstract. As such algebra has direct applications to areas of physics and chemistry. In particular the modern theory of gauge fields in physics uses this area of algebra extensively and this project will study some of these connections at length.

9623051 Tsygan这项拨款支持B. Tsygan教授研究泊松流形变形理论中的各种问题。特别是他将研究代数指标定理，特征类的同调代数和量子上同调，以及黎曼-洛克和Verlinde公式中变形的结构定理的应用。这是一项从代数领域开始的研究，但它超越了代数领域，触及了几何和理论物理。产生这些联系的原因是，代数可以被认为是对抽象对称性的研究。因此，代数在物理和化学领域有直接的应用。特别是现代物理规范场理论广泛使用代数的这一领域，本项目将详细研究其中的一些联系。