Studies in Noncommutative Geometry

非交换几何研究

• 批准号: 9706886
• 负责人: Henri Moscovici
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706886&HistoricalAwards=false
• 关键词: Studies; Noncommutative; Geometry

ABSTRACT Moscovici The proposed research involves employing the unifying power of the concepts and methods of noncommutative geometry to address several problems which affect broad areas of mathematics, ranging from differential topology and geometry to operator algebras and mathematical physics. In collaboration with A. Connes, H. Moscovici will apply their noncommutative local index formalism to the study of diffeomorphism-equivariant characteristic classes, in particular connecting the Gelfand-Fuchs cohomology with the diffeomorphism-equivariant extension of the Atiyah-Singer index theorem. A second theme concerns the application of the straight Chern character construction to the theory of characteristic classes of pseudomanifolds, in order to develop a comprehensive approach encompassing both the classical Chern-Weil theory, in the smooth case, as well as Cheeger's formulae for the combinatorial Pontryagin classes. The third project focuses on the investigation of Connes' notion of noncommutative manifold and should ultimately lead to a complete classification of such objects in terms of a few basic noncommutative operations performed on ordinary manifolds. Noncommutative geometry is a newly emerging field of mathematics, rooted in the time-tested operator formalism of quantum mechanics, which -- in the vision of the renowned French mathematician Alain Connes -- provides a unified geometric framework able to treat the discrete on equal footing with the continuum as well as Einstein's general relativity on a par with the Planck-scale structure of space-time.

莫斯科维奇 拟议的研究涉及采用统一的权力的概念和方法的非交换几何，以解决几个问题，影响广泛的数学领域，从微分拓扑和几何算子代数和数学物理。 与A合作。Connes，H. Moscovici将他们的非交换局部指标形式主义应用于同形-等变特征类的研究，特别是将Gelfand-Fuchs上同调与Atiyah-Singer指标定理的同形-等变扩展联系起来。 第二个主题涉及的应用程序的直接陈省身字符建设的理论特征类的pseudomanifolds，以制定一个全面的方法，包括经典的陈省身-韦伊理论，在顺利的情况下，以及Cheeger的公式组合庞特里亚金类。 第三个项目的重点是研究Connes的非对易流形概念，并最终根据在普通流形上执行的一些基本非对易运算对此类对象进行完整的分类。 非对易几何是一个新兴的数学领域，植根于量子力学久经考验的算子形式主义，在著名的法国数学家阿兰·康纳斯（Alain Connes）的设想中，它提供了一个统一的几何框架，能够将离散的与连续的平等对待，并将爱因斯坦的广义相对论与时空的普朗克尺度结构相提并论。