C* - Algebras, Groupoids, and Noncommutative Geometry

C* - 代数、群形和非交换几何

• 批准号: 0605725
• 负责人: Ping Xu
• 金额: $ 12.74万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605725&HistoricalAwards=false
• 关键词: Algebras; Groupoids; Noncommutative; Geometry

Xu proposes to continue the study of differential stacks and grebes in terms of Lie groupoids. Xu also proposes to investigate some aspects of twisted $K$-theory over differentiable stacks using groupoids and KK-theory of C^*-algebras, based on the theory developed by Tu, Laurent-Gengoux and himself. These include investigating ring structures on twisted K-theory groups, studying the relation between the twisted K^0-group and the Grothendieck group of twisted vector bundles over groupoids, developing twisted equivariant cohomology and studying its relation with twisted equivariant $K$-theory groups under the Chern-Connes character map. The project also aims at the quantization of Lie bialgebroids and quasi-Poisson groupoids.Groupoids are useful tools in studying the symmetry of various geometric problems. They also appear naturally in foliation theory as well as in modern Poisson geometry. Groupoid C^*-algebras, on the other hand, have been studied for more than two decades by operator algebraists, and they play an important role in noncommutative differential geometry. Indeed noncommutative geometry is the study of geometry through operator algebras, which has applications to many areas of mathematics including analysis, topology and geometry, mathematical physics and number theory. A stack, roughly speaking, is a Morita equivalence class of groupoids. (Lie) groupoids relate to (differentiable) stacks like open covers relate to manifolds. Just like there are many ways to describe the same manifold by open covers and gluing data, there are many groupoids describing the same stack. The equivalence relation defined on groupoids is the Morita equivalence. The project, which is centered around the application of Lie groupoids, is to investigate questions motivated from mathematical physics, in particular string theory, by a combination of ideas from algebraic geometry, noncommutative geometry, operator algebras and KK-theory, and Poisson geometry. Thus it promotes further interaction between these fields.

Xu建议在lie群体方面继续研究差分堆栈和格里布斯。 Xu还建议根据Tu，Laurent-Gengoux及其他本人开发的理论，研究使用c^* - 代数的twisted $ k $理论的某些方面。其中包括研究扭曲的K理论组上的环结构，研究扭曲的K^0组与扭曲的矢量捆绑包上的扭曲的旋转矢量捆绑包之间的关系，发展了扭曲的eproivariant共同体，并通过Chern-connes特征图下的扭曲的Equivariant $ k $ - 理论研究了其关系。该项目还旨在量化双质虫和准脉 - poisson groupoids。群是研究各种几何问题的对称性的有用工具。它们也自然出现在叶面理论以及现代泊松几何形状中。另一方面，二十多年来，c^* - 代数已被操作员代数主义者研究了二十多年，并且它们在非共同差异几何形状中起着重要作用。实际上，非交通性几何形状是通过操作方代数对几何形状进行的研究，该几何形状在许多数学领域都有应用，包括分析，拓扑和几何学，数学物理学和数字理论。粗略地说，堆栈是摩托塔等效类别类别的类别。 （Lie）类固醇与（可区分的）堆栈（如开放式盖子）相关，与歧管有关。就像有很多方法可以通过开放式封面和胶合数据来描述相同的歧管一样，有许多群体素描述了相同的堆栈。在群体素定义的等效关系是莫里塔对等。该项目围绕li lie groupoids的应用，是为了调查数学物理学，尤其是弦理论所激发的问题，通过代数几何形状，非共同几何形状，操作员代数和KK理论和Poisson几何形状的结合。因此，它促进了这些领域之间的进一步相互作用。