Higher Structures and Groupoids in Noncommutative Geometry

非交换几何中的高级结构和群形

• 批准号: 1406668
• 负责人: Ping Xu
• 金额: $ 18万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406668&HistoricalAwards=false
• 关键词: Higher; Structures; Groupoids; Noncommutative; Geometry

This project involves problems in noncommutative geometry. The idea of noncommutative geometry is to study geometry using noncommutative algebras, which are mathematical objects that have addition and multiplication, however the multiplication is not commutative: xy does not have to equal yx. The purpose of the project, which is centered around higher structures and noncommutative geometry, is to investigate mathematical problems in these fields motivated by physics. More specifically, it is motivated by a combination of ideas from quantum mechanics, quantum field theory, string theory, deformation quantization, differential geometry, noncommutative geometry, and operator algebras. The interdisciplinary nature of the proposed project promotes further interaction between these fields. The PI continues to disseminate his research by speaking at conferences and seminars and organizing workshops. Some parts of the project are joint work with collaborators from China and Europe. The project thus provides an excellent opportunity for the PI to work with young scientists and to exchange ideas with colleagues from other countries and to promote collaborations.The idea of noncommutative geometry is to study geometry via algebras of functions on noncommutative manifolds. On such a noncommutative manifold, the relevant objects are no longer points in a space, but rather a noncommutative associative algebra, or a differential graded commutative algebra. Many noncommutative spaces can be obtained from groupoids. The PI proposes to continue the study of deformation quantization, differential graded manifolds (DG manifolds), Chern--Connes character, and delocalized twisted equivariant cohomology. The problems include exploring the homotopy Lie algebra structures appearing in connection with a generalization of the Atiyah class to various contexts, and developing a de Rham model of equivariant twisted K-theory using the theory of Lie groupoids and differentiable stacks.

该项目涉及非交通性几何形状的问题。非交通性几何形状的想法是使用非共同代数来研究几何形状，这些代数是具有添加和乘法的数学对象，但是乘法并不是交换性的：XY不必等于yx。 该项目的目的围绕更高的结构和非共同的几何形状，是在这些领域中研究由物理学动机的数学问题。更具体地说，它是由量子力学，量子场理论，字符串理论，变形量化，差异几何形状，非交通性几何形状和操作符代数的思想组合所激发的。拟议项目的跨学科性质促进了这些领域之间的进一步相互作用。 PI通过在会议和研讨会上讲话并组织研讨会来继续传播他的研究。 该项目的某些部分是与中国和欧洲的合作者共同合作。因此，该项目为PI提供了与年轻科学家合作并与其他国家的同事交流思想并促进合作的绝佳机会。非交互性几何形状的想法是通过非公共流形的功能代数来研究几何形状。在这样的非交换歧管上，相关对象不再是空间中的点，而是非交互性的联想代数或差分分级的交换代数。可以从类别素中获得许多非交换空间。 PI建议继续研究变形量化，差分歧管（DG歧管），Chern-connes特征和DELACALIGET TWISTED EDORIVARIANT共同体学。这些问题包括探索与Atiyah类概括到各种环境有关的同质谎言代数结构，并使用lie groupoid和可区分堆栈的理论开发了de rham的模型。