Groupoids, Deformations and Noncommutative Geometry

群形、变形和非交换几何

• 批准号: 1101827
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101827&HistoricalAwards=false
• 关键词: Groupoids; Deformations; Noncommutative; Geometry

The PI proposes to continue the study of problems in noncommutative geometry using tools from deformation theory, Lie groupoids, and operator algebras. They include the study of deformation quantizations,quantum groupoids and twisted cohomology over differentiable stacks. The project also involves studying the Chern-Connes character map for twisted K-theory, delocalized twisted equivariant cohomology, and its ring structure. It also aims to study C*-algebras associated to non-abelian gerbes and 2-groupoids.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 an associative algebra, which may not be commutative. Many ``noncommutative spaces'' are obtained either as deformations of commutative algebras or as convolution algebras of groupoids. The theory of deformation quantization lies on the boundary between classical and quantum mechanics. The mathematical structures of the two theories are very different, so it is challenging to understand how the transition from classical to quantum takes place. Quantization, roughly speaking, is the study and prediction of quantum phenomena, which is normally described by noncommutative associative algebras, from the geometry of their underlying classical counterparts. The Kontsevich formality theorem in a certain sense confirms that such a prediction is indeed possible, which implies that there is a deep interplay between noncommutative algebras and geometry. Groupoid convolution algebras, substituting for algebras of functions (on badly behaved quotient spaces), play a central role in Connes' noncommutative geometry program. They also play an important role in the study of the stringy space-time, or gerbes. The purpose of the project, which is centered around the application of noncommutative geometry, is to investigate questions in these fields motivated by mathematical physics. More specifically, it is motivated by a combination of ideas from quantum groups,representation theory, differential geometry, noncommutative geometry, and operator algebras. Thus the interdisciplinary nature of the proposed project promotes further interaction between these fields.

PI建议使用变形理论、李群和算子代数等工具继续研究非交换几何中的问题。它们包括变形量子化、量子群和可微堆栈上的扭曲上同调的研究。研究了扭曲k理论的chen - connes特征映射、离域扭曲等变上同调及其环结构。它还旨在研究与非阿贝尔gerbes和2-群类群相关的C*代数。非交换几何的思想是通过“非交换流形”上的函数代数来研究几何。在这种“非交换流形”上，相关对象不再是空间中的点，而是一个可能不可交换的关联代数。许多“非交换空间”要么是交换代数的变形，要么是群类群的卷积代数。变形量子化理论处于经典力学和量子力学的边界上。这两种理论的数学结构非常不同，因此很难理解从经典到量子的过渡是如何发生的。量子化，粗略地说，是对量子现象的研究和预测，量子现象通常是由非交换结合代数描述的，来自它们潜在的经典对立物的几何。Kontsevich形式定理在某种意义上证实了这种预测确实是可能的，这意味着非交换代数与几何之间存在着深刻的相互作用。群卷积代数取代了函数代数（在不良的商空间上），在cones的非交换几何程序中起着核心作用。它们在弦时空（gerbes）的研究中也起着重要作用。该项目的目的是围绕非交换几何的应用，研究由数学物理引起的这些领域的问题。更具体地说，它是由量子群、表示理论、微分几何、非交换几何和算子代数的思想组合而成的。因此，拟议项目的跨学科性质促进了这些领域之间的进一步互动。