Groupoids, Deformations and Noncommutative Geometry

PI建议继续使用变形理论，李群胚和算子代数的工具研究非交换几何中的问题。它们包括形变量子化、量子群胚和可微堆上的扭曲上同调的研究。该项目还涉及研究扭曲K-理论的Chern-Connes特征映射，离域扭曲等变上同调及其环结构。 它也旨在研究与非交换格贝和2-广群有关的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。在这样一个“非交换流形”上，相关的对象不再是空间中的点，而是一个结合代数，它可能不是交换的。 许多“非交换空间”是作为交换代数的变形或作为群胚的卷积代数而得到的。 形变量子化理论介于经典力学和量子力学之间。这两种理论的数学结构非常不同，因此理解从经典到量子的过渡是具有挑战性的。量子化，粗略地说，是量子现象的研究和预测，这通常是由非交换结合代数描述，从它们的基础经典对应物的几何。Kontsevich形式定理在某种意义上证实了这样的预言确实是可能的，这意味着非交换代数和几何之间存在着深刻的相互作用。 广群卷积代数，取代代数的功能（在不良行为的商空间），发挥了核心作用，在康纳斯的非交换几何计划。它们在弦时空（gerbes）的研究中也发挥着重要作用。 该项目的目的是围绕非对易几何的应用，研究这些领域中由数学物理激发的问题。 更具体地说，它是由量子群，表示论，微分几何，非交换几何和算子代数的思想相结合的动机。 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。

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.