Operator Algebras and Noncommutative Geometry

该项目利用算子代数，特别是群C*-代数中的工具研究非对易几何中的问题。徐建议在屠维平、劳伦和他自己发展的理论的基础上，利用C*-代数的KK-理论继续研究可微堆栈上的扭曲K-理论。这些问题包括研究真李群胚的卷积代数的周期圈同调，研究群胚上扭曲向量丛的扭曲K0-群与Grothendieck群之间的关系，研究扭曲K-理论的Chern-Connes特征标映射，以及研究整体扭曲上同调的环结构。该项目还旨在研究与Tonon-Abel群和2-群胚相关的C*-代数。Connes意义下的非对易几何的思想是通过非对易流形上的函数代数来研究几何。在这样一个非对易流形上，？相关的对象不再是空间中的点，而是一个可能不可交换的结合代数。然而，经典(交换)几何中的许多概念，包括向量丛、联络、K-理论、(上)同调、椭圆伪微分算子、陈特征标和测度，都可以推广到自然产生于几何情形的非对易环境。在弦论中，时空是由一种被称为GERBES的新的数学结构来建模的。认为弦时空的一个非常有用的方法是把它看作一个？非对易空间？在康奈斯的意义上。这样的非对易空间可以利用某一群胚的卷积代数来构造。这个项目以非交换几何和算子代数的应用为中心，通过结合代数和微分几何、非交换几何、算子代数和KK理论的思想来研究数学物理中的问题，从而促进这些领域之间的进一步互动。

AbstractXuThe project involves studying problems in noncommutative geometry using toolsfrom operator algebras, in particular groupoid C*-algebras. Xu proposes tocontinue the study of twisted K-theory over differentiable stacks usingKK-theory of C*-algebras, based on the theory developed by Tu,Laurent, and himself. The problems include studying the periodic cyclichomology of convolution algebras of proper Lie groupoids, investigating therelation between the twisted K0-group and the Grothendieck group of twistedvector bundles over groupoids, studying the Chern-Connes character map fortwisted K-theory, and studying the ring structure on global twistedcohomology. The project also aims to study C*-algebras associated tonon-abelian gerbes and 2-groupoids.The idea of noncommutative geometry in the sense of Connes 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. Nevertheless, many notions in classical (commutative) geometry including vector bundles, connections, K-theory, (co-)homology, elliptic pseudo-differential operators, Chern characters, and measures can be generalized to noncommutative settings arising naturally from geometric situations. In string theory, space-time is modeled by a new kind of mathematical structure called gerbes. A very useful way to think of the stringy space-time is to consider it as a ?noncommutative space? in the sense of Connes. Such a noncommutative space can be constructed using the convolution algebra of a certain groupoid. The project, which is centered on the application of noncommutative geometry and operator algebras, is to investigate questions motivated from mathematical physics by a combination of ideas from algebraic and differential geometry, noncommutative geometry, operator algebras, and KK-theory, and thus the project promotes further interaction between these fields.