Operator Algebras and Noncommutative Geometry

本项目涉及利用算子代数，特别是广群C*-代数的工具研究非交换几何问题。在Tu，Laurent和他的理论的基础上，Xu提出利用C*-代数的KK-理论继续研究可微堆上的扭曲K-理论。这些问题包括研究真李群胚上卷积代数的周期循环同调，研究群胚上扭向量丛的扭K 0-群与Grothendieck群之间的关系，研究扭K-理论的Chern-Connes特征映射，研究整体扭上同调环的结构。该项目还旨在研究与tonon-abel gerbes和2-广群胚相关的C*-代数。非对易流形 在这样一个？非对易流形？相关的对象不再是空间中的点，而是结合代数，它可能不是交换的。然而，许多概念在古典（交换）几何，包括向量丛，连接，K-理论，（上）同调，椭圆伪微分算子，陈特征，和措施可以推广到非交换设置自然产生的几何情况。 在弦理论中，时空是由一种叫做格贝斯的新型数学结构建模的。一个非常有用的方法来考虑弦时空是考虑它作为一个？非对易空间？在康纳斯的意义上。 这样的非交换空间可以用某个广群的卷积代数来构造。 该项目以非交换几何和算子代数的应用为中心，旨在通过代数和微分几何，非交换几何，算子代数和KK理论的思想组合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.