C* - Algebras, Groupoids, and Noncommutative Geometry

Xu建议继续从李群类群的角度研究微分堆和梯度。Xu还在Tu， Laurent-Gengoux和他自己的理论基础上，提出了利用群类和C^*-代数的kk -理论研究可微堆栈上的扭曲K$-理论的某些方面。这些成果包括研究扭曲K-理论群上的环结构，研究类群上扭曲向量束的扭曲K^0群与Grothendieck群的关系，发展扭曲等变上同调，并研究其在chen - connes特征映射下与扭曲等变$K$-理论群的关系。本项目还致力于李双代数群和拟泊松群的量子化。群拟是研究各种几何问题对称性的有用工具。它们也自然地出现在叶理理论和现代泊松几何中。另一方面，类群C^*-代数已经被算子代数家研究了二十多年，它们在非交换微分几何中起着重要的作用。事实上，非交换几何是通过算子代数来研究几何的，它在数学的许多领域都有应用，包括分析、拓扑和几何、数学物理和数论。粗略地说，堆栈是群类群的森田等价类。（Lie）群类群与（可微）堆栈的关系就像开盖与流形的关系一样。就像有许多方法可以通过打开盖子和粘合数据来描述相同的流形一样，也有许多群类群来描述相同的堆栈。在群类群上定义的等价关系是Morita等价。该项目以李群样的应用为中心，通过结合代数几何、非交换几何、算子代数和kk理论以及泊松几何的思想，研究数学物理，特别是弦理论中产生的问题。因此，它促进了这些领域之间的进一步互动。

Xu proposes to continue the study of differential stacks and grebes in terms of Lie groupoids. Xu also proposes to investigate some aspects of twisted $K$-theory over differentiable stacks using groupoids and KK-theory of C^*-algebras, based on the theory developed by Tu, Laurent-Gengoux and himself. These include investigating ring structures on twisted K-theory groups, studying the relation between the twisted K^0-group and the Grothendieck group of twisted vector bundles over groupoids, developing twisted equivariant cohomology and studying its relation with twisted equivariant $K$-theory groups under the Chern-Connes character map. The project also aims at the quantization of Lie bialgebroids and quasi-Poisson groupoids.Groupoids are useful tools in studying the symmetry of various geometric problems. They also appear naturally in foliation theory as well as in modern Poisson geometry. Groupoid C^*-algebras, on the other hand, have been studied for more than two decades by operator algebraists, and they play an important role in noncommutative differential geometry. Indeed noncommutative geometry is the study of geometry through operator algebras, which has applications to many areas of mathematics including analysis, topology and geometry, mathematical physics and number theory. A stack, roughly speaking, is a Morita equivalence class of groupoids. (Lie) groupoids relate to (differentiable) stacks like open covers relate to manifolds. Just like there are many ways to describe the same manifold by open covers and gluing data, there are many groupoids describing the same stack. The equivalence relation defined on groupoids is the Morita equivalence. The project, which is centered around the application of Lie groupoids, is to investigate questions motivated from mathematical physics, in particular string theory, by a combination of ideas from algebraic geometry, noncommutative geometry, operator algebras and KK-theory, and Poisson geometry. Thus it promotes further interaction between these fields.