C* - Algebras, Groupoids, and Noncommutative Geometry

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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.