The project involves studying problems in noncommutative geometry using tools from operator algebras, in particular groupoid C*-algebras. Xu proposes to continue the study of twisted K-theory over differentiable stacks using KK-theory of C*-algebras, based on the theory developed by Tu, Laurent, and himself. The problems include studying the periodic cyclic homology of convolution algebras of proper Lie groupoids, investigating the relation between the twisted K0-group and the Grothendieck group of twisted vector bundles over groupoids, studying the Chern-Connes character map for twisted K-theory, and studying the ring structure on global twisted cohomology. The project also aims to study C*-algebras associated to non-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.
