Operator Algebras and Noncommutative Geometry

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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.