K-theory, C*-algebras and Index Theory of Elliptic Operators

研究人员将从事三个项目。 将继续研究双曲动力系统与C ~*-代数的K-理论之间的联系。特别地，Smalespace的zeta函数将用稳定代数的K-理论和诱导同态来表示。 这导致了一个通用的程序使用的K-理论的这些C*-代数作为一个替代ordinaryhomology群，在过去的研究hyperbolicsystems。第二个方向是研究群的正合性猜想与Novikov猜想之间的关系。 如果一个群顺从地作用在紧致空间上，那么这两个定理都成立。 研究了约化C ~*-代数的苏坎作用的存在性与逼近性质之间的关系，建立了这两个性质之间的精确关系，并在第三个方向上，得到了正则李群胚的高指数公式。 这将提供一个公式配对的某些循环cocycle光滑卷积代数的广群与类在K理论代表analyticindex的椭圆operator.These项目的一部分，一个通用的程序，看看如何对称，在其各种形式，影响不同部分的数学。 意外的对称性往往有着重要的意义。物理学和工程学中重要方程的解已经被发现，并利用它们的形式在一定的变换下必须保持不变这一事实进行了研究。 希望本项目的结果能有类似的应用。

AbstractKaminkerThe researcher will work on three projects. Investigation will becontinued on the connection between hyperbolic dynamical systems andK-theory of C*-algebras. In particular, the zeta function of a Smalespace will be expressed in terms of the K-theory of the stable algebraand the induced homomorphisms. This leads to a general program to usethe K-theory of these C*-algebras as a replacement for ordinaryhomology groups which were used in the past to study hyperbolicsystems. The second direction involves the study of the relation between theExactness conjecture and the Novikov conjecture for finitely presentedgroups. If a group acts amenably on a compact space then both ofthese conjectures hold. The relation between the existence of suchan action and approximation properties for the reduced C*-algebra will bestudied and precise relations between the two conjectures will be established.In a third direction, work will be done on obtaining a higher indexformula for regular Lie groupoids. This will provide a formula forthe pairing of certain cyclic cocycle on the smooth convolution algebra ofthe groupoid with the class in K-theory representing the analyticindex of an elliptic operator.These projects are part of a general program to see how symmetry, inits various forms, effects different parts of mathematics. It isoften the case that unexpected symmetry has important implications.Solutions to important equations in physics and engineering havebeen discovered and studied by using the fact that their form mustbe unchanged under certain transformations. It is hoped that theresults of the present project will have similar applications.