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

这位研究人员将致力于三个项目。我们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，空间的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，用这些C*-代数的K-理论来代替过去用来研究双曲系的普通同调群。第二个方向是研究有限呈现群的严密性猜想与Novikov猜想之间的关系。如果一个群体在一个紧凑的空间上友好地行动，那么这两个猜想都成立。我们将研究这种作用的存在性与约化C*-代数的逼近性质之间的关系，并建立这两个猜想之间的精确关系。在第三个方向上，我们将致力于获得正则李群胚的一个更高的指数公式。这将为群胚的光滑卷积代数上的某些循环余循环与K-理论中表示椭圆算子的分析指标的类的配对提供一个公式。这些项目是了解对称的各种形式如何影响数学的不同部分的通用程序的一部分。通常情况下，意想不到的对称性具有重要的含义。人们发现和研究了物理和工程中重要方程的解，这是利用它们的形式在某些变换下必须保持不变的事实。希望本项目的成果也能有类似的应用。

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.