K-Theory, C*-algebras and Index theory

此次德俄合作的重点在于C*-代数的K-理论及相关课题。它的范围从扭曲的K-理论，最近的兴趣源于它在弦论和数学物理中的应用，到通过几乎平坦丛以及通过Burnside理论和渐近同态的C*-代数的K-理论的方法，它还涵盖了K-理论方法在边值问题研究中的应用。现代方法对C*-代数的K-理论的研究更深入，可以用来获得对C*-代数结构的重要见解，更广泛地理解非对易几何、拓扑学和概率论的新方面。这里的中心对象是Hilbert C*-模，它将是非对易概率论中条件期望和框架方法的研究重点。这些领域的整体分析部分被扩展到动力学Lefschetz公式的研究，从而为数论提供了一座桥梁。来自德国和莫斯科的首席调查人员在拟议的领域积累了大量经验；这些经验在许多情况下是相辅相成的。为了取得进一步的进展，我们现在计划在德国和莫斯科的不同团体之间建立紧密的联系(并加强现有的联系)。在项目期间，应在对基本原理的概念性理解以及明确的计算结果和应用方面取得进一步进展。为此，有必要对来自莫斯科和德国的参与研究人员进行密集的交流计划。为这一交流提供资金是本提案的主要目的。

In this proposed German-Russian research cooperation, the focus lies in K-theory of C*-algebras and related topics. This ranges from twisted K-theory where recent interest stems from its proposed applications in string theory and mathematic physics over approaches to the K-theory of C*-algebras via almost flat bundles and via Burnside theory and asymptotic homomorphism, it also covers the application of K-theoretic methods to the study of boundary value problems. The methods of modern approaches to the K-theory of C*-algebras carry much further, and can be used to gain important insights into the structure of C*-algebras, and much more generally to understand new aspects of noncommutative geometry, topology, and probability theory. The central object here are Hilbert C*-modules; which will be a focus of research for approaches to conditional expectations and frames in non-commutative probability theory. The global analysis parts of these areas are extended to the study of dynamical Lefschetz formulas, thus providing a bridge to number theory. The principal investigators from Germany and Moscow have build up a large body of experience in the proposed fields; which is in many cases complementary. To make further progress, we now plan to build intensive links (and strengthen the existing ones) between the different groups in Germany and Moscow. During the project, further progress shall be made in the conceptual understanding of the underlying principles and in explicit computational results and application. For this, an intense exchange program for the participating researchers from Moscow and Germany is necessary. Funding for this exchange is the main aim of the present proposal.