K-Theory, C*-algebras and Index theory

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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.