Coarse geometry and applications to the Baum-Connes conjecture

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Coarse geometry is concerned with the study of the large scale structure of metric spaces. It has also applications to non-coarse questions. The most important such application is to the calculation of the K-theory of the C*-algebra of a discrete group G (via the Baum-Connes conjecture). To approach this conjecture, one studies (coarse) spaces associated to G. For these one defines different kinds of C*-algebras whose K-theory is related via certain index maps. Using methods from large scale geometry, one then has to show that this index map is an isomorphism. Then, one has to use the relation of the space to the group to descent to the original Baum-Connes conjecture. New developments have lead to an axiomatic characterization of coarse spaces. To many new groups one should assign interesting non-metric coarse spaces with nice properties. Developing new methods, one should be able to push further the applications to the Baum-Connes conjecture. This should also give new insights in other (currently non-coarse) approaches to this conjecture, which might be special cases of these general coarse geometry methods. As a more concrete example we plan to study Cayleygraphs (or more generally random subgraphs of Cayley graphs) and their rescaling limits. The focus will be on the consideration of classical operators and fields on such graphs, in comparison with associated limiting objects. An example are discrete Laplacians (as they are used in image processing).