Coarse geometry and applications to the Baum-Connes conjecture

粗略几何主要研究度量空间的大尺度结构。它也适用于非粗俗的问题。最重要的应用是计算离散群G的C*-代数的K-理论(通过Baum-Connes猜想)。为了接近这个猜想，我们研究了与G相关的(粗)空间。为此，我们定义了不同类型的C*-代数，它们的K-理论通过一定的指标映射联系在一起。使用大比例尺几何的方法，我们必须证明这个索引映射是同构的。然后，人们不得不利用空间与群的关系来证明最初的鲍姆-康奈斯猜想。新的发展导致了粗空间的公理刻画。对于许多新的群，人们应该分配具有良好性质的有趣的非度量粗空间。开发新的方法，人们应该能够进一步推动鲍姆-康尼斯猜想的应用。这也应该在其他(目前非粗略的)方法中提供对该猜想的新见解，这些方法可能是这些常规粗略几何方法的特例。作为一个更具体的例子，我们计划研究Cayley图(或更广泛地说，Cayley图的随机子图)及其重新定标极限。重点将集中在这类图上的经典算符和场的考虑，以及与相关限制对象的比较。离散拉普拉斯方法就是一个例子(因为它们被用于图像处理)。

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).