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Coarse geometry and applications to the Baum-Connes conjecture

粗略几何及其在 Baum-Connes 猜想中的应用

基本信息

批准号：
23527961
负责人：
Professor Dr. Thomas Schick
金额：
--
依托单位：
Mathematisches Institut (MI)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2006
资助国家：
德国
起止时间：
2005-12-31 至 2010-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/23527961?language=en
关键词：
Coarse geometry applications Baum Connes

项目摘要

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).

粗糙几何学研究度量空间的大尺度结构。它也适用于非粗糙的问题。最重要的应用是计算离散群G的C*-代数的K-理论（通过鲍姆-康纳斯猜想）。为了解决这个猜想，人们研究了与G.对于这些定义不同种类的C*-代数，其K-理论是通过一定的指标映射。使用大尺度几何学的方法，然后必须证明这个索引映射是同构的。然后，人们必须使用空间与群的关系来下降到原始的鲍姆-康纳斯猜想。新的发展导致了粗糙空间的公理化特征。对于许多新的群，人们应该赋予有趣的非度量粗糙空间以良好的性质。发展新的方法，人们应该能够进一步推动Baum-Connes猜想的应用。这也应该给新的见解，在其他（目前非粗糙）的方法，这一猜想，这可能是这些一般的粗糙几何方法的特殊情况。作为一个更具体的例子，我们计划研究凯莱图（或更一般的凯莱图的随机子图）和它们的重标度限制。重点将放在考虑经典的运营商和领域，这样的图形，在与相关的限制对象。一个例子是离散拉普拉斯算子（因为它们用于图像处理）。