K-Theory, Group C*-Algebras, Large Scale Geometry, and Topology

亲爱的乔：约翰和我写了以下几段话，总结了我们的拨款提案。如果你觉得不满意，请告诉我。你的，Nigel ------------------------------------------------------------------在C*-代数理论和拓扑学中，群和空间的大规模几何在不变量的计算中起着决定性的作用。研究人员的目的是探索群边界的几何和拓扑（使用大尺度几何定义）对群的谐波分析和它们的C*-代数k理论的确定的影响。Baum-Connes猜想将群同调与有限子群的表示理论相结合，提出了一种计算简化群C*代数k理论的方法。如果这个猜想是正确的，那么它将在几何学和拓扑学中产生许多影响，并且将Baum-Connes猜想与群的调和分析和边界空间上群的几何作用联系起来，形成一个迷人的思想圈。调查人员将试图弄清这些关系。一个长期的目标是证明Baum-Connes猜想，更重要的是更好地理解它的意义，对于像Gromov的双曲群这样的类。更直接的目标包括澄清这些群的部分猜想形式的现有证明之间的关系，并进一步发展C*-代数k理论，流形理论和控制拓扑之间的联系。尽管用于研究它的工具相当复杂，但大尺度几何背后的思想非常简单：忽略局部、小尺度的数量波动，专注于其大尺度或长期的行为。通过这样做，趋势或品质可能会变得明显，而这些趋势或品质可能会被无关紧要的小规模波动所掩盖。研究人员已经开发出工具来区分几何中不同种类的多维、大规模行为。令人惊讶的是，除了他们的内在兴趣之外，他们的工具还在普通的小尺度几何中找到了应用。

Dear Joe, John and I worked up the following paragraphs summarizing our grant proposal. Let me know if you find them unsatisfactory. Yours, Nigel ------------------------------------------------------------------ The large scale geometry of groups and spaces plays a determining role in the calculation of invariants in C*-algebra theory and topology. The investigators aim to explore the effect of the geometry and topology of group boundaries (defined using large scale geometry) on the harmonic analysis of groups and the determination of their C*-algebra K-theory. The Baum-Connes conjecture proposes a means of calculating the K-theory of reduced group C*-algebras which blends group homology with the representation theory of finite subgroups. The conjecture, if true, would have a number of implications in geometry and topology, and a fascinating circle of ideas is coming into view which links the Baum-Connes conjecture to aspects of the harmonic analysis of groups and the geometry of group actions on boundary spaces. The investigators will attempt to clarify these relations. A long term goal is to prove the Baum-Connes conjecture, and more importantly to understand better its meaning, for classes such as the hyperbolic groups of Gromov. More immediate objectives include clarifying the relationships between existing proofs of partial forms of the conjecture for these groups, and developing further the connections between C*-algebra K-theory, manifold theory, and controlled topology. Although the tools used to investigate it are rather elaborate, the idea behind large scale geometry is very simple: ignore the local, small scale fluctuations in a quantity and concentrate on its large scale, or long term, behaviour. By doing so, trends or qualities may become apparent which are obscured by inconsequential, small scale fluctations. The investigators have developed tools to distinguish between different sorts of multi-dimensi onal, large scale behaviour in geometry. Somewhat surprisingly, aside from their intrinsic interest, their tools have found application in ordinary, small scale geometry.