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

K 理论、C* 群代数、大尺度几何和拓扑

• 批准号: 9800765
• 负责人: Nigel Higson
• 金额: $ 28.04万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800765&HistoricalAwards=false
• 关键词: Theory; Group; Algebras; Large; Scale

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.

亲爱的乔，约翰和我编写了以下段落，总结了我们的拨款提案。 如果您发现它们不满意，请告诉我。 您的，奈杰尔 ---------------------------------------------------------- ------------------- 群和空间的大规模几何在 C* 代数理论和拓扑中的不变量计算中起着决定性作用。研究人员旨在探索群边界的几何和拓扑（使用大尺度几何定义）对群调和分析及其 C* 代数 K 理论的确定的影响。 Baum-Connes 猜想提出了一种计算约化群 C* 代数 K 理论的方法，它将群同调性与有限子群的表示论相结合。 如果这个猜想成立，将会对几何和拓扑产生许多影响，并且一个令人着迷的思想圈正在进入人们的视野，它将鲍姆-康尼斯猜想与群的调和分析和边界上的群作用的几何学联系起来。空间。 调查人员将试图澄清这些关系。 长期目标是证明鲍姆-康尼斯猜想，更重要的是更好地理解其含义，例如格罗莫夫的双曲群。 更直接的目标包括澄清这些群猜想的部分形式的现有证明之间的关系，并进一步发展 C* 代数 K 理论、流形理论和受控拓扑之间的联系。 尽管用于研究它的工具相当复杂，但大规模几何背后的想法非常简单：忽略数量的局部小规模波动，而专注于其大规模或长期行为。 通过这样做，趋势或品质可能会变得明显，而这些趋势或品质会被无关紧要的小规模波动所掩盖。 研究人员开发了一些工具来区分几何中不同类型的多维、大规模行为。 有点令人惊讶的是，除了他们的内在兴趣之外，他们的工具还应用于普通的小规模几何体。