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; 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，约翰和我在以下段落中总结了我们的赠款提议。 让我知道，如果您发现它们不令人满意。 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.研究人员旨在探讨组边界的几何形状和拓扑（使用大规模几何形状定义）对组的谐波分析以及其C*-Algebra K理论的确定。 Baum-Connes猜想提出了一种计算降低组C* - 代数的K理论的方法，该基理论将组同源性与有限亚组的表示理论融合在一起。 猜想，如果是的，则将在几何和拓扑结构中具有许多影响，而引人入胜的思想循环正在展开，这将Baum-Connes猜想与群体的谐波分析以及群体对边界空间的群体行为的几何形状联系起来。 调查人员将试图澄清这些关系。 一个长期的目标是证明鲍姆 - 康涅狄格州的猜想，更重要的是要更好地理解格罗莫夫双曲线群体的阶级。 更直接的目标包括阐明这些群体的猜想部分形式的现有证据之间的关系，并进一步发展C*-Algebra K理论，歧管理论和受控拓扑之间的联系。 尽管用于调查它的工具相当精心，但大规模几何形状背后的想法非常简单：忽略数量的局部小规模波动，而集中于其大规模或长期行为。 通过这样做，趋势或素质可能会变得显而易见，这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形状中不同类型的多二维属，大规模行为。 令人惊讶的是，除了它们的内在兴趣外，他们的工具还发现了普通的小规模几何形状的应用。