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， 约翰和我写了以下几段总结我们的赠款 提议 如果你觉得不满意，请告诉我。 你的， 奈杰尔 ------------------------------------------------------------------ 群体和空间的大尺度几何学在 C*-代数理论和拓扑学中不变量的计算。的 研究人员的目的是探索的几何和拓扑结构的影响， 谐波上的组边界（使用大比例几何定义） 群的分析及其C*-代数K-理论的确定。的 Baum-Connes猜想提出了一种计算K理论的方法， 约化群C ~*-代数，它将群同调与 有限子群的表示理论 这个猜想如果是真的， 在几何学和拓扑学中有许多含义， 一个将鲍姆-康纳斯猜想联系在一起的思想圈正在形成 群的调和分析和群的几何 在边界空间上的作用。 调查人员将试图澄清 这些关系。 一个长期目标是证明鲍姆-康纳斯猜想， 更重要的是，为了更好地理解它的含义， Gromov的双曲群 更直接的目标包括 澄清部分形式的现有证明之间的关系， 这些群体的猜想，并进一步发展联系， C*-代数K-理论、流形理论和控制拓扑之间的关系。 尽管用于调查它的工具相当复杂，但这个想法 大尺度几何的背后很简单：忽略局部的、小尺度的 波动的数量，并集中在其大规模，或长期 术语，行为。 通过这样做，趋势或质量可能变得明显 这些都被无关紧要的小规模波动所掩盖。 的 研究人员已经开发出了区分不同种类 多维的、大尺度的几何行为。 有些 令人惊讶是，除了他们的内在兴趣之外，他们的工具还发现， 在普通的小规模几何中的应用。