CAREER: Geometry of Groups and the Novikov Conjecture

诺维科夫高签名猜想是流形拓扑学中的一个深层次问题，它来自于对酉对偶的K-理论的足够精确的理解，酉对偶被看作是一个非交换空间。 本项目主要研究与此猜想相关的问题，特别是（1）群C*-代数的逼近性质，（2）Hilbert空间中离散群的一致可嵌入性和（3）使用控制方法研究群C*-代数的K-理论。 研究者还将研究与度量空间相关的C*-代数的并行问题。在研究离散群的非交换对偶空间时，特别强调重要的诺维科夫和鲍姆-康纳斯代数，该项目完全符合阿兰-康纳斯的非交换几何计划。 一个反复出现的主题将是把更多的各种想法从几何群论到研究分析性质的群体。 事实上，该项目将提供一个论坛，促进非交换几何和几何群论研究人员之间的持续互动。

The Novikov higher signature conjecture, a deep problem in the topology of manifolds, follows from a sufficiently precise understanding of the K-theory of the unitary dual, viewed as a noncommutative space, of the group in question. This project focuses on issues related to this conjecture, in particular (1) approximation properties of group C*-algebras, (2) uniform embeddability of discrete groups in Hilbert space and (3) the use of controlled methods to study the K-theory of group C*-algebras. The investigator will also study parallel problems for C*-algebras associated to metric spaces.In studying the noncommutative dual spaces of discrete groups, with particular emphasis on the important Novikov and Baum-Connes conjectures, this project fits squarely within Alain Connes' program of noncommutative geometry. A recurring theme will be to incorporate a greater variety of ideas from geometric group theory into the study of analytic properties of groups. Indeed, the project will provide a forum for promoting sustained interaction between researchers in noncommutative geometry and geometric group theory.