Index Theory and the Baum-Connes Conjecture

随着人们对大尺度几何在拓扑学中作用认识的加深，群的大尺度几何特征决定了其么正对偶的小尺度特征。这种效应在阿贝尔群中很容易观察到，这要归功于傅立叶理论和庞特雅金对偶，但对于非阿贝尔群来说，情况更复杂，其么正表示理论太复杂，无法进行直接的描述性解释。然而，Alain Connes非对易几何所提供的关于对偶空间的观点使得对非交换群的这种大尺度到小尺度现象的实例的表述成为可能。此外，代数拓扑学的工具，延续到非对易领域，使得将这一现象提升到群的整体同伦理论结构与它们的约化对偶之间的猜想互易性(由Baum和Connes提出)成为可能。这项研究的目的是为了更准确和更深入地理解K-算子理论中的Baum-Connes猜想及其背后的大到小尺度现象。提出者将研究与群边界、关于群的约化对偶的Soblev理论以及群的Hilbert空间嵌入有关的问题。最近发现的鲍姆-康尼斯猜想变体的反例将被深入分析。尽管用于研究它的工具相当精细，但大尺度几何背后的思想非常简单：忽略几何空间的局部、小尺度特征，专注于其大尺度或长期结构。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小规模的违规行为所掩盖。研究人员和其他人已经开发出工具来区分几何学中不同种类的多维、大规模行为。有些令人惊讶的是，除了它们本身的兴趣之外，这些工具还在普通的、小规模的几何学和其他地方得到了应用。目前的建议集中在群论的几何方面，这些方面被大规模几何所启发。提出者积极参与培养下一代数学科学家。他们领导着宾夕法尼亚州立大学的几何泛函分析小组。他们每周举办两次积极的研究研讨会，他们有8名博士生在他们的直接指导下(其他一些学生定期参加研讨会)。他们目前担任Vigre资助的一名博士后研究员的导师，并将在今年招募第二名研究员，由NSF重点研究资助基金支持。几何泛函分析小组经常接待休假的访客和来访的研究生。除了研讨会外，该小组还继续举办关于当前感兴趣的研究课题的迷你研讨会。本提案中所述的研究将得到几何功能分析小组活动的支持，并作为其活动的一部分进行。

AbstractNigel/RoeA deepening understanding of the role played by large-scale geometry in topology has made it clear that large-scale geometric features of groups determine small-scale features of their unitary duals. The effect is easily observed in abelian groups, thanks to Fourier theory and Pontrjagin duality, but the situation is more involved for nonabelian groups, whose unitary representation theory is too complicated to admit a direct descriptive account. However the perspective on dual spaces provided by Alain Connes noncommutative geometry makes it possible to formulate instances of this large-scale to small-scale phenomenon for nonabelian groups. Moreover the tools of algebraic topology, carried over to the noncommutative realm, make it possible to elevate the phenomenon to a conjectural reciprocity (formulated by Baum and Connes) between the global, homotopy theoretic structures of groups and their reduced duals. The purpose of the research outlined in this proposal is to obtain a more accurate and deeper understanding of the Baum-Connes conjecture in operator K-theory and of the large-to-small scale phenomenon which underlies it. The proposers will investigate issues related to group boundaries, Sobolev theory on the reduced dual of a group, and Hilbert space embeddings of groups. The recent discovery of counterexamples to variants of the Baum-Connes conjecture will be analyzed in depth.Although the tools used to investigate it are rather elaborate, the idea behind large scale-geometry is very simple: ignore the local, small-scale features of a geometric space and concentrate on its large-scale, or long term, structure. By doing so, trends or qualities may become apparent which are obscured by small-scale irregularities. The investigators and others have developed tools to distinguish between different sorts of multi-dimensional, large scale behavior in geometry. Somewhat surprisingly, aside from their intrinsic interest, these tools have found application in ordinary, small-scale geometry and elsewhere. The present proposal focuses on geometric aspects of group theory which are illuminated by large-scale geometry.The proposers are actively involved in training the next generation of mathematical scientists. They lead Penn States' Geometric Functional Analysis group. They run an active, twice-weekly research seminar and between them they have eight doctoral students under their direct supervision (a number of other students attend the seminar regularly). They currently serve as mentors to one VIGRE supported postdoctoral fellow, and will be recruiting a second fellow to be supported by NSF Focussed Research Grant funds this year. The Geometric Functional Analysis group frequently hosts sabbatical visitors as well as visiting graduate students. Besides the seminar, the group runs a continuing program of mini-workshops on research subjects of current interest. The research described in this proposal will be supported by, and carried out as part of, the activities of the Geometric Functional Analysis group.