Index Theory and the Baum-Connes Conjecture

摘要：随着对大尺度几何在拓扑学中的作用认识的加深，群的大尺度几何特征决定了群的酉对偶的小尺度特征。由于傅里叶理论和庞特贾金对偶，这种效应在阿贝尔群中很容易观察到，但对于非阿贝尔群，这种情况更为复杂，其幺正表示理论过于复杂，无法直接描述。然而，Alain Connes非交换几何提供的对偶空间的视角使得这种从大到小的现象在非贝尔群中得以表述。此外，代数拓扑的工具，延续到非交换领域，使得有可能将这种现象提升到群的整体，同伦理论结构和它们的约化对偶之间的推测互易（由Baum和Connes表述）。本提案中概述的研究目的是为了更准确和更深入地理解算子k理论中的Baum-Connes猜想以及其背后的大到小尺度现象。提出者将研究与群边界、群的约化对偶的Sobolev理论和群的Hilbert空间嵌入有关的问题。我们将深入分析最近发现的Baum-Connes猜想变体的反例。尽管用于研究它的工具相当复杂，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，趋势或性质可能会变得明显，而这些趋势或性质可能会被小规模的不规则现象所掩盖。研究人员和其他人已经开发出工具来区分几何中不同种类的多维、大规模行为。有些令人惊讶的是，除了它们固有的兴趣之外，这些工具在普通、小规模几何和其他地方也有应用。目前的建议集中在群论的几何方面，这是阐明了大尺度几何。倡议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功能分析小组。他们举办了一个活跃的，每周两次的研究研讨会，在他们之间，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）。他们目前是一名VIGRE支持的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访问者以及访问研究生。除了研讨会之外，该小组还就当前感兴趣的研究课题开展了一个持续的小型讲习班计划。本提案中描述的研究将得到几何泛函分析小组的支持，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.