Index Theory and the Baum-Connes Conjecture

对大尺度几何在拓扑学中所起作用的深入理解表明，群的大尺度几何特征决定了其酉群的小尺度特征。 由于傅立叶理论和庞特里亚金对偶，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，其酉表示理论太复杂，无法直接描述。 然而，阿兰·康纳斯（Alain Connes）的非对易几何提供了对偶空间的视角，使得我们有可能为非阿贝尔群制定这种从大尺度到小尺度现象的实例。 此外，代数拓扑的工具，进行到非交换领域，使它有可能提升现象的一个相互作用（由鲍姆和康纳斯制定）之间的全球，同伦理论结构的群体和他们的减少的。 本计划的研究目的是为了更准确、更深入地理解算子K理论中的Baum-Connes猜想及其背后的大尺度到小尺度现象。计划者将研究与群边界、关于群的约化对偶的Sobolev理论和群的Hilbert空间嵌入有关的问题。 最近发现的Baum-Connes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。 这样做，趋势或质量可能变得明显，而这些趋势或质量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不同种类的多维，大规模的几何行为。 有些令人惊讶的是，除了他们的内在利益，这些工具已经发现在普通的，小规模的几何和其他地方的应用。 目前的建议集中在几何方面的群论，这是照亮了大规模的几何。提案人积极参与培训下一代的数学科学家。 他们领导宾夕法尼亚州立大学的几何功能分析小组。 他们运行一个积极的，每周两次的研究研讨会，他们之间有八个博士生在他们的直接监督下（其他一些学生定期参加研讨会）。 他们目前担任导师，以一个VIGRE支持博士后研究员，并将招募第二个研究员由NSF重点研究补助金基金支持，今年。 几何函数分析组经常接待休假游客以及来访的研究生。 除了研讨会之外，该小组还就当前感兴趣的研究课题举办了一系列小型研讨会。 本提案中描述的研究将得到几何功能分析组活动的支持，并作为其一部分进行。

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.