Geometry of Groups & Functional Analysis

群的几何

• 批准号: 0100464
• 负责人: Nigel Higson
• 金额: $ 64.44万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2006-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100464&HistoricalAwards=false
• 关键词: Geometry; Groups; & Functional; Analysis

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 State's 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.

摘要 Nigel/Roe 对大尺度几何在拓扑中所扮演的角色的深入理解已经清楚地表明，群的大尺度几何特征决定了其酉对偶的小尺度特征。 由于傅立叶理论和庞特雅金对偶性，这种效应在阿贝尔群中很容易观察到，但这种情况对于非阿贝尔群来说更为复杂，其酉表示理论太复杂，无法接受直接的描述性解释。 然而，阿兰·康尼斯 (Alain Connes) 的非交换几何提供的对偶空间视角使得为非交换群表述这种大规模到小规模现象的实例成为可能。 此外，代数拓扑的工具被转移到非交换领域，使得将现象提升为群的整体同伦理论结构与其约简对偶之间的猜想互易性（由鲍姆和康尼斯提出）成为可能。 本提案中概述的研究目的是为了更准确、更深入地理解算子 K 理论中的鲍姆-康尼斯猜想及其背后的从大到小的尺度现象。 提议者将研究与群边界、群减少对偶的 Sobolev 理论以及群的希尔伯特空间嵌入相关的问题。 将深入分析最近发现的鲍姆-康尼斯猜想变体的反例。尽管用于研究它的工具相当复杂，但大尺度几何背后的想法非常简单：忽略几何空间并专注于其大规模或长期结构。 通过这样做，被小规模的违规行为所掩盖的趋势或品质可能会变得明显。 研究人员和其他人开发了工具来区分几何中不同类型的多维、大规模行为。 有点令人惊讶的是，除了它们的内在兴趣之外，这些工具还应用于普通的小规模几何和其他领域。 目前的提案重点关注大规模几何所阐明的群论的几何方面。提案者积极参与培养下一代数学科学家。 他们领导宾夕法尼亚州立大学的几何泛函分析小组。 他们每周举办两次活跃的研究研讨会，在他们的直接指导下有八名博士生（许多其他学生定期参加研讨会）。 他们目前担任一名 VIGRE 支持的博士后研究员的导师，并将于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客以及来访的研究生。 除了研讨会之外，该小组还针对当前感兴趣的研究主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泛函分析小组活动的支持，并作为其活动的一部分进行。