FRG: Asymptotic and probabilistic methods in geometric group theory

FRG：几何群论中的渐近和概率方法

• 批准号: 0455881
• 负责人: Mark Sapir
• 金额: --
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0455881&HistoricalAwards=false
• 关键词: FRG; Asymptotic; probabilistic; methods; geometric

The main theme of this project is amenability and related concepts(Kazhdan property T, property tau, unitarizability, etc.) and itsapplications in different areas of mathematics from number theory totopology and functional analysis. In particular, the PIs willconcentrate on the following problems: * Classification of amenable groups and associative algebras, * Amenability of Golod-Shafarevich groups and related problems in3-dimensional topology and number theory, * Expander graphs, property tau for lattices in SL(2,C),probabilistic methods in group theory, * The Dixmier Unitarizability problem, * Constructing new examples of finitely presented groups withproperty T, * Linearity of discrete and pro-p-groups, * Asymptotic properties of discrete groups.The PIs are going to organize several conferences and workshops on differentaspects of the projects. The NSF grant will support several graduatestudents and postdoctoral fellows working undertheir supervision. Group theory was born as the theory of symmetry. The work of Gauss,Abel, Galois, Lie and others showed that groups of symmetries carry essential information about solvability of algebraic and differential equations. Group theory plays crucial role in many areas of mathematics and physics. Moreover, recent advances in group theory showed that many areas of mathematics are closely related. In turn, group theory has benefited tremendously from its connections with other areas. About 80 years ago, von Neumann, Banach and Tarski introduced the concept of amenabile group and connected it with basic questions like "Can one assign a weight to any set of points in our space so that the weight is invariant under all symmetries of the space?". The PIs will explore various aspects of amenability of groups and algebras, and deep connections between amenabile groups, number theory and topology.

该项目的主题是顺应性和相关概念（Kazhdan 属性 T、属性 tau、单位化性等）及其在从数论到拓扑和泛函分析的不同数学领域中的应用。特别是，PI将集中于以下问题： * 顺应群和结合代数的分类， * 戈洛德-沙法列维奇群的顺应性以及三维拓扑和数论中的相关问题， * 扩展图，SL(2,C) 中格的性质 tau，群论中的概率方法， * Dixmier 单位化问题， * 构造有限呈现群的新例子 具有属性 T，* 离散群和亲 p 群的线性，* 离散群的渐近性质。 PI 将就项目的不同方面组织多次会议和研讨会。美国国家科学基金会的拨款将支持在他们的监督下工作的几名研究生和博士后研究员。 群论作为对称性理论而诞生。高斯、阿贝尔、伽罗瓦、李等人的工作表明，对称群携带有关代数方程和微分方程可解性的基本信息。群论在数学和物理学的许多领域中起着至关重要的作用。此外，群论的最新进展表明数学的许多领域都是密切相关的。反过来，群论也从其与其他领域的联系中受益匪浅。大约 80 年前，冯·诺依曼、巴纳赫和塔斯基引入了顺应群的概念，并将其与诸如“能否为空间中的任意一组点分配权重，使得权重在空间的所有对称性下保持不变？”之类的基本问题联系起来。 PI 将探索群和代数的适用性的各个方面，以及适用群、数论和拓扑之间的深层联系。