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: 0455922
• 负责人: Gregory Margulis
• 金额: --
• 依托单位: Yale 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 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0455922&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 anddifferential 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.

该项目的主要主题是合理性和相关概念（Kazhdan属性T，属性tau，Unitarizability等）及其在数学理论基托学和功能分析的不同领域中的应用。 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对于离散和Pro-P组的Property t， *线性， *离散组的渐近性能。PIS将在项目的不同方面组织几个会议和研讨会。 NSF赠款将支持在您的监督下工作的几个毕业生和博士后研究员。 群体理论诞生于对称理论。高斯，亚伯，加洛伊斯，谎言等的工作表明，对称群体具有有关代数和差异方程的可溶性的基本信息。小组理论在数学和物理学的许多领域中都起着至关重要的作用。此外，小组理论的最新进展表明，许多数学领域都是密切相关的。反过来，小组理论从与其他领域的联系中受益匪浅。大约80年前，冯·诺伊曼（von Neumann），巴纳克（Banach）和塔斯基（Tarski）介绍了阿米纳比尔（Amenabile）组的概念，并将其与基本问题联系起来，例如“一个人可以将重量分配给我们空间中的任何一组点，以便在空间的所有对称性下重量不变？”。 PI将探索群体和代数的舒适性的各个方面，以及敏捷群体之间的深厚联系，数理论和拓扑结构。