Discrete subgroups of semisimple Lie groups

半单李群的离散子群

• 批准号: 1303121
• 负责人: Alireza Golsefidy
• 金额: $ 14.9万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303121&HistoricalAwards=false
• 关键词: Discrete; subgroups; semisimple; Lie; groups

The PI will continue studying discrete linear groups (specially arithmetic groups) and their interactions with various branches of mathematics, e.g. number theory, homogeneous dynamics and combinatorics. The focus of this project is to expand our understanding of such groups in two different directions: going from finite covolume to infinite covolume (and beyond); going from characteristic zero to positive characteristic. The PI plans to see in what extent the analytical behavior of the congruence quotients of a finitely generated group is dictated by its Zariski-topology. The PI's second goal is to study homogeneous dynamics over a local field of positive characteristic. The analytical properties of the congruence quotients of linear groups have been showed to be extremely useful in various parts of mathematics and computer science. In the past decade they have been used in affine sieve, variation of Galois representations, hyperbolic geometry and group theory. It is clear that extending these results would have immediate impacts in other branches of mathematics. As the second component of this project, the PI plans to work toward the proof of Raghunathan's conjectures for semisimple groups over a local field of positive characteristic. Many mathematicians worked on these conjectures, e.g. Dani, Margulis, Shah, Tomanov, and finally in a series of papers, Ratner completely proved these conjectures over a local field of characteristic zero. As Ratner's results have been extremely fruitful in various parts of mathematics, it is expected that any partial result toward their positive characteristic analogue would have immediate applications.One of the main tools to study an object or a structure is to understand its symmetries. That is the intrinsic reason why group theory is in a close connection with other branches of mathematics and physics. For instance the PI's work on linear groups can give us the precise algebraic conditions to construct explicit families of sparse highly connected graphs known as expanders. Expanders are extremely useful in communication, theoretical computer science (e.g. error correcting codes) and various branches of mathematics. The PI studies also the dynamical systems of algebraic nature and their deep and fruitful connections with other branches of mathematics, e.g. number theory.

PI将继续研究离散线性组（特别是算术组）及其与数学各个分支的相互作用，例如数字理论，均质动力学和组合学。该项目的重点是将我们对此类群体的理解朝两个不同的方向扩展：从有限的Covolume到无限的Covolume（及以后）；从特征零变为积极特征。 PI计划在多大程度上了解有限生成的组的一致性商的分析行为取决于其Zariski-Toprogy。 PI的第二个目标是研究积极特征的当地领域的均匀动力学。线性组的一致性商的分析特性已显示在数学和计算机科学的各个部分中非常有用。在过去的十年中，它们已用于仿射筛，galois表示的变化，双曲线几何和群体理论。显然，扩展这些结果将对数学的其他分支产生直接影响。作为该项目的第二个组成部分，PI计划为Raghunathan在一个积极特征的本地领域中对Raghunathan的猜想进行证明。许多数学家从事这些猜想，例如Dani，Margulis，Shah，Tomanov，最后在一系列论文中，完全证明了这些猜想在当地的特征零领域。由于Ratner的结果在数学的各个部分都非常富有成果，因此预计任何部分对其积极特征模拟的部分结果都将立即应用。研究对象或结构的主要工具之一是了解其对称性。这就是群体理论与数学和物理学其他分支密切联系的内在原因。例如，PI在线性组上的工作可以为我们提供精确的代数条件，以构建稀疏高度连接图的明确家族，称为扩展器。扩展器在通信，理论计算机科学（例如错误纠正代码）和数学各个分支中非常有用。 PI研究还具有代数性质的动态系统及其与其他数学分支的深厚和富有成果的联系，例如数字理论。