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群(特别是算术群)及其与数学的各个分支的相互作用，例如数论、齐次动力学和组合学。这个项目的重点是从两个不同的方向扩展我们对这类群的理解：从有限余体积到无限余体积(以及更远)；从特征零到正特征。PI计划看到有限生成群的同余商的分析行为在多大程度上受其Zariski拓扑的支配。PI的第二个目标是研究具有正特征的局部场上的齐次动力学。线性群同余商的分析性质在数学和计算机科学的各个部分都被证明是非常有用的。在过去的十年里，它们被用于仿射筛子、伽罗瓦表示的变分、双曲几何和群论。很明显，推广这些结果将对数学的其他分支产生直接影响。作为这个项目的第二部分，PI计划致力于证明Raghunathan关于局部正特征域上的半单群的猜想。许多数学家，如Dani，Marguis，Shah，Tomanov都曾研究过这些猜想，最后在一系列论文中，Ratner在特征为零的局部域上完全证明了这些猜想。由于Ratner的结果在数学的各个部分都取得了非常丰硕的成果，可以预见，任何与其正特征相似的部分结果都将立即得到应用。研究对象或结构的主要工具之一是了解其对称性。这就是群论与数学和物理学的其他分支紧密联系的内在原因。例如，PI关于线性群的工作可以给我们精确的代数条件来构造被称为扩展器的显式稀疏高连通图族。扩展器在通信、理论计算机科学(例如纠错码)和数学的各个分支中都非常有用。PI还研究代数性质的动力系统及其与其他数学分支的深刻而富有成效的联系，例如数论。