Discrete subgroups of semisimple Lie groups

PI将继续研究离散线性群（特别是算术群）及其与数学各个分支的相互作用，例如数论、齐次动力学和组合学。这个项目的重点是在两个不同的方向上扩展我们对这类群的理解：从有限协体积到无限协体积（甚至更大）；从特征值为零到特征值为正。PI计划观察有限生成群的同余商的分析行为在多大程度上由其zariski拓扑决定。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。线性群的同余商的解析性质已被证明在数学和计算机科学的各个部分是非常有用的。在过去的十年中，它们已被用于仿射筛，伽罗瓦表示的变化，双曲几何和群论。很明显，扩展这些结果将对其他数学分支产生直接影响。作为该项目的第二个组成部分，PI计划在具有正特征的局部域上证明Raghunathan关于半单群的猜想。许多数学家对这些猜想进行了研究，如Dani， Margulis, Shah， Tomanov等，最后Ratner在一系列论文中在特征为零的局部域上完全证明了这些猜想。由于拉特纳的结果在数学的各个方面都非常富有成果，因此可以预期，任何关于其正特征类比的部分结果都将立即得到应用。研究一个物体或结构的主要工具之一是了解它的对称性。这就是群论与数学和物理学的其他分支密切联系的内在原因。例如，PI在线性群上的工作可以给我们精确的代数条件来构造稀疏高连通图的显式族，即所谓的扩展。扩展器在通信、理论计算机科学（例如纠错码）和各种数学分支中非常有用。PI还研究代数性质的动力系统及其与数学其他分支（如数论）的深刻而富有成效的联系。

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.