Algebraic groups, arithmetic subgroups and geometry

许多几何和算术对象具有对称性，可以用群的数学概念进行编码。通常这样的群具有额外的结构，形成所谓的代数群。PI计划开展几个与代数群及其内部结构相关的研究项目，以及与代数群相关的几何和算术对象，特别是对称空间。此外，PI计划在获奖期间撰写三本书。PI最近的研究包括与Brian Conrad和Ofer Gabber在伪约代数群的分类上的工作，与Andrei Rapinchuk在算术定义的局部对称空间上的工作，以及与Sai-Kee Yeung在复杂代数几何上的工作。根据该奖项，PI建议研究与这些合作相关的新方向。他还将继续研究同余子群问题和马古利斯-柏拉图诺夫猜想（后者给出了定义在全局域上的单连通半单群的有理点群的正规子群的描述）。在另一个方向上，PI将Levi子群与可约p进群的任何不可约容许表示联系起来。他建议研究列维子群在表征的构建和分类中的作用。PI计划写的三本书是一本关于非阿基米德局部域上的约化群的Bruhat-Tits理论，一本是基于他在密歇根大学的研究生课程的李群，还有一本是与Andrei Rapinchuk合作写的同余子群问题。

Many geometric and arithmetic objects have symmetries that can be encoded in the mathematical notion of a group. Often such groups possess additional structure, forming what is known as an algebraic group. The PI plans to work on several research projects related to algebraic groups and their internal structure, as well as their associated geometric and arithmetic objects, especially symmetric spaces. In addition the PI plans to write three books during the duration of this award.Recent research of the PI includes work with Brian Conrad and Ofer Gabber on the classification of pseudo-reductive algebraic groups, with Andrei Rapinchuk on arithmetically defined locally symmetric spaces, and with Sai-Kee Yeung in complex algebraic geometry. Under this award the PI proposes to investigate new directions related to these collaborations. He will also continue his work on the congruence subgroup problem and the Margulis-Platonov conjecture (the latter gives a description of the normal subgroups of the group of rational points of a simply-connected semi-simple group defined over a global field). In another direction, the PI has associated a Levi subgroup to any irreducible admissible representation of a reductive p-adic group. He proposes to investigate the role of this Levi subgroup in the construction and classification of representations. The three books the PI plans to write are one on the Bruhat-Tits theory of reductive groups over nonarchimedean local fields, one on Lie groups based on his graduate courses at the University of Michigan, and one, jointly with Andrei Rapinchuk, on the congruence subgroup problem.