Algebraic groups, arithmetic subgroups and geometry

许多几何和算术对象具有可以用群的数学概念编码的对称性。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有关的研究项目，以及与其相关的几何和算术对象，特别是对称空间。最近的研究包括与Brian Conrad和Ofer Gabber合作研究伪约化代数群的分类，与Andrei Rapinchuk合作研究算术定义的局部对称空间，以及与Sai-Kee Yeung合作研究复代数几何。根据这一奖项，国际和平研究所建议调查与这些合作有关的新方向。他还将继续研究同余子群问题和Marguis-Platonov猜想(后者给出了定义在整体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-ady群的任何不可约可容许表示联系在一起。他建议研究这个Levi亚群在表征的构建和分类中所起的作用。PI计划写的三本书是一本关于非阿基米德局部域上的Bruhat-Tits约化群理论的书，一本关于基于他在密歇根大学研究生课程的Lie群的书，以及一本与Andrei Rapinchuk合作的关于同余子群问题的书。

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.