Algebraic groups, arithmetic subgroups and geometry

许多几何和算术对象都具有对称性，这些对称性可以用群的数学概念来编码。 通常这样的群拥有额外的结构，形成所谓的代数群。 PI计划从事与代数群及其内部结构相关的几个研究项目，以及相关的几何和算术对象，特别是对称空间。 此外，PI计划写三本书在此期间奖。最近的研究PI包括工作与布赖恩康拉德和奥弗加伯的分类伪约化代数群，与安德烈Rapinchuk的算术定义的局部对称空间，并与赛记杨在复杂的代数几何。 根据该奖项，PI建议调查与这些合作相关的新方向。 他还将继续他的工作，同余子群问题和马古利斯-普拉东诺夫猜想（后者给出了一个描述正常的子群的一组合理的点的一个简单的连接半简单的组定义在一个全球领域）。 在另一个方向上，PI将Levi子群与约化p-adic群的任何不可约容许表示相关联。 他建议调查的作用，这列维小组的建设和分类的代表。 这三本书PI计划写的是一个关于Bruhat，山雀理论的还原群体nonarchimedean地方领域，一个李群的基础上，他的研究生课程在密歇根大学，和一个，共同与安德烈Rapinchuk，对同余子群问题。

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.