Normal Subgroup Structure of the Groups of Rational Points of Algebraic Groups and of Their Special Subgroups

该奖项为研究一般代数群以及特殊(主要是全局)域上的代数群的有理点群的正规子群结构提供资金。主要研究人员试图证明关于任意无限域上单代数群的有理点群的有限商可解性的一个新猜想，并计划分析有限维除法代数的乘法群的有限商，以得到它们的合理分类。另一个中心问题是研究全局域上特殊酉群的Marguis-Platonov猜想。首席研究员与Gopal Prasad、Yoav Segev和Gary Seitz合作解决这些问题。他还计划与Gopal Prasad合作一个关于同余子群问题的联合图书项目。与线性群的正规子群结构有关的问题在19世纪有历史渊源(Galois，Jordan，Dixon)，并在20世纪一直是一个活跃的研究领域。最近，在首席研究员Y.Segev和G.Seitz的共同工作中，发现了在一般场上分析各向异性群问题的新技术。这些技术使人们能够使用估值理论的方法来研究一般领域中的群体，这些方法以前只在全球领域的数字理论环境中使用。这种方法适用于同余子群问题的一般研究领域，它与目前应用于数据传输、数据处理和通信系统中的数论中的其他基本问题相联系。

This award provides funding for an investigation of the normal subgroupstructure of groups of rational points of algebraic groups overgeneral as well as over special (primarily, global) fields. Theprincipal investigator attempts to prove a new conjecture onsolvability of finite quotients of the groups of rational points ofsimple algebraic groups over arbitrary infinite fields.He also plans to analyze finite quotients of the multiplicative groupof a finite dimensional division algebra in order to obtain theirreasonable classification. Another central problem is investigationof the Margulis-Platonov conjecture for special unitary groups overglobal fields. The principal investigator collaborates on these problemswith Gopal Prasad, Yoav Segev and Gary Seitz. He also plans to workwith Gopal Prasad on a joint book project in the congruence subgroupproblem.Questions related to the normal subgroup structure of linear groupshave historical roots in the 19th century (Galois, Jordan, Dixon),and have been an area of active research in the 20th century. Recently,new techniques for analyzing this problem for anisotropic groupsover general fields have been discovered in the joint work of theprincipal investigator with Y.Segev and G.Seitz. These techniques enableone to investigate groups over general fields using methods of thetheory of valuations, which were previously used only in thenumber-theoretic setting of global fields. This approach fits intothe general area of investigation of the congruence subgroup problem,which is connected with other fundamental problems in number theory,currently applied in data transmission, data processing and communicationsystems.