Mathematical Sciences: Linear Groups Rings of Fractions of Group Rings and Enveloping Algebras

该奖项支持研究线性群在某些类别的司环和自同构群的自由群和自由可溶群。 主要研究人员最近证明，山雀替代持有的自同构群的无生成自由可溶群。 他将继续这项工作，并研究有限秩自由群的自同构群。 他还将工作扩展他的结果嵌入一个普遍的包络代数到一个司环司环的分数的挠自由幂零群和普遍领域的分数的自由群环。 环是一个代数对象，它同时定义了加法和乘法。这些结构在许多不同的环境中自然出现，并且在数学，计算机科学，工程和物理学中很有意义。这个特殊的项目将环的研究与另一个称为群的代数对象的研究融合在一起。

This award supports research on linear groups over some classes of division rings and to automorphism groups of free groups and free soluble groups. The principal investigator recently proved that the Tits alternative holds for groups of automorphisms of finitely generated free soluble groups. He will continue this work, as well as study groups of automorphisms of free groups of finite rank. He will also work on extending his result on imbedding a universal enveloping algebra into a division ring to division rings of fractions of torsion free nilpotent groups and the universal fields of fractions of free groups rings. A ring is an algebraic object having both an addition and a multiplication defined on it. These structures arise naturally in many different settings and are of interest in mathematics, computer science, engineering and physics. This particular project blends the study of rings with the study of another algebraic object called a group.