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

该奖项支持对某些除环上的线性群以及自由群和自由可解群的自同构群的研究。最近的主要研究人员证明了Tits择一对有限生成的自由可解群的自同构群成立。他将继续这项工作，并研究有限秩自由群的自同构群。他还将致力于将他的结果推广到除环中，将泛包络代数嵌入到除环中，以得到自由幂零群的分式的除环和自由群环的分式的泛场。环是定义了加法和乘法的代数对象。这些结构自然而然地出现在许多不同的环境中，并对数学、计算机科学、工程和物理感兴趣。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结合在一起。

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.