Research in group rings and its applications in combinatorial number theory

点击翻译按钮获取中文摘要

A ring is an algebraic system satisfying certain axioms. The group ring of a group G over a commutative ring K is the ring KG of all formal finite sums and is an attractive and important object of study in algebra. Here group theory, ring theory, commutative algebra, representation theory and number theory come together in a fruitful way. My recent research work has thrown light on the algebraic structures of group rings and their unit groups, and has important applications in coding theory and combinatorial number theory. This will continue to be one of the main streams of my research program for the next 5 years. I will study several open problems and conjectures to group rings, including the first Zassenhaus Conjecture, a long-standing research problem, and the normalizer problem. Recently, I have begun exploring some exciting connections between combinatorial number theory and my work in group rings. Dr. Gao and I are able to use group rings as a main tool to investigate certain zero-sum problems concerning several important invariants such as the Davenport constant and the index of sequences. We have made significant contributions in this research area. Further research is ongoing and we anticipate reporting significant advances in the immediate future. While much of my recent work is in group rings and combinatorial number theory, I am also involved in other areas such as module theory and group theory where I have obtained new results concerning morphic groups.