Grant: Discovery Grants Program - Individual

A ring is an algebraic system with two operations: addition and multiplication. For example, all integers with regular addition and multiplication form a ring. The group ring of a group G over a commutative ring K is the ring KG of all formal finite sums. It 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 work hasshed light on the algebraic structures (and the internal arithmetic ) of group rings and their unit groups, and it 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 conjecturesregarding group rings, including the first Zassenhaus Conjecture- a long standing research problem- and the normalizer problem. I will also investigate the structure of star-clean and reversible group rings. Zero-sum theory is a new branch of combinatorial number theory. Recently I have begun exploring some exciting innovative connections between combinatorial number theory and my work in group rings. Dr. Weidong Gao and I are able to use group rings asthe main tool to investigateseveral open problems in zero-sumtheory concerning several important invariants such as the Davenport constant and the index of sequences.We have already made significant contributions in this research area. Further research is ongoing, and we anticipate reporting significant advances in the immediate future. I am also involved in other areas such as (combinatorial) group theory where I have obtained new results concerning groups with small squaring property, and algebraic coding theory. The outcomes of my research will not only add knowledge to the areas of group rings andzero-sum theory but also inspire other researchers in these areas and other areas to which my research may apply.