This award is concerned with the theory of noncommutative Noetherian rings, particularly with that of connected graded algebras satisfying the Artin-Schelter (AS) regularity condition. The AS regular rings have been a subject of considerable research recently, since they provide genuinely new, geometric techniques for the study of noncommutative rings and are related to other areas of current interest, notably quantum groups. The principal investigator will study specific, important examples of these algebras to develop the abstract theory of AS regular algebras and the related theory of "noncommutative projective geometry". The principal investigator will also work on extending existing results on connected graded, Noetherian PI rings of finite global dimension. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.
