Mathematical Sciences: Regular Graded Noetherian Rings

该奖项涉及非交换Notherian环的理论，特别是关于满足Artin-Schelter(AS)正则性条件的连通分次代数的理论。AS正则环是最近大量研究的对象，因为它们为研究非对易环提供了真正新的几何方法，并且与当前感兴趣的其他领域有关，特别是量子群。主要研究人员将研究这些代数的具体的、重要的例子，以发展AS正则代数的抽象理论和相关的“非对易射影几何”理论。主要研究人员还将致力于推广有限整体维度的连通分次Notherian PI环的现有结果。本研究属于环论的一般领域。环是定义了加法和乘法的代数对象。虽然加法运算满足交换律，但乘法运算不是必须的。乘法不可交换的环的一个例子是整数上的n×n矩阵的集合。非对易环的研究已经成为代数的重要组成部分，因为它对数学和物理的其他分支的意义越来越大。

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.