Mathematical Sciences: Noncommutative Rings

该奖项支持环论的研究，重点是有限群作用下不变量环的表示理论。对于某些特殊类型的不变环R，我们将特别注意Grothendieck群G_0(R)和K_0(R)的结构。G_0(R)的主要目的是用作用群的Brauer特征标来刻画。与K_0(R)有关的特别重要的是Hattori-Stallings迹，更一般的是循环同调中的Chern特征标。虽然这个项目的一部分将涉及经典的交换设置，特别是多项式环上的线性作用和Laurent多项式环上的乘法作用，但也将继续研究某些类型的非交换不变量。这一主题承诺在交换和非交换环论、有限群的表示理论、同调代数和代数K-理论的领域之间有丰富的相互作用。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。***

Lorenz This award supports research in ring theory, focusing on the representation theory of rings of invariants under finite group actions. Particular attention will be given to the structure of the Grothendieck groups, G_0(R) and K_0(R), for certain specific types of invariant rings R. The main objective for G_0(R) is to give a description in terms of Brauer characters of the acting group. Especially important in connection with K_0(R) are the Hattori-Stallings trace and, more general, the Chern characters in cyclic homology. While part of this project will concern the classical commutative setting, in particular linear actions on polynomial rings and multiplicative actions on Laurent polynomial rings, the investigation of certain classes of noncommutative invariants will also be pursued. This topic promises a rich interplay between the areas of commutative and noncommutative ring theory, representation theory of finite groups, homological algebra, and algebraic K-theory. 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. ***