Mathematical Sciences: Noncommutative Rings

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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. ***