Dept Oral Exam

Speaker: Chris Plyley (Western)

"Polynomial Identities on Algebras with Action"

Time: 14:00

Room: MC 108

In order to study the properties of an algebra A, we sometimes endow A with some additional structure in the form of a prescribed action or a grading. Properties of A can often be determined based on the properties of the structure-induced subspaces of A. This tactic is especially effective in polynomial identity theory. For example, classic results state that an algebra with involution satisfies an identity as soon as the subspace of symmetric (or skew-symmetric) elements does, similarly, a group-graded associative algebra is a PI-algebra whenever the identity component of the grading is. Recent results incorporate combinatorial techniques to find quantitative versions of these classic theorems. After providing an introduction into the basics of polynomial identity theory, our first objective in this talk is to describe how to prove a similar result for associative algebras whose induced Lie or Jordan algebras are group-graded. Many types of additional structures on algebras are easily formalized as actions of Hopf algebras on A (for instance, group-gradings or actions by automorphisms). In these cases, a well-known duality between Hopf actions and group-gradings becomes a powerful tool. Other natural actions (for instance, anti-automorphisms) are of interest in PI-theory, but fall out of the range of this duality. Our second objective in this talk is to describe how we can extend this duality to incorporate more general Hopf actions. Following this, we describe a unified Hopf algebra approach to PI-theory that will encompass and improve all of the aforementioned results.

Geometry and Topology

Speaker: Bob Bruner (Wayne State)

"Idempotents, Localizations and Picard groups of A(1)-modules"

Time: 14:30

Room: MC 107

We will start with an introduction to the mod 2 Steenrod algebra, and some structure theory for modules over its subalgebras due to Adams and Margolis. In particular, there is a set of homology functors which detect stable isomorphism, and there are subcategories of modules that are `local' with respect to them.

Next, we specialize to the subalgebras relevant to real and complex K-theory, called E(1) and A(1), where we can give quite precise descriptions of the local modules. The Picard groups of these subcategories are sufficient to detect the Picard group of the whole category and contain modules of geometric interest.

General results obtained along the way allow us to begin to attack the analogous questions for E(2) and A(2)-modules.

Applications include better descriptions of polynomial algebras as modules over the Steenrod algebra, and of the values of certain generalized cohomology theories on the classifying spaces of elementary abelian groups.