Department of Mathematical Sciences
Florida Atlantic University

Boca Raton Algebra Seminar

Spring 2000

Tuesday, 4 April, 2:00, S&E 215

Serre's conjecture

Jim Brewer

Tuesday, 28 March, 2:00, S&E 215

 The Hopf algebra of the umbral calculus, II

Katherine Humphreys

As an example of why looking at the Hopf algebra of the umbral calculus may be a good idea, we look at a suspicious proof of The First Expansion Theorem from "Finite Operator Calculus" and rework it in the language of a Hopf algebra. The First Expansion Theorem is a generalized Taylor series expansion.

Tuesday, 21 March, 2:00, S&E 215

The Hopf algebra of the umbral calculus

Katherine Humphreys

We present some combinatorial problems that can be solved by a sequence of polynomials. Using the umbral calculus and the umbral algebra, we give an explicit closed form solution for a class of these problems. We then look at a problem whose closed form solution is unknown (to us). The Hopf algebra of the umbral calculus, which may contribute to the solution of this problem, will be sketched.

Thursday, 9 March, 11:00 a.m., S&E 215

Normal Forms for Fuzzy Logic

Elbert A. Walker

Tuesday, 7 March, 2:00, S&E 215

 Some Varieties of Modules Constructed from Cozzens' Domain

Markus Schmidmeier

Tuesday, 29 February, 2:00, S&E 215

Covaluated abelian p-groups and equivalence of subgroups

Alan Lebovitz

Let A and B be subgroups of a finite abelian p-group G. Paul Hill has shown that there is an automorphism of G taking A to B if and only if G/A and G/B are isomorphic as covaluated groups in the sense of Liebert.

Tuesday, 22 February, 2:00, S&E 215

 A Recent Comment on V-rings

Markus Schmidmeier

We will consider in more detail the V-rings introduced by J. H. Cozzens in 1970, in particular the skew Laurent polynomial ring K[x,1/x,s] where K is an algebraically closed field of positive characteristic p and  s : a -> a^p is the Frobenius automorphism.

We obtain a detailed description of their category of finite-length modules by using elementary methods from ring theory.

However, as a remark by M. Lehn indicates, Cozzens' result has a geometric interpretation, which gives rise to an independent proof.

Tuesday, 15 February, 2:00, S&E 215

Rings of Similar Structure may have Different Representation Types

Markus Schmidmeier

In their recent talks, Larry Levy and Lee Klingler have presented categories of modules for which they can show that each such category is either of tame or of wild representation type.

It is the aim of my talk to present a small family of non-commutative noetherian rings with the following properties:

*  methods from the representation theory of finite dimensional algebras do apply, hence

*  for many rings in this family, the representation type can be determined, but

*  rings similar in structure may have different representation types.

Tuesday, 8 February, 2:00, S&E 215

Two facts about matrices

Fred Richman

Let k be a field with at least n elements, E an extension field of k, and S a vector space of n-by-n matrices over k. If the vector space over E generated by S contains a nonsingular matrix, then so does S.

Let V be the set of all 2-by-2 matrices over a field k. For A and B in V, the subspace of matrices C in V such that AC = CB is never three-dimensional, but any two-dimensional subspace of V can be written in that form.

Tuesday, 1 February, 2:00, S&E 215

Canonical Forms of Matrices in Algebraic Control Theory

Part II

Lee Klingler

Tuesday, 25 January, 2:00, S&E 215

 Canonical Forms of Matrices in Algebraic Control Theory

Lee Klingler

Tuesday, 18 January, 2:00, S&E 215

Similarity of matrices over extension fields

Fred Richman

It is well known that if two square matrices over a field k are similar over a field extending $k$, then they are already similar over k. The usual proof (I think) relies on knowing everything about the similarity classes of matrices: for example, that they are characterized by the rational canonical form. It does not seem to generalize immediately to simultaneous similarity of pairs of matrices. I will present a proof that does not rely on such a structure theorem and that does generalize to simultaneous similarity.

Tuesday, 11 January, 2:00, S&E 215

 Tameness versus wildness for modules over commutative noetherian rings

Lawrence Levy
University of Wisconsin (emeritus)

Attempts to extend the fundamental theorem of abelian groups by  describing all finitely generated modules over some ring R often collide with an obstacle called "wild representation type". This has been studied for over 20 years, when R is a finite dimenensional algebra (noncommutative), where it has been shown that, in the absence of this obstacle, the desired module-description can usually be achieved.

The main theorem of the present work (joint with Lee Klingler) is  that this tameness-wildness dichotomy occurs for commutative noetherian rings, too: With a suitable description of wildness, every such ring either has wild type or else is "tame", i.e. we can describe its finitely generated modules and their direct-sum relations.

Further Algebra Seminar programs:   Fall 1999, Fall 2000Spring 2001

Last modified: August 30, 2000, by Markus Schmidmeier