Abstract: The Noether-Lasker Primary Decomposition Theorem may be think as a way to extend a version of prime factorization from the integers to a much larger class of rings including the polynomial rings in several variables over a field.
This talk will use the basic homomorphism theorems and some ring and module theory. Most of the mathematics would be covered in a first year graduate algebra course.
Abstract: On the vector space $\Bbb R^{\Bbb N}$ of all sequences of real numbers, one can define a linear operator that, in some respects, resembles a differential operator on the vector space of all differentialble functions. One similarity is that it can be inverted, i.e., allowing "integration" on $\Bbb R^{\Bbb N}$. Furthermore, it satisfies a "generalized product rule". Combining this, one gets an analogue of the integration-by-parts formula, which is very efficient tool for finding and proving certain summation formulas.
Abstract: The steel ball problem is a problem that has perplexed people for many years. The most famous example of this problem is the twelve steel-ball problem. In this case, there are eleven balls of the same weight and one ball of different weight. The goal is to find the odd ball in three weighings or less. If one knows whether the the odd ball is lighter or heavier than others, the solution is fairly straightforward.
Prerequisite: Your curiosity and interest.
Abstract: The Noether-Lasker Primary Decomposition Theorem may be think as a way to extend a version of prime factorization from the integers to a much larger class of rings including the polynomial rings in several variables over a field.
This talk will use the basic homomorphism theorems and some ring and module theory. Most of the mathematics would be covered in a first year graduate algebra course.
Abstract: The Gelfand-Kirillov dimension is a measure of the growth rate of an algebra over a field in terms of any generating set. For a group algebra, it measures the growth rate of the group; For a commutative ``affine'' domain it measures the transcendence degree of the field of fractions.
Abstract: The question to be considered here is the following: If $G_1\times H_1 \cong G_2\times H_2$ with $H_1\cong H_2$, can we ``cancel" and infer that $G_1\cong G_2$? The naive student will unhesitatingly say ``yes," and feel quite content with the response. However, a moment's thought will reveal a counterexample to the above assertion, so we're left to ponder over conditions that might be sufficient to give us an affirmative answer. Largely as a warm-up exercise, I'll start by proving that if everything is abelian, and if $H_1,H_2$ are infinite cyclic, then in fact they can be cancelled off. The main result we want to show is that if $H_1\cong H_2$ is a finite group, then it can be cancelled off, yielding the desired isomorphism. This result appears only to have been first discovered in 1947 by B. J\'onsson and A. Tarski; the treatment we follow is the seemingly rarely quoted paper in the {\em Monthly} by R. Hirshon. The proof is a deceptively simple induction argument that I'll present in the seminar. Finally, we'll see how nicely this applies to obtain uniqueness theorems for the structure theorem of finitely generated abelian groups, which of course, interests me in light of my paper this year in the same {\em Monthly} (February, 1995).
Abstract: Given a finite group $G$ and a finite field $ F $,
let $ R =F[G] $ be the group ring. I will illustrate a general technique
for counting the number of permutations of $R$ that can be realized as
polynomials over $F$. I will apply this technique to count the number of
permutation polynomials of $R$ when $ \mbox{char}(F)= p $ and $ G $ is
a $ p $-group with a cyclic subgroup of index $ p $.
Mark Lesperance
Preliminary Results on Amalgams Involving $\hbox{SL}_2(2^n)$
Abstract: Suppose $G$ is a group generated by two distinct proper finite
subgroups, $P_1,P_2$ with the following
properties
(1) $P_1 \cap P_2 \in Syl_2(P_i), i=1,2$
(2) $P_1 \cap P_2$ contains no nontrivial normal subgroups of $G$
(3) $C_{P_i}(O_2(P_i)) \leq O_2(P_i), i=1,2$
(4) $P_i / O_2(P_i) \simeq \hbox{SL}_2(2^{n_i}), i=1,2$.
In this talk, I will discuss what properties of $G$ these conditions
yield and what this says about the structure of the
$P_i's$.