In this first talk in a series on Tame Congruence Theory, I will motivate the universal algebraist’s notion of abelianness for finite algebras. The goal of this first talk will be Palfy’s Theorem characterizing minimal finite algebras with more than two elements.

We will give a basic introduction to Reverse Mathematics by explaining the overall goal of the subject, and introducing the “Big Five” subsystems of Second Order Arithmetic along with equivalent theorems from various branches of Mathematics.

We will give a basic introduction to Reverse Mathematics by explaining the overall goal of the subject, and introducing the “Big Five” subsystems of Second Order Arithmetic along with equivalent theorems from various branches of Mathematics. This will probably be the first in a short series of talks.

For a polynomial p(X) over a field F, there are two questions that most commonly arise: whether p(X) factors over F, and whether p(X) has a root in F. These questions are no doubt related to one another, but it is not obvious which one is more difficult – or even how we measure “difficulty” in this context. Computability theorists have several different measures of relative difficulty of computation, and we will explain them and examine how they can be applied in this situation. The coarsest of the measures indicates that the two questions are equally difficult, and the finest measure indicates that for algebraic fields, one of these questions can be strictly harder than the other. We will focus on an intermediate measure, where some cyclotomic field theory comes into play, and show that under this measure, the situation is the same as with the finest measure – one of the questions can be strictly harder than the other. To find out which is the harder question, you’ll just have to come to the talk!

Much work has been done recently on the connections between Abelian l-groups and MV-algebras. In particular, it has been shown that there is a bijective correspondence between so-called equational classes of unital Abelian l-groups and subvarieties of MV-algebras. While this correspondence has proven useful, it would be better if one could consider actual varieties generated by classes of unital l-groups. In this paper, we show that a nice bijective correspondence still exists when one considers subvarieties of pA, the variety of all negatively-pointed Abelian l-groups. In fact, the subvariety lattice of pA (excluding the trivial variety) is isomorphic to the subvariety lattice of the variety of MV-algebras.

In 1974, N. Hindman gave a combinatorial proof of the following Ramsey-theoretic statement: for every finite coloring of the natural numbers, there exists an infinite set A such that the set of finite sums of elements from A is monochromatic. The following year, Galvin and Glazer gave a non-elementary but much simpler proof, which we will discuss today, using the semigroup structure on the Stone-Cech compactification ßN. This idea is the basis of the recently announced proof of amenability of Thompson’s Group F