Most of us know that any finitely generated Abelian group is a direct sum of cyclic groups (finite or infinite). When I first saw this theorem, I asked, "What about the rest?" This is not meaningless abstraction: some of our favorite groups do not have this form (the additive group of rationals, for instance).
It turns out that a great deal is known about the structure of Abelian groups in general. Every Abelian group splits into a torsion part and a torsion-free part. The torsion part is completely understood. The torsion-free part can be a good deal more complicated, and it is in the more exotic corners of torsion-free Abelian group theory that we find the few remaining open problems in the classification of Abelian Groups -- and some truly remarkable techniques.
This talk will outline the classification of Abelian groups to the extent that it is known. I will also tell about recent progress on the last frontier, where the tools used include both computable model theory and ergodic theory.
To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert