Boise Set Theory Seminar, December 2013.
Abstract: How hard is it to classify the countable abelian groups? While it is known that the classification of arbitrary countable groups is as complex as that for arbitrary countable structures, the analogous question for countable abelian groups remains open. I will present a partial result of Hjorth which says that the classification of countable abelian groups is fairly complex in that it is not Borel. I will also show how this result is related to a question that arises in the study of conjugacy relations.