The subject of Borel Equivalence relation theory involves deep connections between set theory, particularly descriptive set theory, and classification problems in algebra. The principal theme of the subject is to investigate the complexity of various naturally-occuring equivalence relations, such as the isomorphism relation on finitely generated groups, which arise in other areas of mathematics. It turns out that many of these relations can be viewed as Borel relations on a standard Borel space, and they fit into a hierarchy under the concept of Borel reducibility, introduced by Harvey Friedman. I explained a little about the subject in this MO answer.
Much of the best work in this subject is characterized by deep connections between set theory and algebra.
