Borel reducibility of equivalence relations was introduced by Friedman
and Stanley in 1989. This powerful concept allows us to use methods of
descriptive set theory to compare the complexity of classification
problems from other areas of mathematics.
Our starting point will be the amazing result, due to Hjorth and Thomas in 1998-2001, that the complexity of the classification problem for torsion-free abelian groups of finite rank increases strictly with the rank. Other invariants besides just the rank can be used. For instance, Thomas showed that even once the rank is fixed, the classification subproblems for p-local and q-local groups have incomparable complexities.
In each of these results, the "dimension" of the classification problem plays a crucial role. This leaves open the following natural question, which we will discuss in this talk: To what extent do the dimensions of two classification problems decide their relationship under Borel reducibility?