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?