We show that, in many cases, there is a Borel reduction from the isomorphism relation on a given Fraïssé class to the conjugacy relation on the automorphism group of the Fraïssé limit. Hence, if the former is Borel complete, then so is the latter. The key property is a functorial, Borel form of amalgamation. All relevant notions about Borel redicibility and Fraïssé classes will be defined.

The dynamical and descriptive set theoretic complexity of a countable Borel equivalence relation E can often be understood in terms of the kinds of countable first order structures which are compatible with E in a suitable sense. In this talk I will make this suitable sense precise by discussing the notion of Borel structurability. I will also discuss some recent joint work with Brandon Seward in which we show that the equivalence relation generated by the free part of the translation action of a countable group G on its powerset is structurably-universal among equivalence relations generated by free Borel actions of G.