An introduction to the theory of Borel complexity of classification problems

Talk by Julien Melleray (Université Lyon 1),
at the Antwerp Algebra Colloquium on February 19, 2021.

A common theme throughout mathematics is to try and classify a class of objects according to some notion of isomorphism, and try to produce some nice invariants for this classification. Sometimes, nice invariants cannot exist; and one is led to the idea of comparing complexities of classification problems. Many such problems fit in a framework introduced by Friedman and Stanley in the eighties: Borel complexity of equivalence relations. I will explain this framework and give some examples related to classification of some classes of countable groups up to isomorphism (abelian groups, torsion-free abelian groups of finite ranks, locally finite simple groups...).