This is a talk I shall give at the workshop on Logical Approaches to Barriers in Complexity II, March 26-30, 2012, a part of the program Semantics and Syntax: A Legacy of Alan Turing at the Isaac Newton Institute for Mathematical Sciences in Cambridge, where I am currently a Visiting Fellow.

Many of the naturally arising equivalence relations in mathematics, such as isomorphism relations on various types of countable structures, turn out to be equivalence relations on a standard Borel space, and these relations form an intensely studied hierarchy under Borel reducibility. The topic of this talk is to introduce and explore the computable analogue of this robust theory, by studying the corresponding hierarchy of equivalence relations on the natural numbers under computable reducibility. Specifically, one relation $E$ is computably reducible to another $F$ , if there is a unary computable function $f$ such that $x\mathrel{E}y$ if and only if $f(x)\mathrel{F}f(y)$. This gives rise to a very different hierarchy than the Turing degrees on such relations, since it is connected with the difficulty of the corresponding classification problems, rather than with the difficulty of computing the relations themselves. The theory is well suited for an analysis of equivalence relations on classes of c.e. structures, a rich context with many natural examples, such as the isomorphism relation on c.e. graphs or on computably presented groups. An abundance of open questions remain, and the subject is an attractive mix of methods from mathematical logic, computability, set theory, particularly descriptive set theory, and the rest of mathematics, subjects in which many of the equivalence relations arise. This is joint work with Sam Coskey and Russell Miller.

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