# How hard is it to compute an isomorphism?

## Barbara Csima (University of Waterloo)

### Thursday, October 5, 2017 4:00 pmMONT 214

We say a structure is computable (or computably presented) if it has domain the natural numbers, and if all functions, relations and constants are uniformly computable. This definition is not isomorphism invariant. We can ask: between two computable copies of a given structure, what is this simplest isomorphism that must exist? For example, between any two computable copies of the rationals, there always exists a computable isomorphism, since the usual back-and-forth isomorphism can be found effectively. On the other hand, the difficulty of computing an isomorphism between two computable copies of $(\mathbb{N}, <)$ is exactly that of the halting set, since we need to answer the single-quantifier questions of which element is least, next, etc. In this talk, we present what is known about Turing degrees that arise as degrees of isomorphisms for computable structures.