We say that a Turing degree is high in some context if it is maximally strong in that context. Highness has been studied in various subfields of computability theory for decades: first in degree theory, and then primarily in algorithmic randomness and genericity. Calvert, Turetsky, and I defined a degree to be high for isomorphism if, whenever an isomorphism between two computably presented structures exists, this degree can compute such an isomorphism.
In this talk, I will discuss three possible answers to the following question: When no such isomorphism exists, what type of function might such a degree compute instead? This work is joint with Calvert and Turetsky.