Seminar

In descriptive set theory, the complexities of classes of countable structures are studied. A classical example is the isomorphism problem $\cong_r$ on the class of torsion-free abelian groups of a fixed rank $r$.  Baer gave a simple invariant for $r = 1$. However, jorth showed that $\cong_1 <_B \cong_2$ and Thomas generalized this to show that $\cong_n <_B \cong_{n+1}$. Recently, Paolini and Shelah, and independently Laskowski and Ulrich, showed that the class of torsion-free abelian group with domain $\omega$ is Borel complete.

We compare the class of torsion-free abelian groups and the class of fields in the computability setting. In particular, we show that there is a computable functor from the class of torsion-free abelian groups of rank $r$ to the class of fields of characteristic $0$ with transcendence degree $r$. We will also discuss the (im)possibility of a computable functor that goes in the other direction.

A Computable Functor from Groups to Fields 553 KB application/pdf

A Computable Functor From Groups To Fields