Computability Learning Seminar

Michael Deveau, Department of Pure Mathematics, University of Waterloo

"Computability of Walker's Cancellation Theorem"

We discuss the effectivization of Walker's Cancellation Theorem in group theory, with a focus on uniformity. That is, if we have an indexed collection of instances of sums of finitely generated abelian groups $A_i \oplus G_i \cong A_i \oplus H_i$ and the codes for the isomorphisms between them, then we wish to know to what extent we can give a single procedure that, given an index $i$, produces an isomorphism between $G_i$ and $H_i$. We begin by examining Cohn's classical proof of the theorem and see how much of it is effective. We then show that in a certain case, we are able to show that no computable procedure can exist. We also exhibit some related open problems.

MC 5479