Speaker
Description
Topology can play a surprisingly important role in determining the relationship between different aspects of computability of sets in computable metric spaces. In particular, semicomputable sets with certain topological properties will automatically be fully computable. To express this property, we use the notion of computable type: a space $A$ is said to have computable type if every semicomputable set homeomorphic to $A$ must be computable.
The study of computable type is usually restricted to compact spaces, most notably compact manifolds and simplicial complexes. However, a more general approach can yield similar results for non-compact spaces. Some examples of non-compact spaces with computable type include 1-manifolds and generalized graphs (i.e. graphs with potentialy "infinite edges"), as well as certain (very specific) manifolds of arbitrary dimension.
In this talk, we begin by contrasting two different techniques used to study computable type of non-compact spaces. We focus on the notion of pseudocompactification and show how it can be utilized to obtain more general results than those known so far. In particular, we prove that the infinite cylinder $\mathbb S^1 \times \mathbb R$ (and $\mathbb S^n \times \mathbb R$ in general) has computable type.
