A classification problem consists of a set of mathematical objects equipped with some natural equivalence relation; a solution to such a problem is an assignment of complete invariants.
In this talk we consider the problem of classifying 3-manifolds up to homeomorphism from the perspective of descriptive set theory. We briefly discuss the framework of Borel reducibility, a standard tool for comparing the complexity of different classification problems, and present our recent result which determines the exact complexity of the classification of non-compact 3-manifolds up to homeomorphism.
This is joint work in progress with Vadim Weinstein (Oulu).