Algorithms for 3-manifold homeomorphism

Tuesday, 25-June-2024 

Abstract

Given two mathematical objects, one basic question we can ask is: are they the same? For groups, for example, there is no algorithm to decide this. In practice there is fast software to answer this question for 3-manifolds, and theoretically the 3-manifold homeomorphism problem is known to be decidable. However, our understanding is limited, and known theoretical algorithms could have extremely long run-times. I will discuss some attempts to better classify the complexity of the 3-manifold homeomorphism problem -- in particular, whether it is in NP -- and discuss the important sub-case of Seifert fibered spaces.