The topic of this talk is loosley the phenomenon of fibring over the circle for manifolds, and its group-theoretic twin, algebraic fibring. We will discuss the state of the art, and explain briefly some of the ideas behind the more recent developments, focusing on RFRS groups and manifolds with such fundamental groups. Then we will move on to a more speculative part, where many conjectures about fibring in higher dimensions will be given. The conjectures vary in their level of plausibility, but even the boldest of them might share the fate of Thurston's Virtually Fibred Conjecture, about which Thurston famously said: ``This dubious-sounding question seems to have a definite chance for a positive answer'' [https://arxiv.org/abs/2510.01805]

We motivate and survey the theory of BPS invariants of categories and BPS cohomology of stacks, indicating applications in enumerative geometry and representation theory, as well as recent advances.