The Menasco-Reid conjecture supposes a negative answer to the question: Is there a hyperbolic knot complement which contains a closed embedded totally geodesic surface? Another Kirby problem asks which hyperbolic knot complements admit hidden symmetries? Here a manifold $M$ admits hidden symmetries, if $M$ covers an orbifold $Q$ and  $Q$ is not the quotient of $M$ by symmetries. Historically, there were three knot complements known to have hidden symmetries, and a conjecture Neumann and Reid states these are the only such examples.  After giving some of the relevant background, we will construct examples of knot complements that are counterexamples to both conjectures. Each of these knot complements has the property that it admits a decomposition into geometric prisms.  This is joint work with Jason DeBlois and Arshia Gharazolou and has appeared on the arxiv: arXiv:2507.01263.

In this talk I will give an introductory lecture on constructing Topological Quantum Field Theories (TQFTs) from non-semisimple categories. The main goal of the talk is to give a hint of what is needed to extend the Turaev-Viro and Crane-Yetter TQFTs from the useful setting of semisimple categories to the non-semisimple world.  I will do this from an algebraic and categorical point of view.  In particular, I will discuss what kind of structures are needed in non-semisimple categories to give rise to (2+1)-TQFTs.  Then I will remark that any spherical tensor category (in the sense of Etingof, Douglas et al.) has such structures.  This work is joint with Francesco Costantino, Benjamin Haïoun, Bertrand Patureau-Mirand and Alexis Virelizier and based on arXiv:2302.04509 and arXiv:2306.03225.