The Kervaire conjecture was formulated around 1963 after a conversation between Kervaire and Baumslag. It states that adding a generator and then a relator to a non-trivial group always yields a non-trivial group. To this day, the conjecture remains unproven in its most general form; however, it has been shown under certain additional hypotheses, either on the new relator or on the original group. For instance, the result holds for locally indicable groups and for locally residually finite groups. In this talk, I will explain Klyachko’s proof of the conjecture for torsion-free groups, which uses a funny property of the sphere known as the Car Crash Theorem, and van Kampen pictures. I will also discuss how these techniques were generalised by Fenn and Rourke to study equations over torsion-free groups defined by a large class of words (amenable words).

I will give a short introduction to knotted surfaces in 4-space and discuss some recent developments. First, I will give some motivation, briefly discuss methods for distinguishing knotted surfaces (such as the Khovanov TQFT), and talk about connections with 4-manifolds. Then, I will introduce Artin’s spinning construction, variants of which were defined by Zeeman, Fox, Litherland, and Price-Roseman. Finally, I will specialize to knotted RP^2’s in S^4 and construct a knotted RP^2 in S^4 that cannot be decomposed as the connected sum of an unknotted RP^2 and a knotted S^2. This last result on RP^2’s is joint with Hughes, Kim, and Miller.