Gromov-hyperbolic groups are classically defined geometrically, by the negative curvature of their Cayley graphs. Interestingly, an algorithmic characterization of hyperbolicity is possible in terms of properties of the formal languages of quasigeodesics (geodesics up to bounded error) in their Cayley graphs. Holt and Rees proved, roughly speaking, that these formal languages are regular in the case of hyperbolic groups. More recently Hughes, Nairne, and Spriano established the converse. In this talk, I will discuss progress towards a conjectured strengthening of the result, where we consider context-free quasigeodesic languages. This is based on my summer project, supervised by Joseph MacManus and Davide Sprianoc

The ε-energy is a regularisation of the Dirichlet energy introduced by Tobias Lamm. Like the famous Sacks-Uhlenbeck regularisation this greatly improves the existence and regularity theory. When we take the limit of a sequence of ε-harmonic maps with the parameter ε decreasing to 0 these converge, in the standard bubbling sense, to harmonic maps, which we hope to extract information about. I will talk about some recent results for these sequences, being when we might hope to have no loss of energy and no neck forming and what sort of harmonic maps we can obtain in the limit.

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Skein lasanga modules are a smooth 4-manifold invariant that was introduced by Morrison, Walker and Wedrich using Khovanov homology. This invariant was recently used by Ren and Willis to give the first analysis free proof of the existence of exotic 4-manifolds. However, even for simple handlebodies it remains difficult to compute. A generalisation was introduced by Chen using Knot Floer homology, which in principle should be easier to compute due to cabling formulas for knot Floer homology. I will give a general introduction to lasagna modules assuming no knowledge of Khovanov or knot Floer homology, and then explain some methods, from upcoming work, for computing Floer Lasagna modules.