We show that the group cohomology of simple connected Lie groups, and consequently their lattices and symmetric spaces, with L^p-coefficients vanishes below the rank. This confirms a conjecture of Gromov in the special case of the coefficient module L^p(G). We relate the vanishing of L^1-cohomology to a fixed-point theorem and confirm a conjecture of Farb. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk, I will give a short introduction to group cohomology and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.

A natural energy arising in the modelling of thin ferromagnetic films and liquid crystals is the Dirichlet energy, along with a penalisation term which favours a particular configuration. In this talk, I present recent work which establishes a quantitative analysis for sequences of minimisers to both the Baby-Skyrme model and two regularisation arising from geometry and physics respectively.