ABSTRACT: BPS states in four-dimensional N=2 theories are enumerated by spectral networks on two-dimensional Riemann surfaces. Geometrically they correspond to special Lagrangian discs ending on the spectral networks, more formally they are (conjectured to be) generalised Donaldson-Thomas invariants for an appropriate CY3 category. In this talk I will give a particularly interesting example of these BPS state counts in the E6 Minahan-Nemeschansky theory, where the relevant spectral networks are generated by generalised Strebel differentials. This is joint work with Andrew Neitzke.