The Yang-Mills-Higgs (YMH) equations are a system of nonlinear PDE that arise as variational equations for a gauge invariant energy. In the case of scalar fields these are the Ginzburg-Landau  equations (GLE) which are a macroscopic model for superconductivity that in an appropriate regime can form planar vortex arrays of magnetic flux (Abrikosov lattices) whose existence and stability has been analyzed in terms of  GLE on a surface of genus 1. In this talk we illustrate how this analysis may be extended to vector valued fields by posing the YMH equations on a more general class of surfaces. In this setting functional analysis and PDE theory reduce to classical analysis and ODE theory related to Hilbert's 21st problem. From a broader perspective this also illustrates how gauge and spatial symmetries can intertwine to create patterns.