An oft-quoted idea in solar physics is that coronal mass ejections are, fundamentally, the Sun's way of shedding the magnetic helicity that is continually generated by its interior flows. In this talk, I will show how mathematical models are helping to give us a handle on the build up of magnetic helicity in the Sun's lower atmosphere: how much is injected, where it collects, and how it is ultimately ejected. The talk is aimed at all undergraduate or postgraduate students interested in applying mathematics to real-world phenomena.