Abstract: The geometry of hyperbolic surfaces, the physics of Yang-Mills fields, and the analysis of the Schrödinger equation are very different subjects—so why do the people who study them keep such similar items in their toolkits? As it turns out, they’ve all ended up working in the same geometric space, an example of a Hitchin integrable system. Thanks to this convergence, the same object or construction will sometimes appear in all three subjects, with three different interpretations.