Why does stirring always leave one point of your coffee where it started?

Because of a rule of topology called the Brouwer fixed-point theorem, which says that stirring, swirling or otherwise continuously reshuffling the liquid in a cup can never leave every single point somewhere new: once the liquid comes to rest, at least one point is exactly where it began.

The theorem is supposed to have originated from the mathematician L. E. J. Brouwer watching a cup of coffee. Stirring to dissolve a lump of sugar always seemed to leave a point without motion, and he concluded that at any moment, some point on the surface isn't moving. It's a subtle claim, though: the fixed point isn't necessarily the spot that looks still, since the centre of the swirling turbulence drifts around a little itself. The theorem only guarantees that some point, somewhere, ends up back at its start.

In its simplest form, the theorem states that every continuous function from a closed disk to itself has at least one fixed point, a point that maps back to itself. Brouwer proved the general version for continuous mappings in 1911, building on a differentiable case Jacques Hadamard had proved the year before, and on earlier work on differential equations by French mathematicians around Henri Poincaré and Charles Émile Picard.

The theorem shows up in surprisingly ordinary places. Crumple one of two identical sheets of graph paper and lay it on top of the flat one, keeping it within the flat sheet's edges: at least one point of the crumpled sheet still sits directly above the point with matching coordinates on the flat sheet. Spread a map of a country out on a table inside that same country, and there is always a 'You are Here' point on the map that sits exactly above the real spot it represents.