A mathematical breakthrough made this year is so complex that other mathematicians struggled to understand it, but it has already led to progress in other areas of maths.

In May, Sam Raskin at Yale University and his colleagues published a series of five papers, totalling more than 1000 pages, that proved the geometric Langlands conjecture, in a culmination of decades of work. The problem is actually a simpler formulation of an even grander mathematical project, the Langlands programme, which seeks to make…