When Ciprian Manolescu was working for his PhD in mathematics, he heard about a famous unsolved problem: the triangulation conjecture. It was too difficult for a doctoral thesis, so he worked on a more tractable problem in the same area, known as topology. For the next 10 years he didn’t think about the triangulation conjecture at all. Until one day he realised that the ideas in his thesis show that the conjecture is false.
If his proof holds up under scrutiny—and experts seem to think it will—this is a major breakthrough and a spectacular result. Its importance in mathematics is comparable to that of the Poincaré conjecture, which Grigori Perelman proved a few years ago to great acclaim. And the new methods involved are likely to lead to further advances.
Topology is about properties of mathematical shapes that persist under continuous deformations: squash, stretch, bend, but don’t tear. A cube, for example, can be deformed into a sphere. However, you can’t continuously deform it into a doughnut (the kind with a hole). Topology is fundamental to the whole of mathematics. It provides powerful methods for listing all possible topologically distinct shapes, and working out which of those shapes a given mathematical structure has.