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.
You won’t encounter topology directly in everyday life but that doesn’t mean it’s useless. Topology is a sophisticated tool in mathematicians’ problem-solving kit. Indirect applications abound: satnav, electronics, robots, efficient trajectories for space probes. Mathematics is highly interconnected and significant advances in any area lead to progress in apparently unrelated areas. Moreover, any particular mathematical technique can be applied to many different real-world problems. The wave equation began as a study of violin strings and is now used to detect earthquakes.
Topologists initially studied surfaces—spaces with two dimensions. The standard shapes are like doughnuts but with any number of holes. No holes gives a sphere, one gives a doughnut, and so on. Every surface can be deformed into precisely one of these shapes. The number of holes characterises the topological type of the surface, so it is called an invariant. The pioneers of topology proved this by deforming any given surface into one that can be cut into triangles, and working…