In 1982, Richard Hamilton published a beautiful paper introducing a new technique in geometric analysis which he called Ricci flow. The geometrisation conjecture predicts that any three-dimensional space in which every loop shrinks to a point should have a round metric – it would be a 3-sphere and Poincaré’s conjecture would follow. Thurston made spectacular progress on the geometrisation conjecture, which includes the Poincaré conjecture as a special case. He had many brilliant students who further developed his theories, not least by producing powerful computer programs that could test any given space to try to find its geometric structure. Thurston made a bold “ geometrisation conjecture” that this should be true for all three-dimensional spaces. (To see this geometry on a torus, one must embed it into four-dimensional space!). He observed that known three-dimensional spaces could be divided into pieces in a natural way, so that each piece had a uniform geometry, similar to the flat plane and the round sphere. In particular, he realised that essentially all the work that had been done since Poincaré fitted into a single theme. Thurston made enormous strides in our understanding of three-dimensional spaces in the late 1970s. This is a brief account of the ideas used by Perelman, which built on work of two other outstanding mathematicians, Bill Thurston and Richard Hamilton. Many attempts were made on the Poincaré conjecture, until in 2003 a wonderful solution was announced by a young Russian mathematician, Grigori “Grisha” Perelman. Any loop of string on a 2-sphere can be shrunk to a point while keeping it on the sphere, whereas if a loop goes around the hole in the donut, it cannot be shrunk without leaving the surface of the donut.
![perelman point carré perelman point carré](https://media.newyorker.com/photos/59097509c14b3c606c10872a/master/pass/060828_R15383.jpg)
Donuts to go, pleaseĪ good way to visualise Poincaré’s conjecture is to examine the boundary of a ball (a two-dimensional sphere) and the boundary of a donut (called a torus).
![perelman point carré perelman point carré](https://inteng-storage.s3.amazonaws.com/images/MAY/sizes/henri-poincare_resize_md.jpg)
Poincaré had introduced important ideas in the structure and classification of surfaces and their higher dimensional analogues (“manifolds”), arising from his work on dynamical systems. He asked if the 3-sphere (which can be formed by either adding a point at infinity to ordinary three-dimensional Euclidean space or by gluing two solid three-dimensional balls together along their boundary 2-spheres) was the only three-dimensional space in which every loop can be continuously shrunk to a point.