Poincaré Conjecture… with less detail.

Yes, this was already solved by Grigory Perelman, but I have nothing much, so…

Imagine a spherical item. And, for the easiest solution, let’s use an apple. Wrap the rubber band around the apple. Without tearing it or letting it leave the surface, we can slowly shrink the rubber band until it becomes a single point.

Now, think of a doughnut. Before you get hungry, quickly wrap a rubber band around the doughnut. Now, no matter how hard you try, you can’t shrink the rubber band down to a point without breaking it or making it leave the surface.

The surface of the apple is “simply connected”, while the surface of the doughnut isn’t. Poincaré, about a hundred years ago, knew that a two-dimensional sphere is essentially characterised by this property of simple connectivity, but questioned that of the three-dimensional sphere.

Grigory Perelman already solved it, and the link to the arXiv is here. The upload is by Terence Tao.

Until next time…