ELI5: What is the Poincaré Conjecture?
Imagine you have a giant rubber band. You can stretch it, twist it, and move it around, but you can't cut it or glue it.
Now, imagine you're on a giant ball, like the Earth. If you have a huge rubber band stretched around the Earth, you can always shrink it down to a single point. You can't get it stuck on anything.
But what if you were on a donut? A donut has a hole in the middle. If you stretch your rubber band through the hole, you can't shrink it down to a point without cutting it. It's stuck.
The Poincaré Conjecture is about shapes in three dimensions. It says that if you have a 3D shape that is "simple" enough, then it's basically a sphere (a 3D ball).
What does "simple" mean? It means two things:
- The shape is "closed and bounded", which means it doesn't go on forever and it doesn't have any edges. It's like a ball, not a flat sheet of paper.
- The shape is "simply connected", which means that any loop you draw on it can be shrunk down to a single point. It's like a ball, not a donut.
So, the conjecture says that if a 3D shape has these two properties, then you can always stretch it and bend it (without cutting or gluing) until it looks like a perfect sphere.
This was a very hard problem to prove. It took almost 100 years, but a brilliant mathematician named Grigori Perelman finally solved it in 2003. It's the only one of the seven Millennium Prize Problems that has been solved so far.
