Terence Tao

There were algebraic approaches.

There's various algebraic objects, like things called the fundamental group that you can attach to these homology and cohomology and all these very fancy tools.

They also didn't quite work.

But Richard Hamilton proposed a partial differential equations approach.

So the problem is that you have this object which is sort of secretly a sphere, but it's given to you in a really weird way.

So think of a ball that's been kind of crumpled up and twisted, and it's not obvious that it's a ball.

If you have some sort of surface which is a deformed sphere, you could think of it as the surface of a balloon.

You could try to inflate it.

You blow it up, and naturally, as you fill it with air, the wrinkles will smooth out and it will turn into a nice round sphere.

Unless, of course, it was a torus or something, in which case it would get stuck at some point.

If you inflate a torus,

there would be a point in the middle.

When the inner ring shrinks to zero, you get a singularity and you can't blow up any further.

You can't flow any further.

So he created this flow, which is now called Ricci flow.

which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder, to make it look like a sphere.

And he wanted to show that either this process would give you a sphere or it would create a singularity.

I very much like how PDEs either have global regularity or finite envelope.

Basically, it's almost exactly the same thing.

It's all connected.