Terence Tao

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.

And he showed that for two-dimensional surfaces, if you started something, no singularities ever formed.

You never ran into trouble, and you could flow, and it would give you a sphere.

So he got a new proof of the two-dimensional result.

Yeah, these are quite sophisticated equations on par with the Einstein equations.

It's slightly simpler, but...

Yeah, but they were considered hard nonlinear equations to solve.

And there's lots of special tricks in 2D that helped.