Terence Tao

A surface you can think of as embedded in three dimensions, but a curved free space, we don't have good intuition of 4D space to live in.

And then there are also 3D spaces that can't even fit into four dimensions.

You need five or six or higher.

But anyway, mathematically, you can still pose this question that if you have a bounded three-dimensional space now, which also has this simply connected property that every loop can be contracted, can you turn it into a three-dimensional version of a sphere?

And so this is the Poincaré conjecture.

Weirdly, in higher dimensions, 4 and 5, it was actually easier.

So it was solved first in higher dimensions.

There's somehow more room to do the deformation.

It's easier to move things around to a sphere.

But 3 was really hard.

So people tried many approaches.

There's sort of commentary approaches where you chop up the surface into little

triangles or tetrahedra, and you just try to argue based on how the faces interact with each other.

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.