Terence Tao

I sometimes joke that basically AI has to go through a grad school

And actually, you know, go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes, and learn from that.

All right, so it's a question about curved spaces.

Earth is a good example.

So let's say Earth, you can think of it as a 2D surface.

And just moving around the Earth, there could maybe be a torus with a hole in it, or it could have many holes.

And there are many different topologies a priori that a surface could have, even if you assume that it's bounded and smooth and so forth.

So we have figured out how to classify surfaces.

As a first approximation, everything is determined by something called the genus, how many holes it has.

So a sphere has genus zero, a donut has genus one, and so forth.

And one way you can tell these surfaces apart, probably the sphere has, which is called simply connected.

If you take any closed loop on the sphere, like a big closed loop of rope, you can contract it

to a point while staying on the surface.

And the sphere has this property, but a torus doesn't.

If you're on a torus and you take a rope that goes around, say, the outer diameter of a torus, there's no way, it can't get through the hole.

There's no way to contract it to a point.

So it turns out that the sphere is the only surface with this property of contractibility, up to like continuous deformations of the sphere.

So things that I want to go topologically equivalent to the sphere.

So Poincare asked the same question in higher dimensions.

So it becomes hard to visualize because,