Terence Tao

So we have published things of things that people have been able to prove and conjectures that ended up being verified or maybe counterexamples produced.

But we don't have data on things that were proposed and they're kind of a good thing to try.

But then people quickly realized that it was the wrong conjecture and then they said, oh, but we should actually change

our claim to modify it in this way to actually make it more plausible.

There's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record because it's embarrassing.

We make mistakes and we only like to publish our wins.

And the AI has no access to this data to train on.

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.