Terence Tao

But in 3D, the problem was that this equation was actually supercritical.

It's the same problem as Navier-Stokes.

As you blow up, maybe the curvature could get concentrated in finer and smaller regions, and it looked more and more nonlinear, and things just looked worse and worse.

And there could be all kinds of singularities that showed up.

Some singularities, like there's these things called neck pinches, where the surface sort of behaves like a barbell and it pinches at a point.

Some singularities are simple enough that you can sort of see what to do next.

You just make a snip and then you can turn one surface into two and evil them separately.

But there was the prospect that some really nasty, like, knotted singularities showed up that you couldn't see how to resolve in any way, that you couldn't do any surgery to.

So you need to classify all the singularities.

Like, what are all the possible ways that things can go wrong?

So what Perlman did was, first of all, he made the problem, he turned the problem from a supercritical problem to a critical problem.

I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics.

So he introduced something which is now called Perlman's reduced volume and Perlman's entropy.

He introduced new quantities, kind of like energy, that looked the same at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before.

And then he had to solve, he still had to analyze the singularities of this critical problem.

and that itself was a problem similar to this wave map thing I worked on actually so on the level of difficulty of that so he managed to classify all the similarities of this problem and show how to apply surgery to each of these and through that was able to resolve the Poincare conjecture so

quite a lot of really ambitious steps and like nothing that a large language model today, for example, could.

I mean, at best, I could imagine a model proposing this idea as one of hundreds of different things to try.

But the other 99 would be complete dead ends, but you'd only find out after months of work.

He must have had some sense that this was the right track to pursue because it takes years to get them from A to B. So you've done, like you said, actually, even strictly mathematically, but