Terence Tao

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.

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.