Terence Tao

There's the electromagnetic field, there's things called Young-Mills fields, and there's this whole hierarchy of different equations.

of which Einstein is considered one of the most nonlinear and difficult.

But relatively low on the hierarchy was this thing called the wave maps equation.

So it's a wave which at any given point is fixed to be like on a sphere.

So I can think of a bunch of arrows in space and time, and yeah, so it's pointing in different directions.

But they propagate like waves.

If you wiggle an arrow, it will propagate and make all the arrows move kind of like sheaves of wheat in the wheat field.

And I was interested in the global regularity problem again for this question.

Is it possible for all the energy here to collect at a point?

So the equation I considered was actually what's called a critical equation, where it's actually the behavior at all scales is roughly the same.

And I was able barely to show that...

that you couldn't actually force a scenario where all the energy concentrated at one point.

The energy had to disperse a little bit, and the moment it dispersed a little bit, it would stay regular.

This was back in 2000.

That was part of why I got interested in Navier-Stokes afterwards.

I developed some techniques to solve that problem.

Part of it is that this problem is really non-linear because of the curvature of the sphere.

There was a certain nonlinear effect, which was a non-perturbative effect.

When you looked at it normally, it looked larger than the linear effects of the wave equation.

And so it was hard to keep things under control, even when the energy was small.