Terence Tao

So you can have what's called a wave packet, which is like a very localized wave that is localized in space and moving a certain direction in time.

And so if you plot it in both space and time, it occupies a region which looks like a tube.

And so what can happen is that you can have a wave which initially is very dispersed, but it all focuses at a single point later in time.

Like you can imagine dropping a pebble into a pond and it will spread out.

But then if you time reverse that scenario, and the equations of wave motion are time reversible,

You can imagine ripples that are converging to a single point, and then a big splash occurs, maybe even a singularity.

It's possible to do that, and geometrically what's going on is that there's always light rays.

If this wave represents light, for example, you can imagine this wave as a superposition of photons, all traveling at the speed of light.

They all travel on these light rays, and they're all focusing at this one point.

You can have a very dispersed wave focused into

a very concentrated wave at one point in space and time, but then it defocuses again and it separates.

But potentially, if the conjecture had a negative solution, so what that meant is that there's a very efficient way to pack tubes pointing in different directions into a very, very narrow region of very narrow volume, then you would also be able to create waves that start out, there'll be some arrangement of waves that start out very, very dispersed,

but they would concentrate not just at a single point, but there'll be a lot of concentrations in space and time.

And you could create what's called a blow-up, where these waves, their amplitude becomes so great that the laws of physics that they're governed by are no longer wave equations, but something more complicated and nonlinear.

In mathematical physics, we care a lot about whether certain equations in wave equations are stable or not, whether they can create these singularities.

There's a famous unsolved problem called the Navier-Stokes regularity problem.

The Navier-Stokes equations are equations that govern the fluid flow for incompressible fluids like water.

The question asks, if you start with a smooth velocity field of water, can it ever concentrate so much that the velocity becomes infinite at some point?

That's called a singularity.

We don't see that in real life.