Terence Tao

This is in the two-dimensional plane.

So we understand everything in two dimensions.

So the next question is what happens in three dimensions.

So suppose the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe.

So you want to rotate the telescope to reach every single direction.

And his unrealistic part, suppose that space is at a premium, which it totally is not.

You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky.

How small a volume do you need to do that?

And so you can modify it based on Copic's construction.

And so if your telescope has zero thickness, then you can use as little volume as you need.

That's a simple modification of the two-dimensional construction.

But the question is that if your telescope is not zero thickness, but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta?

So as delta gets smaller, as your needle gets thinner, the volume should go down, but how fast does it go down?

And the conjecture was that it goes down very, very slowly, like logarithmicly, roughly speaking.

And that was proved after a lot of work.

So this seems like a puzzle.

Why is it interesting?

So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry.

For example, in wave propagation, you splash some water around, you create water waves and they travel in various directions.

But waves exhibit both particle and wave type behavior.