Terence Tao

Well, I mean, in your undergraduate education, you learn about the really hard impossible problems, like the Riemann hypothesis, the twin-primes conjecture.

You can make problems arbitrarily difficult.

That's not really a problem.

In fact, there's even problems that we know to be unsolvable.

What's really interesting are the problems just on the boundary between what we can do perfectly easily and what are hopeless.

But what are problems where existing techniques can do like 90% of the job and then you just need that remaining 10%?

I think as a PhD student, the Kakeya problem certainly caught my eye, and it just got solved, actually.

It's a problem I've worked on a lot in my early research.

Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so.

The puzzle is that you have a needle on the plane.

Think of it like driving on a road.

You want to execute a U-turn.

You want to turn the needle around.

but you want to do it in as little space as possible.

So you want to use this little area in order to turn it around.

But the needle is infinitely maneuverable.

So you can imagine just spinning it around its, as the unit needle, you can spin it around its center.

And I think that gives you a disc of area, I think pi over four.

Or you can do a three-point U-turn, which is what we teach people in the driving schools to do.

And that actually takes area pi over eight.