Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

The following is a conversation with Terence Tao, widely considered to be one of the greatest mathematicians in history, often referred to as the Mozart of Math. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed groundbreaking work to a truly astonishing range of fields in mathematics and physics. This was a huge honor for me, for many reasons, including the humility and kindness that Terry showed to me throughout all our interactions. It means the world. This is the Lex Fridman Podcast. To support it, please check out our sponsors in the description or at lexfridman.com/sponsors. And now, dear friends, here's Terence Tao. What was the first really difficult research-level math problem that you encountered? One that gave you pause maybe?

Well, I mean, in your undergraduate, um, education, you learn about the really hard impossible problems, like the Riemann hypothesis, the twin primes conjecture. You can make problems arbitrarily difficult, uh, that's not really a problem. In fact, there's even problems that we know to be unsolvable. What's really interesting, uh, are the problems just at the... on the boundary between what we can do perfectly easily and what are hopeless. Um, but what are problems where, like, existing techniques can do like 90% of the job and then y- you just need that remaining 10%. Um, 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, uh, mathematician Soichi Kakeya, uh, in like 1918 or so. Um, so, the puzzle is that you, you have, um, a needle, um, in s- on the plane, um, or- or think like- like- uh, like driving on- on- on a road or something, and you want to- to execute a U-turn, you want to turn the needle around, um, but you want to do it in as little space as possible, uh, so you want to use this little area, uh, in order to turn it around. So, um, but the needle is infinitely maneuverable. So you can imagine just spinning it around its, um... uh, it's a unit needle. You can spin it around its center, um, and I think, um, that gives you a disc of- of area, I think, pi over 4. Um, or you can do a three-point U-turn, which is what they- we teach people in- in the driving schools to do, uh, and that actually takes area pi over 8. Um, so it's- it's a little bit more efficient than, um, uh, rotation. And so for a while people thought that was the most efficient, uh, way to turn things around. But Bazhukovich, uh, showed that in fact, uh, you could actually, uh, turn the needle around using as little area as you wanted.

Mm.

So .001, there was some really fancy multi, um, uh, um, back and forth U-turn thing that you could- you could do, that- that you could turn the needle around. And in so doing, it would pass through every intermediate direction.