#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI
Episode publishedPodcast: Lex Fridman PodcastLex Fridman Podcast
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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. And now a quick few mention of his sponsor. Check them out in the description or at lexfriedman.com/spons
Well, I mean, in your undergraduate, education, you you learn about the really hard impossible problems like, the Riemann hypothesis, Dupuytren Pram's 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 fairly easily and what are hopeless. But what are problems where, like, 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 certai
this in the two dimensional plane? This is
in the two dimensional plane. Yeah. So we understand everything in two dimensions. So the next question is what happens in three dimensions? So suppose like the Hubble Space Telescope is 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 here's the 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 your needle around in order to see every single star in the sky. How small a volume do you need to do tha
Can you speak to the Navier Stokes? So the existence and smoothness, like you said, millennial price problem. Right. You've made a lot of progress on this one. In 2016, you published a paper, finite time blow up, for an averaged three-dimensional Navier Stokes equation. Right. So so we're trying to figure out if this thing usually doesn't blow up. Right. But can we say for sure it never blows up? Right.
Yeah. So, yeah, that is literally the the the million dollar question. Yeah. So this is what distinguishes mathematicians from pretty much everybody else. Like, if if if something holds 99.99% of the time, that's good enough for most, you know, for for for most things. But mathematicians are one of the few people who really care about whether every like 100%, really 100% of all, situations are covered by by, yeah. So most fluids most of the time, water does not blow up. But could you design a very special initial state that does this?
And maybe we should say that this is a set of equations that govern in the field of fluid dynamics. Yes. Trying to understand how fluid behaves and it's actually turns out to be a really comp you know, fluid is Yeah. Extremely complicated thing to try to model. Yeah.
So it has practical importance. So this clay price problem concerns what's called the incompressible Navier Stokes, which governs things like water. There's something called the compressible Navier Stokes, which governs things like air. And that's particularly important for weather prediction. Weather prediction, it does a lot of computational fluid dynamics. A lot of it is actually just trying to solve the Navier Stokes equations as best they can. Also gathering a lot of data so that they can get they can in initialize the equation. There
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