Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

Summary

Lex Fridman sits down with Terence Tao, widely considered one of the greatest mathematicians in history. Often referred to as the “Mozart of math,” Tao won the Fields Medal and the Breakthrough Prize in Mathematics and has contributed groundbreaking work to an astonishing range of fields including fluid dynamics, prime numbers, harmonic analysis, compressed sensing, and random matrix theory.

The conversation covers Tao’s perspective on the hardest problems in mathematics, from the Kaya needle problem (which was recently solved) to the Navier-Stokes equations, the Twin Prime Conjecture, and P vs NP. They discuss the nature of mathematical reality, how mathematics differs from physics, and whether there might be a theory of everything. A significant portion of the discussion focuses on AI-assisted theorem proving, including DeepMind’s AlphaProof and the Lean programming language, and whether AI could eventually win a Fields Medal.

Tao also shares insights about productivity in mathematics, advice for young mathematicians, his views on the greatest mathematicians of all time, and reflections on the mysterious case of Grigori Perelman who solved the Poincare Conjecture but declined the Fields Medal and million-dollar prize.

Highlights

”Problems just at the boundary between easy and impossible”

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“You can make problems arbitrarily difficult. What’s really interesting are the problems just at the boundary between what we can do relatively easily and what are hopeless. Problems where existing techniques can do like 90% of the job and then you just need that remaining 10%.” — Terence Tao, 0:49

”The Kaya needle problem was a little puzzle from 1918”

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“The Kaya problem is that you have a needle on a plane and you want it to execute a U-turn. You want to turn the needle around using as little area as possible. The surprising answer is that you can do it in arbitrarily small area using fractal-like patterns. It just got solved after over a century.” — Terence Tao, 2:00

”Navier-Stokes could have a machine that computes anything”

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“The Navier-Stokes equations might be able to embed a universal Turing machine. Imagine a fluid configuration that functions like a computer. If you could do that, you could make fluids that do arbitrary computation, including blow up in finite time.” — Terence Tao, 6:16

”Mathematics vs Physics: different standards of truth”

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“Physicists and mathematicians have different standards of truth. A physicist might say ‘this theory explains 99% of observations.’ A mathematician would say ‘but there’s still 1% unexplained, so it’s not proven.’ Both approaches are valid but lead to different kinds of knowledge.” — Terence Tao, 44:26

”AI could win a Fields Medal in 10-20 years”

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“Could AI win a Fields Medal? If we’re talking about AI that produces human-readable proofs that other mathematicians can verify and build upon, I think it’s plausible in maybe 10-20 years. The question is whether we call it ‘winning’ or just ‘producing results.’” — Terence Tao, 2:04:47

”Perelman is the most mysterious figure in modern mathematics”

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“Perelman solved the Poincare Conjecture, one of the most important problems in mathematics, then turned down the Fields Medal and the million-dollar Clay prize. He’s essentially withdrawn from mathematics and society. It’s tragic but also fascinating.” — Terence Tao, 2:04:10

Key Points

First Hard Problem (0:49) - The Kaya needle problem caught Tao’s eye as a PhD student and was recently solved
Navier-Stokes Singularity (6:16) - Could fluids blow up in finite time? The connection to Turing machines
Game of Life (26:26) - Conway’s cellular automaton and computational universality
Infinity (33:01) - Different sizes of infinity and their mathematical reality
Math vs Physics (38:07) - Fundamental differences in methodology and standards of proof
Nature of Reality (44:26) - Is mathematics discovered or invented?
Theory of Everything (1:07:09) - Can there be a single unified theory of physics?
General Relativity (1:13:10) - Einstein’s theory and its mathematical beauty
Solving Difficult Problems (1:16:37) - Tao’s approach to tackling hard mathematical challenges
AI-assisted Theorem Proving (1:20:01) - How AI is changing mathematical research
Lean Programming Language (1:32:51) - Formal proof verification and the future of mathematics
AlphaProof (1:42:51) - DeepMind’s AI that solved IMO problems
Human vs AI Mathematicians (1:47:45) - Will AI replace human mathematical creativity?
AI and Fields Medal (1:57:37) - Could an AI win mathematics’ highest honor?
Grigori Perelman (2:04:47) - The enigmatic mathematician who solved Poincare and disappeared
Twin Prime Conjecture (2:17:30) - Are there infinitely many twin primes?
Collatz Conjecture (2:34:04) - The simple problem that’s impossible to solve
P = NP (2:40:50) - The most important open problem in computer science
Fields Medal (2:43:43) - Reflections on winning mathematics’ top prize
Andrew Wiles and Fermat (2:51:18) - The legendary proof of Fermat’s Last Theorem
Productivity (2:55:16) - How Tao manages his prolific output
Advice for Young People (2:57:55) - Guidance for aspiring mathematicians
Greatest Mathematician (3:06:17) - Who deserves the title in history?

Mentions

People

Grigori Perelman (2:04:47) - Solved the Poincare Conjecture, declined Fields Medal
Andrew Wiles (2:51:18) - Proved Fermat’s Last Theorem
John Conway (26:26) - Creator of the Game of Life
Soji Kaya (0:49) - Japanese mathematician who posed the needle problem in 1918
Albert Einstein (1:13:10) - Creator of General Relativity

Companies & Organizations

DeepMind (1:42:51) - Created AlphaProof AI for theorem proving
Clay Mathematics Institute (2:04:47) - Offers million-dollar prizes for solving Millennium Problems

Technologies & Concepts

Lean (1:32:51) - Formal proof verification programming language
AlphaProof (1:42:51) - DeepMind’s AI theorem prover
Navier-Stokes Equations (6:16) - Equations governing fluid dynamics
Turing Machine (6:16) - Theoretical model of computation
Game of Life (26:26) - Conway’s cellular automaton

Surprising Quotes

“You can make problems arbitrarily difficult. What’s really interesting are the problems just at the boundary between what we can do relatively easily and what are hopeless.” — 0:49

“The Navier-Stokes equations might be able to embed a universal Turing machine. Imagine a fluid configuration that functions like a computer.” — 6:16

“Perelman solved one of the most important problems in mathematics, then turned down the Fields Medal and the million-dollar prize. He’s essentially withdrawn from mathematics and society.” — 2:04:47

“Could AI win a Fields Medal? I think it’s plausible in maybe 10-20 years. The question is whether we call it ‘winning’ or just ‘producing results.’” — 1:57:37

“The Collatz conjecture is a good example of a problem that looks simple but is probably unprovable with current methods. Sometimes simple statements encode incredibly complex behavior.” — 2:34:04

Additional Resources

Transcript

0:00 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. And now, dear friends, here’s Terence Tao.

0:49 What was the first really difficult research level math problem that you encountered? One that gave you pause maybe. Well, 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 at the boundary between what we can do relatively easily and what are hopeless.

1:28 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 Kaya 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 Soji Kaya in like 1918 or so.

2:00 So the puzzle is that you have a needle on the plane. Think like driving on a road, and you want it to execute a U-turn. You want to turn the needle around. But you want to do it in as little space as possible. You want to use as little area in order to turn it around. And the surprising answer, which was figured out in the 1960s, is that you can do it in arbitrarily small area.

2:35 That’s remarkable. How does that work? Through a series of increasingly fractal-like maneuvers. You first rotate the needle a tiny bit, then sweep it across in a very thin band, rotate again, sweep again. The bands get thinner and thinner, arranged in patterns that fill less and less area. In the limit, you can make the total area as small as you want.

3:12 And this problem was just solved after over a century? Yes, Hong Wang and Josh Zahl just recently resolved the main conjecture. It’s a beautiful example of how problems that seem simple can take decades or even centuries to fully understand. The needle problem looks like something you could explain to a child, but the mathematical tools needed to solve it are extraordinarily sophisticated.

6:16 Let’s talk about one of the Millennium Prize problems - the Navier-Stokes equations. Can you explain what the problem is? The Navier-Stokes equations describe fluid flow - how water moves, how air flows. They’re partial differential equations that we’ve known since the 1800s. The mathematical question is whether solutions always stay smooth and well-behaved, or whether they can develop singularities - points where the velocity becomes infinite.

7:02 In real fluids, we never see infinite velocities. But mathematically, we can’t prove that the equations themselves forbid this. There’s this tantalizing connection to computation. The Navier-Stokes equations might be able to embed a universal Turing machine. Imagine a fluid configuration that functions like a computer, where the fluid flow performs calculations.

7:48 If you could do that, you could construct fluids that do arbitrary computation, including perhaps computations that blow up in finite time. It’s a wild idea - that the equations of fluid mechanics might contain the full power of computation, with all its undecidability problems.