Lex Fridman #503: The AI Crisis in Math, Navier-Stokes to Riemann

Three mathematicians join Lex Fridman to make sense of the weeks when AI claimed proofs of Navier-Stokes blowup, a quasi-Riemann hypothesis, the Unique Games Conjecture and a full Lean formalization of Fermat’s Last Theorem. In Lex Fridman Podcast #503, released October 26, 2026, Fridman calls it an emergency episode. His guests are MIT theoretical computer scientist Ankur Moitra, University of Washington fluid dynamicist Steven Brunton and Rutgers number theorist Alex Kontorovich. Over three and a half hours they explain the results, argue about whether AI is creating new mathematics, and ask what is left for human mathematicians and the students who are panicking.

TLDW

An OpenAI release of about 700 AI-generated results included a proof that no zeros of the zeta function lie in the strip with real part above 7/8 (a “quasi-Riemann hypothesis”), a blowup counterexample for the forced version of the Navier-Stokes Millennium problem, and a proof that the Unique Games Conjecture is true. Days earlier, Anthropic produced a 13-million-line Lean proof of Fermat’s Last Theorem with almost no human guidance. The guests agree that autoformalization is essentially solved and that proving well-posed theorems is becoming cheap, but they argue the value of mathematics lies in understanding, asking good questions and building theory. Those parts are still human for now. They also cover how nature computes, why deep learning works, P vs NP, Perelman, the future of prizes and journals, and advice for students facing a broken playbook.

Thoughts

Steven Brunton’s line about Navier-Stokes is the most useful corrective in the episode: “we solved the million-dollar math problem,” but the trillion-dollar engineering problem is untouched. The AI’s counterexample is a precisely forced vortex that stretches infinitely thin in three dimensions, which is a real mathematical achievement and completely non-physical. Fluids are exactly as expensive to simulate on supercomputers and in wind tunnels as they were a month ago. That gap shows where AI is strongest. Brunton later sorts problems on two axes, how verifiable the output is and how bad it is to be wrong. A wrong proof hurts nobody, and Lean checks it in minutes, so mathematics is the perfect domain. Most valuable real-world problems are the opposite: messy, hard to check and high-stakes. The math results are spectacular partly because math is the easiest place for an AI to be spectacular.

Fermat’s Last Theorem matters more than the headline proofs, because of what it does to trust. Alex Kontorovich told a Princeton colloquium on a Wednesday that AI could only formalize mathematics one or two steps from the Mathlib library. On Friday, Anthropic released 13 million lines of Lean proving Fermat from the axioms, with two or three people whose main input was typing “keep going” in all caps. If any human-written proof can now be checked down to the axioms, the bottleneck in mathematics moves from verification to explanation. That is why both Kontorovich and Ankur Moitra predict that papers will change shape: a human story about why the argument works, with a Lean file attached to show that it does. It is also why Kontorovich trusts the quasi-RH proof completely even though he has not finished reading it.

The saddest moment belongs to Moitra. Hours before the release, he asked Subhash Khot, who proposed the Unique Games Conjecture, whether he would read an AI proof of it. Khot said no. The conjecture had organized a whole field for two decades, humans like Dor Minzer and Irit Dinur had made major progress toward it, and Minzer rushed out a paper because of the rumors. Moitra’s worry, which echoes the Terence Tao quote Fridman reads at the start, is not that the answer is wrong. It is that the paths people were walking toward it, and everything they would have found along the way, have been closed off. He adds a second cost: researchers may start keeping their best partial ideas secret, because once an idea is public an AI can exhaust everything it unlocks. A field built on openness could become guarded, at least until it stops rewarding being first.

Moitra’s explanation of why deep learning works deserves wider attention. Training even a four-layer network is cryptographically hard in the worst case, yet much deeper networks train every day. His answer is that “nature abhors computationally hard problems”: proteins that survived evolution are the ones that fold easily, and human language probably evolved to be learnable in the same way. So the right question is not why the algorithm beats worst-case hardness, but what structure in natural data makes the problem easy. The same lens explains “subliminal learning,” where a model fine-tuned on random numbers from an owl-obsessed model starts talking about owls. Concepts share overlapping directions inside the model, and small, coordinated nudges add up like taps on a Ouija board. It is a rare case of theory explaining an alignment problem rather than just naming it.

The advice section is better than usual because it is concrete. Kontorovich sees undergraduate grades splitting into two humps. Students who use AI to quiz themselves and check their understanding are doing much better, and students who ask it for the answers an hour before the deadline are doing worse, so homework scores no longer predict exam scores. His advice to math PhD students is to do what they always have done: learn the hardest, most abstract mathematics they can, because that is what lets them direct and judge the machines. Moitra’s advice works for anyone: everyone is optimizing to win the next stage, so “pretend you’ve already won” and do what you would do then. He took it from Sanjeev Arora’s tenure advice. It is a good answer to anxiety and to an AI that can now win many of the stages for you.

Key Takeaways

OpenAI released about 700 AI-generated results in one night, reportedly from a few thousand attempted problems, a success rate of roughly 10%.
The quasi-Riemann result proves there are no zeros of the zeta function, or of Dirichlet L-functions, with real part between 7/8 and 1. The previous zero-free region had barely moved in 70 years, and the method cannot reach the full hypothesis at 1/2.
Anthropic formalized Fermat’s Last Theorem in about 13 million lines of Lean from the axioms, with two or three people and no domain expertise. Kontorovich concludes that autoformalization of human-written proofs is basically solved.
The Navier-Stokes result is a blowup counterexample for the version of the Clay problem with smooth external forcing. It uses a 3D vortex that stretches and thins like a spinning figure skater until its vorticity becomes infinite.
The Unique Games Conjecture was proved true. Moitra had expected it to be false because its consequences seemed too strong.
A separate Anthropic result showed that 3SUM can be solved asymptotically faster than anyone knew, which reorganizes fine-grained complexity theory.
Kontorovich’s thought experiment, the “solved axiom,” supposes that any problem humans and AI can solve together could be solved by AI alone. None of the three guests believes it, but they argue pure math would survive even then because its value lives in human understanding.
Moitra borrows the “doorman fallacy”: reducing a job to its most visible task. Proving theorems is the doorman’s door. Mathematicians also build theory, ask questions and make connections.
On whether AI is creating new ideas, Kontorovich calls quasi-RH a brilliant chain of known techniques. He prefers Tim Gowers’s “semigroup” image of how far known moves can reach over the dismissive “convex hull.”
Journals were overloaded before AI. The guests suggest Mathlib-style public pull requests, crowdsourced verification and papers that focus on the story while Lean handles the details.
Brunton says insects control flight in turbulence using a few dozen strain-sensing neurons. They are proof that a handful of numbers can be enough to control a chaotic flow.
P vs NP remains the problem Moitra would be most surprised to see fall soon. He suspects many real-world instances are secretly easy, each for its own reason.
Kontorovich wonders whether the 2026 Fields Medals were the last, and asks what prizes are for once solving major problems is no longer rare.
All three urge students to learn fundamentals without AI, stay curious, tolerate boredom and find intrinsic reasons for their work.

Chapters

1:56 A Crisis, or a Flashlight?

Fridman reads Terence Tao’s lament about years of patient student work disrupted overnight. Kontorovich sees a golden age, trading a candle for a flashlight. Moitra feels excited but intellectually lonely, because the blueprint he gave young researchers no longer works. Brunton is hopeful as an engineer, and notes that knowing a solution exists can also speed people up.

10:13 The Solved Axiom and the Doorman Fallacy

Kontorovich compares the moment to centaur chess, where humans eventually just accepted the engine’s top move, and asks whether math would die in that world. He argues no, because pure math matters only when it lives in human heads. Moitra adds that when well-defined problem-solving gets cheap, everything orthogonal to it becomes more valuable. Brunton stresses asking the right questions.

26:24 Can AI Create New Math?

Moitra warns against anthropomorphizing AI, citing research showing that adversarial examples are features, not bugs, and notes that the OpenAI proofs do not read like human work. The group debates whether the results sit inside the “convex hull” of known ideas or extend outward like tiles in a semigroup. Moitra points out that even settled answers like 3SUM redirect whole fields, and Kontorovich notes that an AI result on factoring would end e-commerce overnight.

47:14 Journals, Mathlib and the Future of the Paper

Kontorovich uses Perelman’s proof of Poincaré to explain why verification is slow and communal on purpose. Moitra wants new ways to absorb AI results without breaking referees. Kontorovich describes Lean and Mathlib, and argues that papers should separate the engineering of a proof from the story, the way he taught real analysis backwards, starting with intuition.

1:05:56 The Quasi-Riemann Hypothesis and a 13-Million-Line Fermat

Kontorovich explains the zeta function as signal processing on the primes, and the Riemann hypothesis as every instrument in the orchestra playing at the same volume. He was at the National Museum of Mathematics gala, whiskey in hand, when the release landed. He describes the Fermat formalization that broke his predictions two days after his colloquium, and the new 7/8 zero-free strip. Moitra adds a 1950s play in which God’s answer about the Riemann hypothesis is not enough for von Neumann.

1:29:49 Navier-Stokes and the Trillion-Dollar Problem

Brunton explains Navier-Stokes as conservation of momentum for fluids, and finite-time blowup through the simple equation x’ = x². The AI counterexample uses smooth forcing to build a stretching vortex with infinite vorticity. It is an important result, but it changes nothing about simulating or designing real wings and hearts, and turbulence still lacks the handful of summary numbers that gases have.

1:55:45 How Nature Computes

Moitra brings the algorithmic lens to nature, from slime molds to Kamal Jain’s line “If my laptop can’t find it, then neither can the market.” Brunton contrasts weather, where everything matters, with fruit flies, which control turbulent flight on a few numbers from strain-sensing neurons. Moitra notes that outfielders run curved paths because robust strategies beat perfect plans.

2:04:39 The Unique Games Conjecture Is True

Moitra walks through SAT, the PCP theorem and hardness of approximation, and why Unique Games would settle the best approximation algorithm for a huge class of problems. He describes the human progress before the release, Subhash Khot’s refusal to read an AI proof, and his worry about young researchers flooding arXiv. His question for young mathematicians: are you more curious than you are smart?

2:39:16 P vs NP, Worst-Case Analysis and Why LLMs Work

Moitra explains worst-case analysis through the secretary problem and caching, and argues that modeling real-world structure is the next frontier. He says natural data evolved to make hard problems easy, explains subliminal learning with a Ouija board image, and Brunton proposes a verifiability-versus-stakes grid for deciding where AI should be trusted.

3:03:57 Perelman, Prizes and Advice for Young People

The group considers what Grigori Perelman would make of the moment and whether Fields Medals still make sense. Kontorovich describes AI as a tutor that finds each learner’s zone of proximal development, and notes the bimodal grades. The advice is to learn the fundamentals yourself, stay curious, be bored sometimes, find intrinsic purpose and value the people around you.

Notable Quotes

“I think we’re entering, if we’re not already in, a period of some kind of golden age of mathematical knowledge after millennia of having a candle that we are walking around the woods, and this is as far as we can see. Now we have a flashlight.”
Alex Kontorovich, on how he feels about the moment

“When one thing becomes cheap, like problem-solving for well-defined things, it makes the other things that are orthogonal to it all that much more valuable.”
Ankur Moitra, on what mathematicians are for

“And I believe most of their involvement was, ‘Keep going,’ all caps.”
Alex Kontorovich, on the humans behind the Lean proof of Fermat’s Last Theorem

“Math is not the fact, it’s the why. It’s always been the why.”
Alex Kontorovich, after Moitra’s story about von Neumann and the Riemann hypothesis

“We solved the million-dollar math problem. There’s a trillion-dollar engineering problem, which is, like, how do you predict these things in a fast, efficient way and do this inverse design?”
Steven Brunton, on what the Navier-Stokes result means for engineering

“I guess my version of it is that in some sense, nature abhors computationally hard problems.”
Ankur Moitra, on why deep learning works

“The best way to live a rich and fulfilling life is just to pretend you’ve already won, and then figure out what you would do in that scenario.”
Ankur Moitra, on advice for anxious students

Watch the full conversation here: AI Crisis in Math, Lex Fridman Podcast #503. A full transcript is on Lex Fridman’s site.