#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory
Episode of Lex Fridman Podcast · October 26, 2026 · 3 h 43 min
Lex Fridman and mathematicians Alex Kontorovich, Ankur Moitra and Steven Brunton discuss AI's claimed proofs of quasi-Riemann, Navier-Stokes, unique games and what it means for mathematics.
Summary
In this episode, framed as an emergency discussion, Lex Fridman spoke with number theorist Alex Kontorovich, theoretical computer scientist Ankur Moitra and fluid dynamicist Steven Brunton about a wave of AI-generated mathematical results, including OpenAI's release of some 700 problems with claimed solutions (among them a quasi-Riemann hypothesis, a forced Navier-Stokes blow-up construction and the unique games conjecture) and an Anthropic-produced 13-million-line Lean formalization of Fermat's Last Theorem. The guests described mixed feelings: Kontorovich saw a coming golden age, Moitra described intellectual loneliness and concern for students, and Brunton was largely hopeful while noting deep uncertainty.
Kontorovich introduced a hypothetical 'solved axiom' and argued that even in such a world pure mathematics retains value because it lives in human minds; none of the guests said they believed the axiom holds today. Moitra invoked the 'doorman fallacy' to argue that proving theorems is only the most visible part of a mathematician's work. The group debated whether AI results lie within the 'convex hull' of known ideas, with Kontorovich preferring Tim Gowers's semigroup framing and Brunton arguing that even interpolation lifts a valuable fog. They also discussed strain on journals, Lean and Mathlib, and how papers may become stories accompanied by formal proofs.
Each guest explained a problem from his field: Kontorovich the Riemann hypothesis and why the AI's 7/8 zero-free strip method cannot reach 1/2; Brunton the Navier-Stokes blow-up result, which he said leaves practical turbulence engineering unchanged; and Moitra the PCP theorem and unique games conjecture, which he had doubted. Moitra argued P vs NP remains far out of reach and that natural data's structure explains why deep learning succeeds, and he discussed subliminal learning and AI alignment. Brunton urged a verification 'firewall' for high-stakes AI use.
The conversation closed with Grigori Perelman, the future of the Fields Medal, education in the AI era, and advice for students, including Moitra's suggestion to 'pretend you already have tenure' and Kontorovich's emphasis on family and community.
Chapters
- 0:00
Introduction and sponsors
Fridman frames the episode as an emergency discussion of AI's claimed solutions to famous math problems and introduces the three guests.
- 9:53
Feelings about the crisis moment
The guests describe their reactions to the AI results, ranging from hope to intellectual loneliness, and discuss Terence Tao's lament.
- 18:12
Will AI solve or kill mathematics?
Kontorovich proposes the 'solved axiom' and Moitra introduces the doorman fallacy to argue mathematicians do far more than prove theorems.
- 41:07
Can AI create new mathematics?
The group debates convex hull versus semigroup framings of AI results, field-altering consequences, and factoring and cryptography.
- 54:46
Papers, journals, Lean and Mathlib
Discussion of overloaded peer review, collaborative models, Lean and Mathlib, and how papers may become stories backed by formal proofs.
- 1:13:27
The Riemann hypothesis and quasi-RH
Kontorovich explains the Riemann hypothesis, the shock of the quasi-RH proof and the Fermat's Last Theorem formalization, and why the full hypothesis needs new ideas.
- 1:37:46
Navier-Stokes, turbulence and how nature computes
Brunton explains the Navier-Stokes equations and the blow-up construction, while the group discusses turbulence, insect flight and natural computation.
- 2:12:35
Unique games conjecture
Moitra explains satisfiability, the PCP theorem and the unique games conjecture, and his reaction to its AI proof.
- 2:27:28
How mathematicians think and whether math is discovered
The guests describe their working styles and mentorship, then discuss whether mathematics is invented or discovered and the history of infinity.
- 2:47:13
P vs NP and why deep learning works
Moitra discusses worst-case analysis, the secretary and paging problems, why natural data makes deep learning tractable, and subliminal learning.
- 3:11:54
Hard problems, Perelman and the future of prizes
The guests name problems they expect to remain open, discuss Perelman's refusals and whether Fields Medals will continue, and AI as a learning tool.
- 3:25:41
Advice for young people
The guests advise students to stay curious, learn fundamentals, act from intrinsic motivation, and value human community.
Key points
What was said, by whom, with the exact words. Timestamps jump to the moment.
- Alex Kontorovich
Kontorovich argues that AI is ushering mathematics into a golden age of knowledge rather than merely a crisis.
“I think we're entering, if we're not already, in a period of some kind of golden age of mathematical knowledge”
- Ankur Moitra
Moitra says the traditional career blueprint for mathematics students has been invalidated by AI, making advising harder.
“you know, the playbook and the blueprint for success that we had for so long is now thrown out the window”
- Steven Brunton
Brunton argues that knowing a solution exists is a strong motivator for humans to rediscover it quickly.
“At the same time, if we found out that another country had invented teleportation or time travel, I guarantee we would reinvent it in the next five years”
- Alex Kontorovich
Kontorovich argues that pure mathematics remains valuable even if machines can prove theorems, because its value lies in human understanding.
“I think the value of pure mathematics is that it lives in the heads of human beings”
- Ankur Moitra
Moitra argues, via the doorman fallacy, that reducing mathematicians to theorem-provers misses their broader role and that panic is unwarranted.
“So even though this is a momentous time, I don't think it's the right time to hit the panic button”
- Alex Kontorovich
Kontorovich judges that the AI's quasi-Riemann hypothesis proof combined well-known techniques rather than introducing a genuinely new idea.
“at least in the case of quasi RH, I would not say it's a genuinely new idea. It's a beautiful combination of a lot of things that were very well known”
Linked moment · 52:58Skip to 52:58 using a listen link in the info box.
Steven BruntonBrunton argues that even if AI only recombines human knowledge, this should not be seen as derogatory, and he thinks it is more extrapolatory than that.
“I don't think we should be interpreting that as derogatory at all. I do think it is more extrapolatory than that”
- Alex Kontorovich
Kontorovich reports that the Lean formalization of Fermat's Last Theorem involved only a few humans who mostly told the AI to continue.
“I believe it was two or three humans. And I believe most of their involvement was keep going”
- Alex Kontorovich
Kontorovich concludes after the Fermat's Last Theorem formalization that autoformalization of human-written proofs into Lean is essentially solved.
“can you get a sorry free proof of something that human beings have written down on paper? I think that problem is”
- Alex Kontorovich
Kontorovich says the method behind the AI's quasi-RH proof fundamentally cannot reach the critical line, so the full Riemann hypothesis needs a new idea.
“that method sort of fundamentally can't get down to one half. So a new idea is needed”
- Ankur Moitra
Moitra worries that mathematicians may become more secretive with partial ideas because AI could rapidly exploit them.
“maybe people become a little bit more secretive with those ideas than they were in the past”
- Steven Brunton
Brunton says the Navier-Stokes blow-up result leaves practical fluid engineering unchanged, contrasting a million-dollar math problem with a trillion-dollar engineering problem.
“you know, 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”
Listen from 1:57:21 · Navier–Stokes existence and smoothness
- Ankur Moitra
Moitra admits he had at times doubted the unique games conjecture because its consequences seemed too strong.
“I did find myself on the side of the doubters because unique games just sounded too nice to be true”
- Ankur Moitra
Moitra argues deep learning succeeds on worst-case-hard problems because natural data has evolved structure that makes them easy.
“I guess my version of it is that in some sense, nature abhors computationally hard problems”
- Steven Brunton
Brunton argues that AI is ideal for low-stakes, verifiable tasks but that society needs a firewall for high-stakes decisions.
“we need a firewall of what needs to actually be verified”
- Ankur Moitra
Moitra bets that many practical instances of worst-case-hard problems are secretly easy, while hard instances for cryptography still exist.
“But it could be that in practice, a lot of the problems that we model in the worst case are secretly easy”
- Alex Kontorovich
Kontorovich says mathematicians already questioned this summer whether the Fields Medal might no longer be awarded, which he hopes is not the case.
“Already this summer, we were thinking, is this the last time that a Fields Medal is awarded? I sure hope not”
- Alex Kontorovich
Kontorovich argues, citing calculators and times tables, that students should master skills even when machines can perform them.
“just because a calculator can do it, you're going to be much better off if you can also do it yourself”
- Alex Kontorovich
Kontorovich reports that undergraduate grades have become bimodal, split between students who use AI to learn and those who use it to get answers.
“What we're seeing at the undergraduate level is a bimodal distribution”
- Steven Brunton
Brunton warns that greater intelligence in AI systems does not imply greater morality.
“Like super intelligence does not imply super morality. And I think that's a huge concern with AI systems”
- Ankur Moitra
Moitra advises young people to act as if they had already succeeded and pursue what they genuinely value.
“Pretend you already have tenure. What would you now do?”
People
In the conversation
- Lex FridmanAmerican podcaster and AI researcher; same host, second diarization label covering the interview segments
- Alex KontorovichNumber theorist at Rutgers University working on the Riemann hypothesis and the Lean proof assistant
- Ankur MoitraMIT mathematician and theoretical computer scientist, visiting the Institute for Advanced Study
- Steven BruntonUniversity of Washington mathematician and fluid dynamics expert
Discussed
- Terence TaoMathematician cited as critical of the effects of AI bulk-solving on mathematics
- Andrew WilesMathematician who proved Fermat's Last Theorem
- Grigori PerelmanRussian mathematician who solved the Poincaré conjecture
- Kevin BuzzardMathematician leading a funded project to formalize Fermat's Last Theorem in Lean
- Subhash KhotTheoretical computer scientist who introduced the unique games conjecture
- Dor MinzerMIT theoretical computer scientist who co-proved the two-to-one games theorem
- Irit DinurTheoretical computer scientist who co-proved the two-to-one games theorem
- Tim GowersFields Medal-winning mathematician who described AI results as a semigroup of known moves
- Richard FeynmanPhysicist in an anecdote about quitting the National Academy of Sciences
- John von NeumannMathematician whose essay 'Can We Survive Technology?' Leahy cited
- Bernhard RiemannMathematician who formulated the Riemann hypothesis
- Aleksander MądryFormer MIT colleague of Moitra known for work on adversarial examples
- Virginia Vassilevska WilliamsMIT computer scientist who wrote up an Anthropic result in fine-grained complexity
- Sanjeev AroraTheoretical computer scientist who advised Moitra to act as though he already had tenure
- Max TegmarkMIT physicist and AI safety researcher who spoke on Sanders' panel
Topics
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From the publisher
Ankur Moitra, Steven Brunton and Alex Kontorovich are three mathematicians who can speak to this incredible moment in the history of mathematics, when AI solved some of its hardest problems: Navier-Stokes, the quasi-Riemann hypothesis, the unique games conjecture and hundreds of other problems. Ankur Moitra is a mathematician at MIT (visiting the Institute for Advanced Study in Princeton) working in theoretical computer science, who can speak to the unique games conjecture, complexity theory and the mathematics of how AI learns. Steven Brunton is a mathematician and fluid dynamics expert at UW, who can speak to Navier-Stokes, turbulence and machine learning for fluid dynamics. Alex Kontorovich is a mathematician at Rutgers specializing in number theory, who can speak to the Riemann hypothesis, Lean and machine-checked proofs. Thank you for listening ❤ Check out our sponsors: https://lexfridman.com/sponsors/ep503-sc See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc. Transcript: https://lexfridman.com/math-ai-crisis-transcript CONTACT LEX: Feedback – give feedback to Lex: https://lexfridman.com/survey AMA – submit questions, videos or call-in: https://lexfridman.com/ama Hiring – join our team: https://lexfridman.com/hiring Other – other ways to get in touch: https://lexfridman.com/contact EPISODE LINKS: Ankur’s Website: https://people.csail.mit.edu/moitra/ Alex’s X: https://x.com/AlexKontorovich Alex’s Website: https://sites.math.rutgers.edu/~alexk/ Steven’s X: https://x.com/eigensteve Steven’s YouTube: https://youtube.com/@Eigensteve Steven’s Website: https://eigensteve.com Lean: https://lean-lang.org/ SPONSORS: To support this podcast, check out our sponsors & get discounts: Wispr Flow: AI-powered voice dictation app. Go to https://wisprflow.ai/lex Fin: AI agent for customer service. Go to https://fin.ai/lex LMNT: Zero-sugar electrolyte drink mix. Go to https://drinkLMNT.com/lex Blitzy: AI agent for large enterprise code…
Summary, chapters and key points were generated from the episode transcript on November 8, 2026. Quotes are checked word for word against the transcript.