This episode of Lex Fridman was divided into 13 topics. There were three guests:
Steven Brunton, who studies fluid dynamics at the University of Washington
Ankur Moitra, who works on theoretical computer science at MIT
Alex Kontorovich, who studies number theory and Lean at Rutgers University.
The video discusses the Navier–Stokes equations, the Riemann hypothesis, and computational complexity theory, prompted by AI’s claims over the past few weeks of having successively solved long-standing open problems in these fields.
Three mathematicians, drawing on their respective fields, have dissected several questions: How significant are these results, and is AI creating new ideas or merely reorganizing existing methods? Once “problem-solving” can be handed over to machines, what, then, remains for mathematicians…?
I watched the video and felt quite discouraged—the breakthrough is real, but the boundaries remain uneven. Moitra noted that AI was reportedly tested on thousands of problems, with a success rate of only about 10%, and what got released were all the successful cases. “Solving math problems” and “understanding math” are becoming decoupled. The doorman fallacy reminds us that human value lies in posing questions, inventing concepts, and telling compelling stories.
And the Navier–Stokes equation’s label as a “million-dollar math problem and a trillion-dollar engineering challenge” best illustrates that mathematical breakthroughs do not necessarily translate into solutions for real-world problems. An amusing anecdote: Kontorovich had just said on Wednesday that it would take “at least three years,” only to be proven wrong by 13 million lines of Lean code by Friday. The reversal came so quickly that the week hadn’t even ended yet.
The lesson here is not that experts are untrustworthy, but that right now, everyone is updating their own models in real time. In the face of uncertainty, stay curious, master genuine skills, and cherish human connections.
As soon as Lex spoke, he labeled this episode an “emergency broadcast.” In just a few weeks, the machine claimed to have solved the Navier–Stokes equations, the Riemann hypothesis, the unique game conjecture, and hundreds of other problems. He invited three mathematicians, jokingly calling it a “mini multi-agent swarm composed of human brains,” to jointly interpret what had just transpired.
Press and hold the screen to move the small window.
AI-Induced Mathematical Crisis: From the Navier–Stokes Equations to the Riemann Hypothesis I. Introduction
Lex started by apologizing, saying it wasn’t very professional to ask about feelings right off the bat, and then passed the question on to Alex.
Kontorovich: I’m reluctant to call it a “crisis”; crises often bring many opportunities. For thousands of years, we’ve been like people walking through a forest with a single candle, able to see only a small patch right in front of us. Now, we finally have a flashlight in our hands.
Lex didn’t let him downplay it, pulling up his reaction on Twitter: “If a human had done this, they’d have won the Fields Medal on the spot.” He even added a remark—original text likely included an exclamation like “holy shit.”
Moitra’s feelings are somewhat more complicated. He said that the announcement of the results that night wasn’t entirely unexpected, as rumors had been circulating within the community for weeks that the two biggest open problems in his field—the Unique Games Conjecture and RL = L—were both addressed in that set of papers.
Moitra: AI is like a world-class expert who’s always on call, explaining other fields to me in my own language—truly remarkable. Yet I’m increasingly unsure what kinds of problems to assign to my students and postdocs to help them gain recognition. I’m excited, but intellectually, I also feel lonelier than before.
Immediately afterward, Lex read a passage by Terence Tao: “The years of patience and deliberate, painstaking research undertaken by graduate students and postdocs were so easily disrupted by a single public release,” he said, adding that he felt “appalled” by this.
It seems that Brunton is the most optimistic of the three. He said, “I’m an engineer; my focus is on artificial hearts and aircraft. Our ability to solve real-world problems is far greater. As for the meaning of life, it never lies in fame or winning a Fields Medal—it resides instead in curiosity, helping others, and striving to understand the world.”
II. Will AI “solve” mathematics, or will it “kill” mathematics?
Lex asked everyone to present the strongest arguments for each of the two possible futures. Kontorovich first broke down “solving mathematics” and “destroying mathematics” into two separate issues, then recounted the story of chess.
Kontorovich: In the early years after chess engines first emerged, human–machine cooperative play was popular. Later, as the engines became too strong, all humans could do was “accept the first move, accept the first move,” which lost its appeal, and now no one plays that way anymore.
He and his colleagues “axiomatized” this argument, proposing a “solved axiom”: any problem that can be solved through human–AI collaboration can also be solved by AI alone.
However, he later clarified that he does not believe we have reached that point: “If I can prove a theorem but still don’t understand why the proof works, then it’s far from over.”
Moitra cautions against fixating on a single objective function. He recounts that when he lived in New York, he often chatted with the doorman in his building. If you reduce the doorman’s job to “opening doors,” you might assume that installing an automatic door could replace him—this is known as the “doorman fallacy.”
Mathematicians are no different: proving conjectures is merely a byproduct of deep thinking, and we are still developing new theories and inventing new concepts.
Lex immediately used Perplexity to find out what a doorman actually does and came across the entry “highlighting brand value.” Moitra chimed in with a smile: “Then I’ve got way too much in common with doormen—I don’t highlight brand value, so I honestly wouldn’t even know what I’m doing.”
Kontorovich: Until five seconds ago, “being able to prove profound new theorems” was almost entirely tied to what we truly sought—“beauty, understanding, and depth.” Now, with the push of a button, theorems can be generated, and those two concepts have become decoupled.
Brunton: Going a step further than answering questions is asking the right questions.
Moitra: I completely agree. I’d just like to add one point: asking the right questions itself requires deep mathematical insight—don’t underestimate that.
3. Does it truly bring new ideas, or is it just going in circles?
Lex asks: Is AI just stringing together the threads from thousands of papers?
Kontorovich: I haven’t read all 700 papers. Take the quasi-Riemann hypothesis as an example—every step in its proof employs techniques already known to humanity, yet the long chain of reasoning is truly astonishing: “Wow, it’s so beautiful—how did you come up with this?”
He used an analogy: think of all the ideas humans have already come up with as a balloon, with the interior of the balloon representing the “convex hull”—that is, the various combinations of known methods.
Some people say that all of AI’s achievements are just bubbles, and he thinks that’s not entirely wrong. He also noted that as far back as twenty years ago, others were already exploring this direction; two years ago, breakthroughs were made again with the theta function and the cubic Gaussian sum. AI is indeed following this path, “and many people are probably regretting it now.”
Moitra: The key to the balloon lies in its boundary. Calling it pejorative stems from the fact that the boundary appears unchanged. Yet merely knowing that a certain conclusion holds is enough to transform the entire field.
He gave the easiest-to-understand example.
This example isn’t among OpenAI’s achievements; it’s the result of Anthropic, compiled into a paper by its colleagues Virginia Vassilevska Williams and Josh Alman: the 3SUM problem, a frequent interview question, can actually be solved faster than everyone thought. In fine-grained complexity theory, there’s an entire set of implications along the lines of “if it’s hard, then all such problems are hard,” which has suddenly undermined the foundational assumptions.
Kontorovich naturally envisioned a scenario: if one of those 700 results were “polynomial-time factorization of large numbers,” credit cards would become unusable, and e-commerce and encrypted communications would all be doomed.
Brunton takes a more positive stance: even if AI is merely interpolating within the bounds of human knowledge, its ability to cut through the fog and equip mathematics with a kind of radar—enabling it to see what lies beyond the horizon—is already remarkable and should not be dismissed.
Kontorovich still insists on one point: humans occasionally introduce entirely new definitions, which feels like the emergence of a new boundary point—quite different from simply recombining old ideas.
IV. Journals, arXiv, and those 13 million lines of Lean
Lex drew an analogy to the open-source community: maintainers have been overwhelmed by a tidal wave of merge requests. Kontorovich said that even before AI, mathematical journals were already overburdened.
Kontorovich: Our approach to seeking truth has always been deliberately slow. After Perelman posted his proof on arXiv, he traveled the world giving three-hour impromptu lectures—perhaps one hour outlining his own line of thought, with the remaining two hours devoted to others asking, “On page 247, Lemma 412—I don’t understand it.”
Moitra, for his part, argues, “Don’t waste a good crisis”: papers don’t have to be the authors’ final, exclusive drafts, and AI proofs can also be collaboratively verified by the community. When asked about “Lemma 57,” he admitted that he “doesn’t remember it at all”; what he really wanted to convey was just that key idea.
Next, the discussion turns to Lean, an “interactive theorem prover” that enables computers to verify each step of the logic line by line.
Kontorovich recounted one of the most dramatic stories of this episode.
Kontorovich: I gave a talk at Princeton’s Department of Mathematics, where I explained in detail Lean and Mathlib (Lean’s public math library). I also pointed out that as soon as AI ventures three or four steps beyond what’s already available in the library, things fall apart: it defines the same object for two distinct purposes, and the two sets of definitions simply cannot be reconciled.
Fermat’s Last Theorem is far from Mathlib; Kevin Buzzard, with a million-dollar grant, plans to spend five years on it. I think it will take at least another three years. The report is due on Wednesday.
On Friday, Anthropic released 13 million lines of code and, starting from the axioms, proved Fermat’s Last Theorem. Every sentence I mentioned in my report was incorrect.
Lex: How many humans are involved?
Kontorovich: As far as I know, there are two or three people whose main job is to keep saying “continue.”
That Friday happened to fall on Labor Day, so he took his child to the beach and stayed off his phone for a few days, “throwing his child into the sea to play.” He said he had already experienced a similar psychological shock once before, about a month ago.
His conclusion is: the problem of automated formalization has essentially been solved, but the code generated by AI is riddled with ad-hoc hacks and still falls far short of the “standardized” quality seen in Mathlib.
Moitra concluded by emphasizing a point he holds dear: mathematics is not “a tide that rises higher and higher with each theorem.” When grappling with a good problem, the problem, in turn, challenges you—and it is precisely through this process that you learn.
V. The Riemann Hypothesis: From Whiskey to 7/8
Kontorovich: Riemann discovered something truly astonishing: a function whose inputs and outputs are both complex numbers actually governs the distribution of prime numbers.
You can think of it this way: the microphone breaks my voice down into a string of numbers, and the speaker reconstructs it. That’s exactly what the zeta function does with prime numbers.
Lex asked him what he was doing at the moment he saw the results.
Kontorovich: At a dinner at the National Museum of Mathematics in New York, I was holding a whiskey in one hand, chatting with people, and constantly refreshing the page, because I knew the announcement would be released around 6 p.m. Eastern Time.
At the time, I was chatting casually with a journalist and guessed that around three to four hundred results would be returned; later, I learned there were 700. Then I saw it: the “almost Riemann hypothesis”? I simply couldn’t process that information.
He then explained the significance of this achievement: the Riemann Hypothesis states that all non-trivial zeros of the zeta function lie on the critical line where the real part is 1/2.
In the past, we only knew that there was a progressively thinner zero‑free strip along the line where the real part is 1, and it hadn’t changed in seventy years. At most, he had expected this strip to become slightly thicker.
However, the AI directly provided an entire infinite vertical strip: there are no zeros in the real part between 7/8 and 1. Another proof reaches approximately 11/12.
Kontorovich: But this method essentially cannot reach 1/2. To prove the full Riemann Hypothesis, new ideas are still needed.
Moitra chimed in with a short anecdote: In the 1950s, a mathematician wrote a play in which John von Neumann, after his death, went to heaven and asked God whether the Riemann Hypothesis was true. God replied that all the non-trivial zeros lie on the critical line at 1/2. Unsatisfied, von Neumann wanted to know why.
Kontorovich: Mathematics has never been about facts; it’s about “why.” Simply telling me that a conclusion is true contributes nothing to my understanding.
Brunton says the allure of prime numbers lies in the fact that even a three-year-old can understand them.
Kontorovich immediately recounted his own son: when he was two or three years old, playing with 20 building blocks, he could arrange them into a 4×5 rectangle or a 10×2 rectangle. Later, one block rolled under the sofa, leaving only 19 pieces, and no matter how he tried, he couldn’t form a rectangle—so frustrated that he was beside himself.
Brunton added, "It still bothers me to this day."
VI. Navier-Stokes: Vortices in Figure Skating
Lex brought up the story Brunton had told about the “soup” that made him fall in love with fluid dynamics, and then asked him why the Navier–Stokes equations are so difficult.
Brunton: Ultimately, it’s the fluid‑based equivalent of F=ma—namely, conservation of momentum. The equation for a simple pendulum is easy to write, but adding another pendulum makes it much more complicated, whereas a fluid consists of an infinite number of pendulums. While the equations are straightforward to formulate, truly understanding how fluids behave is extremely challenging.
Kontorovich posed a physicist’s question: General relativity’s equations develop singularities, which were later shown to correspond to real black holes. So, if the Navier–Stokes equations break down, will that reveal some new property of water, or merely indicate that the equations fail at the atomic scale? Brunton offered an “engineer’s answer”: these equations are excellent models at a certain scale.
Brunton: The simplest example of a blowup is dx/dt = x²; as long as there’s quadratic feedback, the solution will shoot to infinity in finite time. The issue lies in viscosity—internal friction akin to that of honey—and whether it can always dampen the system before blowup occurs.
This time, it has been proven for the version of the Clay problem that includes a smooth external force.
Brunton: A carefully designed external force counteracts certain terms in the equation, forming a vortex whose radius shrinks ever smaller—much like a figure skater pulling in their arms to spin faster—and the rate of contraction is just right, ensuring that the energy remains finite while the velocity approaches infinity. This can only occur in three dimensions.
Lex asked what this means for engineering.
Brunton: Everyday fluid engineering has hardly changed at all; fluids remain just as difficult to simulate as they were one or two months ago. I like to quip that we’ve solved a million-dollar math problem, but there’s still a trillion-dollar engineering challenge: how to predict flow quickly and accurately, and then use that insight to design shapes.
He said the real big mystery is turbulence: large vortices, medium-sized vortices, and small vortices, each scale having its own influence—just like those flowing waters depicted by Leonardo da Vinci.
Gases can be fully described by a few parameters—temperature, pressure, and density—but turbulence lacks such a set of numbers. Lex asks: Will the answer come from mathematics, or should we let AI observe fluids for a while?
Brunton: Both are necessary. Before every major breakthrough in physics, there is first a coordinate transformation. Using massive amounts of data to find the simplest coordinate system is perfectly suited for AI.
VII. How Does Nature “Compute”?
Moitra took over the conversation: Which computational problems can nature solve natively?
He quoted Kamal Jain: “My laptop can’t compute it, and neither can the market.” The Nash equilibrium is notoriously difficult to calculate, which raises doubts about whether it truly explains how people actually engage in strategic interactions.
Brunton cited insects as an example: they don’t hold PhDs in fluid dynamics, nor do they run supercomputers, yet they can fly steadily even in gusty winds and rain.
When the wings are bent by the wind, neurons in the shoulder muscles perform the necessary calculations and make corrections on the spot—faster than the time it takes for signals to travel to the brain and back.
Moitra is a baseball fan, and he recalls a paper: outfielders don’t run in a straight line toward the ball’s landing spot; instead, they follow a slightly curved path, adjusting their course as they run based on the ball’s trajectory.
Moitra: Solutions found naturally tend to be very robust. The real challenge isn’t the optimization itself, but rather formulating the problem correctly in the first place.
Before taking a break, Lex tossed out what he called a “fleeting idea”: perhaps the Navier–Stokes singularities only come into play when traversing two connected black holes. He joked that, just in case, it might turn out to be the only time he was ever right.
8. The Unique Game Conjecture: The author says he will not read it.
Lex asked Moitra to explain the Unique Game Conjecture.
Moitra: Let’s start with SAT: given a set of “or” clauses, such as “x1 or not x3 or x7,” find an assignment of true or false values that makes all of them true. If someone gives you the answer, it’s easy to verify; but finding it yourself can be quite difficult.
He then discussed the PCP theorem, which made the front page of The New York Times in the 1990s: the theorem shows that a proof can be written in a specific format, and a verifier, by randomly sampling just a few clauses, can with very high probability detect a dishonest prover. The Unique Games Conjecture is an even stronger version; if it holds, it would settle the question of “how well” a wide range of problems can be approximated.
He recalled how opinions in the 2010s swung back and forth until colleagues such as DorMinzer and IritDinur proved the weaker 2-to-1 version, “sending shockwaves through the entire field.”
Moitra: To be honest, I used to be on the skeptics’ side, because it sounded too good to be true.
The most remarkable thing was a coincidence: on the day of the announcement, while in New York, he stopped by NYU to visit Subhash Khot, who had proposed this conjecture, and casually asked, “If they released a positive proof, would you read it?” Khot replied, “No.”
Moitra worries that the human‑centric approach, which once held promise, will be abandoned, leaving people disheartened. That evening, at the Museum of Mathematics, an exhibition of Escher’s works was opening, and he kept recalling that the original motivation for doing mathematics was simply to have fun—to explore and understand the world.
Moitra: Young people should ask themselves: Is your curiosity greater than your intelligence, or is your intelligence greater than your curiosity? Curiosity must come first.
9. How do mathematicians think, and is mathematics an invention or a discovery?
They talk about how Olympic champions often fall into a slump after winning gold. Thirty years ago, there was a documentary about Wiles—the man who proved Fermat’s Last Theorem—showing him sitting at his desk, recalling the moment he had his epiphany. He said he would never experience such a moment again in his life, and as he spoke, his voice choked up until he could go no further.
Kontorovich: Can you imagine getting an answer with just the click of a button? Absolutely not.
Moitra described his approach to mentoring students: he deliberately poses questions that lie “ten to twenty degrees” outside his own field, encouraging students to read key papers and then explain them to him. Through this process, he helps them identify their own perspectives and uncover new research questions.
Now AI can also explain a field clearly in its own language. “It’s a bit sad to lose that human element,” he added. He also noted that students, overwhelmed by the AI wave and concerns about whether there will still be jobs in the future, dare only think about matters within the next three to six months.
Brunton: I’ll stick to a single engineering problem, but I also make sure to leave myself the freedom to glance around. The chaos and complexity of adjacent fields like biology and materials are often where inspiration comes from.
When asked whether “mathematics is invention or discovery,” Brunton replied, “It’s both”: we create formal tools, and the fact that they work so remarkably well is itself a miracle. He says he still struggles with the question of whether integers really exist.
Kontorovich went on to add: It took humanity thousands of years to discover zero; Western mathematicians only accepted negative numbers after they had already embraced complex numbers; calculus forced humanity to confront infinity for the first time, and it wasn’t until the 19th century—when Cauchy and Fourier proved mutually contradictory theorems—that “infinity” was finally compelled to become a specialized field. Many first-year graduate courses today are still shaped by that very crisis.
10. PvsNP: In the worst case, it’s like deciding on a marriage based on the worst possible outcome.
Moitra: PvsNP is probably the hardest to shake. We can’t even prove that SAT requires more than “reading the input once,” despite widespread belief that it needs exponential time.
Lex offered an analogy: in complexity theory, “hardness” refers to the worst-case scenario—much like basing your decision about whether to marry someone on “what’s the worst that could happen.” In the context of interpersonal relationships, that’s probably not a good approach.
Moitra used the secretary problem to explain “online algorithms”: candidates arrive one by one, and you must decide on the spot whether to hire them, with the goal of selecting the best candidate possible. He also cited the example of computer caches: laptops are constantly predicting which data will be accessed next. He believes that an important direction is to determine why real-world inputs do not correspond to the worst-case scenario—a “reverse problem” that requires coarse-graining the world to just the right level of abstraction.
Kontorovich added an example: Pick a large number at random; factoring it is easy—half of them are even. The real challenge lies in those special numbers that result from multiplying two large prime numbers.
11. Why LLMs Work Well: The Owl-Loving Model
Lex once again delves into why deep learning works. Moitra admits that no existing theoretical bound can explain LLMs; the key lies in the fact that natural data is “very friendly.”
Moitra: Protein folding is, in the worst case, a computationally intractable problem, yet nature performs it every day. My view is that evolution itself has selected for proteins that fold easily, just as our languages have evolved to make such problems easier.
Then, he described an experiment that sent chills down one’s spine:
Moitra: Give model A a system prompt that says, “You absolutely love owls and bring them up whenever you get the chance,” then have it output only a string of numbers. Then use those numbers to fine-tune model B; as a result, B starts mentioning owls in every conversation. This is called “subconscious learning.”
He explained using his low-rank model: concepts are linearly embedded as vectors within the model, and when the dimensionality is insufficient, these vectors are not fully orthogonal; thus, a small adjustment in one direction can subtly spill over into other directions.
He compared it to the Ouija board he used to play with as a child: each person’s hand merely gives a gentle nudge, and when they come together, the pointer spells out words.
Lex, as the host, brought the conversation back on track.
Our civilization is very fragile, and this hidden “owl” tendency must be understood.
Brunton presented a two-axis framework: one axis measures the trustworthiness of AI outputs, while the other assesses the cost of errors. “If AI produces an incorrect proof, no one will shut down the power because of it.” However, high-risk domains must have robust verification firewalls in place.
12. Problems that AI cannot yet solve, and Perelman
Lex asked again: What is the problem most unlikely to be solved in the next one or two years?
Moitra: I’m still betting on P vs NP.
Even if P=NP, if the algorithms are so slow as to be impractical, the impact would be limited. He quoted the opening line of “Anna Karenina”: “Happy families are all alike; every unhappy family is unhappy in its own way.”
His concern is that many problems are actually easier than we think, while difficult instances each have their own unique challenges.
Kontorovich: The quasi-Riemann hypothesis has already left me deeply impressed, but it is not the Riemann hypothesis.
Lex brought up Perelman again—the man who said, “The proof is correct; that’s all the recognition I need,” and declined the Fields Medal.
Kontorovich: I couldn’t have predicted at all what Perelman would do. I didn’t expect him to turn down the Fields Medal, nor did I expect him to decline the one-million-dollar Clay Prize.
He said the significance of the award lies in recognizing “what we consider important.” These days, he chats with AI about mathematics every day and is learning much faster than before. In the past, he couldn’t understand research papers and had to rely on others to help him skip the introduction and jump straight to Lemma 3; now, he can simply feed the paper to an LLM and ask it directly.
Lex’s concern is more everyday: When middle-school homework starts getting harder, why does the child still have to do it on their own? Kontorovich replies that calculators have been around for decades, yet we still teach multiplication tables.
Kontorovich: How happy is a child the first time they learn to tie their shoelaces? You might think it’s just a shoelace. No, it’s the greatest thing in the world.
Final advice for young people
Kontorovich: For PhD students in mathematics, my advice is simple: keep doing what you’ve always done—study the hardest, most abstract math your mind can handle. I’ve also noticed that some students hand their notes to an AI and say, “Test me,” and their grades improve dramatically; others say, “I’ll turn in the assignment in an hour—just give me the answers,” and that’s a whole different story.
Brunton: Superintelligence doesn’t mean super morality. Stay curious, and master the fundamentals like linear algebra and calculus—even if computers can compute faster than you.
Lex added: Curiosity is being stolen by the attention economy—the very engine that sparks your curiosity is also what keeps you endlessly scrolling through your phone.
It was Moitra’s turn to speak last, and he joked, “As the final speaker, I guess all the good ideas have already been taken.”
Moitra: When I was a freshman, I used a small notebook to map out my four-year course schedule, just so I could take photonics in my sophomore year—though I had no idea what photonics even was; I just thought it sounded cool.
In today’s society, starting in high school, you’re taught to optimize your AP courses and extracurricular activities, get into prestigious universities, skip classes for interviews at cutting-edge labs, and even invest in stocks… You win every step of the way—so what? Sanjeev Arora once told me, “The easiest way to secure a tenured position is to pretend you’ve already got it, and then do the things you’ve always wanted to do.”
Lex admitted that he was speaking with three “survivors” and summarized Moitra’s advice on confronting fear as follows: Imagine you’ve already won—what would that victorious person do now?
Finally, Brunton wraps things up: “We’ve spent roughly thirty years experimenting to see how much machines can do; now that machines can do so much, he hopes we’ll realize that we actually prefer working alongside people.”
Lex went on to say that when AI makes intelligence affordable, remember that intelligence and those puzzles are merely the icing on the cake; what truly matters is life itself—and the people in it.