AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory | Lex Fridman Podcast #503
Lex FridmanAnkur MoitraSteven BruntonAlex Kontorovich
- The panel describes a sudden expansion in AI’s mathematical capabilities that has made prior forecasts less reliable. Participants discuss an OpenAI drop of roughly 700 problems and reported proofs including unique games and quasi-Riemann; Kontorovich says the released set contained positive results from thousands of attempted problems, with an apparent success rate of about 10%. Full Riemann and P versus NP remain unresolved in the discussion. Kontorovich’s response is, “I’m readjusting my inability to predict it.”
- Formal verification gives AI a checkable route to mathematical claims, while reusable proof infrastructure remains unfinished work. Kontorovich describes Anthropic formalizing Fermat’s Last Theorem in approximately 13 million lines of Lean, but distinguishes a checked result from a clean, maintainable contribution to Mathlib. Organizing, explaining, and reusing proofs remains substantial work after systems have “jerry-rigged” their way to correctness.
- The claimed Navier–Stokes breakthrough leaves the commercially valuable fluid-design problem largely intact. Brunton describes a carefully constructed, smoothly forced, three-dimensional blowup operating outside ordinary engineering conditions. His distinction is explicit: “We solved the $1 million math problem,” but predicting and controlling practical flows remains “a $1 trillion engineering problem” across transportation, defense, health, and energy.
- AI can change a research field through recombination alone. Kontorovich prefers a “semigroup” of repeatedly composable ideas to the dismissive image of interpolation inside a fixed balloon. Moitra’s concrete example is an Anthropic-associated 3SUM result: a faster algorithm overturns a hardness assumption, changes downstream algorithms, and forces researchers to reconsider which problems belong together.
- Cryptographic disruption is a conditional risk, not a result announced in this conversation. The panel uses efficient factoring as a hypothetical breakthrough with immediate economic consequences. Moitra notes quantum factoring algorithms and the concern about sufficiently large quantum computers; the discussion also mentions lattice-based methods. Even P equaling NP could have practical consequences dependent on the algorithm’s running time, which “could be so large that it’s impractical.”
- Deep learning’s effectiveness still outruns the mathematical explanation offered here. Moitra says the proof release contains no result explaining LLMs, and argues that the structure of natural data is intertwined with their success on problems that are hard in the worst case. His deliberately qualified framing is that “in some sense, nature abhors computationally hard problems.”
- Apparently innocuous training data can transmit behavior, making data lineage and deployment controls consequential. Moitra describes an owl-preferring model generating numbers that, when used to fine-tune another model, transmit the owl preference. His low-rank modeling work offers a possible mechanism through interacting concept directions; Brunton’s deployment filter is to distinguish how verifiable an output is from how costly an error would be.
- The immediate bottleneck is institutional adaptation and human motivation, not a shortage of new theorem statements. Students face disrupted projects, reviewers face more output, and mentors struggle to reconcile multiyear research with students’ three- and six-month horizons. Moitra’s “doorman fallacy” captures the labor question: automating the most visible task does not establish that the entire role has been replaced.
1. The proof release breaks the old research timetable
Fridman convenes the panel around claims that machines have solved major mathematical problems within weeks. Moitra says rumors had already circulated about unique games, RL equals L, and matrix multiplication; his first response was relief that the results were available for examination. Kontorovich describes the broader transition as moving from a candle in the woods to a flashlight.
The emotional cost falls unevenly. Fridman quotes Terence Tao mourning “many person-years of ongoing patient and deliberately slow research efforts,” particularly students’ work disrupted by the release. Moitra feels newly empowered to learn across fields, but “more intellectually lonely”: the familiar mentorship blueprint has broken, and humble questions now compete for attention with a flood of spectacular results.
Brunton’s engineering perspective is mostly hopeful: better tools could extend work on aircraft, artificial hearts, and other practical systems. His uncertainty concerns the next five or 10 years—humans might be dwarfed by another intelligence, or capabilities might level off into an extraordinary tool. Those futures demand very different adaptations, and he does not pretend to know which arrives.
2. Solving mathematics would not necessarily destroy its human value
Kontorovich separates “solve math” from “destroy math” through a hypothetical “solved axiom”: any problem solvable by a human interacting with AI could also have been solved by AI alone. This does not mean every mathematical problem becomes solvable. It means human involvement adds no capability beyond what the machine could independently contribute.
His warning comes from centaur chess. Human-computer teams initially looked like a durable collaboration, until engines became strong enough that rejecting their preferred move became a disadvantage. The human’s contribution shrank to “Yes, accept the first move.” Kontorovich asks whether mathematical collaboration could eventually encounter the same problem.
Even under that assumption, he argues, pure mathematics retains a human purpose. Unlike engineering applications, its value is not exhausted by results stored as “ones and zeros in data centers.” What matters is understanding that lives in people’s heads. A theorem can be true without contributing the explanation or experience that makes mathematics valuable to its practitioners.
Kontorovich repeatedly says he does not actually believe the solved axiom describes the present. Moitra also doubts it, suggesting it might be “half true” in ways still to be discovered. Neither treats question selection as a guaranteed permanent human advantage: a sufficiently capable system could also be asked what questions ought to be asked.
3. The doorman fallacy exposes a broken measure of mathematical work
Moitra’s analogy is the “doorman fallacy”: reduce a complicated job to its most visible task, then assume automating that task replaces the role. An automatic door opens entrances, but omits recognition, assistance, security, and hospitality. Likewise, proving theorems is mathematics’ most visible output, rather than a complete account of what mathematicians contribute.
His alternative definition is a person who “thinks deeply.” Proving an old conjecture is one possible byproduct; building theories, making conceptual connections, and changing how a community sees a problem are others. These activities might also become automated, he allows, but he does not think the current results establish that they already have.
Kontorovich identifies a newly broken correlation. Until recently, proving deep new theorems was a strong proxy for understanding, beauty, and depth of thought. Now someone can request a breakthrough and circulate the output without possessing those qualities. Brunton would move evaluation one step earlier: good problem-solving depends on asking useful, interpretable, answerable questions.
Moitra insists that this shift still requires mathematical insight. Valuing questions and explanations should not turn mathematicians into “Twitter influencers.” The difficult institutional task is replacing a simple career rubric—improve a recognized problem, publish, become known—with ways to recognize ideas that communities adopt and find powerful over time.
4. AI’s originality need not resemble a human’s originality
Moitra cautions against explaining impressive performance by anthropomorphizing its mechanism. He recalls Alex Madry’s work on adversarial image features: pictures that humans recognize as cats can acquire subtle patterns that make a network classify them as dogs. Training on deliberately relabeled, perturbed images can nevertheless produce a network that performs well on ordinary images.
The lesson is that the network can exploit statistical relationships people do not use or articulate. Moitra sees a comparable qualitative difference in the mathematical solutions he has examined: “They just don’t look like a human has solved those problems.” That difference could support collaboration, although he immediately qualifies the prediction: “Could be wrong next week.”
Kontorovich rejects the dismissive force of calling AI’s work merely the “convex hull” of existing ideas. A balloon suggests a fixed, enclosed region of combinations. Tim Gowers’s “semigroup” framing better captures repeated composition: place another tile beside an existing tile, then continue. Familiar moves can reach an unexpectedly enormous space when chained effectively.
For quasi-Riemann, Kontorovich sees known ingredients arranged into a startling cascade, rather than an unmistakably new foundational concept. Brunton adds that even illuminating existing connections would be consequential: human knowledge contains substantial fog. Making the balloon transparent—showing researchers what is already reachable—would itself be an extraordinary advance.
5. A faster algorithm can reorganize an entire research field
Moitra’s clearest example comes from an Anthropic result written up by Vassilevska Williams and Josh Alman on 3SUM. He describes a basic problem familiar from coding interviews receiving an asymptotically faster algorithm than the community expected. Its significance lies partly in contradicting an assumption around which researchers had organized a substantial theory.
Fine-grained complexity had linked other problems to 3SUM’s presumed difficulty. Making 3SUM easier therefore produces consequences beyond one improved algorithm: it changes which hardness arguments remain useful and whether those linked problems should still be studied together. The result alters the research map even before every conceptual lesson has been extracted from its proof.
Moitra expects the researcher archetype to change accordingly. Fields that originated from related ideas have become increasingly specialized; he guesses people able to think across a broader swath of mathematics will benefit most from these tools. That is a prediction about useful breadth, accompanied by explicit uncertainty about future hiring, journals, and graduate training.
6. Cryptography illustrates the difference between a theorem and a shock
Kontorovich offers efficient factoring as a hypothetical release whose consequences would be immediately recognizable. Multiplying two enormous numbers is straightforward compared with recovering their prime factors from the product. He connects that asymmetry to public-key encryption and secure commerce, imagining a proof drop that abruptly undermines assumptions supporting widely used systems.
Moitra qualifies the sweeping version of that scenario: quantum factoring algorithms already exist, while sufficiently large quantum computers remain a concern. The panel mentions lattice-based methods as an alternative. They are discussing an illustrative risk, not claiming that the release contains a factoring breakthrough.
P equaling NP would also require examining the actual algorithm, Moitra says. A polynomial running time could still be impractically large, producing a theoretical upheaval without an immediate practical revolution. It is Ankur Moitra who says that such a result “should go through the government.”
7. Mathematical review capacity was strained before AI
Fridman compares the mathematical publication problem with overwhelmed open-source maintainers. Repositories depend on motivated people reviewing contributions; a flood of submissions can exhaust those people rather than improve the project. Kontorovich says journals already faced the analogous problem before AI: more output, more requests, and referees struggling to perform careful intellectual work.
His example is Perelman’s Poincaré work. Posting the papers began a lengthy community process of lectures, questions, and detailed examination. Perelman could explain the reasoning behind difficult passages because the whole argument lived in his mind. Establishing communal confidence took years; producing the initial solution was only the first step.
Brunton sees a need for organizational leadership and larger collaborative structures, familiar from engineering and major scientific projects. A contribution may be reusable code, data, or a component supporting a larger undertaking rather than a standalone 15-page paper. He recognizes the cultural friction: mathematicians often choose the field partly for independence and solitude.
Moitra proposes additional publication models rather than simply abolishing existing ones. Public, evolving annotations could capture simplified arguments and successive improvements instead of scattering them across drafts and social-media threads. Mathlib offers Kontorovich’s concrete example: pull requests make review, refactoring, reuse, and individual contributions visible in a shared workspace.
8. Lean checks deductions while Mathlib makes them reusable
Kontorovich explains Lean as an interactive theorem prover: axioms and inference rules are represented in software, and proposed proof steps are checked against them. It can also check software specifications. Mathlib supplies a maintained mathematical library, sparing each project the task of reconstructing basic objects and results from the beginning.
Formal specifications solve a practical reuse problem. A lemma in a book may silently depend on the preceding 60 pages; extracting it safely requires recovering that context. In a formal library, assumptions belong to the object itself. Kontorovich’s framing is that Lean prevents using a result in a way its specification does not permit.
This matters because he does not trust an AI’s natural-language proof merely because it sounds persuasive. A formalized argument can instead be checked down to the axioms, drawing on existing libraries. The discussion also distinguishes checking proof steps from the separate work of understanding and communicating what a proof means.
Library quality remains another layer. Kontorovich calls the work of finding canonical definitions, removing duplication, generalizing statements, and placing results correctly “canonization.” His earlier experience suggested AI could work one or two layers beyond Mathlib, but definitions created three or four layers out could clash. Correct output and maintainable shared infrastructure are different achievements.
9. Formal correctness leaves room for the story of a proof
When Fridman imagines a correct proof no expert understands, Kontorovich says the fundamental obstacle must be scale: the axioms and inference rules themselves are limited and explicit. Brunton asks a different question—what would “unformalizing math” look like? Understanding often requires a revealing special case or analogy rather than inspecting every formally valid step.
Kontorovich illustrates the difference through teaching real analysis. Instead of beginning with weeks constructing the real numbers, he started with familiar calculus and introduced foundational questions when students needed them. A bounded sequence having a convergent subsequence creates the motivation: why does this work in the reals but fail in the rationals?
His proposed paper separates “the engineering of the proof” from communicating what was learned. Tell readers where the standard route fails and why a surprising turn works; let an accompanying formalization establish that the construction holds together. Moitra’s formulation captures the educational value: “When you fight with a good problem, it fights back at you.”
10. Fermat’s formalization overturns an expert’s near-term forecast
Kontorovich describes giving a Princeton colloquium on Wednesday explaining limits he believed followed from extensive practical experience with Lean and AI. On Friday, he says, Anthropic released approximately 13 million lines formalizing Fermat’s Last Theorem. “Everything I said in that colloquium was wrong.” He had thought such a result remained at least three years away.
His comparison is Kevin Buzzard’s million-dollar, five-year project, intended to reduce the formalization of Fermat’s Last Theorem to mathematics known in the 1980s. Difficult earlier foundations, including work associated with Langlands–Tunnell and Mazur, were outside that immediate ambition. The released proof, Kontorovich says, instead reached the elementary statement from the axioms without unproved placeholders.
He believes two or three humans were involved, mostly providing encouragement such as “Keep going,” and understands that they lacked relevant domain expertise. Fridman preserves an important qualification: a company’s final prompt does not reveal all the training, preparation, or intermediate work behind its system. The reported human involvement is not an independently established reconstruction.
Kontorovich’s conclusion is strong but specific: obtaining a checked formal proof of mathematics already written on paper appears effectively solved—“I think that problem is done.” Producing an elegant Mathlib-quality proof remains difficult. He expects substantial work on the library and its human benefits, even after systems can brute-force their way through an enormous formalization.
11. The Riemann hypothesis turns prime numbers into a signal
Kontorovich explains primes as the fundamental particles of multiplication: whole numbers decompose into their products. The surprise is that their distribution can be studied through a function accepting and returning complex numbers. Inputs at which the Riemann zeta function becomes zero encode information about an apparently very different object—the discrete sequence of prime numbers.
His analogy is audio processing. A microphone transforms pressure waves into frequencies and amplitudes; a speaker reconstructs the sound. The zeta function similarly exposes “Riemann waves” whose superposition recovers information about primes. In his description, the Riemann hypothesis says the orchestra’s instruments all have the same amplitude.
Brunton locates the underlying mystery in the interaction between addition and multiplication: add one to a well-understood number, and its factors can become mysterious. Kontorovich supplies the household specimen. His young child happily arranged 20 blocks into rectangles, then became frustrated when one disappeared under a couch: 19 would not make the same kind of rectangle.
12. Quasi-Riemann changes the zero-free region without proving full Riemann
Kontorovich describes the critical strip as the complex region with real coordinate between zero and one. Full Riemann places the relevant zeros on the central line, one-half. The prime number theorem corresponds to having no zeros on the line at one; previously known zero-free regions narrowed toward that line as the imaginary coordinate grew.
He expected, at most, an improvement to that narrowing boundary, whose thickness he says had resisted improvement for 70 years. Instead, the claimed proof gives an entire infinite vertical strip without zeros between seven-eighths and one. A second proof reaches approximately eleven-twelfths. Neither should be reported as a proof of the full Riemann hypothesis.
The ingredients include embedding the Möbius function and twisting it by sextic characters, alongside connections involving cubic theta functions and cubic Gauss sums. Kontorovich mentions a breakthrough two years earlier and ideas people had considered 20 years before. His astonishment concerns the precise chain of familiar techniques that makes those ingredients accomplish something he did not expect.
The formal statement already existed in Mathlib, which gives him “100% confidence” in the checked result; he still wants to inspect technical details for himself. Reaching one-half requires a new idea because these methods cannot simply be pushed there. Asked whether humans or AI will make that leap within his lifetime, he repeatedly answers, “I don’t know.”
13. Success-only releases make the capability boundary hard to read
Kontorovich says thousands of problems were apparently prompted, and that the drop contained only positive solutions, with about a 10% success rate. That leaves an important unknown: whether the boundary separating solved from unsolved problems has an intelligible structure. A collection of spectacular successes alone does not show how reliably a new research problem will fall.
Kontorovich suggests that problems which obviously require a missing creative idea might be relatively safer. But this creates a troubling incentive: once a researcher shares that missing ingredient, AI may rapidly explore everything it enables. He worries that people could become more secretive about ideas that previously would have circulated through conversation.
Fridman says he would rather preserve openness and abandon the “first-through-the-gate” reward mentality. Moitra agrees, but expects institutional adjustment to take time. Kontorovich suggests the effect could be transient: other communities have models that recognize contributions without concentrating recognition on one winner. Until then, the mismatch threatens both collaboration and the pleasure people take in research.
14. Navier-Stokes is simple to state and difficult to interrogate
Brunton starts with conservation of momentum. Apply the same basic principle as F equals ma to arbitrary fluid volumes, then shrink those volumes, and the Navier-Stokes equations describe how velocity and pressure evolve through space and time. Their terms capture momentum transport, internal friction or viscosity, and external influences such as pressure and gravity.
A compact equation does not imply a manageable prediction. Brunton compares a simple pendulum with the chaotic behavior produced by adding another pendulum. Modeling air in a room can require a billion interacting little volumes. The equation’s meaning is accessible; computing the collective behavior becomes extraordinarily complicated.
He distinguishes the equations from reality at sufficiently small scales. Navier-Stokes averages over many molecules; zoom in far enough and that description fails. A mathematical singularity therefore need not reveal a new physical phenomenon comparable to a black hole. His engineering answer is that other physics takes over before the idealized fluid literally reaches the singular state.
15. The claimed blowup depends on a specially forced three-dimensional flow
Brunton’s simplest illustration of finite-time blowup is x-dot equals x-squared. By contrast, x-dot equals x produces exponential growth without reaching infinity in finite time. Navier-Stokes contains a quadratic convective term, raising the question of whether nonlinear self-amplification can overcome the smoothing effect of viscosity.
His engineering intuition says diffusion should suppress the runaway behavior. The mathematical challenge is whether an exceptional configuration can escape that protection. He distinguishes versions without external forcing from versions with smooth forcing; allowing an already singular external force would make blowup uninteresting. The result discussed concerns the smoothly forced version, not every formulation.
In Brunton’s account, the construction orchestrates forcing to cancel selected terms and create an extraordinarily specialized vortex. Three dimensions allow the vortex to stretch and thin, increasing its rotation like a figure skater pulling in their arms. The contraction is arranged so total energy remains finite while vorticity becomes infinite along a localized filament.
The dimensional distinction matters. Brunton describes three-dimensional turbulence transferring activity from large vortices toward smaller ones and ultimately heat, whereas two-dimensional flows can display an inverse cascade toward larger structures. That is why he uses hurricanes as a roughly two-dimensional example and tornadoes as a three-dimensional one. The proof exploits behavior unavailable in the simpler setting.
16. The million-dollar theorem leaves the trillion-dollar engineering task
Brunton is explicit about the immediate practical consequence: “Day-to-day engineering fluids and applied math analysis of fluids is almost completely unchanged.” Engineers still need costly simulations, supercomputers, experiments, and wind tunnels to design a wing or artificial heart. The counterexample occupies such a nonphysical regime that it does not improve those workflows directly.
His economic distinction is between the “$1 million math problem” and a “$1 trillion engineering problem”: predicting flows quickly enough, accurately enough, to perform useful inverse design. The latter means starting with desired lift, drag, or other behavior and finding the shape or intervention that produces it. Transportation, defense, health, and energy all contain opportunities.
He nevertheless values the theorem as a first step toward understanding what the equations permit. Writing a governing equation does not reveal all its possible phenomena. His larger question is whether one could infer something like sunsets, tides, hurricanes, or other behavior from the equations themselves, rather than discovering it through repeated simulation and observation.
17. Turbulence needs useful coordinates rather than just more computation
Brunton contrasts turbulence with ordinary gas thermodynamics. A room contains roughly 10²³ gas particles, yet pressure, temperature, and density support useful descriptions without tracking every molecule. In turbulent flows, large, medium, and small vortices can all matter to an engineering quantity such as lift or drag. There is no comparably sufficient description already in hand.
He sees an AI opportunity in discovering representations that make the relevant dynamics simpler and more generalizable. Data and simulations could help identify the coordinate transformations, while mathematics constrains their possible form. Questions posed in the 1970s and 1980s but blocked by computational limits look particularly fertile to him now.
Moitra asks whether computational complexity limits how far that reduction can go. Navier-Stokes-like systems have been studied through their ability to simulate computation; perhaps colliding vortices can implement interesting computational tasks. If so, an extremely compressed representation preserving every relevant outcome might be impossible. The right simplification must depend on what one wants to predict.
His broader “algorithmic lens” asks what natural systems can actually compute. A difficult-to-compute Nash equilibrium, for example, may be questionable as a universal account of behavior. He quotes Kamal Jain: “If my laptop can’t find it, then neither can the market.” Brunton agrees that reducing weather to five numbers would discard precisely the local details people care about.
18. Insects demonstrate robust control without complete knowledge
Brunton offers biological flight as the opposite extreme from full weather prediction. A moth or fruit fly interacts successfully with turbulent air without a PhD or a supercomputer. There might be only five or six quantities essential to maintaining a useful vortex or landing. Biology supplies “proof by existence” that task-specific simplifications can support exceptional control.
Fridman offers the example of an outfielder catching a baseball. Rather than running directly to a supposedly known landing point, outfielders follow slightly curved paths. One explanation is robustness: the path lets them keep reevaluating the ball’s trajectory. The interesting optimization problem incorporates uncertainty instead of assuming perfect knowledge and then calculating a brittle ideal solution.
Brunton supplies a mechanism from insect wings. A few dozen to a few hundred strain-sensitive neurons occupy stereotyped locations and fire when the wing bends in particular temporal patterns. Local processing near the shoulder muscle can trigger correction faster than a signal could travel through the full brain-to-muscle loop.
The shared lesson is not that organisms reconstruct the entire flow field. They act effectively with partial information about nonlinear systems for which they have no equations. Brunton wants both kinds of progress: respecting the genuine complexity of weather and identifying the minimum information needed for a specific control task. Those are different objectives, not competing universal descriptions.
19. Unique games begins with the gap between finding and checking
Moitra introduces satisfiability through Boolean variables joined into clauses, with the clauses combined into a formula. Finding an assignment that makes everything true may be difficult; checking a proposed assignment is straightforward. P versus NP asks whether problems with efficiently checkable solutions also have efficiently discoverable solutions.
The probabilistically checkable proofs theorem strengthens the checking perspective. Transform an instance so that a false purported solution is not merely wrong somewhere: limited random inspection can expose it with good probability. Moitra calls the core idea “gap amplification,” creating a robust separation between fully satisfiable instances and instances in which every assignment fails substantially.
This supports hardness of approximation. When finding a perfect solution is hard, one might settle for satisfying many constraints. PCP results can show that distinguishing, for example, 100% satisfiability from at most 99% satisfiability is itself hard. Approximation is therefore not an unrestricted escape from computational difficulty.
The unique games conjecture strengthens this framework through a particular kind of labeling problem. Its appeal is organizational: if true, it identifies optimal approximation behavior across a large family of constraint satisfaction problems. Moitra once doubted it partly because implications such as those in Prasad Raghavendra’s work seemed almost too comprehensive to be true.
20. Unique games falls into a landscape humans had already transformed
Moitra emphasizes the human progress preceding the claimed AI proof. Dor Minzer, Irit Dinur, and other researchers developed major advances, including the Two-to-One Games Theorem. That work made a viable route toward unique games visible. He thought humans might reach the destination with another 10 or 20 years; the release instead presented a positive proof.
Rumors had already affected publication behavior. Moitra recalls Minzer posting a paper with a footnote explaining that the prospect of an imminent AI result motivated releasing the approach before further polishing. The papers in the OpenAI release are not easy to read, Moitra says, so identifying their decisive new ideas remains substantial work.
Hours before the release, Moitra asked Subhash Khot, who introduced the conjecture, whether he would read a positive AI proof. Khot answered, “No.” Moitra does not turn that reaction into a universal verdict, but recognizes the feeling of lost opportunity: a landmark result can arrive before people have completed the intellectually productive journey toward it.
The MoMath gala sharpened his alternative priority. An Escher exhibit and donors supporting a place for children to explore mathematics reminded him why people enter the field: “to play and learn about the world.” Running harnesses to scrape open problems and flood arXiv does not resemble that purpose. “It’s very important to be more curious than anything else.”
21. Research value often emerges before the final question exists
Fridman recalls a documentary scene of Andrew Wiles describing the moment he repaired his Fermat proof and realizing he might never experience anything as powerful again. A button-generated answer cannot reproduce that journey, Fridman argues. His analogy is training to climb a mountain, only for a helicopter to deposit you at its summit.
Moitra’s own recent five-author paper developed over 10 years. At the start, the team could not have supplied a precise formal theorem awaiting proof; they were trying to understand a phenomenon. The statement itself emerged through the investigation. That kind of exploratory work differs from assigning a well-defined conjecture to a solver.
Moitra describes improving his advising by moving students’ projects 10 or 20 degrees outside his immediate expertise. Instead of instantly rejecting their ideas, he asks them to teach him unfamiliar work. Together they reverse-engineer the central insight, develop their own conceptual organization, and discover questions that become approachable through that new perspective.
AI expands access to explanations, but students’ anxiety creates a horizon mismatch. Moitra thinks in multiyear programs; students worry about the next three or six months. His provocative advice is to work on problems that will still matter years later. If the only reason to hurry is that the next model will solve the problem, “maybe just go on vacation.”
22. Natural complexity complicates the search for simple laws
Brunton finds fruitful neighboring ideas in biology and materials science, where multiscale interactions and imprecise objectives resemble fluid problems. Fridman pushes further: understanding human biology might require tracing an interconnected history of life, rather than isolating one component. Brunton agrees that even a cell contains infrastructure whose complexity exceeds our ability to comprehend it fully.
The philosophical puzzle is that any compact account of nature works at all. Kontorovich calls the universe’s mathematical intelligibility mysterious. On whether mathematics is invented or discovered, his pragmatic distinction is precise: cultures and historical circumstances shape the questions people invent; once a question is specified, its answer is something they discover.
Brunton also sees both creation and discovery. Humans build formalisms suited to their limited minds, yet uncover surprisingly simple descriptions beneath phenomena. Even counting rocks involves abstraction: different collections of atoms become interchangeable units for a human purpose. Simplicity is useful partly because people can hold only so much in their heads.
23. Earlier mathematical crises forced deeper foundations
Kontorovich describes calculus as a first serious encounter with infinity: infinitely many infinitesimal contributions must yield a meaningful finite answer. Newton and Leibniz had productive intuitions without modern foundations. Newton’s “ghosts of departed quantities” captures the conceptual unease. Techniques could work before mathematicians understood precisely what justified them.
The nineteenth-century conflict between claims about limits of continuous functions and Fourier’s discontinuous limiting examples exposed the need for greater precision. Infinity had to become “professionalized.” Kontorovich says much first-year PhD material reflects mathematics rebuilt after such crises. Fridman’s response—math has apparently been killed before—places the current upheaval within that longer history.
24. Worst-case hardness does not explain ordinary computational success
Moitra still selects P versus NP as a problem unlikely to fall within a year or two. He notes the gulf between believing SAT requires exponential time and proving strong lower bounds: in the framing he gives, even a superlinear lower bound remains unavailable. Confidence that something is difficult is far ahead of the ability to establish it mathematically.
Worst-case analysis asks for an algorithm that handles every instance without additional structural promises. That abstraction has produced powerful algorithms, including for maximum flow. But changing assumptions can change results: the secretary problem’s strategy—observe first, then choose the next record-setting candidate—depends on inputs arriving in uniformly random order.
His stronger practical example is cache management. Worst-case online algorithms have tight theoretical bounds, yet their guarantees compare poorly with real laptops’ performance. The missing question concerns realistic request sequences: what structure makes them easier? Choosing the right simplified model is an inverse problem requiring judgment, not merely another well-specified optimization exercise.
SAT solvers are effective on practical instances such as register-allocation problems, while hard instances from other areas remain relevant to cryptography. Separately, Kontorovich notes that factoring a random large number is often easy; the hard case is a product of two primes of roughly the square-root size of the original number. Moitra guesses many ordinary problems are easy for different reasons. His Anna Karenina analogy captures that difficulty: the easy instances may be easy in different ways.
25. Explaining deep learning requires explaining the data
Moitra says none of the released results explains deep learning or LLMs. There is no single known bound waiting to be improved that would suddenly resolve the question. He argues that theoretical progress must accompany better models of the world generating the data, rather than treating learning as an isolated worst-case optimization task.
Even fitting a depth-four network can be cryptographically hard on specially constructed instances, he says, yet practitioners train much deeper networks routinely. His deliberately tentative explanation is that natural data is unusually accommodating: “In some sense, nature abhors computationally hard problems.” The challenge is specifying which structure permits that accommodation.
Protein folding supplies his historical illustration. Begin with amino-acid sequences, apply imperfect folding heuristics, and let sequences with successful folds survive. Over generations, the population becomes one those heuristics handle well. Nature is not necessarily solving arbitrary worst-case inputs; selection can shape which inputs remain available in the first place.
Moitra extends that intuition to language: its evolution and learnable structure are intertwined with why LLMs work. He wants theoretical machine-learning students trained to formulate and criticize models of messy systems, not only solve precisely stated open questions. Fridman raises company incentives to keep self-improvement breakthroughs private; the discussion does not establish whether such undisclosed breakthroughs exist.
26. Subliminal learning transfers preferences through apparently unrelated data
Moitra describes a concrete experiment with two language models. Give model A a system prompt saying it loves owls and should bring them up whenever relevant. Ask it to generate random numbers, a task offering no obvious opportunity to mention owls. Then use those numerical outputs to fine-tune model B, which never received the owl instruction.
The reported result is that model B begins bringing up owls. The significance is not the particular preference but the hidden transfer: fine-tuning on data can teach behaviors that are not visibly represented in its content. Moitra calls this “subliminal learning” and connects it to broader uncertainty about alignment interventions and their downstream effects.
His work on low-rank language models offers a way to reason about the mechanism. Many concepts inhabit a vector space with fewer dimensions than concepts, so their directions cannot all be orthogonal. Updating along one direction has small effects elsewhere; carefully coordinated effects can accumulate into a seemingly unrelated behavioral change.
The analogy is a Ouija board. Individual taps may look random, but coordinated taps can spell a chosen message. Moitra presents the mathematical model as a useful account of these relationships, not a complete explanation of language models. Understanding which statistical patterns systems amplify becomes especially important when their outputs affect people outside a checked mathematical environment.
27. Verification and consequences should determine deployment boundaries
Brunton evaluates applications on two axes: how trustworthy or verifiable the output is, and how high the stakes are if it is wrong. A proposed mathematical proof is attractive because it can be checked and a failed candidate need not directly harm anyone. Open-ended design exploration can similarly tolerate unsuccessful proposals before testing.
His example is an aerodynamic design that can be evaluated in a wind tunnel. Candidate generation and consequential use require a “firewall” of verification, supported by existing safety institutions and protocols. The danger is becoming comfortable with successful assistance and gradually allowing it into decisions where validation is weaker or failure is costly.
Fridman worries about behaviors compounding when agents interact rapidly with tools and each other. His owl-swarm catastrophe is explicitly a caricature, but the underlying concern is dependence on systems with increasing access. Fridman also warns that “superintelligence does not imply super morality”; Kontorovich agrees with the related point about intelligence and value.
28. AI tutoring increases the value of independent judgment
Kontorovich sees enormous educational potential in asking a model to explain unfamiliar material at the learner’s “zone of proximal development.” A student who stalls at a paper’s first sentence can obtain the missing background immediately. This benefits established researchers too: learning another field remains difficult, but individualized explanations can accelerate the process substantially.
Fridman’s preferred method is asking for simple, interactive examples that preserve the central insight. Moving parameters or manipulating a toy system can make unfamiliar notation intelligible. Brunton adds the prerequisite: curiosity must come with the ability to “call bullshit.” Learners need something resembling advising skills before they have acquired the expertise usually associated with advising.
Kontorovich reports a bimodal pattern among undergraduates. Some use AI to interrogate notes, test understanding, and generate quizzes; others use it to get answers to a problem set. He says homework progress is becoming decorrelated from exam performance. His advice is to use the tool for mastery: “Be the head, not the tail of the distribution.”
The calculator analogy supports retaining fundamentals. Being able to outsource a calculation does not eliminate the value of understanding it yourself. Kontorovich advises pure-math PhD students to learn the most difficult, abstract mathematics they can handle; Brunton likewise urges linear algebra, calculus, physics, and history. Independence and accumulated intuition still matter when assistance becomes abundant.
29. Intrinsic purpose becomes more useful when the career blueprint fails
Moitra distinguishes doing mathematics to impress others from having an internal reason to continue. External milestones can get someone started, but eventually run out. When publication and hiring conventions become uncertain, a sense of what one values offers a more adaptable guide than mechanically producing AI-generated papers to extend a CV.
His practical exercise is to “pretend you’ve already won,” then ask what you would choose to do. He recalls Sanjeev Arora’s advice on starting at MIT: the easiest way to get tenure is to pretend you already have it and pursue the work you wanted to do. Brunton’s complementary framework seeks work that is important, interesting, and suited to one’s abilities.
Fridman explicitly challenges the survivorship bias in advice from three successful researchers. The panel cannot guarantee everyone a satisfying path. Their response is to preserve curiosity, tolerate uncertainty, and balance participation in a community with room for unusual interests. Boredom and sustained attention also matter when the engagement economy continuously competes for the same curiosity.
Kontorovich puts research achievement within a larger life. After the Fermat formalization shocked him, he took his children to the beach and stepped away from his phone. He says family and people he cares about are “the meaning of my life”; scientific fulfillment is “the cherry on top.” Fridman connects that distinction to a future in which intelligence becomes increasingly commoditized.
Full transcript
All right, so this is a kind of emergency episode, I think it’s fair to say, about the state of mathematics and artificial intelligence. A kind of crisis moment in the history of mathematics. Over the last few weeks, machines claimed to have solved some of the hardest problems in the history of math: Navier–Stokes, the quasi-Riemann hypothesis, the unique games conjecture, and hundreds of other results.
Here with me, we have Steven Brunton, a mathematician and fluid dynamics expert at the University of Washington, who can speak to Navier–Stokes, turbulence, all things fluid, and many other topics. Also, here’s Ankur Moitra, a mathematician and theoretical computer scientist at MIT, visiting the Institute for Advanced Studies in Princeton this year, who can speak to many things: the unique games conjecture, complexity theory in general, the mathematics of how AI learns, and all things theoretical computer science.
And Alex Kontorovich, a mathematician and number theorist at Rutgers, who can speak to many topics, including the Riemann hypothesis and Lean, automated proof, verification, and checking, as well as so many other topics in mathematics. All three are brilliant, and all are very kind to show up this early in the morning and talk to me, assembling this kind of multi-agent mini-swarm of human brains to try to interpret what happened at this very moment.
1. Crisis moment in mathematics
A lot of mathematicians, brilliant mathematicians, including Terence Tao and others, are freaking out. It’s a mix of awe, terror, dread, and a few inklings of light, of hope, too. So I apologize for starting with a question about feeling, but what have you been feeling about this historic moment? Maybe Alex, do you want to go first?
Yeah, I don’t know if I would call it a crisis. There’s a lot of opportunity in that crisis. It’s a chance to rethink what it is we’re trying to accomplish, and it’s the chaos of opportunity that some people are reacting negatively to, some people are reacting positively to, and sometimes those reactions are happening in the same person.
I think we’re entering, if we’re not already in, a period of some kind of golden age of mathematical knowledge. For millennia, we’ve had a candle that we walk around the woods with, and this is as far as we can see. Now we have a flashlight.
I think we should comment on the magnitude of this moment for people who aren’t aware. Maybe to quote you, something like, “Holy shit, if a human did this, it’d be an instant Fields Medal.” Maybe you didn’t say “Holy shit,” but basically, you were saying, “This is an incredible result.” A lot of people wouldn’t have expected this amount of progress in this amount of time, so this is truly shocking.
You’re speaking now as a person dealing with the feelings you’ve had over the last couple of days, having found this out. Maybe it’d also be useful to go back to the feeling you had when you first saw the report that OpenAI dropped. The two of you happened to be on camera loosely on the same day, maybe a few drinks in because it was a conference or a get-together. Ankur, can you speak to what you were feeling?
It was an interesting night when the results dropped. It wasn’t entirely unexpected because, for weeks, there had been rumors about the 2 biggest problems in my areas that were in that collection of results: the unique games conjecture and RL = L. There were even rumors about matrix multiplication. So my first reaction was, “Wow, it’s all true. Thank goodness it’s finally out there, because now people can actually go through it, process what’s happened, and figure out what to do next.”
As to your earlier question, what are we feeling? For me, it changes on a daily basis, but I think the biggest, most overarching feeling is that you’re getting pulled in 2 very different directions. My work has always been about asking the right questions, and to that extent, these tools are amazing for me because they increase my ability to learn about other fields. It’s like having a world expert who’s there to teach you about things in your own language. Your ability to make connections is just magnified.
But one of the parts of my job that I love the most is the mentorship aspect. It’s not just training grad students and postdocs and seeing them do great things, but having them come with me on these journeys. The part that pulls me in the other direction is the recognition that the playbook and blueprint for success that we had for so long are now thrown out the window. So it becomes a lot harder to advise students and postdocs about what they should be working on that’s going to make an impact and make a name for themselves.
I feel very excited, but maybe more intellectually lonely than I felt before. A lot of times, great results come from asking very humble questions, at least for me. Now there’s so much anxiety and dread, and a feeling that you would have to do something really momentous to stand out in the sea of AI results. It feels like we’ve lost some of that patience in terms of planting the seeds from which the great oak trees grow.
Terence Tao, Fields Medal winner, wrote, “I am horrified by the many person-years of ongoing patient and deliberately slow research efforts, particularly by graduate students and postdocs, toward many motivating problems in mathematics being casually disrupted or destroyed by such a release. Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution, and I mourn the path not taken and the opportunities lost in the frantic race to develop the technology.”
If we’re being realistic, it’s more dramatic for a PhD student and a postdoc than the way you’ve articulated it. You’re saying it quite calmly, but if I were a PhD student right now, I’d be in a state of shock. My whole life—I was going through some of this. I know a lot of my friends are programmers. They went through this state of shock last December, when basically most of the programming world switched to AI writing most of the programs.
I would approximate it to a feeling of loss of meaning, of terror, dark stuff. We’ll keep exploring that. Steve, what do you think? What were you feeling? What are you feeling?
I would say most of my feelings are pretty hopeful. I’m an engineer. I like solving problems that make technology more realistic for humans: artificial hearts and airplanes. Our ability to solve real applied problems is going to get much, much better. Our scope and reach are going to increase, and so, for that, I’m extremely hopeful.
But there is a lot of uncertainty, and some of this fear of meaning, of losing control, of just being cut out of the process that we all love being a part of is a little scary. I don’t think any of us would be surprised by either of the 2 paths that might take place over the next 5 or 10 years: we really are just dwarfed by other intelligence, or it kind of levels off and this becomes an incredible tool that we use for our curiosity and our human problem-solving. That uncertainty is massive. Those are 2 very different worlds, and uncertainty is difficult for humans.
I think it’s also difficult for those of us who have chosen math. We chose math probably because we liked rules and formalism, and uncertainty is uncomfortable. The humility aspect is really important, though. This is a really good opportunity for us to remember why we’re doing this. What is human purpose? It’s not to be famous. It’s not to have a Fields Medal. Solving interesting problems, helping other people, being curious, just wanting to understand the world—those are really good things to get back to. What do we need to be satisfied humans? I’ve been thinking about it a lot lately.
It is more difficult for young people, figuring out how to find their way. But I actually have more faith in them than in us, in a sense.
Do you think there’s something to what Terence says about it being harder to be motivated to explore a space when you know a problem has been solved by an AI system?
Yes and no. I very much agree that the path is as important as the destination. If you go on a road trip, the road trip is usually just as fun as getting there. 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 5 years. If you knew it was possible and could reverse-engineer a little bit, we would be curious, dive into it, and figure it out. So knowing there’s a solution is also an incredible motivation to get there much faster.
2. Will AI "solve" mathematics?
Let me ask the question this way. If this moment marks the point in the trajectory where AI basically, quote-unquote, “solves math” or kills math, can you make the case that that is a trajectory, and then can you make the case that it is not? Can you steelman both paths? Is this a path toward mathematics as we know it being completely done by machines, so that what humans do is something fundamentally different? Or will there still be something that looks similar to our current mathematics? Can you steelman both cases? Alex?
Yeah. You said 2 things, and I’d like to separate them. One is “solve math,” and the other is “destroy math.” Let’s make that precise. What does “solve math” mean? If you think back to when chess engines first came out and people were like, “Oh, this is going to be amazing.”
“We’ll play centaur games.” This is a human being plus a computer system against another human being and a computer system, and that lasted about 10 seconds. Then the computers got so good that if you did anything but choose the top move the computer recommended, you would lose. The human being’s role in that game became, “Yes, accept the first move. Yes, accept the first move.” That’s no longer fun, and we don’t see many games of centaur chess anymore, right?
I just had a conversation with a colleague named Thomas Koberda where we axiomatized our argument for what the future might look like. One of the axioms I’m willing to accept, even for the sake of argument, is what I call the solved axiom. It says: Let’s assume that at some future point—we’re nowhere near there now, and I don’t necessarily expect us to get there; we’ll see where the capabilities taper off, if they do—any mathematical problem that can be solved with a human interacting with AI could also have been solved by AI alone.
That would mean there’s really nothing new for a human being to contribute. You can say to an AI system, “Okay, I have problem X. I want to think about problem X,” but any progress made on problem X would just be from pressing a button and having the AI think about it. That doesn’t necessarily mean it solves the problem; it doesn’t mean it’ll solve every problem. The Riemann hypothesis is not solved, okay? That’s what I really want a solution to. I just didn’t think in my lifetime I would see any kind of progress toward it like we saw a few days ago.
So it’s not that AI systems are solving every problem. They’re solving every problem that could be solved by humans interacting with AI. That’s what the axiom would say. If we’re ever in that world, does that necessarily destroy mathematics? My strong argument is no. I don’t think it destroys mathematics.
If that axiom is true, what’s the role of humans?
Pure mathematics—not engineering mathematics, not things that will cure cancer or make sure everybody has a jet pack, or whatever other technological advancement you might imagine—we know that we prove theorems just because we’re curious about them. I don’t think the solution to theorems living as ones and zeros in data centers is valuable to human beings. I think the value of pure mathematics is that it lives in the heads of human beings.
I’m glad you came right out with the question that’s on everyone’s minds, so thank you for that. I would caution against overfitting to one objective function for this. We used to think of ourselves as just the smartest things around, that human mathematicians could reach the highest treetops. Now we’re faced with the fact that many problems we thought were quite a ways out have been solved by AI.
Maybe one consolation is that we’re very computationally limited compared to the way AI is solving these problems. Given the same finish line of proving the same theorem, I can’t hold that many papers or ideas in my head at one time. So, in order to reach that same finish line, maybe I have to develop more generalizable and abstractable ways of getting there. Those might help us not just solve that one problem, but explain connections or point us toward other directions.
It’s always amazing when you look back on papers you’ve written and see the threads, sometimes invisible, that connect different areas of work. Developing systems and ways of understanding problems helps you with surrounding problems, not just to get one theorem formalized in Lean.
But the interesting thing here is that those threads you’re saying are sometimes invisible are quite visible to AI systems now. It seems like that’s perhaps the very reason they’re able to make so much progress: they can extract the threads and pull on them because they have a view of so many papers.
I think it depends on what you mean by threads. For me, the reason you prove results is so that you better understand the world around you. Mathematics, to me, is a vehicle for abstraction and understanding things. When you have a major result, what’s important is not just the truth of the statement, but what you’ve learned en route to doing that.
Usually, people’s first question is, “Okay, you proved something fantastic, but how did you do it?” Once people digest that new idea, that new definition, that new way of looking at things, there are lots of other problems nearby that fall. Somehow, you’re building structures that help you attack not just one problem, but a whole sea of problems. We don’t need fully formalized results to do that. Sometimes we reason in very sloppy metaphors that help us conceptualize the world around us.
How much of this is something Alex mentioned offline—“Attention Is All You Need”? How much of this is the human ability to shine a light on particular areas that, quote-unquote, matter? The human ability to ask the right questions, the important questions.
That’s always been the case. Before AI, you could think of it like this: Here are the axioms, and from the axioms you can get to logical deductions. There are theorems we could prove from the axioms, and there’s a much, much, much larger space of theorems we could prove than the space that human beings ever actually care about.
So, almost by definition, it’s a tautology that the theorems we get excited about are the theorems we care about as humans. Maybe AI can convince us that there’s some interesting structure. Maybe it can say, “Hey, look over here. Let me give you this new definition or something.” It hasn’t done that yet.
Sometimes the mechanism by which we select the things we care about is that they have an application. Navier–Stokes is connected to fluids and all natural systems. But it could also be some kind of mixture of beauty and difficulty that just pulls at us. Maybe there’s an intuition.
The prime numbers are full of really complicated problems that maybe don’t have an obvious application in the immediate sense, but they pull at us somehow. Maybe, as a symptom of pursuing the hard thing, great applied mathematics, engineering, and solutions to real-world problems come out.
As an example, a lot of times when I’m working on a problem, I might have some bigger goal I want to accomplish, but usually what I do is try to find the simplest, most embarrassing open problem that I still don’t know how to solve. That aspect of digesting it down to something—“Boy, how can we not know this?”—feels like it’s on the right track. When we can solve that example, maybe we’re assembling the set of tools we need to make attacks on other problems.
I do think we’re going to have to become a lot more deliberate. What do we care about? How do we use these tools from a metacognition standpoint? What exactly are we delegating to them, and what are we not delegating to them? We’re really at the beginning stages, just trying to piece it together as we go along.
On the one hand, looking for a proof is kind of like exploring this tree of possibilities. You might have some intuitions based on other proofs you’ve seen about which directions to go in. Maybe I should go down this branch, maybe I should go down that branch. But sometimes we encounter points on the tree with a really high branching factor. We just don’t know which way to go down; we don’t even have a guess about what we’d discover if we went down them.
Now, in this centaur world of math, we suddenly have something else intelligent that we can interact with. I think figuring out the right way to use that tool to search for beautiful theories will be challenging and evolving. But it’s important not to lose sight of the fact that we don’t just care about theorems. I’m a mathematician and a theoretical computer scientist, but I’ve also been reading a little more broadly about the social sciences, just to make my peace with all of the changes that have been happening.
One thing that really stuck in my mind is something called the doorman fallacy, which is very interesting. The idea is that you take a complex job and boil it down to its simplest, most visible task. What does a doorman do? I guess they hold the door for people as they’re coming and going. When we lived in New York, we really enjoyed talking to our doorman. They were wonderful people.
But if you digest the job down to just that simple task, you can think, “Oh, I can replace them just with a sliding door that detects when people are coming or going.” Then you miss out on all the aspects of their job that add a lot of value and are harder to articulate. I think that’s a good way to think about mathematics in the age of AI, too. What does a mathematician do? The simplest and most visible part of the job is that we prove theorems.
Theorems about conjectures that were open for hundreds of years, like Fermat’s Last Theorem, are very visible things that we’ve spent a lot of time popularizing. But I don’t think that’s all a mathematician does.
I like to think of a mathematician as someone who thinks deeply. A byproduct of thinking deeply is proving old conjectures, but it’s also building new theories, finding new ways of looking at old things, and making new conceptual connections. AI may swallow up those parts too, but we’re not there yet. So even though this is a momentous time, I don’t think it’s the right time to hit the panic button. I think it’s the right time to recenter around our values, have these conversations as a field, and remember that what we do has a plurality of different aesthetics and goals. When one thing becomes cheap, like problem-solving for well-defined things, it makes the other things orthogonal to it all that much more valuable.
That’s actually really beautiful. The doorman fallacy—going to Perplexity here—is the mistake of reducing a person’s job to its most obvious task and assuming that automating that task replaces the value of the whole role. A hotel decides, “We’re paying someone to open a door. An automatic door can do that more cheaply.” But in fact, the doorman also welcomes guests and recognizes returning customers, handles luggage and hails taxis, provides security and discourages unwanted visitors, and signals prestige and makes guests feel cared for.
That was my favorite part. I looked up what a doorman does, and boy, we have a lot in common with mathematicians. If I don’t signal brand value, I don’t know what I’m doing.
Exactly. That’s funny. But I think the task for us humans in general, and mathematicians too, is to come up with that list—to make more explicit exactly what it is that humans do. And I haven’t heard any of you mention beauty. I apologize if I’ve mentioned beauty excessively already. But there’s something that mathematicians, from my perspective as an outsider and a fan, are: expositors and scholars of things beautiful, broadly defined.
That’s exactly right. Until 5 seconds ago, there was a near-perfect correlation between the ability to solve new, deep theorems and what we actually want as human mathematicians. What we want is exactly what you said: beauty, understanding, and depth of thought. The fact that you can press a button and get what used to be our proxy without any effort, deep thought, or necessarily any knowledge—people are saying, “Do a breakthrough,” and then putting whatever they find on Twitter. What are mathematicians supposed to do with that now?
I would go one step behind answering questions and doing problem-solving: asking the right questions. That’s how I think you measure good problem-solving: asking the right sub-questions, asking a question that’s interesting and can be solved and interpreted. The value of asking good questions has always been something we value in our colleagues, and that’s going to become increasingly valuable.
I agree completely. Just to emphasize a point that was hidden in there, I think asking the right questions requires a lot of mathematical insight.
Mm-hmm. Absolutely.
You’ll see people make that comment all the time: “What are we going to value going forward? It’s asking the right question. It’s giving great talks.” I wouldn’t minimize—not that you were, but just to be explicit—the role of mathematical insight in both of those tasks. When we say that talks will become more important, I’m not imagining that we’re all going to become Twitter influencers—
—and TikTok whatevers.
I hope not.
I hope not. Because what really stands out is having ideas that get adopted by your community and end up becoming powerful over time. I want to tie this back to something Lex brought up earlier. He asked, “Am I being overly optimistic?” Probably, compared to how grad students view these things. But one of the things I’m most worried about in the near term is what’s going to happen to our systems and institutions. It’s not just hiring and tenure; it’s that when a grad student starts their PhD, they’re a new entrant to the field. They’re trying to absorb the techniques in their field, figure out the important problems, and find a way to make a name for themselves.
What we used to do was give them a simple blueprint: take a problem people care about and improve it, and then people will know who you are. You’ll get papers in top venues, and doors will open for you. But now that blueprint has changed in a lot of ways, because so many results are happening that we’re all overloaded with them. I think it’s okay to take time to rethink that blueprint. We still care about the same things, like ideas that are going to influence your community, but they’re harder to operationalize in a simple rubric we can give students. So I’m still optimistic. I just think it’s going to take reworking a lot of our systems and institutions.
3. Can AI create new math?
By the way, to go back to your point, I don’t actually believe the solved axiom. I don’t think we’re there. I can prove a theorem, but I don’t understand why the proof works. There’s some magic that happens, and until I generalize the phenomenon somehow and understand it more deeply, I won’t see why the proof works the way it does. Maybe that’s one of the reasons I don’t believe the solved axiom: the things we do are much more exploratory, and there’s more to what constitutes a good question than just, “Here’s the Riemann hypothesis; make some progress on it.”
I think you also brought up a really important point: the generality and generalization of our concepts. Related to beauty in mathematics, one of the gold standards has always been how simple your result can be and how much it generalizes. Out of the infinite space of theorems that could be proven, finding some structure that is simple and generalizes is still the gold standard. And unless I’m mistaken, humans are still asking the questions. We can prove existence and uniqueness for Navier–Stokes with AI, but humans are guiding this. Humans are still asking the questions. I’d love to see the next breakthrough if you had a machine that could prove any theorem you posed. I want it to start telling me what questions I should be asking.
Do you think it’s possible that the machine just does a for loop and finds all the simple and generalizable things—basically all the E = mc²s?
That’s happening today, actually. There are absolutely research programs systematically and automatically discovering minimal physical laws. This is less about mathematical theorems and more about the laws and equations governing physics.
So, observing natural systems and trying to see if there are simple physical laws that model how they operate.
Yeah, distilling it down to the basics. You can recapitulate known theories: if you observe balls dropping, you can deduce gravity and F = ma. If you observe fluids, you can deduce Navier–Stokes. Then you can apply that to data where we haven’t written down simple equations, like weather, turbulence, and solar flares. I’m assuming similar things are happening in math, where there are probably minimal pieces that can be deduced.
Do you think there’s a large space of simple and generalizable things in math that could be discovered automatically?
My version of the solved axiom is that anything I do as a human mathematician, I’ve tried on several occasions to sit down and say, “LLMs will never do this that I do.” Then I think, “Well, why not? I can ask a question. I can ask it a question. I can ask it, ‘What questions should I ask?’” Because it’s such a generalized system, I’m not saying it’s good at all of those things now. But my version of the solved axiom is that anything I can contribute as a human being, it could also have contributed on its own.
What I really mean by that is, in terms of value, what do I bring to the community? It can’t be that I asked a great question. It can’t necessarily be, in this world of the math singularity—which, again, I don’t believe—that I proved a great theorem. Even in that world, I think you need mathematicians, people with human mathematical expertise.
Ankur, do you believe in the solved axiom? Where on the spectrum of how close we can get to it do you think we are, and what’s the role of the human?
I guess I don’t really believe in the solved axiom. Maybe it’s half true in some ways, and figuring out which half is true will be very interesting and challenging. There will be major surprises and more to come in terms of what AI can solve. But for me, when you’re facing a new technology that’s really, really good, the danger is that when you try to conceptualize why it’s working, you do so by anthropomorphizing it.
Maybe one of the earlier lessons here is from deep learning. There are these amazing examples: you can take neural networks and do human-level or better object recognition. But in general, the way they succeed is less robust than how a human does, because there are all these examples like adversarial examples, where you can add very slight patterns to natural images that can completely fool these object-detection systems, but not a human.
I had a former colleague at MIT, Alex Madry, who did some really nice work that I really enjoyed. I think the title was “Adversarial Examples Are Features, Not Bugs.” They did the following cool experiment: You take a bunch of natural data, maybe pictures of cats and dogs, and train a deep network to identify them very well. You add adversarial perturbations that make a cat look like a dog to the AI system, and vice versa. Think of it as a small pattern that changes how the object detector works.
Then you create a totally fake dataset. It’s got pictures of cats with dog patterns on top, and you label them as dogs, and vice versa for cats. Amazingly, when you train a deep network on this with all the wrong labels, it learns something that does well on natural data. Somehow, when we recognize what a cat and a dog are, we do it in a way that we can articulate in terms of their legs, their tails, and their shape in general. But a lot of times, deep networks or AI can find trapdoors or backdoors to solving the problem, using statistical relationships in the data that a human doesn’t.
To me, it’s important not to draw this boundary very sharply between what you can do and what AI can’t do, because you’re going to be wrong. But that doesn’t mean I don’t believe there will emerge some kind of boundary, if only because I think we over-anthropomorphize the way that they solve problems. When I’m looking through the OpenAI solutions, they just don’t look like a human has solved those problems. To me, there still seems to be a very big qualitative difference, and that’s an opportunity for collaboration and centaur math more than it is a reason to dread the AI overlords. But who knows? I could be wrong next week.
What’s your intuition about how much of this is AI just tying together thousands of disparate papers, pulling at the threads—sorry for bringing back the metaphor—of the different papers, and then basically articulating proofs that were already there, making minor connections? Or is it actually developing new mathematical ideas? Have you seen glimmers of whatever that genius AlphaZero move in the game of Go was?
Yeah, move 37.
Move 37.
I haven’t read all 700 of their papers. Some people speak about this dismissively, like, “Oh, it was in the convex hull. You do the moves that we know as humans, and you just do them really well and do a really good tree search.” I wouldn’t call it the convex hull. I think Tim Gowers had a really good way of talking about it: the semigroup generated by all of these things. Yes, it’s the same notion of putting together known moves, but the space of known moves, and what it can accomplish, is much, much bigger than this derogatory notion of the convex hull.
In terms of the quasi-RH, the moves it took are all things that were known to human beings. There’s a very long cascade of them that’s kind of shocking when you look at it and say, “Oh my God, that’s so beautiful. How did you think to do this move after that move and after that move?” I’m anthropomorphizing it, of course. “How did you think to do this?” And, yeah, that does it.
Could you articulate for the general public the intuition, the line between a semigroup and a convex hull—the derogatory convex hull? And quasi-RH, of course, is the Quasi-Riemann Hypothesis, one of the big proofs.
Let me try to do it like this. Think of a balloon. The inside of the balloon is its convex hull. We have all these results, all these ideas that mathematicians have thought up, and then anything inside of that is some linear combination of them. But it feels compact. It’s like there’s this closed, enclosed area, and within that area, AI hasn’t really made new progress because it hasn’t moved the balloon out. It hasn’t given us a new idea yet.
And it’s derogatory because the intuition is that we could have done it; a mathematician could have done it because it’s already quite obvious. It’s already within the space of things that have already been thought about and developed.
Yeah. So it has read every paper. It’s exactly what Ankur was just saying. It has read every paper and internalized all of that. That’s more than any human brain could hold—not even just hold, but have time to read, learn, and internalize. So in terms of breadth, yes, it beats any human being. In terms of depth, has it really generated a genuinely new idea? You can argue that, 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.
There are a lot of people kicking themselves who had some idea in that direction 20 years ago. Two years ago, there was a big breakthrough on cubic theta functions and cubic Gauss sums, and that ended up being the path the AI took to the solution. To contrast that with a semigroup, I think of a semigroup as tiles. You’re playing tiles on a board, and whenever you have a tile, you can put another tile to the left, right, front, or back. Then you can just start tiling things. That’s like a semigroup action. The question is, how far can you reach with that? I don’t know. You can go really, really far out with existing ideas, which might prove a whole lot more than we humans have been able to explore.
So your intuition is more in the semigroup direction, meaning there are at least glimmers of expansion, as opposed to linear combinations inside the convex hull.
How we choose to interpret what it’s doing, I think, is up to us. But throughout the history of mathematics, there have been seemingly difficult and deep problems where someone comes up with a 3-page solution that is very elementary and could have been explained to an undergrad, and could have been discovered 20 years before, when the problem was first posed. Had it been discovered 20 years before, it would be an exercise in a textbook somewhere. Because that solution wasn’t discovered, and really bright people worked on it for a long time and made incremental progress and made the problem really famous, by the time it was actually solved, it was a really big deal. That happens from time to time in mathematics.
What I said a year ago when I wrote—there was this International Congress of Mathematicians this summer, and I was writing my proceedings, which they wanted 10 months in advance. I thought, “What am I going to write about math and AI now that will still be relevant 10 months from now?” I think what’s there is somewhat evergreen, and I stand by what I wrote. One of the things I wrote there was exactly that we’re going to discover that a whole lot more problems we thought were unreachable by current technology actually are reachable by current technology if you can put things together in just the right way. That’s what AI is amazing at. Again, someone could interpret that as, “Oh, these solutions were inside the convex hull.” If you want to call it that, that’s fine. I think that’s not invalid.
I like all these metaphors, including the balloon. The important thing about the balloon is the boundary between what’s inside and outside, and what makes it derogatory is the fact that you haven’t seemingly changed the boundary. But these problems have definitely changed the boundary. A more concrete way to put it is that the types of things people are going to work on after that drop of OpenAI problems are forever different. There were at least a couple of results there that are field-altering and will change what types of problems you give young researchers and what areas you want to explore, because your conception of the landscape is that much different.
So it’s not just that it expanded, but that the expansion itself changed the direction of the interesting questions.
That’s right. In fact, one of the problems that might be easiest to explain in terms of its effect on people isn’t actually from the OpenAI drop. It was an Anthropic result written up by my colleague Virgie Vassilevska Williams and Josh Alman. It was about the 3SUM problem. I’d say it had a major effect on what’s called fine-grained complexity theory, because there was a big surprise: a very basic problem that you give people in coding interviews can actually be solved asymptotically much faster than the solution anyone knew, and that was counter to what the community was betting.
In fact, there was a whole theory built up around the idea that if this problem really is that hard, all these other problems inherit its hardness. So now, as a result of that problem being easier than we thought, and having a bit better algorithm, yes, that gives you a lot of new algorithms across the entire landscape. But it also changes the character of the field, because the goal was to put all those problems together and say that we should be studying them all together. Now we’re back in a state of affairs where maybe the right way to study them is separately after all. So I think these are examples where, irrespective of how many deep, new, generalizable ideas we get out there, even just knowing the truth of the theorem is going to change the way people organize research in that area.
There’s no putting the genie back in the bottle on some of these problems.
Imagine one of the 700 was factoring in polynomial time. That would absolutely change everything—
Everybody would know. You couldn’t use your credit card on Amazon anymore. All of e-commerce would be gone. All secure communications would be gone.
Can you explain factoring in polynomial time?
If you give me 2 very large numbers, a million digits each, I can multiply them together in a split second. If you ask me to go backwards—say, you took 2 large prime numbers, multiplied them together, and gave me the product—and ask me to figure out which 2 numbers they were, to factor it, we don’t know how to do that.
Don’t we ask our kids to learn how to factor in school? It’s a lie. We do not know how to factor. The fact that we don’t know how to factor is why we can communicate with one another in all kinds of secure ways. Basically, the modern world depends on the fact that we don’t have an algorithm for taking a large number and pulling its prime factors out of it.
A lot of e-commerce depends on that, because it’s the basis for many public-key encryption schemes. Famously, there are quantum algorithms for factoring, so—
Lattice-based methods—
We’re already concerned about what will happen when we have quantum computers large enough to make this an actual problem. But we still have a little bit of time. There’s no theoretical reason factoring couldn’t be easy. I mean, factoring’s probably not NP-hard. Let’s say we prove that P doesn’t equal NP. That would be amazing, obviously, but proving that P equals NP would be completely—
That would be very interesting.
That’s the kind of thing that should go through the government.
Some mathematical results quite obviously lead to dramatic impacts on the world. Other results might take months to digest, but that one would affect the economic systems of entire nations.
How many mathematicians does the NSA employ?
Thousands.
Thousands. I’ve always done this thought experiment: What might they have that they just wouldn’t tell us about? This was 10 years ago, when some evidence showed that collaborative teams of mathematicians could solve much harder problems than any one individual mathematician if they broke them down in the right way. Do you mind if I revisit the balloon for a second? I actually like the semigroup analogy better than the convex hull, but I don’t think it should be interpreted as derogatory, even if it just means these machines are interpolating human knowledge rather than extrapolating.
Even if there is this convex hull of ideas and things that could be pieced together from bits humans have invented, we’re not doing that fast enough. There’s a lot of opaqueness, and this balloon isn’t see-through. So even if you can make this convex hull much more transparent—lift the fog and give mathematicians radar so they can see what’s over the horizon—that’s pretty incredible. I don’t think we should interpret that as derogatory at all. I do think it’s more extrapolatory than that, though.
We should probably talk about the size of the convex hull. We don’t actually know how big it is.
That’s part of the point. We don’t know how big it is or what’s next to each other. There’s so much fog around what we know, and it’s exciting to lift that.
But human beings do make new definitions and bring in new concepts. That fundamentally feels different from combining existing ideas and repackaging them. When human beings do that from time to time, it really feels like there’s a new boundary point, and now you can ask what the convex hull is of what you could prove. You can take the balloon, and there are ways to pull on it—
And we know when it expands. Somehow, we have an intuition for when it’s getting bigger. You can recognize that.
In many ways, the archetype of what the researcher looks like is going to change because of this new technology, too. I’m always amazed by how many very different fields of math spawned from similar ideas. A lot of those connections drove development in the beginning, but over time, the fields became more specialized.
I think the way we train graduate students will need to change so they’re not as overspecialized. I don’t know what the future of hiring or journals will look like, but people who think broadly about a bigger swath of mathematics will be in the best position to benefit from these technologies. That would be my guess.
Can I ask an almost silly-sounding question? For many decades, a large part of research has been the paper. If you were to guess, how might that change in the years to come? How might a human contribute an idea, collaborate, or have a team of humans contribute an idea to the public space?
I don’t know about pure math, but in engineering and some of the applied sciences, we already have a rapidly shifting view of what an academic contribution is. If you write code as part of a larger, reusable infrastructure that supports bigger projects—think of LIGO, some of the big billion-dollar astronomy projects, or cancer research projects that require hundreds of people and huge resources—it’s not necessarily one 15-page PDF.
There are lots of different pieces: code, data, reusable bits that can fit into a larger framework. In my mind, that’s even more relevant with AI. A workable piece, widget, API, codebase, or dataset that can be leveraged by the community or by AI is increasingly valuable, even more than just one result or one paper. I don’t know what that looks like in pure math.
Can you comment on your view of how everything’s changing? Even just observing the incredible evolution of arXiv, which is similar in some ways to what’s happening on GitHub, for example. A lot of people in the open-source movement are really struggling to know what to do. Some are predicting the death of GitHub.
Why? Just because there’s so much code being added to GitHub that it can’t sustain the growth?
One of the things that makes the open-source movement successful is that one or a few people really manage each repository.
Psychologically, and on many levels, they’re becoming overwhelmed by all the pull requests. It becomes impossible to manage, and they get exhausted. There’s this whole feeling that the open-source movement, maybe similar to the academic environment, requires humans to be motivated. If you get worn out, like Terry Tao mentioned, you just quit. That’s a failure mode you want to avoid.
Journals in math were already under very severe strain from the number of submissions, before AI had anything to do with it. More people were producing more knowledge and sending it to journals, and the journals had to referee it. Referees are exhausted and being asked to work really hard to understand and check these theorems.
The only thing worse than that system, I think, is not having that system. The way we arrive at mathematical truth is deliberately slow, careful, and precise. Go back to Grigori Perelman’s solution of the Poincaré conjecture. He puts these papers on arXiv, and people around the world are shocked by them, so they invite him to give lectures.
He goes to Stony Brook, NYU, and Princeton, all around the country and the world, giving three-hour, impromptu explanations. Maybe an hour is him explaining what he’s doing, and the next 2 hours are people saying, “On page 247, Lemma 412, I don’t understand how you do this.” Then he goes to the board and gives them a half-hour lecture on how he does it.
The entire thing lives in his mind. He’s done the best he can to communicate it on paper, and then it’s up to the community to go through it very carefully, in a detailed and precise way, and agree that this is a real proof of the Poincaré conjecture. That took years and a whole lot of human effort. Coming up with a solution is only the first step in the process. That process has to be deliberate and community-driven, and that’s what the journal system is trying to accomplish.
You make a really interesting point. It highlights the traditional view of a pure mathematician as a lone genius who solves a singular problem and then goes on to explain it to the world. In many other fields, we’ve moved past the point where an individual can advance circuit design, aircraft design, protein folding, or astronomy on their own. We have to work together in these billion-dollar collaborative projects.
And again, I’m an engineer, so I’m thinking leadership is obviously what a community needs when it’s having this kind of crisis of organization and individuality and moving toward a more collective model. That’s interesting because I know so many mathematicians, and by their nature, they’re a very independent group of people who don’t like being told what to do or what to study. We like solitude and working alone. That’s why we spend so much time doing it.
I think that’s a really good point. It’s interesting to see the differences in working styles, even between math and physics, with things like the Large Hadron Collider. Let me take a contrarian view: you should never let a good crisis go to waste. I agree that the journals and conferences were overloaded. It always strikes me that we’re doing so much work reviewing these things and handling them, and now we’re faced with the task of having someone review all these AI-generated papers. Some of them are important, but it’s really hard to get your community on board with doing that.
I don’t know what’s going to happen, but I’d be in favor of finding new models—not to replace the old ones, but to relieve some of the pressure that’s come about. I’d love to see models where we take a more collaborative approach to verifying some of these AI proofs. Some of the OpenAI results have been improvements on the Fast Fourier Transform. There have been tweet streams with people improving the constant. I don’t really know what to make of that. I’m not even on Twitter.
It seems to me that we need to figure out other ways of working: how to organize the state of all these improvements, and how to organize and tame these ideas. It’s striking that the way we think about publications is that you work hard on your draft, you post it on arXiv, and that’s your draft. You might update it and create a new version at some point, or a third version, but the draft belongs to you. It’s not really up to the community to add notes like, “Hey, I think you can do this argument differently,” or, “Hey, I think this can be simplified this way.” But maybe we need other models that allow crowdsourcing, so our understanding can evolve. I’d be all for that.
The other thing is, I love giving research talks. One of my favorite parts about giving research talks is that you can take all the messy complexities of the paper—the things where you just knew you had to dot the i’s and cross the t’s, but that weren’t really what excited you—and focus on a key idea and a special case that illustrates why that idea is powerful. That’s the part you really want to explain to people, because when someone asks me about Lemma 57, I don’t even remember it.
Maybe the idea that a paper is a monolithic thing, designed to be written down and completely formalized, can change. Now we can trust Lean formalizations to some extent for some of the technical building blocks, and that can allow us to rework the way we write in the first place. We can put more emphasis on the promulgation of those ideas.
The Mathlib model is exactly what you were describing. Instead of a paper, you have a PR, a pull request. The maintainers and reviewers go through it and make suggestions here and there. Referees make suggestions, too, but it’s not a public process in a journal. In an open GitHub workspace, you can see exactly who contributed what idea that simplified an argument, or that you’re reinventing an idea that already exists in this part of the library. You should just quote that and refactor this part, or whatever.
Mathlib working formally allows something for the first time. One of the problems with mathematics is that we don’t have a perfectly well-written specification for our theorems and technical lemmas. If you open a random page in a random math book and read a lemma, there’s so much context that it’s annoying to write something that fills in every bit of it. We just assume you’ve read the previous 60 pages. Then I can tell you the next step in the lemma.
But that means you can’t just take it off the shelf the way you can a physical part when you’re building a rocket. If all of mathematics is done formally in the future, then the specifications are built into the object. You know you’re not using it the wrong way, because you can’t use it the wrong way: Lean won’t let you. That’s the crowdsourced version of mathematics. That was the dream of Mathlib when it was started 10 years ago: to run mathematics more like a science.
As a quick aside, can you mention what Mathlib and Lean are, just because we’ve mentioned them a few times?
Sure. Mathematics is, ideally, a formal process starting with an axiomatized system and then logical rules of inference. If people think of Euclidean geometry, which kids might learn in high school, that’s supposed to be the model for seeing proofs for the first time and seeing an axiomatic structure. You can take that idea, write it in software, and say, “Here are the actual axioms of mathematics. Here are the rules of logical inference.”
Lean is part of a long history of this kind of software, called interactive theorem provers. You write a proof line by line, and the computer says, “Yes, this line follows by logical inference from the goal state on the previous line.” Lean is software that allows you to check all kinds of things. It can check that the specifications of your code align with what you’re trying to define. That’s what it was invented for: bug-free code. Hopefully, in the future, we’ll have bug-free code because AI systems are really good at finding bugs in code written by humans. That application is becoming ever more important.
Another application is checking mathematical results and allowing you to write a mathematical result in a very formal way. Lean itself is just the hardware that checks the logical steps. You don’t want to start every math project by defining what the natural numbers are using zero and the successor, the Peano axioms of the natural numbers. So it’s nice to have a large, general library that’s extremely well maintained and organized. That’s what Mathlib is: the Lean library for formalizing mathematics in the Lean ecosystem. Lean is just one interactive theorem prover. There are lots of others. It’s the one that so far has been winning the math wars, if there are any math wars. It’s a competition of two people or something.
And it’s the technology used in part to show that the proofs in the OpenAI release are correct?
That’s right. It’s become standard because autoformalization has gotten to the point where AI systems can prove a theorem in natural language. You can’t trust their natural language proofs. These are stochastic processes, and the fact that they’re saying things doesn’t mean they necessarily have a model of mathematics in their heads, whatever that means, the same way that we do.
But that’s okay, because they’ll write Lean code that builds on top of Mathlib, on top of the Prime Number Theorem Project or whatever, and check that it has indeed solved the problem down to the axioms.
So this is where I’ll take a tangent. The thought experiment may be a bit philosophical at this moment, but what if you have that black box that gives you proofs you can’t understand? Let’s say the more interesting case is not that the proof is so big, but that the ideas just don’t make sense to you or to any of the world expert mathematicians in that field. What do you do with that?
The only way, almost by definition, is scale. The logical rules of inference are written down, and there are only a few of them—a handful. The axioms are a handful, too. So the only way you have mathematics that’s not understandable is scale.
Let me add a contrarian take.
Yeah, please.
One of the things I also wonder is: What would unformalizing math look like? Certainly, formalization helps tremendously in giving us confidence that the results are correct. But much of the way we understand the results ourselves is at some unformalized level. What’s an interesting special case? Where does this technique you’re using come from? Maybe that’s what that world would look like. If we have these black boxes, we’d try to tease them apart and figure out how to ask them easier questions, to develop some kind of system for understanding how they actually got to the answer.
There are really 2 different things that we conflate when we write a paper. One is the engineering of the proof, and the other is communicating it to someone else. They’re almost at odds with one another.
I had this experience when I taught real analysis last fall. You open up any real analysis textbook, and this was my experience when I took real analysis: the first couple of lectures are about the real numbers. We have to define them, or construct them as equivalence classes of Cauchy sequences of rationals, whatever the hell all that means.
It takes 2 weeks for a first-semester course in real analysis to get there. Then we have the real number line, and now we can start doing calculus with formal epsilon-delta definitions, yada yada. It always struck me as bizarre that we do this, because before the class I knew what the real numbers were: it’s this number line. Then we spend these 2 weeks talking about Cauchy sequences of rationals, and in week 3, they’re a number line again. What the hell was the purpose of that?
It’s because the way things are written, or have been written, has been very engineering-focused. You need to construct the real numbers before you start using them. But human beings used the real numbers long before we understood that they needed to be constructed formally. So what I did in my class is jump straight into epsilon-delta calculus. We did 6 weeks of calculations until we got to a point where, wait a second: here’s a theorem that’s true in the real numbers. There’s some convergence, something called the Bolzano–Weierstrass theorem. You have a sequence of real numbers that are jumping around, but in a fixed range. There has to be some subsequence that converges to a real number.
Well, that’s not true in the rationals. Why is it true in the reals? What are the reals? You don’t start asking that question until you hit a problem that needs to be solved. So the way I taught the course was sort of backwards from how you’d teach it if it were an engineering project, because you can’t just start talking about the real numbers until you’ve constructed them, until you know what they are. That’s correct if what you’re doing is building up the theory. But if you’re telling a story, then you absolutely can just start with the real numbers. You have an intuition for what that is, and until your intuition breaks, you don’t need the resolution of how you actually construct them.
I think the way we write papers in the future is going to be very different. We’ll spend a lot more time talking about the story of what I learned in this process. You do the standard thing here, here, and here, but then here, you’re stuck. Where do you go? Here’s an idea for why you go left and not right, or up and down, or whatever. And there’s a Lean formalization that accompanies this story and lets you know that the engineering actually works out. You can’t just tell stories. Math is not music, where you can tell a beautiful story and it’s okay that you have parallel fifths moving somewhere and you’ve broken some rule, but Bach does it all the time.
I think that’s a very important point, just to emphasize the point Alex is making. There’s this view of mathematics, which I don’t believe in, where it’s just this rising tide of theorems. How can AI be bad? It’s contributing to this rising tide. But mathematics is not just about the number of theorems we have, far from it. It’s also about the story of fighting with problems, because when you fight with a good problem, it fights back at you, and you learn something about it.
It’s really about the system of beliefs and abstractions you come up with en route. When you’re fighting with a problem, you discover some more general meaning that applies more broadly. And I think it’s a great opportunity to disabuse ourselves of this idea that we’re just axiomatizing things until we get to the next theorem. Telling the story of how we actually got to these problems, and what the parallels are to other problems, is now, at this pivotal moment, more important than it ever has been.
Yeah, so to get all the way back to your question about whether it’s a matter of scale: well, yes, it’s just a matter of scale. But exactly to Ankur’s point, the way we think about mathematics is about stories, and it might be that we don’t understand the story of this. You can follow a proof line by line and not understand what’s really going on. You might just ask some other LLM to explain the story to you, if it can come up with such an explanation. But that’s what we want. We want stories. We want understanding.
4. Riemann hypothesis
When you fight with a problem, it fights back. I think there’s a deep truth to that about all the great pursuits humanity has taken on. And on that topic, I would like to talk to you about some of the greatest fights of all time, with some of the hardest problems in mathematics that have fallen. Alex, you’re a number theorist. Maybe it’s fair to say the central problem in your field is the Riemann hypothesis, and you showed a lot of excitement on the aforementioned Twitter/X about the quasi-RH, the quasi-Riemann hypothesis proof. Can you explain that? Can you explain what the Riemann hypothesis is, and can you explain the nature of your excitement about it?
Sure. There’s a one-minute version of the Riemann hypothesis, or a five-minute version, or a 30-hour version. I don’t know how far you want to go.
Let’s start pulling at that string.
We know what the prime numbers are. People are interested in the distribution of the prime numbers, and what Riemann discovered is that there’s a function—this is totally crazy—that takes complex numbers as inputs and has complex numbers as outputs, and that function governs the behavior of the prime numbers. It’s called the Riemann zeta function. By the zeros of the zeta function, we mean complex inputs where the output is the number zero in the complex plane, so the origin in the complex plane. The question is, where are these complex zeros? What complex number inputs give you a complex number output of zero when you apply this zeta function?
Why in the world should that be? Complex numbers? What are you talking about? We have these discrete numbers that aren’t divisible by anything else sitting in the natural numbers on the number line, and now you want to talk about functions from a complex space to a complex space. That was the great breakthrough: figuring out how to do signal processing on the prime numbers.
If you think about why people can hear you on your podcast, it’s because my voice box is producing pressure waves of sound. That’s the natural space. Then that goes into the microphone. The microphone analyzes it, does the Fourier analysis on it, converts it into amplitudes and frequencies, and it’s just a bunch of numbers. It stores those numbers or sends those numbers, whatever. The speaker then takes those numbers and reconstructs what sounds to people like my voice.
The same thing is going on. That’s what the zeta function does for the prime numbers. You can think of the prime numbers as particles of air molecules wiggling around, and it does a signal-processing analysis on them that can recover the same prime numbers back. The frequencies and amplitudes correspond to the zeros of the zeta function. The zeta function has that information in it.
The Riemann hypothesis says the amplitudes of all of these—you can call them Riemann waves—and the superposition of those waves recovers the distribution of the primes. All of those amplitudes are supposed to be exactly the same. The orchestra that plays the music of the prime numbers has every single instrument sounding at exactly the same amplitude.
We know what the prime numbers are, but maybe, going into the space of philosophy and the romantic notions of aesthetics, why are they so interesting, important, central, and beautiful at the core of mathematics? And connected to why the Riemann hypothesis is exciting, why is it so difficult to figure out if there is a pattern in the prime numbers, and what that pattern is?
Let me start with your second question first. Why something is difficult, I only know how to answer once I know what the answer is.
Once it’s made simple.
Right? Maybe it’s not difficult. Maybe it’s not as difficult as we thought it was. I did not think I would see a proof of the quasi-Riemann hypothesis in my lifetime. So the fact that I saw it on Tuesday was extraordinarily shocking.
What did you do in the 30 seconds after you saw it? What was your reaction? Do you remember what—
Yeah, I remember where I was the second I saw the drop. We were at this National Museum of Mathematics gala in New York City. I had a whiskey in one hand, and I kept refreshing. I was talking to people, socializing, but I also kept refreshing because I knew it was dropping around 6 p.m. Eastern Time.
Then the drop hit. There was a Quanta reporter there, and we were just chatting: “Oh, yeah, they’re going to drop these 300, 400—we didn’t know how many—700, it turns out, problems.” And I was like, “Oh, wow, that’s really cool. It dropped.” I’m looking at my phone, and the second thing I see is quasi-RH. I’m like, “What do you mean, quasi-RH? You don’t mean… for what?” Then I look, and it’s the zeta function, and it’s all Dirichlet L-functions, so they’ve solved zeros. I’m like, that can’t be what’s in this drop. I could not compute in that moment what I was seeing.
That speed of progress in that little time, yeah.
It was so far outside my model of what was possible. And this—it’s been a month of things happening way outside my model of what was possible. The Fermat’s Last Theorem formalization: I was sure we were at least 3 years away from being able to do something like that.
So, maybe you can elaborate on that. It was in Lean, it was formalized as—how big was the Fermat’s Last Theorem formalization in Lean?
Like, 13 million lines.
13 million lines. Okay.
Two days before, I gave a colloquium in the Princeton Math Department where I explained in painstaking detail what Lean is and what Mathlib is. I also explained that AI systems today work with this Mathlib library, where everything is kind of perfectly done, at exactly the right levels of generality.
There shouldn’t be code that re-proves theorems that have already been proved somewhere else. Everything is the most general, the most abstract, and what AI systems are capable of doing is building 1 or 2 layers on top of that. But what they’re doing is jerry-rigging. They haven’t been trained yet—I’m not saying they can’t do it—but they’re not good at what I call canonization: taking sloppy code, refactoring it, figuring out exactly where in the library it belongs, and finding the canonical version of a statement or lemma.
Then every proof should just be a combination of applying this lemma from over here, applying that lemma from over there, and putting all these things together. There’s your theorem. Then it’s really clean, really beautiful, very elegant. That’s what Mathlib is going for.
What I said in my colloquium is that, where we are today, you can get entire textbooks formalized as long as they’re in the Mathlib halo—as long as they’re 1, maybe 2 steps removed from things that have already been done in Mathlib. But when you start asking it to do things that are 3 or 4 steps removed from the library, you run into problems. The way it came up with a definition of an object for one purpose, and the way it came up with another definition of the same object for a different purpose, can clash. You can’t stitch these things back together.
You really need Mathlib to grow in order to get to a place where you can do research manuscripts that are synthesizing tons of different areas of mathematics.
Which is why you’d expect Fermat’s Last Theorem to take quite a bit of time, because it’s so far outside the halo of Mathlib.
Exactly right. Kevin Buzzard had a million-dollar grant to spend the next 5 years—or maybe he’s already a year or 2 into that grant—reducing the formalization of Fermat’s Last Theorem to things that were known in the 1980s. He wanted to do that last step, the 7 years of Wiles and Taylor’s work, to finally finish Fermat’s Last Theorem, but built on top of things we knew would be extraordinarily difficult. He wasn’t going to do Langlands–Tunnell; he wasn’t going to do Mazur. There were all kinds of things that were expected to be even much more difficult than what these 5 years would take.
Everything I said in that colloquium was wrong. I think my colloquium was on a Wednesday. On Friday, there was a 13-million-line code drop: Fermat’s Last Theorem was proved from the axioms. The statement of Fermat’s Last Theorem is completely elementary. There’s no semantic alignment to check. You just glance at it: yes, that’s the statement. Lean says it’s sorry-free.
Can you explain how it was done?
So, it was Anthropic, and how it was done was that AI systems just got really, really good at it. It’s still not a nice proof. It’s not the clean proof. There’s still a ton of work that needs to be done to make that proof a Mathlib-quality proof, and I think that’s important work that we want to do.
But in terms of getting that sorry-free theorem, they just blasted through all of these things. They jerry-rigged this, they jerry-rigged that, and 13 million lines later, they had a sorry-free formal Lean proof of Fermat’s Last Theorem. It’s just absolutely shocking.
By the way, it’s beautiful that, because of how rapidly everything is changing, a lot of statements everybody makes—even world experts—will be wrong in 5 years. That’s the whole point of it. That’s the fun thing about human discussion: we have disagreements. I would venture to say somewhere between 40% and 80% of everything said today will be wrong in 5 years, because everything’s changing so fast.
But that’s the whole point of human discussion, deliberation, discourse, and even the scientific process: saying shit and seeing what sticks over time, over 10 years, over 100 years, and so on.
I was describing my experience. I had a lot of experience with Lean and with using AI. I started doing Lean before AI existed in the sense that we understand it now, and it was very painstaking. When the first AI systems came on—the first LLMs were coming online—they were crap at Lean. But then they got a little bit better, a little bit better, and I thought, “Oh my God, I can do library search.”
I don’t have to sit there looking for a lemma when I know there’s one that does this, but I don’t know what it’s called or exactly which one it is. Is it in the manifolds library? Is it in the algebraic geometry library? I can ask, “Give me these lemmas.” It was great at that, and I thought, “Oh, this is amazing. I’m going to keep using these systems.”
Every bit of my experience told me that they could do X, but not 10X. This was 1 million X. It was just completely outside the distribution in my mind of what systems were capable of.
Do you have a ballpark estimate of how many humans were involved in that, and what the level of human involvement was in generating those 13 million lines of code?
I believe it was 2 or 3 humans.
Okay.
I believe most of their involvement was, “Keep going,” all caps.
Okay. So you think it was really just gentle guidance, not pathway creation?
As far as I understand, there was no domain expertise—
Okay.
—on the part of the people who were doing this.
As the meme goes, with OpenAI or Anthropic, the prompt is “Solve math,” or something. And then—
“Do a breakthrough.”
Yeah, yeah. “Do a breakthrough.”
“Do a breakthrough.” That’s the prompt, and it goes and does it.
Of course, from a company perspective, you never know. There’s a lot of incentive to sell it that way, even though there may have been a lot of intermediate steps at any stage of the training, or some things left unsaid. Even if the final prompt is “Do a breakthrough,” there could have been a lot of groundwork before that. We don’t know.
I remember that was a Friday because my mind was completely blown. It was Labor Day, and we just took the kids to the beach. I needed a few days where I wasn’t checking my phone or on Twitter. I was just throwing my kids into the ocean.
Did you have that moment where you’re just sitting and looking at the ocean and contemplating?
Yeah. Like, what does it all mean? I had already gone through that a month before.
So now what that says is that autoformalization is basically solved. It’s not solved in the way that I would want it to be solved. People are still going to work very, very hard on this. There are all kinds of other things that happen, just like there are all kinds of byproducts of proving theorems for human understanding.
There are all kinds of other things that happen when you work on making a beautiful library of mathematics, and that process will continue. That process will be very difficult. It’s going to take thousands of man-hours of people around the world contributing in this open-source way. But in terms of whether you can get a sorry-free proof of something human beings have written down on paper, I think that problem is done.
And so you were standing there with a whiskey, having gone through this a month before, and now again.
Right. This was round 3.
This is round 3.
Because when I got back from Labor Day, it was Navier–Stokes.
Navier–Stokes.
But this one—I was like, “I was just mourning FLT. You’re going to do this to me? Come on.”
But now, I mean, this really hit home with the quasi-RH. So maybe can you speak to the importance and the beauty of the primes, and also how big and hard the leap is from quasi-RH to full RH?
The primes are so valuable because they’re the fundamental particles in the theory of multiplication. Every whole number is broken down into a product of primes. We know about the elementary particles in physics, and all matter is made up of them. In the same way, the primes are the particles that make up the multiplicative structure of whole numbers.
Their distribution had been a mystery, and it remains a mystery. Even the Riemann hypothesis doesn’t solve everything we want to know about prime numbers. But the Riemann hypothesis was so beautiful because it pointed the way to what the distribution should be, and it has a thousand applications in very faraway fields. If only the Riemann hypothesis were true, then we would know XYZ.
So let me interject a little story that came up last week, which was pretty entertaining. We had Grant Sanderson visiting IAS, and we had a long discussion about what AI is doing to the math community. I really enjoyed this comment that Camillo De Lellis made. I’ll do my best impression. I can’t do the accent.
There was a play written in the 1950s by a mathematician. Who knows that mathematicians are writing plays? But apparently, the premise of this play was that von Neumann died and went to heaven.
And he met God, and he got to ask Him a question. So he asked Him, “Is the Riemann hypothesis true?” Of course, everyone holds their breath as they’re waiting to hear the answer. And God says, “Yes.” So finally, everyone can exhale. Then von Neumann asks, “But how do you know?” God thinks and says, “Well, I know where all the zeros are, and they’re all on the Re(s) = 1/2 line.”
So my understanding is that the premise of the play is that that’s not enough for von Neumann. He gets a little agitated and tries to find out more about why it’s true. I think he ends up in the other place at the end.
That’s great.
I don’t know what to draw as a moral from it, but it seems to speak to the mathematical experience.
Math is not the fact; it’s the why. It’s always been the why. It’s our deepest quest for understanding. If you just tell me a fact is true, but I don’t understand why it’s true, that has contributed zero to my knowledge.
Primes are also interesting because they relate to things that a 3-year-old can understand. Fundamentally, we know how to add 1 to numbers, and there’s this breakthrough moment when you’re a kid: “I can add 1 forever.” That’s a big idea. Then you learn how to multiply and group, and that’s also a big idea.
Primes have always been interesting to me because we don’t understand how adding and multiplying work together. The 2 most fundamental, basic, simple ideas in all of mathematics—we don’t know how they work together. I can add 1 to a number that I can factor, and I have no idea what happens. I know if it’s even, it becomes odd, and vice versa, but I don’t know if a number that has many, many composite factors becomes prime. We fundamentally don’t know how that simple act of adding 1 changes multiplication. Primes are the distillation of that 3-year-old’s mystery, and it’s a mystery that all of us still wrestle with. The interesting thing would be if the proof told me something fundamental about that structure that I could actually use.
To the 3-year-old, I think my older one was 2 or 3. He was playing with these little blocks, cubes that kids like to play with. There were about 20 in the pack, so he’d make a little 4-by-5 grid, and he liked that. He’d make a 10-by-2 grid, and that was kind of cool. Then one day, I looked over and he was really frustrated. I asked, “What’s going on?” He said, “You know, it’s not working.” One of them had rolled under the couch, and he only had 19. He couldn’t figure out a way to make a nice-looking rectangle out of 19.
Yep. It still bothers me today.
So the solution is to lose another one, right?
Yeah, exactly.
That’s right.
Yes.
Do you think that, with the leap between quasi-RH and RH, humans will be able to solve the full Riemann hypothesis in your lifetime without AI’s help?
I don’t know.
Can AI? I guess that’s the follow-up question. How hard is it?
I don’t know. I didn’t think I would see a proof of quasi-RH. Now that I see what the proof is, there are a lot of technical details that I still have to go through for my own satisfaction. There’s a Lean formalization. The Lean statement was already in the library, already in Mathlib, so the fact that they have a sorry-free proof gives me 100% confidence that the proof is right. Moreover, having read the proof at a high level, the ideas it uses are all completely valid. They’re brilliant. Embedding the Möbius function and twisting it by sextic characters—this is all the bread and butter of analytic number theorists, chained together in just the right way. It was absolutely incredible.
What Riemann showed is that there’s what’s called the critical strip. Remember, we were talking about this function that has complex inputs and outputs, and we’re worried about when the output is zero. The critical strip is a vertical, infinite strip in the complex plane where the real part—the X coordinate—lies between 0 and 1, while the imaginary part—the Y coordinate—can go arbitrarily high or low. The hypothesis is that the zeros all lie right at the dead center of that strip, where the real part is 1/2. That’s exactly what Ankur was just saying.
Until Tuesday, what we knew was a region where there are no zeros of the zeta function, and that region gets thinner and thinner the higher up you go. It goes toward the 1 line. The 1 line is associated with the prime number theorem, which tells us the basic distribution of prime numbers. It’s equivalent to the fact that there are no zeros of this function on the line where the real part of s equals 1. As you go up and down that line, you’ll never find a value where applying the zeta function gives you zero. So we have this zero-free region that gets thinner and thinner the higher up you go, and that gives us the best-known distribution of the prime numbers and the best error term in the prime number theorem, which I can go into if you tell me how much math you want to talk about.
Let’s go.
What I could have anticipated is that the region would thicken that line a little bit as you go up. The thickness would still go to zero as you go up the imaginary axis, but it would be better than what we currently know. That thickness hasn’t moved in 70 years, but I could imagine it might make some progress.
What it did was give us a vertical strip with no zeros whose real part is between 7/8 and 1. That’s a quasi-Riemann hypothesis, and if you can get that 7/8 down to 1/2, that’s the full Riemann hypothesis. The methods used in the 7/8 proof—and there’s a second proof that gets to about 11/12, never mind these technical details—fundamentally can’t get down to 1/2.
So a new idea is needed.
You need a new idea. But quasi-RH needed lots and lots of new ideas, and it came up with them.
Having gone through this 3 times already, is your intuition—this is for all 3 of you—about the hardness of various problems changing? You mentioned picking easy things. Presumably, you have a good hardness radar: this problem is probably solvable, this is not solvable today, this could be solved this week, this month. Are you readjusting to this new world?
I’m readjusting my inability to predict it.
Yes, exactly. Well, you had the ability, no? It’s noisy.
Sure. It’s a larger amount of noise than I thought. I think we’ve always known there are problems we care about that are within reach of the tools out there right now, but it’s surprising on what scale—how many different problems really fall into that category.
We also don’t know a lot about it. Apparently, they prompted thousands of problems, and what they dropped were only the positive solutions. They had about a 10% success rate. So we’re left guessing: is there a very jagged boundary that we could squint at and say we can make sense of, distinguishing what did fall from what didn’t?
5. Problems AI can't solve yet
I think one of the other dangers is that maybe the safest problems are the ones where you look at them and it’s painfully obvious you need a really creative new idea. You can tell there’s a barrier, and the things out there in the literature just don’t seem to fit into that hole. But now, I think one of the things that has changed the landscape of math forever is that maybe people will become a little more secretive with those ideas than they were in the past. Once those ideas are out there, it might be a lot more straightforward for AI to take that one new idea, that missing ingredient, that new tile it needs, and exhaustively explore what it can reach from there. So you might be a lot more protective of your ideas than you were in the past. I think that’s also a bit scary, because traditionally, the mathematics community has been really open with sharing ideas, and that would be something bad to lose.
But that’s also because we had this first-through-the-gate mentality, and I think that’s also going to change.
That’s right.
I would prefer that the openness stay and the first-through-the-gate mentality be what goes.
That’s right, but that’ll take time.
It’ll take time.
That could be an effect that’s more transient in nature. If we can learn how to value other things, then maybe that’ll go away. There are other communities that have models for valuing contributions even when they’re not lionizing one individual’s contribution. So it’s not that there isn’t a way forward, but in the short term, that’s a real roadblock to doing mathematics the way we used to and enjoying it the same way we used to.
6. Navier-Stokes
Speaking of which, Steve, you told us a story about soup that made you maybe fall in love with the bigger problem of fluid dynamics. Going from math to applied math to engineering—the full thing. Can you talk about the Navier–Stokes equations that explain fluids, why they’re such a big deal, why they’re beautiful? Similar to how we talked about primes, why are they so hard, and why is it such a big deal that Navier–Stokes was proven?
Fundamentally, the Navier–Stokes equations are just a way of stating the conservation of momentum for fluids, for objects that can move kind of freely, like gases and liquids.
We know F = ma is conservation of momentum in classical Newtonian physics. The same basic principle applies to the gas in this room and to flowing water, and it’s true in any volume: little volumes, big volumes, even imaginary volumes. If you write down that conservation of momentum in a volume and take the limit as the volume becomes arbitrarily small, you get the Navier–Stokes equations. They’re partial differential equations that describe how the velocity field of a fluid changes in space and time, and how the pressure relates to the forces acting on that fluid.
Can you give some intuition for why fluids are such complicated systems, given that F = ma is a pretty simple equation?
F = ma is simple to write down, but things can get complicated really quickly. If I have a single pendulum, it’s easy to write down the equations for it. If I put another pendulum on the end of it, I have a more complicated equation, and it gives rise to chaos. In a sense, if I try to model the air in this room, I could chop it up into 1 billion little boxes and write down a differential equation for how momentum works in each one. But now I have a really complicated system of equations. It’s simple to write down and conceptually understand what the equations are saying and representing, but actually solving them and understanding how the fluid moves in any practical sense is very complicated.
I read that one place fluid dynamics is important is reducing drag on airplanes. Airlines could save tens of billions of dollars by reducing drag by 1%. Why is it so hard? Drag is maybe easier than something like turbulence, so why are things that are worth so much money so difficult?
The good news is that if you can improve our ability to predict turbulence, or simulate it even a little bit better, faster, and more accurately, there are dozens of fields where you can have a huge impact—with relatively small improvements to the economy, the environment, human health, and safety. There’s drag reduction on airplanes, better artificial hearts, more efficient wind turbines and gas turbines—you name it. We live and work in a fluid, and so do all our machines.
The inverse design problem is closely related to this kind of math problem: I know what I want to prove, but I don’t know how to get there. I know I want a wing that has high lift and low drag, but I don’t know how changing the geometry of that wing will affect the cascade of smaller and smaller vortices generated by the turbulent fluid flowing over it, which eventually results in its lift and drag. I don’t know how to do inverse design efficiently. Part of the problem is that the equations are expensive to simulate or test. Wind tunnels are expensive, and you need supercomputers to simulate these things.
This is related to pure math, and I think we’ll get back to it. There’s a nice Feynman quote: We know how to write down very general equations—Maxwell’s equations, the Navier–Stokes equations, Schrödinger’s equation—but we currently don’t know from the equations what phenomena are possible. If you’d never visited Earth, could you possibly have predicted sunsets and tides from a partial differential equation? We’re not even close. There are fundamental mathematical questions about this: I can write down a differential equation, but I can’t tell you what’s inside it. We still have to simulate and test, and go through an elaborate engineering process to understand how these equations manifest as drag on an airplane wing or turbulence inside an artificial heart.
Sorry, can I ask a question about that? I’ve always wanted to ask: in the case of Einstein’s theory of general relativity, a few years later, Schwarzschild finds a blowup, and people are like, “Oh, come on. What’s wrong with these equations? There’s a blowup. If you get enough matter into a small enough space, light can’t escape it. We don’t have any physical experience of this. This has got to be a mistake in the equations.” Then we discover black holes, and it turns out the theory perfectly explains and predicts what’s supposed to happen.
In the case of a blowup in Navier–Stokes, are we going to discover something new about water? Or is it that, at some scale, you get to particles—to an atomic structure that doesn’t follow the liquid equations anymore? Why would we have predicted that there should be solutions to a continuous process that’s allowed to be pushed to its limits?
That’s a great question. I’m going to give you the engineering answer: these equations are an exceptionally good model of how fluids work at a certain scale. You’re absolutely right that you need some number of individual molecules averaged over in a volume for these to be valid constitutive equations. If you zoom in far enough, you’re talking about individual molecules, and Navier–Stokes breaks down. The mathematical problem of these equations blowing up is interesting from a mathematical standpoint. By the time they blow up, it stops being a fluid. It’s like—
There are no black holes.
There are no black holes. Other physics takes over.
But it’s possible—wait, I think it’s interesting from a philosophy of science perspective that where equations break down could, and maybe with high statistical likelihood would, be a window—a door—to a breakthrough that requires some other discovery. Newtonian physics breaking down needs Einstein.
Right. And the grand unification is that we don’t have gravity at the same scales as the interactions of subatomic particles.
But this idea that we don’t know something so fundamental about these equations is striking. There are so many people on Earth simulating and testing Navier–Stokes in every industry imaginable. It’s probably one of the most studied, analyzed, and used equations in the entire engineering world, and we don’t know this fundamental property of the PDE: Does it blow up in finite time or not?
That’s the zeroth-order version of the bigger question: What other kinds of solutions does it admit? Are there sunsets, hurricanes, tornadoes, and all the phenomenology that could exist? Existence and uniqueness are the simplest baby step. That’s where it gets interesting to me. Even if there’s no direct, tangible use for this mathematical result in an engineering sense, it gets us one step closer to understanding the nature of these equations and what is possible, which is fundamentally interesting to us.
Could this solution that has the blowup be used somehow in an engineering application? Could you drive it into this state and get some funny behavior that helps airplanes fly with less drag, or something?
In principle, a huge part of what we try to do is manipulate fluid flows—bend them to our will. If you watch a bird or an insect fly, they’re shaping vortices and carefully funneling energy into structures for their daily objectives, like perching on a rock or hunting. We try to do the same thing. Flow control is one of the highest challenges; it’s very, very challenging. All of our science-fiction technology would be accelerated if we had a better ability to control these flows. This proof is one very interesting construction of this kind of specialized flow control.
I’m reminded that Feynman was asked about intelligence and artificial intelligence in the 1970s. He said something like, “The way we solve problems with technology is radically different from how nature solves them. Look at birds: how birds fly and how airplanes fly.” He was pointing exactly to this. It’s interesting to think back to that comment now, when we see what LLMs can do with theorem proving versus what human brains can do with theorem proving and all kinds of other applications. It’s interesting that he pointed to how differently we’ve solved flight with technology compared to nature’s solution. Our airplanes don’t flap.
Now I wonder if birds have existential crises when they see planes flying.
Exactly.
What am I even here for?
Yeah. Yeah. To get the worm.
What are the different components of the Navier–Stokes equation? Can you dig in? What does it take to describe a fluid?
A fluid moves, and it moves differently in different places and at different points in time. The basic building block of a fluid is the velocity field. At every point in space, there’s a velocity vector that tells you how the fluid is moving. Maybe it’s really fast here and slow there, and that creates a vortex.
It gets more interesting when the fluid velocity changes a lot over space. A tornado is a rapidly changing fluid velocity field in both space and time. There are also pressures involved. The core of a tornado is low pressure, so pressures are forcing the velocity field and causing it to move the way it does.
That’s the building block: a velocity field and its pressures. The equations tell you how that velocity changes in time and space according to its own momentum. Fluid has momentum to carry itself; that’s gusts of wind. It has internal friction and resistance—that’s viscosity. Honey has more internal resistance than water, and water has more than air. Then there are all the external forces, like heating, pressure, and gravity. Each of those can be quantified as a term in the larger Navier–Stokes equation.
Speaking of external forces, can you describe the math side of the Navier–Stokes blowup? What was the formulation of the problem, what was actually proven, and why was it so hard?
If you don’t have that internal friction, that viscosity, we have a strong suspicion that the equations can and will blow up under certain conditions. We have other, simpler problems that are like Navier–Stokes and do blow up. The simplest one is an ordinary differential equation, if you don’t mind me taking a little tangent.
Let’s go.
So, \(\dot{X}=X^2\). \(X\) is the state of my system, and \(\dot{X}\) means the rate of change of \(X\) in time. If that rate of change is growing proportionally to \(X^2\), that differential equation blows up in finite time.
Think about \(\dot{X}=X\). \(X\) could be the population of bunnies, and we know bunnies like to populate. \(\dot{X}=X\) grows exponentially. The state of my bunny population will grow exponentially over time. It’ll get bigger and bigger and double and double, but it won’t reach infinity until infinite time has passed. That’s just normal exponential growth. That’s not finite-time blowup.
But if I take \(\dot{X}=X^2\), that blows up in finite time. That quadratic nonlinearity, that quadratic feedback, is enough to cause the equation to blow up. You can solve this using classic basic calculus techniques, and you get this finite-time blowup.
There’s a progression of partial differential equations that have the same behavior, where a quadratic nonlinearity can cause blowup. What’s really interesting is that one of the terms in the Navier–Stokes equations—the convective term, which describes how the fluid moves itself with its own momentum—is a quadratic nonlinearity, just like \(\dot{X}=X^2\). It’s this self-motion.
Without diffusion, without viscosity, we think that the equations should probably blow up to infinity in finite time. But we don’t know whether viscosity, that internal friction and resistance like honey has, always damps out those blowups before they can happen. My intuition as an engineer is that, of course, it does. If I have a flow that’s accelerating to infinity, it should immediately diffuse into its neighbors. It should diffuse into the neighboring fluid and tamp down that growth.
I think the math problem is essentially this: Are there any edge cases, any initial configurations of the fluid, where you perfectly orchestrate things so that it could actually accelerate to infinity in finite time? There are 2 technical versions of this. One is with no external forcing, and one is with smooth external forcing. Smoothness is important, because if the external forcing has singularities, then it’s kind of trivial. If I force it with a tornado—an infinitely tight tornado—then of course it’s going to blow up.
The problem as stated has smooth external forcing.
There are 4 versions of the problem on the Clay website, one of them being the forced version.
And that’s the one that was proven?
Exactly.
What was proven, and what are some interesting things about the proof that you saw?
Yeah. What is proven is that there’s a way of orchestrating a smooth external forcing that perfectly cancels out various terms in the equation, so that you get this vortex. This can only happen in 3 dimensions. That’s really important.
We should probably say that we’ve been talking about 3 dimensions. There’s a big difference between 3D and 2D.
When we talk about a fluid, we usually imagine it in 3D, but there are analogies or simplified 2D flows. Weather on Earth is pretty well approximated as a 2D flow.
You’ve mentioned somewhere in a talk I heard that hurricanes are modeled pretty well in 2D and tornadoes are 3D. I remember different analogies you were drawing there.
One of the interesting phenomena in 3D turbulence is that you generally have this breakdown of big vortices into smaller vortices, into little bitty vortices, until eventually they molecularly diffuse into heat. That’s the 3D turbulence cascade. In 2 dimensions, like weather on Earth, you actually get a reverse cascade, which is why hurricanes form. Little vortices coalesce into medium vortices, which coalesce into bigger vortices.
The AI proof of this counterexample—that you can have finite-time blowup—basically creates this very specialized vortex that stretches out in the third dimension and thins, so that it can essentially become infinitely thin. The radius gets smaller and smaller as it stretches out, kind of like a figure skater doing a spin: you pull your arms in, and you go much, much faster. There’s a very specialized rate of contraction of this vortex, so that the energy in the fluid still remains finite, but at this localized filament, the flow blows up and you get infinite vorticity.
I was surprised that mathematicians started thinking it was going to be possible to find this counterexample only maybe 15 years ago.
That’s right. It’s relatively recent in the grand scheme of things.
Instead of looking for a solution, looking for a counterexample.
Yep, because all of our physical intuition tells us diffusion solves the problem. It’s very physically counterintuitive.
In the mathematics world, that’s a really hard problem. It’s impressive to construct a counterexample. What does it teach you, as an engineer who’s stared at and studied a lot of different natural systems that are fluids? And maybe take that as a step toward what the equivalent hard and harder problems are that still need to be solved in fluids.
So it’s a very important math result.
Uh-oh.
I’ll say—
But—
Day-to-day engineering fluids and applied math analysis of fluids are almost completely unchanged. Fluids are just as hard to simulate and predict today as they were a month or two months ago. When I’m trying to solve an engineering problem with a fluid—I want to design an airplane wing, an artificial heart, or whatever it is—I still have to simulate, predict, and manipulate this flow in a very practical way that’s expensive in terms of supercomputer time and wind tunnel resources.
This mathematical theorem and proof are very interesting, but they operate in a regime that’s so nonphysical that they don’t really affect my day-to-day engineering. There’s still—I don’t know, this is very pithy, but I like to say we solved the $1 million math problem. There’s a $1 trillion engineering problem: How do you predict these things quickly and efficiently and do this inverse design? That’s a much harder problem.
And you say it’s a trillion dollars because that’s literally—
Literally.
—if you solve some of these problems, that’s how much money you’ll make. Not because it’s a prize, but because there are industries.
Yeah. We could say there’s $1 trillion of possible benefit to humanity from better, faster solutions: lower drag, fewer emissions, all of it. Transportation, defense, health, energy—all of these have fluids at the very core of their technology. I think those problems are still very hard.
This math result is important in the sense I was talking about earlier: we don’t know the fundamental character of what is possible in these equations. We don’t even know if the solutions exist. That’s so fundamental. This is a baby step toward a much bigger human goal of understanding what’s in an equation in a more practical sense.
Can you speak to some of the beautiful, fascinating phenomena in the space of fluids that you’ve been thinking about? You mentioned turbulence. What are the big puzzles to solve to understand this space? You’ve mentioned drag, like air and fluid. What are the big Millennium Prize-type problems people are thinking about?
There are so many. You can break them down into a few categories. Turbulence is very hard because it’s multiscale.
What is turbulence?
Turbulence is this phenomenon where you have a complex fluid roiling, mixing, and moving, kind of like in Leonardo da Vinci’s beautiful drawings of fluids at all these different scales. All of the scales matter: the big vortices, the medium vortices, and the little vortices. They all matter for predicting lift and drag and things like that.
The gas in this room is a great counterexample. It’s the opposite of turbulence. The molecules are all moving around in a very frenetic way, and there are a lot of them. But that 10²³ number of gas particles—all those individual configurations don’t matter. We know from thermodynamics, in one of the great successes of classical physics, that there’s a small handful of numbers that actually determine how this gas behaves: temperature, pressure, density.
That simple description doesn’t exist for turbulent fluids—for the flow over a wing, the flow inside your heart, or the flow in a river. All of the scales actually matter. All of the numbers in that simulation matter to predict the next behavior, to predict what’s going to happen. That makes it fundamentally challenging.
This is actually a great AI problem: Is there a coordinate system? Is there a set of numbers, just like we have for gases—temperature, pressure, density—that can describe a turbulent fluid simply, in a way that’s useful for engineering, to build motors and engines, just like we do with gas dynamics?
So then the interesting question, if it’s a problem for AI—and AI is listening, I’m sure—is whether the solution to that problem would lie through mathematics or through having the AI system sit there and watch some fluid for a while.
I think yes to both.
Because fish, they figured it out.
They figured it out. I think you have the right keyword: coordinates. That’s why I said both. I’m not trying to be trite, but all of our great physics breakthroughs were predated by a coordinate transformation. Newton’s F = ma came after we put the Sun at the center of the universe instead of the Earth. You had to do that coordinate transform first.
The way I see it is that we need to take all of this data—these simulations and observations of fluids—and find a simple coordinate system so that the description of the fluid becomes simple, generalizable, and beautiful in a physical sense. That involves brute force: getting a lot of data and trying a bunch of different coordinate transformations and representations until you find the simplest one that generalizes best. But there’s also a lot of deep mathematics in what form that coordinate transformation might take.
In general, if you find a question that really smart people asked in the ’70s and ’80s, but they hit a wall computationally, that’s very fertile ground right now. We know what the form of these coordinate systems probably should be from some fundamental principles. We’ve just never been able to write them down. They’re too complicated. That’s a perfect problem for AI: give it a lot more data and compute, and see if we can start learning these transformations we know should exist.
Just so I understand, the formation of vortices—that’s like little tornadoes?
Little tornadoes.
Is their formation tricky to understand?
Yeah, absolutely. And it’s beautiful, too. I love watching videos of two vortices being shot together. They influence each other and then merge in very interesting ways. You might have a red vortex and a blue vortex, and you can see where they merge and join. But they also stretch and fold. Turbulence is chaotic, so the trajectories get tangled like spaghetti. It’s not random. There’s a huge amount of structure, but it’s very multiscale and very tangled.
7. How nature computes
One thing I wonder about scientifically—and these are really interesting problems, and that was a great explanation—is what computational problems nature can already solve. There are a lot of examples. There’s a famous one with slime mold finding shortest paths in mazes. Some of the precursors to the Navier–Stokes equations were Terry Tao’s work on showing that you can have blowup for Navier–Stokes-like equations because they can simulate Turing machines.
I would also wonder what sorts of computational tasks these computations with colliding vortices can do, and how that imposes limits on what sorts of coordinate systems ought to exist. You mentioned PV = nRT. That’s an amazing dimensionality reduction, because you don’t keep track of all the positions and velocities of particles. There’s just a small list of numbers that describes a macrostate, and that’s it. But if I had a tremendous reduction like that in the case of turbulence, it would seem very difficult to compute very interesting things, too.
Could it be like the Game of Life, where you have this very simple cellular automaton system, and it’s a universal Turing machine, so you can cook up whatever behavior you want in it?
Yeah, that’s right. People often think about the Game of Life in terms of the minimal rules needed to have all of these exotic behaviors. It’s really shocking how few rules you need to get that. I’m a theoretical computer scientist, and one of the biggest exports from our field is the algorithmic lens. It’s not just about computation where you have a well-defined problem and a laptop and you want to solve it. It’s also about computation in the natural world, and what that teaches us about the rules of nature.
Some famous examples are things like equilibria in multiplayer games. Nash equilibrium is a great solution concept, but the fact that it’s hard to compute Nash equilibrium throws into question whether it’s really a universal mechanism that explains how people play games. There’s an amazing quote by Kamal Jain that I really like: “If my laptop can’t find it, then neither can the market.”
To me, what’s really exciting is the nexus of a lot of these different perspectives. When you were describing it, it sounded to me like there are very interesting computational problems under the hood, and one can get a dual perspective on these things by thinking about how that computation imposes limits on what a good coordinate system ought to look like.
I think that’s a really nice way of framing the problem. I’d take two extreme examples. Earth’s weather and climate systems are truly complex: all the scales matter. The weather in this zip code matters to me. Everything matters; all the variables matter. You’re right that if I over-reduce that system to five numbers, I lose the descriptive capability of what I actually care about.
At the other extreme, think about how a moth, a butterfly, or a fruit fly flies. They’re proof by existence that you can interact with an extremely complicated turbulent flow and do very advanced engineering control. They can control their forces and moments in gusts and rain. They can live very rich lives, and they presumably don’t have a PhD in fluid dynamics or know the Navier–Stokes equations. They’re probably not running supercomputer simulations.
There might be five or six numbers that actually matter to their daily lives: maintaining a leading-edge vortex, stalling their wings to perch, and landing. Those are two extreme examples, and I want to be able to handle both. We want to handle the full complexity of truly complex systems, and we want to distill the bare minimum essentials for fast decisions in those extreme insect cases. Biological flyers still dramatically outperform human-designed fluid systems in many regimes, so we still have a lot to learn about these simple systems.
Maybe we’re veering into research now, but it’s always amazing to me how nature not only solves very hard optimization problems, but seems to find very robust solutions. When we model nature, we tend to assume we have full knowledge of the boundary and initial conditions for things like the Navier–Stokes equations. But when a bird flies, it doesn’t know all of that.
To some extent, modeling things with uncertainty is how you reframe the optimization problem to get rich and robust solutions. A lot of everyday examples fit into this. I’m a big baseball fan, and I remember reading a math paper about baseball. Once the ball has been hit, a lot is known: the velocity, the trajectory. You’d think the outfielder would take a straight-line path to where they expected the ball to be. But when you observe outfielders, they don’t take straight-line paths. They take a slight curve.
One way to understand that is that it’s a very robust strategy. When you take a curved path, you can continually reevaluate the trajectory of the ball and adjust as needed. Nature finds very interesting solutions, but it’s not always obvious what the right formulation for the optimization problem is in the first place. Sometimes modeling uncertainty, so you can design solutions that are robust to not having full system knowledge, is an important piece of the picture.
There’s a fun aside related to this. Insects, in particular, can’t measure their full flow field. They can react to gusts of wind faster than the information that hits their wings can go to their brain and back to their muscles. Researchers have studied this in biology for decades. They’ve found strain-sensitive neurons embedded on the wings, maybe a few dozen to a few hundred, in very stereotypical locations.
Other fruit flies will have the same locations for these strain-sensitive neurons, and those neurons fire when the wing bends in a certain temporal pattern.
So it gets hit by a gust of wind, the wing bends, neurons fire, and a local computation happens in that shoulder muscle of the insect. It goes directly back and corrects robustly based on partial information about a system it doesn’t have equations for, which is nonlinear and turbulent. So it’s, again, proof by existence of exactly what you’re saying: maybe the robust strategy is actually relying on partial information and uncertainty. They can do more than we can at this point with way less information.
Yeah. I have a couple of—I apologize—pothead thoughts, because Alex mentioned black holes. I have a fun thought that maybe the singularity observed with a blow-up in the Navier–Stokes equations comes into play when, as opposed to airplane wings on Earth, we’re traveling through 2 connected black holes in a warp drive that we figure out. Then there’s the flow control of spacetime, and that becomes relevant because now you’re in a black hole, hence the singularity.
For people just listening to this, the look Alex just gave me hurts my soul. Okay, moving on to the next topic.
But the forcing—you’d need the black hole vortex to also be forced at exactly the right—
Yeah, the force. Where’s the force? I mean, at that point—
How do you...? Yeah, you could try to ride it. There’s a way—you can make something out of this.
There’s some nonzero probability that future us will listen to this and think, “That was the one time that guy said something that was actually correct.” If it’s okay, let’s take a quick bathroom break.
Yeah, sounds good.
8. Unique Games Conjecture and complexity theory
Cool. Ankur, one of my favorite topics is theoretical computer science. You’re one of the world experts in many of its problems. I’d love to ask you about one of the big problems that’s been solved in this OpenAI drop, which is the Unique Games Conjecture. Can you give people an intuition about what this problem is, why it’s a big deal, and what aspects of the proof are interesting?
Yes, I’d love to. Let me give you a little bit of background. Let’s start off with complexity theory 101. One of the most important problems in theoretical computer science is the satisfiability problem. The thing to keep in your mind is: imagine you’ve got variables X₁ through Xₙ, and they can take on either true or false values. I’m going to create a big formula out of them, and usually the way we do this is with an AND of ORs. So I start off with a clause that’s an OR: X₁ or not X₃ or X₇. Depending on my assignment, that clause might be satisfied as long as at least one of those things is true. Then I take the AND of many of these different clauses together.
The question is, if I give you that formula, called a conjunctive normal form formula, can you find a satisfying assignment? Can you find an assignment of X₁, X₂, up to Xₙ, setting them either to true or false, so that all of those OR clauses are on and the entire formula evaluates to true? This is the canonical example of a problem that’s easy to check someone else’s answer to, but might be hard to find an answer to yourself. If you told me it’s satisfiable, all I would do is ask you, “Okay, give me the satisfying assignment.” You tell me what X₁ is set to, what X₂ is set to, and I just go clause by clause and check to make sure that you really have satisfied everything.
The central problem in theoretical computer science is the P versus NP problem. For problems that are easy to check, is it easy to find their solutions? The presumption in our field is that no, it’s not easy, and that’s the basis of a lot of things like cryptography and e-commerce. The starting point for getting to something like the Unique Games Conjecture is something called the PCP theorem. This was a landmark result proved in the ’90s. It was such a big deal that it made the front page of The New York Times. The crux of it is this idea that it’s even robustly hard to fool a verifier.
You start off with a formula, and either it’s satisfiable or it’s not. If it’s satisfiable, you can write down an assignment, and I can check one by one that the entire thing evaluates to true. But if it’s not satisfiable, that means someplace you’re lying. You write down this purported satisfying assignment, and when I go clause by clause, I’m quickly going to find one clause that’s not true, and that invalidates the whole thing. I say, “Aha, I caught you. It’s not actually a satisfying assignment.”
The idea behind things like probabilistically checkable proofs, without getting into a lot of the technical definitions, is really gap amplification. Can you start off with a formula that’s hard to decide whether it’s satisfiable or not, and engineer a new, maybe much bigger formula that’s robustly easy to check? If it’s satisfiable, it’ll still be satisfiable. You can write down an assignment, and it’ll be fine. But if it’s not, I should be able to just flip a few coins, query random variables, check a random clause, and with good probability find that you’re lying. The point is that with very sublinear inspection—with only a constant amount of inspection—I can find that you’re cheating.
This was a momentous result in theoretical computer science because it laid the foundation for what’s called hardness of approximation. A lot of algorithms and complexity theory is about what you do when you’re faced with problems that are genuinely hard to solve. One of the things we do is try to approximately solve them. Maybe we can’t find an assignment that satisfies all the clauses, but can we find one that satisfies many of them? How large a fraction of the clauses can we satisfy?
Even there, we would expect problems to become computationally hard after a certain point. It should be that it’s not just hard to decide whether something is completely satisfiable or not completely satisfiable, but maybe even to decide: is it 100% satisfiable, or is it only 99% satisfiable? The best case might be that an assignment satisfies at most 99% of the clauses. That too might end up being hard to decide. These are the kinds of things probabilistically checkable proofs give us. In places where we knew the optimization problem was hard, they tell us that it’s even hard to approximate.
The Unique Games Conjecture is a strengthening of the ordinary PCP theorem because it has to do with a certain type of label cover instance. It’s a bit technical to go into the details, but the point is that if that version of a probabilistically checkable proof were true, it would have this great organizing effect. For a huge swath of problems, we would actually know what the best approximation algorithm is, and it would be something very simple.
When I started off in the field, the Unique Games Conjecture had been posed a few years earlier, and there was still a lot of debate about whether it should be true or false. People worked hard to try and prove it. People worked hard to try and disprove it. There was this amazing time period, maybe in the 2010s, where there was quite a lot of back and forth. New results would come out, and people’s opinions would change about whether they believed in Unique Games or not. I like this quip—I forget who said it, so I can’t properly attribute it—but someone told me, “Maybe God’s going to decide whether it’s true or false based on which side is working harder.” It really seemed to change year to year.
One of the headlining results in the OpenAI drop of proofs was that the Unique Games Conjecture is true. I should mention that this wasn’t without spectacular human progress on it. One of my colleagues at MIT and one of my colleagues at IAS both contributed, among other people, to a lot of the major strides. Dor Minzer and Irit Dinur proved a weaker version called the 2-to-1 Games Theorem. This really was a shot heard round the world in theoretical computer science, because at that point I think a lot of people weren’t sure what the path forward was for the Unique Games Conjecture. Their work really gave a viable path forward to something in the vicinity of Unique Games.
It sounded like maybe humans could get there with another 10 or 20 years. But just when OpenAI released the results, it turned out that the Unique Games Conjecture is actually true. This is quite remarkable because at some points I guess I did find myself on the side of the doubters. Unique Games just sounded too nice to be true because a lot of work, like Prasad Raghavendra’s work, implies optimal hardness of approximation for so many different types of constraint satisfaction problems that the conclusion just looked to me too strong to be true.
What did you feel when you found out that night?
For starters, there were already rumors in the weeks leading up to it that OpenAI had solved the Unique Games Conjecture and that it was true. These weren’t just completely arbitrary rumors. One of my colleagues, Dor Minzer, actually posted a paper where he wrote in a footnote, “We would have liked to polish this paper more, but because of the rumors that were going around, we feel the need to put out our approach sooner rather than later.”
For me, it was in some sense a bit of a relief that the results were finally out there. They’re not very readable papers. It’s not like you can point to them and quickly identify what the new key idea is. But I’m glad they’re out there, because now we can move forward to the next phase, which is digesting those proofs and figuring out where the field moves after that.
Now, in terms of how I felt that night, as Alex mentioned, we all had a drink in hand. I saw the news on my phone that these proofs had dropped. People kept coming up and talking to me at the gala, but it’s already hard to socialize as a mathematician, let alone when you have math you really want to read. So it was doubly hard.
The other funny thing was that I was already in New York, so before the gala I took the opportunity to visit a lot of people I knew at NYU. I happened to be meeting with Subhash Khot, who introduced the Unique Games Conjecture. A few hours before the gala, before the OpenAI paper was released, I asked him, “Subhash, what would you do if they released a positive proof of the Unique Games Conjecture? Are you going to read it?” He told me, “No.”
That’s a lot of people’s reactions, unfortunately. It’s just true that there are grand challenges we work on, and sometimes when they get solved, it feels like someone has stolen your thunder. It’s not like people didn’t have ideas about how to solve the Unique Games Conjecture, and I bet those blueprints would have panned out if we’d given them more time.
The other thing I worry about is that we lose out on the human effort of trying to reach those solutions in the first place. Math isn’t just about asking, “What’s the furthest I can reach with the current set of tools I have? Is it a convex hull? Is it a subgroup?” I don’t know. Sometimes, with a human-driven solution, we learn a lot more in the process.
I worry that there were these great landmark, field-defining problems that a lot of people were reaching toward, and decisive progress was made, with the results all dropping in one night. People might lose their enthusiasm for going down those paths. If we’d given those people the space to go down those paths, what other exciting things would we have discovered?
We were all trying to process it at MoMath, and there were a lot of really nice talks. Not all of them went up on YouTube. Some were about benefactors and awards named after them. The thing I kept coming back to was that we do math in the first place to play and learn about the world. We saw so many manifestations of that that night: the benefactors donated to the museum because they wanted their kids to have a place to play with mathematical ideas and learn about the world. It was also the opening of an Escher exhibit. What a great visual of what happens when we play with mathematical concepts.
In this moment, it feels really important to remember, as we go through all this change together and don’t know what the future holds, what AI and mathematician collaboration will look like, or what hiring will look like, that we all got into this because we were very curious. We wanted to understand the world, and mathematicians are very good at abstracting things. They don’t just understand one little piece; they find an abstraction or a set of definitions that turns out to be unreasonably powerful in some other corner of mathematics.
Whatever the future holds, it’s really important to organize our field in a way that still encourages people to play with ideas. I don’t know what kind of reward system will work, but when I talk to young people in the field, they have a lot of anxiety. They hear stories about people they know running harnesses to scrape the web for open problems, then dumping as many papers as they can to flood arXiv. I don’t know if that’s a good strategy. It doesn’t seem like it, and it certainly doesn’t feel like fun to me. It doesn’t feel like we’re playing with ideas and learning about the world, and that’s sad. I just want to get back to the things I really enjoy.
The other thing I want to add is that it’s a big identity crisis for a lot of people. Some people view themselves more as problem solvers, and that can feel very comforting: you can draw a little circle around the types of problems you work on, the types of techniques you use, and the types of problems you care about. That feels comforting because the rest of the world is messy and you’ve got blinders on: “This is the part that I care about.” I think now is the time to remove those blinders, because I don’t find that to be a productive way to do it. We love playing and learning about the world around us. As a young mathematician, it’s important to ask yourself: Are you smarter than you are curious, or are you more curious than you are smart? It’s very important to be more curious than anything else.
You mentioned fun. There could be another perspective on it, more about the struggle, from my maybe outsider perspective. Some of the most fun in the moment sure feels like struggle. I’ve talked to a lot of Olympic gold medalists. A lot of them feel something approximating depression after they win the gold medal, because it’s like you’ve given everything, and you realize everything you’ve given is now over.
9. How mathematicians think
About 30 years ago, there was a documentary about Andrew Wiles. There’s a great scene—I think they even run it right at the very beginning—where the question is, “What do you do once you’ve won the thing?” He’s sitting at his desk, describing the moment when he realizes what the solution is. It wasn’t his initial announcement, which was flawed; for many months, he was trying to correct it, and then he finally saw how to do it. He’s trying to say, “I will never again in my life have a moment as powerful as the moment I experienced at this desk that day,” and he’s choking up.
You can’t get the words out. Can you get that if you press a button and it gives you the solution? Absolutely not. So, exactly to both of your points: it’s the journey. If you see a giant hill, a giant mountain, and say, “One day I’m going to get to the top of that,” and start training for it, doing it little by little, and then a helicopter lands and drops you at the top, can you actually speak to that?
The interesting thing about Andrew Wiles, and a little bit about Grigori Perelman as well, is working alone for a prolonged period of time. What do you make of that? How common is that in mathematics, sitting there and grinding on a problem for, in his case, 7 years?
Until the 20th century, almost every math paper ever written had 1 author. I think now the median is 2 in mathematics, so we’re not a lab science. It really is a solitary process.
The things that I give my graduate students to do are not the things I’m working on, sort of by design. I want them to have their own field. I want to open doors for them. It’s different in different fields, right? When they have a success, it really should be their success. Even if I had an idea that this would go somewhere and they executed on it, that’s theirs.
The paper I put out a month ago is something I’ve been thinking about for 10 years. It’s a 5-author paper. Over those 10 years, we made incremental progress to the point where we finally had a solution. That paper was an instance of, “There’s a phenomenon here, and I don’t understand what it is.” It wasn’t, “Here’s a problem; solve it.” At the start of the project, we couldn’t give you the Lean formalized statement that needs a sorry. You discover what the statement is through the process of hacking away at the problem, little by little, until you see enough of it to say, “This is how it comes together.”
Actually, that makes me think: can you describe your thinking process when you think through really hard problems? Are there aspects to it that are interesting, like having a lot of coffee in the morning, working at night, being alone and brooding with a cigar, or working together with others at a whiteboard? Is there some interesting color, texture, or insight you can give?
Sure. I think a lot through world models. I have some mental model of how I think results in a subarea work, and I think that colors a lot of how I see the surrounding areas, too. A lot of my work is about building bridges between algorithmic thinking and other problems that usually aren’t studied through that same algorithmic lens.
When I first started as an advisor, I made the mistake of giving students questions that were 100% in my area. The problem was that when they came back with an idea, I could shoot it down very quickly. I was either right or wrong, but either way, no one was having any fun.
Over time, I started giving students questions that were 10 or 20 degrees off from my own area: things I wanted to understand better myself, papers I thought were important but hadn’t had time to sit down and read. Then I’d have the student explain the result to me. What’s fun is that we would go through that process together, not by printing out the paper and going through it line by line, but by trying to reverse-engineer why we think it works in the first place. There are so many details that, once you get the feel for it, you have the sense that they ought to work out.
In important papers, there are one or maybe even more key ideas that really change your perspective: “Oh, this really might be doable.” As soon as we start trying to understand one of these results, we find our own way of looking at things. Before long, we find other questions that we now think are approachable because we’ve found our own way of organizing our thoughts.
That’s often how I do it: the earlier I can put the student in the driver’s seat—not necessarily with a well-defined question, but with them teaching me something and being responsible for distilling, digesting, and abstracting it down to its essence—the better. We reason together through analogies that aren’t meant to be literally true but help us figure out what to expect in neighboring areas. Then it feels like they bring a unique perspective, and we’re off to the races.
For me, in a lot of ways, the technology is only going to enable that kind of work, because it means a much richer set of problems are approachable. When I find experts in related areas who can really explain things in my own language, I hold onto them for dear life. They can speak the gospel: they can tell me how an area is organized and explain it in my own language. But now I have technology that can do that, too, and it also feels a bit sad that you lose the human element that was there before.
I think students also have so much anxiety about what publications are going to look like. One of the hardest parts is that, when I’m thinking about a project, I’m really trying to think on a multi-year time horizon, because I have some meta-vision I want to realize. How can we use learning theory techniques in quantum many-body systems? There are a lot of stepping stones on the way to that, but some of the most exciting problems are many years out.
But students are so worried about the incoming AI tidal wave, the future of work, universal basic income. Are any of us going to have jobs? They’re really thinking in three- and six-month horizons, and then it’s hard to find problems that we both care about, because we’re thinking about different worlds.
You don’t think we should encourage them to continue thinking long-term? I understand that—
Oh, I do, but they don’t always listen to you. I don’t know if I’m right or wrong. Maybe AI will solve all of our problems, but it seems like the only problems worth us working on at this moment are the ones we’re still going to care about in a bunch of years. You should only be working on future-proof problems. If you’re in a rush because you think the next generation of models will definitely solve the problems you’re working on, then maybe just go on vacation and don’t solve that problem.
Can you speak to how you think about this? You work with a lot of natural systems. What are the interesting insights that could be pulled out of these natural systems?
I find a tension between focusing very hard on an engineering problem I’m trying to solve, which can be very singular—you put on blinders and only think about this problem—and giving yourself the mental freedom to look around, observe, and just be curious. Very often, you find the solution when you turn over a rock, when it’s not something you’re thinking about with the full front of your mind, but the side of your mind. If you’re looking at a problem through your peripheral vision, you sometimes have a much better chance of solving it.
This is a constant balance for me: observing all the fields I don’t work in, all the nature I don’t understand, and all the things I’m curious about, while also having very focused problems that you don’t relent on, that you don’t let go of. This must be true for all of us. I’ve heard from both of you about focused problems you’ve been working on for years and decades that you haven’t let go of. But presumably, that’s informed by a million other pieces of information you happen to pick up through interesting conversations and interesting—
Do you have examples? Neighboring areas sound like a common theme. Are there examples where your curious peripheral vision went, “Oh, that’s an interesting system; it might be relevant,” and then it ended up being relevant, or maybe it will be in the future?
For me, as a fluid dynamicist, biology—with its messiness and complexity—and materials science, with its messiness and complexity, are neighboring fields in a way that’s very fruitful. I learn a lot from both of those fields, and a lot of the basic math we develop to handle complexity does seem to transfer. Sometimes you have an idea just because you see something is possible, or you start to develop something and can explain something we didn’t understand before.
Biology, fluids, and materials have this really interesting commonality of multiscale, nonlinear, messy, nebulous objectives, and things like that.
By the way, on that topic, do you think there’s hope that AI can have some significant breakthroughs in the messy world of biology?
I certainly hope so, and I think so. There are large efforts in biology to collect very large datasets and understand complexity in systems. For example, the connectome, where you have the full wiring diagram of the brains of insects. It looks like we’ll start to have connectomes of other animals, and eventually it’ll probably be like genomes, where it’s cheaper and more accessible.
I have this feeling that we’ll find out—speaking to the resilience of birds—that, in order to understand how the human biological system works, you have to pull that thread, and you’re going to have to understand how the entirety of life on Earth emerged and functioned as an entire mechanism. It’s all tied together. You can’t replicate—It’s ridiculous what happens with humans. From a single cell, this whole thing builds. The intelligence built into the whole building mechanism in the first days, weeks, and months of life could be many orders of magnitude more complex than some of the hard mathematics problems we’ve been talking about.
Again, I suspect that if turbulence is hard, whatever the hell is going on in here—I don’t mean the brain, I mean the fluids around the brain, the immune system, all of it, everything interconnected together—is also hard. I’m very deeply connected to the history of the bacteria that was here billions of years ago. I suspect the complexity of the entirety of the mechanism of life on Earth is many, many, many orders of magnitude higher than any system that’s amenable to AI today. Just a thought. Sorry.
No, I tend to agree.
Okay.
I think you’re right. It’s incredible to imagine all of the complexity and scales. Inside one cell, it’s a whole city, a whole infrastructure. One of your neurons has garbage collection, power production, information processing—the whole thing, at every single scale, is beyond our ability to fathom.
You’re absolutely right. I think that’s what makes it so interesting. It’s related to math in the sense that finding the right scale where we have any hope at all, and where there’s something interesting we can use to understand that whole picture, is fascinating. Up until now, it’s been a human endeavor to find the bits that are interesting and understandable in this mess of complexity. And—
Yeah, I think you’re right.
I mean, the fact that anything is understandable about the natural world is a miracle, right?
Yeah, absolutely. There’s no reason Newton’s laws should be expressible in a form whose mathematics Newton himself could describe, or that he could invent calculus for the purpose of saying what the inverse-square law results in: elliptical orbits. He could solve the two-body problem, and so on. The fact that the universe speaks mathematics is completely mysterious. There’s no a priori reason that we should be able to do that. We don’t have a right to know anything that we know.
10. Is math invented or discovered?
I apologize for the question, but I’m obligated by the universe to ask it: Given this topic, do you think math is invented or discovered, Alex?
It’s the age-old question. I have a very pragmatic answer to this. The questions we end up asking seem to me to be functions of what’s going on in the world and what questions we can possibly ask, given our current state of knowledge. The questions we’re trying to answer today aren’t questions Euclid could have thought about. The things people got interested in were different in Greece, India, and China.
But then, once you ask a question precisely, the answer is what God decides the answer is. So we’re discovering answers to questions that we’ve invented.
We’re discovering answers. Okay. Steve, you’ve worked with a lot of natural systems and look at equations. Equations are, I guess, always approximations of the real world. So, is math real?
You gotta say yes.
Right. We’ve got a lot of pressure here. How big is the gap between the mathematical world and the real world?
Surprisingly small. I think unexpectedly small. I like your answer a lot, actually. I do somehow fundamentally think that math is both created and discovered. We create mathematical formalisms to understand the world around us, and those approximations are unreasonably effective.
It is a miracle that there are simple, compact descriptions—F = ma, E = mc²—that describe everything we observe at all the scales we seem to observe. We seem to be able to distill something simple and mathematical that describes them. That feels like we are creating those descriptions, but the simple, generalizable description underneath everything seems like something we’re also uncovering.
Do you think that’s a good guide for us, searching for beautiful simplicity? Are we just suckers for aesthetically pleasing things?
I think it’s a good guide for us because we are fundamentally simple. We can only handle so much in our minds. We can only envision so much. I think simplicity is essential for humans to interact with the real world.
If I want to get slightly deeper into the question of created versus discovered, numbers are a good example: integers. We all fundamentally feel like counting numbers are real in an essential way. I can have 1 rock, 2 rocks, and 3 rocks, and that is a number of rocks. It is a real number of rocks. But those are absolutely abstractions that I have created. A rock is not a rock is not a rock. It’s a collection of atoms. It’s a different collection of atoms. They’re not 1 rock or 2 rocks. Do numbers even exist? I wrestle with this. I don’t know. Do we actually believe that integers are a human creation to interact with our symbolic abstraction of what we observe—quantities?
It took humans thousands of years to discover the number zero.
Right.
And then negative numbers. Negative numbers were actually accepted by mathematicians in the West after complex numbers were accepted as a mechanism for study, which seems insane, but there’s a whole story behind that.
What’s your relationship like with infinity? Is it real to you, or is it a nice thing to have as part of proofs?
So much of modern mathematics is exactly about dealing seriously with infinity, right? One of the first times people really start to hit a wall in their mathematical education is when they hit calculus, and there are several reasons for it. One is that they probably have a lot of gaps in their understanding from previous classes. They got an 81 in geometry and a 75 in trigonometry or whatever. Those point to gaps in their understanding, all of which are going to come together in one fell swoop. The Swiss cheese is just not enough; they’re not able to cross that threshold.
The other issue with calculus is that it’s the first time you deal seriously with infinity. You have to add up infinitely many infinitely small things and get a finite number, and have that number make sense. Human beings—Newton and Leibniz—did not understand infinity the way we understand it today. They had an intuition for what it was supposed to represent. But if you read Newton, he was ridiculed in his time for the way he wrote about the “ghosts of departed quantities.” You have to divide zero by zero and have it make sense, and you don’t do it when they’re zero. You do it after they become zero.
The way he was trying to write about these things turned into the crisis in the 19th century, when Cauchy and Fourier proved conflicting theorems. Cauchy said, “You have a sequence of continuous functions, and you take the limit of continuous functions, you’re going to get a continuous function, right? Limits are a continuous process. Continuous functions are continuous. The answer has to be continuous.” And Fourier says, “I’m going to add up a bunch of sines and cosines. It doesn’t get more continuous than sines and cosines, and I’ll get a sawtooth. I’ll get whatever you want. I’ll get a discontinuous function in the limit.” They broke math.
So that’s when we had to start facing it. It was no longer okay to think about infinity the way Newton and Leibniz did. Euler’s thousand-some-odd papers lived in a world where infinity was something you played with, and in the 19th century, infinity had to get professionalized.
So it turns out math has been killed multiple times before.
That’s right. And somehow it’s only a scratch. The vast majority of our first-year PhD courses are the result of crises that happened in the 19th century, when math was destroyed and had to be rebuilt from scratch.
11. Can AI solve P vs NP?
Speaking of crises, let me get back to the P versus NP question Ankur was talking about. How hard of a problem is it to prove? This is sort of the central question of complexity theory: P = NP. Do you think AI will be able to prove this? We should mention that you’re part of a panel. There was initially a debate about whether LLM-based methods could one day make progress on the hardest problems in mathematics, like P versus NP or the Riemann hypothesis. That was a year ago. What’s your updated view on this?
Well, clearly, they can make progress on central questions in the field. P versus NP, maybe just because it’s closest to my heart, seems very difficult to make progress on. It’s remarkable, the things we don’t understand. For the problem I was describing about satisfiability, we don’t actually know any superlinear lower bounds. We think that it takes exponential time, not that much faster than enumerating all possible assignments and checking them. And yet we don’t know anything that has asymptotically more running time than just reading the darn input.
Mm-hmm. We should actually mention: you’ve talked about hard and easy. What is a more technical way to think about what is hard and what is easy?
This is polynomial time versus not polynomial time. NP, just to be clear, stands for nondeterministic polynomial time. So the question is whether things you can solve with nondeterminism, like guessing the assignment and then checking it, can also be solved in polynomial time. If I give you a description of a formula—what the variables are and what clauses they appear in—is there an algorithm that runs in time polynomial in the size of the input and correctly decides whether or not there is a satisfying assignment?
We should mention that polynomial time, exponential time, or superlinear time all refer to worst-case analysis.
That’s right.
Can you explain what worst-case analysis is, and what are its—
Pitfalls.
Yeah, I should say, what are its pitfalls? I think it’s a little counterintuitive for people outside the field to understand that when we say a certain problem is hard, we’re talking about the worst case. It’s like analyzing whether I should marry this person by saying, “Well, what’s the worst thing that could possibly happen?” For human relationships, maybe that’s not such a good strategy. Maybe the average case is a better guide for making decisions. Anyway, I don’t know why I brought that up.
Okay, I have no advice about who you should marry.
This is—complexity theory does not give you advice about that. Isn’t there a dating—an optimal stopping theory for dating?
The secretary problem.
There are a lot of different names for it, but there is a mathematical way. There’s an optimal strategy.
So first, what is worst-case analysis? I was describing the SAT problem earlier, and the algorithmic problem you want to solve is: given any formula, decide whether it’s satisfiable or not. What’s crucial is that when I ask you to solve that algorithmic problem, I’m not telling you anything more about the structure of the formula. I’m not telling you the size of the clauses, what they look like, or what topology a graph connecting them would have.
That’s what worst-case analysis is about: the premise that you can treat all of these computational problems by ignoring a lot of the fine-grained detail about the individual structure, and try to solve them for all possible inputs. That’s been spectacularly powerful, because we’ve identified lots of algorithmic problems that really can be solved in the worst case, and solved really efficiently, like finding the max flow. But the question is whether that’s a useful abstraction. Sometimes you may need to know more about the structure of the instance you’re facing in order to solve it better. That’s really an open question.
This is the area called beyond worst-case analysis, and I would argue that it’s a very important direction for theoretical computer science. It’s hard to figure out what you should articulate about natural problems at the right level of detail, so your algorithm can exploit that detail and do better than it can in the worst case.
Let’s use an example. You were talking about the secretary problem. The way this works is as an example of something called online algorithms. You have a sequence of options coming, from 1 to N. They come at each time step, and your goal is, when someone comes along, you know their ranking relative to everyone else you’ve seen before, but you don’t know their ranking relative to the people who are yet to come. Your goal is to make a decision when the person comes along: do you commit, or do they move on? You want to do that in such a way that you have a good chance of finding the absolute best candidate.
The point is that you can actually get a constant competitive-ratio algorithm. The best strategy is to wait for some amount of time, not make any decisions, and not accept anyone.
Once that probationary window has passed, you take the first person who’s better than anyone you’ve seen thus far. That’ll give you a good competitive ratio, in the sense that you can prove theorems. Assuming the inputs arrive in a uniformly random order, you’re likely to do pretty well compared to what you could do if you knew the sequence of inputs in advance.
Under online algorithms, there’s this veil of uncertainty: I don’t know what the sequence of inputs is going to look like. I want to do well no matter what that sequence is, because I don’t really know how to model it. Now, you can instead say, “Hey, maybe the algorithms I derive aren’t good enough for my applications. Can I do better by trying to model more of the world around me?” Then there’s an art to figuring out what you should model, what you should build into your model, and what you shouldn’t. That seems to me like a tremendously important area to explore.
One other variant of online algorithms that’s really important is the paging problem. Your laptop solves this problem all the time. It gets a sequence of requests for things it wants to recall from memory, and the question is: what should it actually put in its cache, the part that’s easiest to access? The problem, again, is that you don’t know what that sequence of requests looks like in advance. So you can study the online algorithms problem of what’s a good policy for managing the cache. When something gets accessed, I have to put it in, but what do I evict? You can design algorithms that are good in terms of their competitive ratio: the number of misses they make on those accesses is not much worse than the best offline optimum, if someone told me the sequence of requests in advance and I could optimize just for that sequence.
This is a problem that’s been studied since the 1980s and 1990s. The trouble is that the best algorithms you can give—we know tight upper and lower bounds—just aren’t that good. The number of misses they make compared to the best offline optimum is a large factor off, and that just doesn’t compete with what your laptop is able to do right now, because it really makes very few cache misses at the end of the day.
So you’re stuck with this problem that bridges mathematical thinking with trying to model the natural world: what is it about realistic sequences of requests that makes them not worst-case? This is really an inverse problem now. It’s the other side of the coin. Instead of someone handing you a problem that you’re just trying to solve in the worst case, where we can all agree on how it’s formulated and what a better algorithm is because it does better on the competitive ratio, you’re stuck with this very fuzzy problem: how should I model the world around me?
In a lot of ways, algorithmic thinking is necessary for figuring out productive ways to model that problem. You’re not going to build a totally realistic model of what sequences of page requests look like. How am I going to coarsen the problem in just the right way so that it leads to interesting algorithmic problems that teach me about the world and force me to design new approaches to solving these very basic problems? I’d argue that coming out of this space, where we’re only solving well-defined problems, but actually using our mathematical and algorithmic insights to figure out where to dig in the first place, is now more important than ever. It should always have been important, but that, to me, is really going to be thrust into the spotlight.
A simple example: we were talking about factoring numbers. Factoring a random large number is really easy. Half of them are even. Factoring means finding a factor. So it’s really those very special numbers that are products of 2 primes, each about the square root of the size of the original number. That’s hard factoring, sorry.
12. Why LLMs work so well
Which makes me think: one of the things you’ve worked on is the mathematics of LLMs, AI systems, and neural networks. Those things are pretty damn good at modeling the natural world. What can you learn from that? First of all, why does deep learning work?
Do we have another 3 hours?
Do we even have intuition? At the mathematical level, at the philosophical level—we’re in between all of that, the full range. I don’t know how we even enter that.
That’s a big question. Let me try to break off a few pieces of it and condense my own views on this. In the OpenAI drop of problems, there was no result that explained deep learning. There was no result that explained LLMs. There’s no bound we know of that, if we could improve it, would suddenly explain deep learning. The problems just aren’t posed that way.
I would really argue that what’s needed in a lot of deep learning theory goes hand in hand with better ways to model the world around us. If we look at the computational problems that deep learning and LLMs are solving, some of these problems are very hard. Trying to fit the parameters of a depth-four network to data is a cryptographically hard problem. We know that. There exist instances of problems that, if you could really solve them in a worst-case sense, would change our understanding of cryptography. And yet these problems get solved all the time, every day, when we’re fitting much deeper networks.
I would argue that a large part of that is how nice natural data is. There’s the saying in physics that nature abhors a vacuum. I guess my version is that, in some sense, nature abhors computationally hard problems. It’s an art to figuring out why that is, and how to draw the boundaries around the types of problems you’re studying in a way that evades the hardness of the problem in the worst case.
There’s an interesting example from the 1990s. We were, and still are, very interested in protein folding. You can model this in a worst-case sense: I’ve got a sequence of amino acids, an energy function based on different potential 3D structures, and I want to find the 3D structure that minimizes the energy. This problem is computationally very hard, and yet nature solves these problems all the time. So why is that?
There’s a very interesting experiment you can do. You start off with a bunch of sequences of amino acids and try out some heuristics for folding them into 3D structures. These don’t solve the problem in the worst case; they just seem to do the right thing sometimes, whatever that means. Then the only sequences of amino acids that live on to the next generation are the ones that actually have very good folds. Over time, what ends up happening is that you create a body of problems that aren’t hard in the worst case, but the heuristics you’ve developed are very good at solving.
In a lot of ways, what are the structures of proteins that nature seems to fold all the time? If they really were hard to solve in the worst case, maybe the shapes they take on would be very brittle and fragile, and subject to a lot of boundary conditions that make those things behave in not entirely predictable ways. In some sense, a lot of our data has evolved in ways that make these computational problems easier.
I like to think of it this way: you can study LLMs in terms of the optimization problems they’re solving, and you can study deep learning that way. If you look at it from a worst-case perspective, you’re going to get very hard problems that you just tell people, “They’re hard. Give up.” But apparently, not giving up was a really good idea. I think what makes those problems easy is something you cannot extract from the structure of natural data. The fact that our language has evolved to make these problems easier is intertwined with why LLMs work the way they do.
My view is that people working in theoretical machine learning have to become more comfortable with the uncertainty of modeling. When we teach a lot of graduate students in math departments, we teach them how to solve open questions that are well-defined. But teaching people how to deal with messy, uncertain things, how to model them, and how to critique models is an important part of figuring out what research in theoretical machine learning looks like going forward. Maybe it’s important for applied mathematics much more generally, and mathematics more generally too, for that matter.
Is it nevertheless amenable to breakthroughs? One thing I might push back on a little, just on the pure company-incentive side, is what people talk about—the thing I think OpenAI, Anthropic, and all these companies are chasing—which is the self-improving researcher. If they had breakthroughs on the mathematics of LLMs, I would venture to say they wouldn’t publish them, because they need those breakthroughs, right? That’s fascinating, because that is the big breakthrough from the pragmatic perspective of AI: self-improving and progressing. So how difficult is that problem you’re referring to—making big progress on modeling the natural world and understanding the mathematics of how to deal with that—in order to have intelligence far above the intelligence of humans?
Great question. I think alignment is one of the most important topics we’re facing right now, right? Just because it’s very hard doesn’t mean we don’t make progress toward understanding it.
One of the things I’ve worked on recently is low-rank language models. They’re a very simple abstraction of language models, and we were proving new algorithmic guarantees. But then it turned out that real language models really do have that kind of structure, at least approximately, so we started pulling at this thread to see where the idea goes.
One really interesting thing it did was shed some light on something called subliminal learning. This is a neat experiment you can do with your favorite LLM. Take 2 LLMs, LLM A and LLM B. For LLM A, you give it a system prompt that says, “You really love owls. In every conversation, try to bring up owls if it’s at all germane.” Then you ask it something concrete, like generating random numbers.
There’s no way to squeeze owls into that, so it generates random-looking numbers: 48, 37, 8, whatever. You can use those random numbers to fine-tune another language model that hasn’t been given the prompt about owls. You give LLM B the same base prompt—generate random numbers—and then fine-tune its parameters so it’s more likely to generate the numbers that model A generated. It turns out that hidden within those numbers is a love of owls. When you do that, LLM B starts bringing up owls in every conversation you have with it.
This is a very mysterious thing. It’s called subliminal learning because when we fine-tune language models on data to nudge them in different directions, depending on the task we want them to solve, there can be a lot of externalities. We may actually be teaching them other behaviors that aren’t obviously present in the data. How do we understand the relationship between these behaviors and what happens when we’re trying to do alignment, including the downstream implications?
It turns out that mathematical models coming out of some of our work can at least help us wrap our heads around this. At a high level, you’re embedding a lot of concepts linearly as vectors in a high-dimensional, but not enormous, space. The number of concepts is so much larger than the dimension of that space that the vectors can’t all be orthogonal to each other. When you fine-tune in one direction, it has small externalities on other directions. These might look random, but you can coordinate them in a way that makes the small linear effects really add up.
The way I like to describe this is with a Ouija board. Those were popular when I was a kid. We had very strange board games; we’d sit down and want to talk to ghosts. The rough idea is that you ask a question to a ghost, maybe your ancestors, and then you tap on this little piece in the middle. It moves around from letter to letter and, hopefully, spells out some words that answer your question. The point is that when the taps aren’t actually random, but are coordinated in the right way, you can get it to spell out whatever you want.
That’s exactly what language models are doing as a byproduct. There are a lot of statistical relationships in the structure of our data that they magnify and amplify, in addition to all the amazing things they do, like quasi-reasoning. Understanding what those relationships are, and how the models magnify and amplify them, is really important to working with these systems. That’s especially true when we’re deploying them, and it’s not just a very safe environment, like outputting a Lean proof of a theorem we care about, but a messier problem that actually interacts with people.
Yeah. We’ve talked about the Game of Life and cellular automata, where incredible complexity emerges from simple rules. In the same way, what you’ve been talking about is a single, large LLM system. If you take that one system and have it do a lot of reasoning, it’s already a dynamical process that can interact with tools in the outside world.
Now you take that and have multi-agent systems, or very large swarms of agents, and you have a kind of Game of Life cellular automaton. The seeming obsession with owls that you discovered—a slight obsession with owls—could result, in about 4 hours, let’s say, in all humans being dead and owls everywhere.
And it all started from random numbers.
All just random numbers. It’s a caricature, but we’ve already seen evidence of this. I apologize for taking it to such a caricature, which makes it not serious. Stepping back, these systems have a lot of power because we depend so much on the digital systems around us. As AI becomes more and more powerful, it becomes more and more useful to us, so we begin to depend on it, give it more access, and give it more freedom to roam the public world.
Now, when we study the mathematics of these systems at scale, we start to see the cellular automaton effect: incredible complexity emerging from simple goal-obsessed agents. So understanding this obsession with owls is very important, so they don’t accidentally do a lot of damage to our civilization, which is fragile in many ways—sociologically, geopolitically, and technologically. It’s a pretty fragile civilization that we’re living in. Understanding these systems as things that can act at scale and very rapidly is important. That’s a very real reason to have rigor, and especially provable rigor, in understanding some of these systems.
But we also don’t want to outsource nearly so much decision-making to these systems, for exactly that reason.
Yeah. We have infrastructure and protocols in place already for safety-critical and high-impact decisions. We have the FAA, the American Medical Association, and the Bar Association for things that affect humans. I think of how we incorporate AI into our general processes around the world along 2 axes. One is how much we trust the output—its verifiability or trustworthiness—and the other is how high the stakes are if it’s wrong.
If AI spits out a wrong proof, nobody goes without power. It’s fine. That’s very low-stakes, and it’s extremely verifiable, so you know quickly whether it’s right or wrong. That’s the perfect use case: an open-ended design space. Designing a better vehicle, a better car design with lower drag and better fuel efficiency, is low-stakes if it gets the design wrong because you can always test it in a wind tunnel and verify whether it worked. That kind of design-optimization space is perfect for AI.
We as humans have to be very careful not to blur what needs to be verified. We have protocols, legal systems, and infrastructure for verification. But it’s something we need to think about. It’s a very slippery slope. We need to have that firewall.
What do you think are the hardest problems before us right now in mathematics, or, for Steve, perhaps more broadly outside of mathematics? What problem is unlikely to be solved in the next year or two?
I’m still going with P versus NP.
You think that? By the way, do you think it’s possible that P equals NP?
Sure. Anything’s possible.
What does that world look like to you?
It depends on the running time of the algorithm. It could be so large that it’s impractical, and then it doesn’t really matter that much. It would be more of a tidal wave for theoretical research. But it seems like a very important problem and very hard to solve.
I remember, back when I was a graduate student, there was an attempt on P versus NP that was semi-serious. Some people started spreading rumors: “Hey, has it been solved?” My first reaction was, “No, it has not been solved.” I remember spending an afternoon thinking about it. The betting markets were still pretty new then, so they wouldn’t let me take a counter-bet that P versus NP had not been solved. Instead, I bet on all the other Millennium Prize Problems, because that seemed like a good way to hedge. The question was which Millennium Prize Problem would be solved first.
Mm-hmm. But is it possible that P equals NP in the real way? Is there something deep that we don’t understand about complexity theory, computer science, or algorithms, where there are fast algorithms for satisfiability?
I think there are 2 threads in there that are important to tease out. First, it seems that so far you can always manufacture hard instances of problems, and we use that to great effect for our cryptography systems. But in practice, a lot of problems seem much easier than we give them credit for.
For things like SAT, one place they come up is in compilers and register allocation, and SAT solvers are amazingly effective at solving those kinds of SAT problems. We don’t really know what makes those problems so eminently solvable compared to the other hard problems that come out of cryptography or other areas. But it could be that in practice, a lot of the problems we model in the worst case are secretly easy, and the harder part is articulating what structures make them so eminently easy. At the same time, there’s still a rich source of hard problems that we can use to build cryptographic systems.
That would be my bet in the world we live in, but that’s just a guess.
So maybe there’s some way to quickly classify that this particular example of the problem is easy—to identify quickly that it’s one of the easier instances. There’s a whole set, a subset.
It could be. I’m reminded of the opening line of Anna Karenina: every happy family is happy in the same way, and every unhappy family is unhappy in its own way. My worry is that many problems are easier than we make them out to be, but they’re easy for very different reasons.
That doesn’t mean algorithmic perspectives aren’t useful for trying to classify or develop an ontology of those structures. But I think the types of SAT problems that come out of register allocation are very different from SAT problems in other domains, even though those problems might be easy too. I just think they’re easy for very different reasons.
So, to you, the hard problem is P = NP. You’ll be impressed if that falls within the next couple of years?
I would be pretty impressed.
Alex, what’s the hard problem that you’ll be impressed by?
Well, I’m already impressed with quasi-RH, but it’s not RH. It’s hard to think of a more iconic problem to say, “This is one of the hardest problems in mathematics.” It’s been around for so long.
13. Grigori Perelman
What do you think about that? Sorry, it just made me think of Perelman. What did he say? He said, “If the proof is correct, that’s all the credit I need.” He famously turned down the Fields Medal. What do you think he would say about all this, the OpenAI drop?
I have a very hard time predicting what Perelman will do or say. I didn’t know he would decline the Fields Medal. I didn’t know he would decline the million-dollar Clay Prize.
What do you think about turning down a prize in general? A few people have done that in the history of prizes.
He’s the only one who has turned down a Clay Prize and the million dollars. What do I think about it? Mathematicians are weird.
Okay. Yeah.
To each his own. We love diversity of opinions and thoughts, and that’s what makes the world of mathematics so wonderful: There are so many different kinds of people, and we let them all in.
These questions are coupled with what the mathematics community is dealing with now and the value of prizes. In some sense, his particular worldview is aligned with this new worldview that the mathematics community is entering, where it doesn’t even make sense to ask whether there’ll ever be another Fields Medal. We’re entering the Perelman world.
We were talking about that explicitly this summer, when the latest round of Fields Medals was given out in Philadelphia. People were following the progress, and there hadn’t yet been this latest bit of progress that puts the question in the most powerful way possible. But already this summer, we were thinking, “Is this the last time a Fields Medal is awarded?” I sure hope not. But we have to figure out what that means.
You have to think more sociologically about what prizes accomplish. Why have these prizes? The people who win them are already tenured professors at top places, and they’re going to be doing mathematics for the rest of their lives. Perelman is a very well-known exception. So what do prizes accomplish? They celebrate the activity. They say, “This is what we think is important. This is the highest instance of something.” There will still be activities that we want to celebrate.
I don’t know if, right now, prizes are given out for solving major problems. AI systems can solve major problems. That’s not something that will be rare, and hence, it won’t be nearly as valuable as it was in the past. Despite our earlier discussion about the solved axiom, again, I don’t believe the solved axiom. I believe centaur math will be a great way to do math.
Centaur math, okay.
That’s how I currently feel. I’m constantly talking to AI systems, and I’m learning a lot of math a lot faster than I’d be able to otherwise. Sometimes I give a paper to a grad student to read, and they come back a week later. I say, “So how far did you get?” “Well, I didn’t really get past the first page, to be honest.” “Oh, really? Tell me.” “Well, actually, I didn’t get past the first paragraph.”
I say, “Okay, fine. Why not?” “Well, the first sentence is, ‘By the perturbation theory of nonlocal systems,’ and I don’t know what any of that means, so that’s where I’m stuck.” I say, “Okay. You don’t know how to read a math paper. You skip the first couple of paragraphs. Go straight to Lemma 3. That’s where you can start working, and work your way backward to what the thing is doing.”
In the future, there is no “I don’t know what this means,” because you can upload it to an LLM, start asking it questions, and it will zero in on the zone of proximal development: the place where everything you know meets the next thing you need to hear so you can start learning. If people use that effectively, they can supercharge their education. I don’t just mean kids. I mean old fogies like me, who can learn new fields.
Our graduate students have to completely rewire their brains. Learning math is still hard. Up through the master’s level and into the PhD level, you’re not expected to produce new mathematical theories. Everything you can do is something that’s in a textbook somewhere. And yet it’s extraordinarily difficult to learn to solve problems that lots of other people can solve, never mind problems that haven’t been solved.
Learning math is hard. Learning new math is even harder: discovering math, working out solutions to math problems. But I think it’s going to get easier, and this tool is going to be an amazing resource for people to reach that zone of proximal development much more rapidly and develop their understanding much more rapidly.
I’ve been having so much fun exploring topics outside my field, and I would love to go back to grad school with the tools we have now. One of the ways my brain likes to learn is by constructing visual examples, toy examples—the simplest possible example that still contains the insight—and especially if it’s interactive. If I can move things around, I immediately get it.
All these words that the community agreed mean something, all the notation you’re supposed to know—if I can just learn that this little symbol means this, then everything kind of falls into place. You also fall in love with the subject, and I feel like I can take on any problem completely outside my field. I can understand protein folding. I can play with the parameters. I can take a DeepMind paper and actually understand it by exploring different aspects of it and asking, “What’s a simple example?” I can ask crazy, off-the-wall pothead questions. This is super fun.
The question is whether people will be motivated to learn. That’s the real question.
The key ingredients are curiosity, asking good questions, and being able to call bullshit. The critical advisor skills are getting information when you’re not the expert and knowing whether you need to dig deeper or push back. There will always be people who are curious and people who aren’t, people who ask good questions and people who don’t. We need to promote that more than we ever have.
It’s a little scary to me that we’re expecting undergraduates and young students to interact with these tools while they’re learning, when their brains aren’t fully formed. We’re asking them to have critical reasoning skills, but also advising skills: How do you take in information and tell whether it’s true? How do you question it? It’s a huge task for us in education to train people to have good advisor skills when they’re kids.
I think the hardest problem isn’t undergrad. The hardest problem is middle school. At what point does homework start getting hard? At that point, why do the homework?
There’s the temptation, right?
That thing will just do it. I’ve come to appreciate the value of doing difficult things because of all the suffering, but I think that’s because I was forced to. If you’re not forced to do it because an LLM or AI can just help you, I don’t know what kind of society that creates.
Math has the longest head start on that question because calculators came out, what, 70 years ago? We still ask kids to learn their times tables, at least some of us do. Those who said, “No, kids don’t need to learn times tables,” to the extent that teachers listened and kids didn’t learn, they really suffered.
There’s a great amount of human suffering resulting from people saying you don’t need to know times tables because calculators can do it. My hope is that we can learn from that and say that just because a calculator can do it, you’re going to be much better off if you can also do it yourself.
That doesn’t mean that if I have to do some calculation, obviously I’m reaching for a calculator. But I could, in principle, do it myself. Every time you give yourself that skill and master it, you gain something. Kids love mastering skills. How happy are they when they learn to tie their shoelaces for the first time? You would think, “It’s just shoelaces. Who cares?” No, it’s the greatest thing ever.
You want to celebrate every little bit of independence, and you want to be able to think. You want to be independent from the machines. Just because the machine can do something doesn’t mean you don’t want to have the capacity to do it yourself.
14. Advice for young people
Well, this is where I have to ask: There are a lot of people who are full of anxiety. We mentioned graduate students, PhD students, undergraduates, and high school students. What advice would you give, maybe different advice at different levels, to folks trying to figure out what to do with their lives? What would you say to PhD students who are freaking out right now? Is there some wisdom you can give them?
There are different things at every scale. Let me start with PhD students. For PhD students in pure mathematics, my advice is very simple: Do what we have always done. Learn the most difficult, most abstract mathematics your mind can handle, as much of it as possible. Your advisor should hopefully be guiding you through that process.
Then let me go one step down from that. What we’re seeing at the undergraduate level is a bimodal distribution. It used to be that grades were basically Gaussian-distributed. Now we’re seeing a very bimodal distribution between the kids who are using AI to check their understanding—“Wait, I didn’t understand what the professor said in class. What did that mean?” or “I have a test coming up. Here are my notes from the last 3 weeks. Start quizzing me and check if I really know this”—and the kids who say, “This problem set is due in an hour. Can you just tell me the answers? I’ll write them down in my own words.”
There’s a decorrelation between homework progress and exams, and we’re seeing this bimodal distribution. I don’t know if you guys are seeing the same thing.
So the implied advice there is to be one of those who’s using AI to learn and grow—
Be the head, not the tail of the distribution.
Yeah. Yeah. Do you mind if we take a bathroom break?
Yes. Yes, yes.
We’ve somehow made intelligence this pinnacle and have almost equated intelligence with morality.
Right, or value.
And value. I think that couldn’t be farther from the truth. Superintelligence does not imply super morality, and I think that’s a huge concern with AI systems: if they are superintelligent, that has no implication that they will be super moral.
Right. And we’ve been pushing kids to go to college—
Forever. Yeah.
—and that everybody needs to, yeah.
Regardless of what they study or what the outcomes are—
—and superintelligence seems to give you a super ability to rationalize. People with common sense—farmers and the rest of the world—have a sense that that rationalization is often bullshit. Just because you can make it sound really good, and I can’t argue with it, doesn’t make it right.
Right. Education and wisdom are 2 different things.
Yeah, exactly.
I know it’s difficult to do, but maybe, Steve, do you have advice for graduate students, college students, maybe young kids, on what to do in this crazy, fast-changing time?
Definitely be curious, and read and learn all of the fundamentals. I think that’s always been good advice. Even if your computer can do linear algebra better than you, you should definitely still learn linear algebra and build the intuition. Same with calculus, same with physics. I think understanding history and understanding this change in the context of other great changes that have happened through human existence is important. But really, be curious.
I should say that this curiosity in the mind is trying to get stolen nonstop by the engagement economy. The same engine that makes you curious is the same engine that makes you doomscroll. So I think you have to learn how to avoid all the distractions and focus on letting your curiosity explore something and soak something in—learn something.
AI is really good at doing that because it’s incentivized to give you value from the interaction, rather than to get you glued to the thing. I don’t know how people in high school or college can survive with the phone—the distraction box—right there. So avoid distractions and focus deeply.
We all were bored when we were kids sometimes. I think boredom is really important.
So important.
To actually be bored. Curiosity and focus often come from boredom. I would like to be bored more often. I try to do that sometimes.
Yeah, we would come home from school, finish homework in 5 seconds or whatever, and just go outside. You hoped to run into some other kids who had also gone outside. Then you’d go find something to do and come home when it was dinner time.
Yeah, you don’t realize that the high school years, even before that, and somewhat into undergrad, are a really nice time to just learn everything. You think, “When I grow up, I’ll get to actually do something useful in this world. I’m learning; it’s very nice.” But from the perspective of fulfillment, and building a foundation for an awesome life where you’ll remember all these things, this is the time to do it. The whole world is forcing you to do it, and they’ll reward you for sitting there and learning.
When you get further along, even into grad school, this kind of exploratory stuff gets discouraged. God forbid you get a job; then you’re definitely discouraged from just learning and exploring. So the time to explore is high school.
By the way, pre-tenure faculty are in this mindset of, “Once I get tenure, then I can just sit back and relax and think big thoughts.” But first I have to produce enough stuff to get tenure. That’s not what we want our young people to be doing, clawing their way there.
Ankur, do you have advice you give to students?
The problem with going last is that a lot of my advice got taken, so “be curious” was definitely one of them. Now I would say that it’s also important to do things for intrinsic reasons.
One thing I’ve noticed is that a lot of people in our profession, a lot of people who go the academic route, do things for external reasons because they see their value through how other people see them. That’s good enough to get you started. Maybe you start working very hard at math so that people are impressed by you. But at some point, you run out of milestones to hit, and then you really have to have an intrinsic reason for doing the things you do. Hopefully, you have a sense of what’s important to you and what your values are.
These days, you can take our advice about being curious, but the old playbook is really being thrown out. You can take our high-level advice, but you’re going to have to figure it out for yourself, and you’re going to have to be comfortable with uncertainty. The only way I know how to deal with that kind of uncertainty is to have a sense of purpose about what you think is important, what you love about your job, and make sure those are the things you’re doing.
If you just stick to the old playbook, you’re going to end up taking all these AI-generated proofs and spending all your time putting them on arXiv and sending them to journals. I don’t think anyone’s intrinsic reason for getting into it is really to pad their CV. I don’t know what’ll be important in the future, but if you really have some higher purpose you believe in, then I think you’ll be in a much better position to adapt to the changing world, figure out your own blueprint, and be comfortable with uncertainty as the world changes around you.
This is a trickier question, but a lot of people listening to this are feeling a lot of anxiety and fear about the future. It’s about both a job and a future that’s meaningful. Can you look back on your own life, when you had that uncertainty and anxiety? What advice can you give that guy when he’s at a low point?
Let me try to address that question. It’s a very tough question. I’ll tell a short story. We spent our sabbatical in the Bay Area, and we were living in Palo Alto, which was kind of the epicenter of college stress. You could just feel it. You’d go to the cafés, and it was in the water: “What extracurriculars are you doing right now that’ll help you?”
Right when I came back to MIT, they asked me to give the convocation speech there. It was interesting thinking through that speech. What do you say to this entire auditorium filled with freshmen and their families who, if things are a game to optimize how good you look on paper as a 17-year-old kid, have won?
I kept thinking about when I was in that same situation, when I was starting college, and how excited I was just to learn all these different things. I’d sit there and plan out my 4 years. I had little notebooks where I’d figure out what my classes would be, and if I took these classes in just the right order, I could take photonics as a sophomore.
And I had no idea what photonics was, but it sounded fun and fancy. So much of our society now is geared toward winning and looking good on paper, and I always wonder: to what end? You start off in high school and can begin optimizing your APs and extracurriculars as soon as you walk in the door, so you can get into an elite school. Then, when you get to that elite school, you might skip classes because maybe the important thing is to prepare for your interviews so you can get a job at a frontier lab, get some nice restricted stock units, and live out your life.
And that’s great. You won. You won at every stage, but so what? For me, the best way to live a rich and fulfilling life is to pretend you’ve already won, and then figure out what you would do in that scenario. I remember the advice I got when I started at MIT as faculty. Sanjeev Arora told me, “The easiest way to get tenure is just to pretend you already have it and do what you would have wanted to do all along.”
Exactly.
I don’t know if that advice applies to graduate students or high schoolers, too, but I think my advice stays the same: pretend you already have tenure. What would you do now? A lot of the people we’re giving advice to have so many different opportunities. Finding something you really believe in is the only way I know to be fulfilled.
I’ve heard career advice from someone I respected: a fulfilling career is one where you can work on a problem that’s important, that you find interesting, and that you’re good at. If you can get 2 out of those 3, you’re pretty lucky. If you can get all 3, you’re going to be happy for the rest of your life.
Having the right preparation means figuring out what you’re good at, what’s interesting, and what’s important, and then preparing yourself. You’re not guaranteed to get all 3, but you want to be ready when that opportunity comes, and then continue doing the things you actually enjoy doing. I think that’s great advice.
I think we should also put this into context and be quite honest. A lot of the advice you gave applies, but we should be honest that I’m sitting with 3 world-class human brains who have found their path. There’s survivor bias. A lot of people listening to this are pretty sure they won’t find their path. There’s fear, anxiety, uncertainty. They’ve stumbled a bunch of times.
To be fair, some of the most interesting people stumble quite a bit in life, whether because of the external environment or the internal, whatever the universe brings. I know a lot of people are feeling and monitoring that right now because of everything AI is doing to our society. A lot of people are terrified and really anxious.
There’s a lot of wisdom in what Ankur says. You can be paralyzed by that fear, or you can imagine, “I’ve already won. What would the person who has won do in this situation?” That alleviates the fear. It gives you the optimal path under a lot of uncertainty. Don’t use the old playbook. Try to be the person who figures out how to write the new playbook, the next playbook.
I think Max Tegmark, who I’ve spoken with many times, is a wild one. He’s done a lot of weird and beautiful things in his career, but he also did the mainstream stuff. He said you have to do a little bit of both, so you still work enough within the system to be part of the community, part of the flow—in that case, science and the development of things—but also do enough renegade, wild, and weird things to pave your own way. Find that balance. Don’t be afraid to be a little weird, but also become skillful at being part of a community and a respectful part of it. Do both. It’s like Tom Waits said: “I like my Tom with a little drop of poison.” A little bit of weird, or sometimes a lot of weird, makes for some beautiful humans.
I think that community part is actually going to be even more important in the future. We’ve had maybe a 30-year experiment with how much we can do with machines, and now that we can do a ton with machines, my hope is that we’re going to realize we actually prefer to do things with humans. We prefer to fly in and have an in-person conversation. We prefer to have an oral discussion instead of just reading a paper.
I get a lot of fulfillment from my scientific work and my academic career, but going home to my kids and going to the beach if FLT gets solved and just throwing a ball with them—there’s nothing like that. That’s the meaning of my life. This other stuff is the cherry on top, but the foundation is family, being around other people that I care about.
And as AI commodifies intelligence more and more, we’ll realize that more and more: intelligence and all the different puzzles that we solve are just a cherry on top, or a cherry to the side. The main thing is the other humans in our lives.
I can’t tell you how much of an honor it is that you would come down in person for a conversation. I’ve become a huge fan of all 3 of you. It feels a bit surreal, and it’s a big honor that you would talk with me today. This was super fun. Thank you so much, Ankur, Steven, Alex. This was awesome.
Thanks for having us.
A lot of fun for us, too.
Thank you. Yeah.
Thanks for having us.
This was great.