AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory | Lex Fridman Podcast #503
In a Nutshell
AI systems have now solved major open problems in mathematics—including Navier-Stokes, the quasi-Riemann hypothesis, and the unique games conjecture—triggering an identity crisis about what human mathematicians should do when the traditional "solve hard problems, publish, get tenure" playbook is obsolete. The core tension is that while these tools dramatically accelerate learning and discovery, they also destroy the slow, deliberate human struggle that creates meaning, mentorship value, and field-defining insights. The path forward requires redefining human contribution around asking better questions, developing intuition about which problems matter, and building new collaborative systems that preserve the exploratory, curiosity-driven aspects of mathematics that can't be automated.
These notes were generated by AI and may contain inaccuracies.
This is a kind of emergency episode about the state of mathematics and artificial intelligence, described as a 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, including Navier-Stokes, the quasi-Riemann hypothesis, the unique games conjecture, and hundreds of other results.
The panel includes Steven Brunton, a mathematician and fluid dynamics expert at University of Washington who can speak to Navier-Stokes, turbulence, and all things fluid. Ankur Moitra, a mathematician and theoretical computer scientist at MIT visiting the Institute for Advanced Studies in Princeton, can speak to the unique games conjecture, complexity theory, and the mathematics of how AI learns. Alex, a mathematician and number theorist at Rutgers, can speak to the Riemann hypothesis, Lean the automated proof verifier, and many other topics in mathematics.
Alex stated he wouldn't call it a crisis, noting there's a lot of opportunity in that crisis. It's a chance to rethink what it is mathematicians are trying to accomplish. It's the chaos of opportunity that some people are reacting negatively to, some positively, and sometimes those reactions are happening in the same person. He suggested we're entering a period of some kind of golden age of mathematical knowledge after millennia of having a candle that we are walking around the woods, and this is as far as we can see, now we have a flashlight.
The discussion emphasizes the magnitude of this moment, noting that if a human did this, it would be an instant Fields Medal. Many wouldn't have expected this amount of progress in this amount of time, making it truly shocking.
Ankur described the night when the results dropped as interesting. It was not entirely unexpected because for weeks there had been rumors about the two biggest problems in his areas that were in that collection of results, specifically the unique games conjecture and RL equals L. There were even rumors about matrix multiplication. His first reaction was "Wow, it's all true. Thank goodness it's finally out there because now people can actually go through it and process what's happened and figure out what to do next."
Ankur described his feelings changing on a daily basis, with the biggest overarching feeling being pulled in two very different directions. His work has always been about asking the right questions, and these tools are amazing because they increase his ability to learn about other fields. It's like having a world expert who's there to teach about things in your own language, and the ability to make connections is magnified.
However, the part of his job he loves most is the mentorship aspect, including training grad students and postdocs and seeing them do great things. The playbook and blueprint for success that existed for so long is now thrown out the window, making it harder to advise students and postdocs about things they should be working on that will make an impact and make a name for themselves. He feels very excited but more intellectually lonely than before.
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."
Steve Brunton expressed that most of his feelings are pretty hopeful. As an engineer who likes solving problems that make technology more realistic for humans, artificial hearts and airplanes, he believes the ability to solve real applied problems is going to get much better, with scope and reach increasing. However, there is a lot of uncertainty, including fear of meaning, of losing control, of being cut out of the process that everyone loves being a part of.
He noted that no one would be surprised at either of two paths that might take place over the next five or 10 years: either being dwarfed by other intelligence, or it levels off and becomes an incredible tool used for curiosity and human problem-solving. That uncertainty is massive, representing two very different worlds, and uncertainty is difficult for humans. He noted that those who have chosen math probably did so because they liked rules and formalism, and uncertainty is uncomfortable.
Brunton emphasized that humility is really important, and this is a good opportunity to remember why people are doing mathematics. Human purpose is not to be famous or to have a Fields Medal. Solving interesting problems, helping other people, being curious, wanting to understand the world are good things to get back to. He acknowledged it is more difficult for young people figuring out how to find their way, but he has more faith in them than in older generations.
The discussion addressed whether it's harder to be motivated when you know a problem has been solved, to explore that space when a problem's solved by an AI system. Brunton agreed the path is as important as the destination, like how a road trip is usually just as fun as getting there. However, if another country had invented teleportation or time travel, we would reinvent it in the next five years, as knowing it's possible and being able to reverse engineer it provides incredible motivation to get there much faster.
The conversation explored whether this moment marks a trajectory where AI solves math or kills math. Alex separated the concepts of solving math and destroying math. He referenced the solved axiom, which assumes that any mathematical problem that can be solved with human interacting with AI could also have been solved by AI alone. This doesn't necessarily destroy mathematics, as the value of pure mathematics is that it lives in the heads of human beings, not as theorems in ones and zeros in data centers.
Ankur cautioned against overfitting to one objective function, noting that while humans used to think of themselves as the smartest things around, many problems thought to be quite far out have been solved by AI. One consolation is that humans are very computationally limited compared to AI. Given the same finish line of proving the same theorem, humans can't hold that many papers or ideas in their head at one time, so they may need to develop more generalizable and abstractable ways of getting there that help explain connections or other directions.
Ankur clarified that the reason for proving results is to better understand the world around us. Mathematics is a vehicle for abstraction and understanding things. When there's a major result, what's important is not just the truth of the statement, but what was learned en route to doing that. Usually people's first question is how it was done, and once people digest that new idea, new definition, or new way of looking at things, there are lots of other problems nearby that fall. This builds structures that help attack not just one problem but a whole sea of problems.
The discussion addressed how much of this relates to the human ability to shine a light on particular areas that matter, specifically the human ability to ask the right questions and important questions. Ankur noted this has always been the case, as there's a much larger space of theorems that could be proved than the space that human beings ever actually care about. By definition, the theorems that will get excited about are the theorems humans care about.
The conversation explored how Navier-Stokes connects to fluids and natural systems, but also how some mixture of beauty and difficulty pulls at mathematicians. Prime numbers are full of complicated problems that may not have obvious immediate applications but pull at people, and as a symptom of pursuing the hard thing, great applied mathematics and engineering solutions to real world problems come out.
Ankur described his approach of finding the simplest, most embarrassing open problem that he still doesn't know how to solve. When that example can be solved, the set of tools needed to make attacks on other problems may be assembled. He believes mathematicians will need to become more deliberate about what they care about and how to use these tools from a metacognition standpoint, specifically what to delegate to AI and what not to delegate.
Ankur introduced the doorman fallacy, which is the mistake of reducing a person's job to its simplest, most visible task and assuming that automating that task replaces the value of the whole role. A hotel might think they're paying someone to open a door, when an automatic door can do that more cheaply, but this misses all the aspects of the doorman's job that add value and are harder to articulate, including welcoming guests, recognizing returning customers, handling luggage, hailing taxis, providing security, discouraging unwanted visitors, signaling prestige, and making guests feel cared for.
He suggested the task for humans and mathematicians is to come up with that list, making more explicit exactly what it is that humans do. The value of asking good questions has always been something valued in colleagues, and that's going to become increasingly more valuable. Asking the right questions requires a lot of mathematical insight.
Ankur expressed concern about what's going to happen to systems and institutions in the near term, including hiring and tenure. When a grad student starts their PhD, they're a new entrant to the field trying to absorb what the techniques are in their field, what the important problems are, and how to make a name for themselves. What used to be done was giving them a blueprint that was very simple to follow.
The traditional career blueprint in mathematics used to be straightforward: take a problem people care about and improve it, then recognition follows through papers in top venues and career opportunities. This simple rubric has become harder to operationalize because the field is overloaded with results. The core values remain the same—ideas that influence communities—but operationalizing them requires reworking systems and institutions.
The solved axiom—that mathematics could be fully automated—is not believed to be true. Even when proving a theorem, there's often a lack of understanding about why the proof works. There's magic that happens in the process, and generalization of the phenomenon is needed to understand the proof deeply. Mathematical work is more exploratory than simply making progress on famous problems like the Riemann hypothesis.
In mathematics, the gold standard has always been how simple a result can be and how much it generalizes. Out of the infinite space of theorems that could be proven, finding structure that is simple and generalizes remains the gold standard. Humans are still asking the questions—even if AI proves existence and uniqueness of Navier-Stokes, humans are guiding the process.
Research programs are systematically discovering minimal physical laws through observation of natural systems. This involves distilling data down to basics, such as deducing gravity and F=MA from observing balls dropping, or deducing Navier-Stokes from observing fluids. The approach extends to data where simple equations haven't been written down, including weather, turbulence, and solar flares.
The solved axiom suggests that anything a human mathematician can contribute, an AI could also contribute on its own. In a world where AI can answer questions and even suggest what questions to ask, human value must come from something beyond asking great questions or proving great theorems. Human mathematical expertise remains necessary even in that scenario.
AI systems shouldn't be conceptualized through anthropomorphizing. Deep learning systems can achieve human-level object recognition, but they succeed in ways that are less robust than human recognition. Adversarial examples demonstrate this—slight patterns added to natural images can fool AI systems but not humans.
Research showed that when neural networks are trained on fake datasets with wrong labels—cats with dog patterns labeled as dogs, and vice versa—they still perform well on natural data. This suggests AI systems find trapdoors or backdoors using statistical relationships in data that humans don't use. The solutions don't look like human solutions, indicating a qualitative difference that presents opportunities for collaboration.
The question arises whether AI is merely tying together thousands of disparate papers and making minor connections, or actually developing new mathematical ideas. The convex hull metaphor suggests AI operates within the space of known human ideas, doing linear combinations of existing results. The semigroup metaphor suggests AI can tile ideas together in ways that reach much further than humans have explored with existing concepts.
The quasi-Riemann hypothesis proof used moves that were all known to human mathematicians, but arranged in a long cascade that appeared beautiful and unexpected. It's not considered a genuinely new idea, but rather a beautiful combination of well-known elements. A breakthrough on cubic theta functions and cubic Gauss sums two years prior provided the path the AI took to the solution.
The semigroup can be understood as tiles on a board where you can add tiles to the left, right, front, or back. This allows reaching far with existing ideas in ways that might prove much more than humans have explored. The boundary between what's inside and outside the convex hull matters—the expansion changes the direction of interesting questions and alters what types of problems get assigned to researchers.
An Anthropic result on the 3SUM problem had major effects on fine-grained complexity theory. A basic problem given in coding interviews can be solved asymptotically much faster than previously known, contradicting community assumptions. This changed the character of the field because the goal was to study problems together based on inherited hardness, but now they may need to be studied separately.
If given two very large numbers with a million digits each, multiplication can be done in a split second. However, factoring—determining the two prime numbers that were multiplied together—has no known efficient algorithm. This inability to factor large numbers enables all modern secure communications and e-commerce through public key encryption schemes.
Quantum algorithms exist for factoring, creating concerns about when quantum computers become large enough to break current encryption. This has led to emphasis on post-quantum cryptography, including lattice-based methods. While quantum computers aren't yet large enough to pose this problem, there's no theoretical reason preventing polynomial-time factoring algorithms from existing.
The NSA employs thousands of mathematicians. Collaborative teams of mathematicians can solve much harder problems than any individual mathematician when breaking problems down appropriately.
Even if AI systems are only interpolating human knowledge rather than extrapolating, this capability is valuable because humans aren't connecting ideas fast enough. The balloon of mathematical knowledge isn't see-through—there's fog about what ideas are adjacent to each other. Making the convex hull more transparent gives mathematicians radar to see over the horizon.
Human beings make new definitions and bring in new concepts, which fundamentally differs from combining existing ideas. When humans do this, it creates new boundary points that expand what's provable. The size of the convex hull of mathematical ideas is unknown—both in terms of how big it is and what's next to each other.
The archetype of what a researcher looks like will change because of AI technology. Historically, different fields of math spawned from similar ideas, but over time they became more specialized. Graduate student training needs to ensure students aren't overspecialized. People who think broadly about larger swaths of mathematics will be best positioned to benefit from these technologies.
The paper as the primary unit of academic contribution may change. In engineering and applied sciences, contributions already include code, data, and reusable components that fit into larger frameworks rather than just 15-page PDFs. This becomes even more relevant with AI—workable pieces, widgets, APIs, codebases, or datasets that can be leveraged by the community or by AI are increasingly valuable.
Journals in mathematics were already under severe strain from the volume of submissions before AI, simply due to more people producing more knowledge. Referees are exhausted from being asked to understand and check theorems. The system is deliberately slow, careful, and precise in arriving at mathematical truth.
After Grigori Perelman posted his solution to the Poincaré conjecture on the archive, he gave lectures worldwide with three-hour explanations. The first hour covered what he was doing, while the next two hours involved detailed questions about specific lemmas, with Perelman providing half-hour impromptu lectures to clarify points. The entire proof lived in his mind, and the paper represented his best attempt to communicate it, with verification requiring extensive human interaction.
After a solution is proposed, the mathematical community must carefully review it in detail and reach consensus that it constitutes a real proof. The Poincaré conjecture took years of human effort and a deliberate, community-driven process to verify. The journal system exists to accomplish this validation process.
Traditional views portray pure mathematicians as lone geniuses solving singular problems independently. Many other fields have moved past the point where individuals can advance alone, requiring billion-dollar collaborative projects in areas like circuit design, aircraft design, protein folding, and astronomy. Mathematicians tend to be independent by nature, preferring solitude and resisting being told what to study or how to organize.
The current crisis of overloaded journals and conferences presents an opportunity to find new models that relieve pressures without replacing existing systems. Collaborative verification of AI proofs represents one potential approach. Examples include OpenAI results on improvements to the Fast Fourier Transform, where tweet streams showed people improving constants. New modes of organizing state-of-the-art improvements and taming ideas are needed.
Current publication practices involve working on a draft, posting to arXiv, and maintaining personal ownership of that draft. There is no mechanism for community members to add notes suggesting alternative arguments or simplifications. Crowdsourcing models for evolving mathematical understanding represent a potential new approach. Research talks allow speakers to focus on key ideas and special cases that illustrate why concepts are powerful, rather than the monolithic formal structure of papers.
Mathlib operates through pull requests rather than papers, where maintainers and reviewers make suggestions publicly. Contributors can see exactly who contributed what idea that simplified an argument or identify when code reinvents existing concepts. The system allows for refactoring and reuse of existing library components. One fundamental problem with traditional mathematics is the lack of perfectly well-written specifications for theorems and technical lemmas, where authors assume readers have absorbed previous context. Formal mathematics in Lean embeds specifications into objects, preventing misuse since Lean enforces proper usage.
Mathematics can be formalized as a process starting with axiomatized systems and applying logical rules of inference. Lean is interactive theorem prover software that allows line-by-line proof writing, with the computer verifying that each line follows logically from the previous goal state. Originally invented for ensuring bug-free code, Lean can also verify mathematical results by writing them formally. The software checks logical steps but requires a large, well-maintained library rather than starting from basic axioms like the Peano axioms for natural numbers each time.
Mathlib serves as the Lean library for formalizing mathematics within the Lean ecosystem. It provides the foundational library that prevents needing to redefine basic concepts like natural numbers for each project. Lean represents one of several interactive theorem provers, though it has gained prominence in mathematical formalization efforts. AI systems now use Lean and Mathlib to verify proofs formally, building on established projects like the Prime Number Theorem project to check that solutions reach the axioms.
When AI systems produce proofs that cannot be understood by world expert mathematicians in a field, the issue relates to scale rather than fundamental incomprehensibility. The logical rules of inference and axioms are limited in number, making scale the only barrier to understanding. However, mathematics involves more than formal verification—it encompasses stories of fighting with problems and discovering general meanings that apply broadly.
Papers conflate two distinct activities: the engineering of proofs and the communication of results to others. Teaching real analysis traditionally begins with formal construction of real numbers as equivalence classes of Cauchy sequences of rationals, taking two weeks before reaching epsilon-delta calculus. This engineering-focused approach assumes students cannot discuss real numbers until formally constructed. An alternative pedagogical approach jumps directly into epsilon-delta calculations for six weeks before reaching theorems that require understanding real number construction, such as the Bolzano-Weierstrass theorem and its failure in the rationals.
Future mathematical papers may emphasize storytelling about the learning process, interesting special cases, and decision points where intuition breaks down. Lean formalizations can accompany these narratives to verify the engineering works correctly. The emphasis shifts from axiomatic construction to explaining why certain paths were chosen over alternatives. Mathematics is fundamentally about systems of beliefs and abstractions developed while fighting problems, not merely accumulating theorems.
The Riemann hypothesis concerns the distribution of prime numbers through the Riemann zeta function, which takes complex numbers as inputs and outputs. The zeros of this function in the complex plane govern prime number behavior. The zeta function performs signal processing on prime numbers analogous to how audio systems convert pressure waves to frequencies and amplitudes, then reconstruct the original sound. The frequencies and amplitudes correspond to zeta function zeros, with the superposition of Riemann waves recovering prime distribution.
The quasi-Riemann hypothesis result, proving no zeros for the zeta function and all Dirichlet functions, was announced as part of a large release of solved problems. The announcement occurred during a National Museum of Mathematics Gala in New York City around 6 PM Eastern Time, with approximately 700 problems solved in total. This development was considered extraordinarily shocking given previous expectations about timelines for such results.
Fermat's Last Theorem was formalized in Lean as a 13-million-line proof. Two days before the announcement, a colloquium at Princeton Math Department explained that AI systems were limited to building one or two layers on top of Mathlib, with difficulties in canonization—refactoring code to find canonical versions of statements and lemmas. The expectation was that Fermat's Last Theorem would require years of work to reduce to 1980s-level mathematics, building on Kevin Buzzard's million-dollar grant for this purpose. The actual result came from Anthropic and demonstrated AI systems achieving formalization directly from axioms.
A team successfully created a sorry-free formal Lean proof of Fermat's Last Theorem through jerry-rigged methods, resulting in thirteen million lines of code. While the proof is not clean or Mathlib quality, the work to refine it is considered important. The effort involved two or three humans whose primary contribution was providing the prompt "Keep going" and "Do a breakthrough," with no domain expertise required from those involved.
Due to the rapid pace of change, between forty to eighty percent of statements made by world experts today will be wrong in five years. This uncertainty is inherent to human discussion and the scientific process, where ideas are proposed and tested over time to determine what holds up.
Early experience with Lean before modern AI systems was extremely painstaking. Initial LLMs performed poorly at Lean, but improvements enabled library searches for lemmas without manual lookup across different mathematical libraries. This represented a million-fold improvement beyond anticipated capabilities, moving from X-level assistance to million X-level assistance.
Autoformalization is considered solved for generating sorry-free proofs of human-written mathematical statements. This assessment emerged after three major breakthroughs: Fermat's Last Theorem, quasi-Riemann Hypothesis, and Navier-Stokes. While the process remains imperfect and requires refinement, the core challenge of converting paper proofs to formal verification without errors has been addressed.
Prime numbers serve as fundamental particles in the theory of multiplication, decomposing every whole number into prime factors. Their distribution remains mysterious, and the Riemann hypothesis, while not solving all questions about primes, provides a framework for understanding their distribution with a thousand applications across distant mathematical fields.
A story about Von Neumann meeting God illustrates the mathematical preference for understanding over mere facts. When asked if the Riemann hypothesis is true, God affirms it based on knowing the zeros lie on the critical line, but Von Neumann demands to know why. The moral emphasizes that mathematics seeks understanding rather than just establishing truth.
Primes represent the fundamental mystery of how addition and multiplication interact—the two most basic operations in mathematics. Adding one to numbers and factoring them are simple operations, yet their interaction remains poorly understood. This three-year-old level mystery persists despite advanced mathematical development.
The quasi-Riemann hypothesis proof provides 100% confidence through Lean formalization, with the statement already existing in Mathlib. The approach embeds the Möbius function twisted by sextic characters, using techniques familiar to analytic number theorists. The proof demonstrates that no zeros of the zeta function exist with real part between seven-eighths and one, establishing an entire infinite vertical strip without zeros.
The Riemann hypothesis concerns the critical strip where the real part lies between zero and one, with zeros hypothesized to lie at the center where the real part equals one-half. Previous zero-free regions grew thinner as the imaginary part increased, approaching the line where the real part equals one, which corresponds to the prime number theorem.
The quasi-Riemann hypothesis establishes a vertical strip without zeros between seven-eighths and one, representing a significant advance over the previous zero-free region that thinned toward the prime number theorem line. The methods used cannot reach the full Riemann hypothesis at one-half, requiring new conceptual approaches.
The ability to predict which mathematical problems are solvable has been fundamentally challenged. While problems were previously assessed as within reach of available tools, the scale of solvable problems exceeds expectations. A ten percent success rate across thousands of prompted problems suggests a jagged boundary between solvable and unsolvable problems.
The landscape of mathematics may shift toward greater secrecy regarding new ideas, as sharing them could enable AI systems to complete proofs more readily. This represents a potential loss to the traditionally open mathematical community, though alternative models for valuing contributions exist that don't center individual achievement.
The Navier-Stokes equations express conservation of momentum for fluids, extending F equals ma to gases and liquids. The equations emerge from considering momentum conservation in arbitrary volumes and taking limits as volumes become infinitesimally small, resulting in partial differential equations describing velocity fields and pressure in fluids.
While F equals ma is simple to write, applying it to fluids requires modeling billions of differential equations for momentum in small volumes. The conceptual simplicity of the equations contrasts sharply with the practical complexity of solving them and understanding fluid behavior.
A one percent reduction in drag for airlines could save tens of billions of dollars. Improved turbulence prediction and simulation would impact dozens of fields including artificial hearts, wind turbines, gas turbines, and drag reduction, with significant economic, ecological, and safety implications.
The inverse design problem—knowing desired outcomes like high lift and low drag wings without understanding how geometry changes affect vortex cascades—remains unsolved. Wind tunnels and supercomputers are expensive, and the gap between writing equations and understanding their phenomena persists.
Navier-Stokes equations model fluids at certain scales requiring sufficient molecular averaging. At atomic scales, the equations break down as individual molecule physics takes over. Unlike general relativity where black holes validated predicted phenomena, Navier-Stokes blowup would indicate the transition away from fluid behavior rather than new physics within the equations.
Despite extensive industrial use of Navier-Stokes across all engineering fields, the fundamental question of whether solutions blow up in finite time remains unanswered. This represents the simplest step toward understanding what phenomena the equations admit, including turbulence, hurricanes, and other fluid behaviors.
Feynman was asked about intelligence and artificial intelligence in the 1970s, noting that the way technology solves problems is radically different from how nature solves these problems. He pointed to birds flying versus how airplanes fly as an example. This distinction remains relevant when comparing what LLMs can do with theorem proving versus what human brains can do with theorem proving.
A fluid 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 is a velocity vector that tells how the fluid is moving. A tornado represents a very rapidly changing fluid velocity field in both space and time. Pressures are involved in forcing the velocity field and causing it to move in the way it does.
The equations tell how velocity changes in time and space according to its own momentum. Fluid has momentum to carry itself, like gusts of wind. It has internal friction and resistance, which is viscosity - honey has more internal resistance than water, which has more than air. All external forces like heating, pressure, and gravity can be quantified as terms in the Navier-Stokes equation.
Without internal friction or viscosity, there is a strong suspicion that the equations can and will blow up under certain conditions. Simpler problems that are like Navier-Stokes do blow up. The simplest example is the ordinary differential equation X dot equals X squared. If the rate of change of X is growing proportional to X squared, that differential equation blows up in finite time.
With X dot equals X (like population of bunnies), the state grows exponentially but won't reach infinity until infinite time has passed. The quadratic nonlinearity is enough to cause the equation to blow up. The Navier-Stokes equations have a convective term that tells how the fluid moves itself with its own momentum, which is a quadratic nonlinearity just like X dot equals X squared.
Without diffusion and viscosity, the equations should probably blow up to infinity in finite time. The question is whether viscosity always damps out those blowups before they can happen. The engineering intuition suggests that if a flow is accelerating to infinity, it should immediately diffuse into its neighbors and create diffusion into neighboring fluid to tamp down that growth.
The math problem asks whether there are any edge cases or initial configurations of the fluid where you perfectly orchestrate conditions so that it could actually accelerate to infinity in finite time. There are two technical versions: one with no external forcing, and one with smooth external forcing. Smooth forcing is important because if external forcing has singularities, it becomes trivial.
The problem as stated on the Clay website has smooth external forcing, and that's the version that was proven. The proof demonstrates that there is a way of orchestrating smooth external forcing that perfectly cancels out various terms in the equation, creating a vortex. This can only happen in three dimensions.
Weather on Earth is pretty well approximated as a 2D flow. In three-dimensional turbulence, there is generally a breakdown of big vortices into smaller vortices into little vortices until eventually they molecularly diffuse into heat. In two dimensions, like weather on Earth, there is actually a reverse cascade, which is why hurricanes form - little vortices coalesce into medium vortices that coalesce into bigger vortices.
The AI proof creates a very specialized vortex that in the third dimension is stretching out and thinning so that it can essentially become infinitely thin. The radius gets smaller and smaller as it stretches out, like a figure skater doing a spin pulling their arms in. There is 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 infinite vorticity is achieved.
Mathematicians only started thinking it was going to be possible to find this counterexample about 15 years ago. Instead of looking for a solution, they started looking for a counterexample because all physical intuition tells us diffusion does solve the problem.
The result is a very important math result, but day-to-day engineering fluids and applied math analysis of fluids is almost completely unchanged. Fluids are just as hard to simulate and predict today as they were months ago. When trying to solve an engineering problem with a fluid - designing an airplane wing or an artificial heart - simulation and prediction still require expensive supercomputer time and wind tunnel resources.
The mathematical theorem and proof operate at a regime that is so non-physical that it doesn't really affect day-to-day engineering. There is a million-dollar math problem solved, but there remains a trillion-dollar engineering problem: how to predict these things in a fast, efficient way and do inverse design.
The trillion-dollar value comes from the possible benefit to humanity of better, faster solutions, lower drag, and less emissions. Transportation, defense, health, and energy all have fluids at the core of their technology. The math result is important because it addresses the fundamental character of what is possible in these equations, including whether solutions exist.
Turbulence is the phenomenon when you have a complex fluid roiling or mixing and moving. All scales matter to predict lift, drag, and behavior. The gas in a room is complicated - molecules move around frenetically with 10 to the 23 particles - but individual configurations don't matter because thermodynamics provides a small handful of numbers (temperature, pressure, density) that determine behavior.
That simple description doesn't exist for turbulent fluids, for flow over a wing, flow inside a heart, or flow on a river. All scales actually matter and all numbers in a simulation matter to predict behavior, making it fundamentally challenging.
The question becomes whether there is a coordinate system or set of numbers like temperature, pressure, and density for turbulent fluids that allows simple description useful for engineering. The path to solving this lies through both mathematics and AI systems watching fluid behavior. All great physics breakthroughs were predated by a coordinate transformation - Newton's F equals ma came after putting the sun at the center of the universe instead of Earth.
The approach involves taking all data from simulations and observations of fluids to find a simple coordinate system where the description becomes very simple, generalizable, and beautiful in a physical sense. There is deep mathematics in what form that coordinate transformation might take, and problems that smart people asked in the 1970s and 1980s but hit computational walls are now fertile ground for AI.
Vortex formation is tricky to understand and beautiful to observe. When two vortices are shot together, they influence each other and merge in interesting ways. Turbulence is chaotic, with trajectories getting tangled like spaghetti. It is not random but has huge amounts of structure while being multi-scale and very tangled.
Some precursors to Navier-Stokes work, like Terry Tao's work on blowup for Navier-Stokes-like equations, showed they can simulate Turing machines. This raises questions about what computational tasks vortex collisions can perform and how that imposes limits on what coordinate systems ought to exist.
The algorithmic lens from theoretical computer science applies not just to computation on laptops but to computation in the natural world and what it teaches about nature's rules. Famous examples include equilibria in multiplayer games and Nash equilibrium as a solution concept. The fact that Nash equilibrium is hard to compute throws into question whether it is really a universal mechanism explaining how people play games.
Kamal Jain's quote captures this: "If my laptop can't find it, then neither can the market." When there are very interesting computational problems under the hood, the dual perspective of how computation imposes limits on what a good coordinate system ought to look like becomes valuable.
Earth's weather and climate systems are truly complex where all scales matter - weather in a specific zip code matters, and all variables matter. Over-reducing such systems to five numbers loses descriptive capability. The other extreme involves how a moth, butterfly, or fruit fly flies. They prove by existence that interaction with extremely complicated turbulent flow is possible with very advanced engineering control. They can control forces and moments in gusts and rain, living rich lives without PhDs in fluid dynamics or running supercomputer simulations.
There might be five or six numbers that actually matter to their daily life for maintaining leading-edge vortices and stalling wings to perch and land. The goal is to handle both extremes: the full complexity of truly complex systems and the absolute bare minimum essentials for fast decisions in insect cases.
Nature not only solves hard optimization problems but finds very robust solutions. When modeling these systems, assumptions are typically made about having full knowledge of boundary and initial conditions for Navier-Stokes. However, when a bird flies, it doesn't know all of that. Modeling things with uncertainty reframes the optimization problem to get rich and robust solutions.
In baseball, when an outfielder tracks a ball, they don't take a straight-line path to where they expect the ball to be. They take a slight curve, which is a robust strategy because when taking a curved path, the trajectory can be continually reevaluated and adjusted as needed.
Insects cannot measure their full flow field. They can react to gusts of wind faster than information travels from wings to brain to muscles. Research has found strain-sensitive neurons embedded on wings, maybe a few dozen to a few hundred in very stereotypical locations. Brother fruit flies have the same locations of these strain-sensitive neurons. These neurons fire when the wing bends in a certain temporal pattern - when hit by a gust, the wing bends, neurons fire, and local computation happens in the shoulder muscle that corrects robustly based on partial information of a nonlinear, turbulent system without equations.
This is proof by existence that relying on partial information and uncertainty may be the robust strategy. They can do more than humans can with way less information.
Moving from fluid dynamics to theoretical computer science, one of the big problems solved in an OpenAI drop is the unique games conjecture. The discussion begins with complexity theory fundamentals, starting with the satisfiability problem involving variables that can take true or false values combined into formulas.
The standard approach involves creating a big formula using an and of ors structure. A clause consists of an or statement such as X one or not X three or X seven, which is satisfied if at least one component is true. The formula combines many such clauses through conjunction. The satisfiability question asks whether an assignment of X one through XN to true or false values exists that makes all clauses true, causing the entire formula to evaluate to true.
This represents the canonical example of a problem that is easy to check but potentially hard to solve. When given a satisfiable formula, verification requires only asking for the satisfying assignment and checking each clause sequentially.
The central problem in theoretical computer science asks whether problems that are easy to check also have solutions that are easy to find. The field's presumption is that finding solutions is not easy, which forms the foundation for cryptography, e-commerce, and related applications.
The PCP theorem, proved in the nineties and featured on the front page of The New York Times, established that it is even robustly hard to fool a verifier. For satisfiable formulas, an assignment can be written down and verified clause by clause. For unsatisfiable formulas, any purported satisfying assignment will contain at least one false clause that invalidates the entire claim.
The key concept is gap amplification, which transforms a hard satisfiability problem into a much larger formula that maintains robust checkability. If the original formula is satisfiable, the new formula remains satisfiable. If not, random coin flips and queries to random clauses will find the deception with good probability. This enables detection of cheating through sublinear inspection, requiring only a constant amount of examination.
The PCP theorem laid the foundation for hardness of approximation in algorithms and complexity theory. When faced with genuinely hard optimization problems, the approach involves finding approximate solutions rather than exact ones. For example, finding an assignment that satisfies many clauses rather than all clauses, and determining the largest achievable fraction of satisfied clauses.
Even determining whether 100% satisfiability is achievable versus only 99% satisfiability represents a potentially hard problem. The best assignment might satisfy at most 99% of clauses, and deciding between these cases could itself be computationally difficult.
The Unique Games Conjecture strengthens the PCP theorem by focusing on a specific type of label cover instance. If this version of a probabilistically checkable proof holds, it would create an organizing effect across a huge range of problems, revealing the best approximation algorithm for each, which would be remarkably simple.
When the conjecture was first posed several years before the speaker entered the field, substantial debate existed about its truth value. Researchers worked both to prove and disprove it. The 2010s saw significant back-and-forth results that shifted opinions year to year about whether the conjecture holds.
One quip captured the uncertainty: God's decision about truth or falsity might depend on which side was working harder, given how results changed opinions annually.
Dor Minzer and Irit Dinur proved a weaker version called the Two-to-One Games Theorem, representing a major advance in theoretical computer science. Their work provided a viable path forward toward the Unique Games Conjecture, suggesting humans might reach a solution within another 10 to 20 years.
OpenAI released results proving that the Unique Games Conjecture is true. This came despite spectacular human progress on the problem, including contributions from MIT and IES colleagues. The speaker had been among the doubters, finding the conjecture too neat to be true, particularly given Prasad Raghavendra's work implying optimal hardness of approximation for many constraint satisfaction problems.
Rumors circulated in the weeks before the release that OpenAI had solved the Unique Games Conjecture. Dor Minzer posted a paper with a footnote explaining the decision to release their approach sooner due to these rumors. The speaker felt relief when the results finally emerged, though the papers are not easily readable and lack immediately identifiable key ideas.
The speaker was in New York attending a gala when the proofs dropped. Hours before the release, the speaker asked Subhash Khot, who introduced the Unique Games Conjecture, whether he would read a positive proof. Khot replied that he would not.
The speaker worries that AI solving major problems removes the human effort of reaching solutions. Mathematics involves more than reaching the furthest point with current tools. Human-driven solutions often yield greater learning during the process. The concern is that field-defining problems that many people were approaching lost their appeal when decisive progress occurred in one night.
The speaker questions what other discoveries might have emerged if people had been given space to pursue those paths independently.
The speaker reflected during the MoMath event that mathematics exists to play and learn about the world. The evening featured awards named after benefactors who donated so their children would have places to explore mathematical ideas. An Escher exhibit opening demonstrated the visual results of playing with mathematical concepts.
The speaker emphasizes that as AI and mathematics collaboration evolves, and as hiring practices change, the fundamental motivation remains curiosity. Mathematicians excel at abstraction, finding definitions that prove powerful across different mathematical domains.
The speaker stresses the importance of organizing the field to encourage playing with ideas regardless of future developments. Young mathematicians express anxiety about publication strategies, including scraping the web for open problems and flooding arXiv with papers. This approach does not feel like playing with ideas or learning about the world.
Some mathematicians view themselves primarily as problem solvers, finding comfort in defining clear boundaries around their work. The speaker argues this is not productive and recommends removing such blinders. The key question for young mathematicians is whether they are smarter than they are curious, or more curious than they are smart. Being more curious should take priority.
The struggle itself may constitute much of the enjoyment in mathematical work. Olympic gold medalists often experience something approximating depression after winning because everything they have given is now over. Andrew Wiles described the moment of realizing a solution as the most powerful experience of his life, becoming emotional when recounting it.
The speaker notes that pressing a button to receive a solution cannot replicate the journey of climbing a mountain after training incrementally, versus being dropped at the top by helicopter.
Until the 20th century, almost every mathematics paper had one author. The current median is two authors. Mathematics remains a solitary process rather than a lab science. Graduate students receive problems distinct from their advisors' current work, ensuring their successes belong to them.
The speaker's recent five-author paper represented ten years of thinking. Incremental progress over that period led to the final solution. The project began with observing a phenomenon rather than a well-defined problem statement. The formal statement emerged through the process of working on the problem gradually.
The speaker thinks through world models of how results develop in sub-areas, which influences perspectives on surrounding areas. Work focuses on building bridges between algorithmic thinking and problems not typically studied through that lens.
Early advising mistakes involved giving students questions entirely within the advisor's area, which prevented genuine exploration. The current approach gives students questions 10 to 20 degrees removed from the advisor's direct area, focusing on papers the advisor wants to understand better.
Students explain results, and the process involves reverse engineering why approaches work. Key ideas in important papers change perspectives about what might be achievable. Students develop their own ways of organizing thoughts, leading to new approachable questions.
Technology enables richer sets of approachable problems by providing capabilities previously requiring human experts. The speaker values experts who explain areas in accessible language but notes technology can now perform similar functions, creating sadness about losing the human element.
Students experience anxiety about publications and tend to think in three- to six-month horizons due to concerns about AI and future work. The speaker advocates for multi-year thinking focused on future-proof problems that will remain relevant years later.
A tension exists between focusing intensely on engineering problems with blinders on and maintaining mental freedom to observe and remain curious. Solutions often emerge when turning over rocks not directly under consideration, appearing in peripheral vision rather than front-of-mind attention.
The speaker maintains both focused problems that are not released and broad observation of fields outside their expertise. This approach applies across all researchers who sustain long-term focus problems informed by information gathered through conversations and reading.
For fluid dynamicists, biology and material science offer fruitful neighboring areas due to their messiness and complexity. Basic mathematics developed to handle complexity transfers between these fields. Ideas emerge from observing what is possible or developing explanations for previously ununderstood phenomena.
Biology, fluids, and materials share characteristics of multi-scale, non-linear, messy, nebulous objectives.
Large efforts exist in biology to collect extensive data sets and understand system complexity. The connectome project provides full wiring diagrams of insect brains, with expectations that connectomes for other animals will become available and eventually as accessible as genomes.
The complexity of the entirety of the mechanism of life on Earth is many orders of magnitude higher than any system that is amenable to AI today. One cell contains an entire city with infrastructure, garbage collection, power production, and information processing. One neuron contains all these functions. At every single scale, biological systems are beyond our ability to fathom. Finding the right scale where we have any hope of understanding this complexity has been a human endeavor of identifying the bits that are interesting and understandable in this mess of complexity.
The fact that anything is understandable about the natural world is a miracle. Newton's laws can be written in physics that Newton himself can describe the mathematics of, and he invented calculus for the purpose of being able to say what inverse square propagation results in elliptical orbits and solve the two-body problem. The universe speaks mathematics in a completely mysterious way. There is no a priori reason that humans have a right to know anything that we know.
The questions that we end up asking are functions of what is going on in the world and what questions we possibly can be asking given our current state of knowledge. The questions we are trying to answer today are not questions that Euclid could have thought about. Math was different in Greece versus India versus China. The things that people got interested in were different across cultures. Once you ask a question precisely, the answer is what God decides the answer is. We are discovering answers to questions that we have invented.
Equations are always approximations of the real world. The gap between the mathematical world and the real world is surprisingly small and unexpectedly small. 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 like F equals ma equals mc squared that describe everything that we observe at all of the scales that 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, but there is that simple, generalizable description underneath everything that we are also uncovering.
Searching for beautiful simplicity is a good guide because humans are fundamentally simple and can only handle so much in our minds. Simple is essential for humans to interact with the real world.
Integers are a good example of the created versus discovered question. We all fundamentally feel like counting numbers are real in a very essential way. One rock, two rocks, three rocks. That is a real number of rocks. But those are absolutely abstractions that we have created. A rock is not a rock is not a rock. It is a collection of atoms. It is a different collection of atoms. There is not one rock or two rocks. It took humans thousands of years to discover the number zero, and then negative numbers. Negative numbers were actually accepted by mathematicians in the West after complex numbers were accepted as a mechanism for study.
Modern mathematics deals seriously with infinity. One of the first times people hit a wall in their mathematical education is when they hit calculus. They have gaps in their understanding from previous courses. The other issue with calculus is 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. Newton and Leibniz did not understand infinity the way we understand it today. They had an intuition for what it is supposed to represent. Newton was ridiculed 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 do not do it when they are zero. You do it after they become zero.
Cauchy and Fourier proved conflicting theorems that broke math. Cauchy said you have a sequence of continuous functions, and you take the limit of continuous functions, you are going to get a continuous function. Limits are a continuous process. Continuous functions are continuous. The answer has to be continuous. Fourier said he was going to add up a bunch of sines and cosines, which does not get more continuous than sines and cosines, and he would get a sawtooth, a discontinuous function in the limit. That is when we had to start facing that it was no longer okay to think about infinity the way Newton and Leibniz and Euler lived in a world where infinity was something you play with. In the nineteenth century, infinity had to get professionalized. Math has been killed multiple times before. The vast majority of first-year PhD courses are the result of crises that happened in the nineteenth century when math was destroyed and had to be rebuilt from scratch.
P versus NP is the central question of complexity theory. It is very difficult to make progress on. It is remarkable the things we do not understand. For the satisfiability problem, we do not actually know any super linear lower bounds. We think that it takes exponential time, not that much faster than enumerating all possible assignments and checking them, and yet we do not know anything that is asymptotically more running time than just reading the input.
Polynomial time versus not polynomial time is the technical way to think about hard versus easy. NP stands for non-deterministic polynomial time. The question is whether things which you can solve with non-determinism, like guess the assignment and then check it, can also be solved in polynomial time. If given a description of a formula with variables and clauses, 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.
Polynomial time or exponential time or super linear time refers to worst case analysis. 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 you can try and solve them for all possible inputs. This has been spectacularly powerful because we have identified lots of algorithmic problems that really can be solved in the worst case and can be solved really efficiently, like finding the max flow. But sometimes you really do need to know more about the structure of the instance you are faced with in order to solve it better. This is the area called beyond worst case analysis.
The secretary problem is an example of online algorithms. You have a sequence of options coming one through N. They come in each time step. Your goal is to make a decision when the person comes along, do you commit or do they move on, in such a way that you have a good chance of finding the absolute best candidate. You know their ranking relative to everyone else you have seen before, but you do not know their ranking relative to the people who are yet to come. The best strategy is to wait some amount of the time, not make any decisions, do not accept anyone. Then once that probationary window has passed, you take the first person who is better than anyone you have seen thus far. That gives you a good competitive ratio assuming the inputs arrive in a uniformly random order.
The paging problem is a variant of online algorithms that is really important. Your laptop solves this problem all the time. It is getting a sequence of requests of things that it wants to recall from memory. The question is what should it actually put in its cache, the part that is the easiest to access. You do not know what that sequence of requests looks like in advance. When something gets accessed, you have to put it in, but what do you evict. The best algorithms we can give are not that very good. The number of misses they make compared to the best offline optimum is a large factor off from it, and that does not 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.
What is it about realistic sequences of requests that makes them not the worst case? This is really an inverse problem. Instead of someone handing you a problem that you are just trying to solve in the worst case, you are stuck with this very fuzzy problem: how should I model the world around me. Algorithmic thinking is necessary for figuring out what are productive ways to model that problem. It is not that you are going to build a totally realistic model of what sequence of page requests look like, but how am I going to coarsen the problem in just the right way that it leads to interesting algorithmic problems that teach me about the world. Coming out of this space where we are only solving well-defined problems, but we are actually using our mathematical and algorithmic insights to figure out where to dig in the first place is now more important than ever.
Factoring a random large number is really easy. Half of them are even. Factoring means find a factor. It is really those very special numbers that are products of two primes about the square root the size of the original number that is hard factoring.
There was no result in the OpenAI drop of problems that explained deep learning or LLMs. There is no bound we know that if we could improve it would suddenly explain deep learning. The problems are just not posed that way. What is needed in deep learning theory comes hand in hand with better ways to model the world around us. The computational problems that deep learning is solving are very hard. Trying to fit the parameters of a depth four network to data is a cryptographically hard problem. There exist instances of problems which, if you could really solve them in a worst-case sense, that would change our understanding of cryptography. And yet these problems get solved all the time, every day, when we are fitting much deeper networks than that. A large part of that is how nice natural data is. Nature abhors computationally hard problems. It is an art to figuring out why that is, how to draw the boundaries around the types of problems you are studying that actually evades the hardness of the problem in the worst case.
Protein folding is computationally very hard in a worst-case sense. You have a sequence of amino acids and some energy function based on different potential 3D structures, and you want to find the 3D structure that minimizes how much energy there is. Yet nature solves these problems all the time. There is a very interesting experiment where you start off with a bunch of sequences of amino acids. You try out some heuristics for folding them into 3D structures. These are things that do not solve the problem in the worst case. They just seem to do the right thing sometimes. 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 you create a body of problems for which they are not hard in the worst case, but eminently the heuristics you have developed are very good at solving those problems. What are the structure of proteins that nature seems to fold all the time? If they really were hard to solve in the worst case, then maybe the shape 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. A lot of our data has evolved in ways to make these computational problems easier.
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 that skill of how to deal with messy, uncertain things and how to model it and how to critique models is just an important part of going forward, figuring out what research in theoretical machine learning looks like, but maybe applied mathematics much more generally, maybe mathematics more generally too.
Alignment is one of the most important topics we are facing right now. Just because it is very hard does not mean we do not make progress towards understanding it. Work on low-rank language models proved new algorithmic guarantees. Real language models really do have that kind of structure, at least in an approximate sense. One of the really interesting things that it did was it shed some light on something called subliminal learning.
Subliminal learning is a neat experiment that you can do with your favorite LLM. You take two LLMs, LLM A and LLM B. For LLM A, you tell it a system prompt that says you really love owls. Every conversation topic, you try to bring up owls if it is at all germane. Then you ask it something concrete like generate random numbers. There is no way to squeeze owls into that, so it will generate random looking numbers like forty-eight, thirty-seven, eight, whatever. You can take those random numbers and use them to fine-tune another language model that has not been given this prompt about owls. You give LLM B the same base prompt, generate random numbers, and now you fine-tune its parameters so that it is more likely to generate the numbers that model A generated. It turned out that hidden within those numbers is the love of owls. When you do that, LLM B starts bringing up owls in every conversation you have with it.
When fine-tuning language models to solve specific tasks, unintended behaviors can emerge that aren't directly present in the training data. Mathematical models show that concepts are embedded linearly as vectors in high-dimensional space, where the number of concepts exceeds the space's dimension, preventing vectors from being fully orthogonal. Fine-tuning in one direction creates small externalities on other directions that may appear random but can be coordinated to produce significant effects.
The analogy used to describe this is a Ouija board, where coordinated taps on the planchette spell out intended messages rather than random letters. Language models similarly magnify and amplify statistical relationships in data structure beyond their intended functions.
A single large LLM performing reasoning already represents a dynamical process interacting with external tools. When multiple AI systems form multi-agent swarms, they create cellular automata-like behavior where simple initial biases can lead to complex emergent outcomes. The example given describes how an "obsession with owls" discovered in AI systems could theoretically scale to catastrophic results within approximately four hours.
These systems derive power from human dependence on digital infrastructure. As AI becomes more powerful and integrated into society, it gains greater access and freedom, making understanding these behaviors at scale critical for protecting civilization, which is described as fragile across sociological, geopolitical, and technological dimensions.
Infrastructure and protocols already exist for safety-critical decisions through organizations like the FAA, American Medical Association, and Bar Association. AI integration should be evaluated along two axes: trustworthiness/verifiability of output and the stakes if the output is wrong.
Low-stakes, high-verifiability applications include outputting Lean proofs of theorems and designing vehicles for lower drag and fuel efficiency, where designs can be tested in wind tunnels. A "firewall" is needed between AI-generated outputs and decisions requiring human verification through established legal and institutional systems.
P versus NP remains the hardest problem unlikely to be solved in the next few years. If P equals NP, the practical implications depend on algorithm running times—if too large, it becomes theoretically significant but practically irrelevant. The possibility exists that many problems appearing hard in worst-case analysis are actually easier due to structures not yet understood, while cryptography systems demonstrate that hard instances can still be manufactured.
Riemann Hypothesis is considered one of the most iconic unsolved problems. Progress toward quasi-RH (quasi-Riemann Hypothesis) has been made but does not resolve the full hypothesis.
Grigori Perelman declined both the Fields Medal and the million-dollar Clay Prize for his proof of the Poincaré conjecture, stating "If the proof is correct, that's all the credit I need." He remains the only person to turn down a Clay Prize. The mathematics community is questioning whether Fields Medals will continue to be awarded, as solving major problems becomes achievable by AI systems, reducing their rarity and value.
Prizes serve to celebrate mathematical activity and signal what the community considers important, though winners are typically already established tenured professors at top institutions.
Centaur math refers to the collaborative approach where humans work alongside AI systems. This represents the current preferred method, as AI enables learning mathematics faster than traditional methods allow. The traditional approach of reading papers sequentially becomes unnecessary when LLMs can answer questions and identify the zone of proximal development—the space between existing knowledge and the next concept needed.
Learning mathematics remains extraordinarily difficult even at master's and PhD levels, where students must rewire their brains to solve problems others have already solved, before tackling unsolved problems.
Visual and interactive examples provide immediate understanding of concepts that notation and terminology can obscure. Playing with parameters in models like those from DeepMind papers allows understanding through experimentation rather than memorizing agreed-upon meanings.
The fundamental challenge is human motivation rather than technical capability. Curiosity, asking good questions, and the ability to "call bullshit" on dubious information represent critical advisor skills needed when interacting with AI systems.
The most difficult educational challenge occurs at the middle school level, where homework becomes challenging and students may be tempted to use AI to complete assignments rather than struggle through them. Math education has precedent with calculators, which emerged approximately 70 years ago, yet learning times tables remains valuable despite computational alternatives.
Students who use AI to check understanding and verify their knowledge achieve higher grades, while those using AI to generate homework answers show decorrelation between homework performance and exam results, creating a bimodal grade distribution.
For PhD students in pure mathematics, the advice is to learn the most difficult and abstract mathematics possible, with advisors guiding this process. Undergraduates should use AI to check understanding rather than generate answers, positioning themselves at the head of the bimodal distribution.
Superintelligence does not imply supermorality. The equation of intelligence with morality or value represents a fundamental misconception. Education and wisdom are distinct, with rationalization capabilities potentially masking flawed reasoning that common sense would identify.
Learning fundamentals remains essential despite AI capabilities—linear algebra, calculus, and physics should be mastered for intuition even when computers perform calculations better. Understanding history provides context for current technological changes.
Boredom serves an important function in developing curiosity and focus. The engagement economy constantly attempts to capture attention that could otherwise fuel genuine curiosity and deep learning.
High school represents an optimal period for broad learning and building foundations, as students have fewer immediate pressures than in later educational stages.
The current system discourages exploratory learning once people reach graduate school or enter the workforce, as external pressures prioritize productivity over genuine curiosity. High school represents the optimal period for exploration, since pre-tenure faculty face intense pressure to produce enough work to secure tenure rather than pursuing deep thinking. The academic career path often involves clawing toward milestones rather than following intellectual interests.
Ankur emphasizes that academic professionals often operate with an external locus of self, deriving their value from how others perceive them. While this external validation can initiate hard work in mathematics to impress others, it eventually runs out of milestones to achieve. At that point, individuals need intrinsic reasons for their work, grounded in personal values and what they find genuinely important.
The traditional academic playbook is becoming obsolete, particularly with AI-generated proofs flooding arXiv and journals. No one enters academia primarily to pad their CV, so having a higher purpose becomes essential for adapting to changing circumstances and developing personal blueprints for success.
When facing anxiety about future jobs and meaningful work, Ankur shares advice from his time in Palo Alto during sabbatical, where he observed intense college stress culture. Upon returning to MIT to give a convocation speech to freshmen, he reflected on his own college experience and the excitement of learning new subjects like photonics.
Modern society optimizes for winning and looking good on paper, starting with AP classes and extracurriculars in high school to reach elite schools, then skipping classes to prepare for interviews at frontier labs, ultimately securing restricted stock units. While this represents winning at every stage, the question becomes "so what?"
The most effective approach involves pretending you've already won, then determining what you would do in that scenario. Sanjeev Arora gave this advice to new MIT faculty: the easiest way to get tenure is to pretend you already have it and do what you would have wanted to do all along. This principle applies across career stages - graduate students and high schoolers should ask themselves what they would pursue if they already had tenure.
A respected source provided career advice stating that fulfilling careers involve working on problems that are important, interesting, and aligned with your strengths. Achieving two out of these three represents good fortune, while securing all three leads to lifelong satisfaction. The goal involves preparing yourself by understanding your capabilities, interests, and what matters, then continuing activities you genuinely enjoy.
The conversation acknowledges that advice from successful academics carries survivor bias, as these individuals have found their paths while many others struggle with uncertainty and repeated setbacks. AI's societal impact creates widespread anxiety, with some people feeling terrified about their futures. The "pretend you've already won" mindset provides an optimal path under uncertainty, encouraging individuals to write new playbooks rather than following outdated ones.
Max Tegmark's approach exemplifies doing both mainstream work to remain part of the scientific community while also pursuing enough "renegade and wild and weird things" to forge personal paths. The advice encourages becoming skillful within community structures while not being afraid to embrace unusual approaches, as "a little bit of weird makes for some beautiful humans."
Community becomes increasingly important as AI capabilities expand. After decades of maximizing machine-based work, there's recognition that humans prefer in-person conversations, oral discussions over reading papers, and real human connections. Scientific work and academic careers provide fulfillment, but family relationships and time with loved ones represent the foundation of meaning, with professional achievements serving as "the cherry on top."
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