#503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory

Lex frames the episode as an emergency discussion after AI systems claimed hundreds of major mathematical results, including Navier–Stokes, the quasi-Riemann hypothesis, and the Unique Games Conjecture. He brings together Alex Kontorovich, a Rutgers number theorist; Ankur Moitra, an MIT mathematician and theoretical computer scientist visiting Princeton’s Institute for Advanced Study; and Steven Brunton, a University of Washington mathematician and fluid-dynamics expert. Lex says mathematicians, including Fields Medalist Terence Tao, are reacting with a mixture of awe, terror, dread, and hope. The central question is whether this is a crisis that destroys mathematics or the beginning of a new mathematical golden age.

Alex rejects calling the moment simply a crisis, describing it as “the chaos of opportunity” and comparing humanity’s former mathematical candle to a new flashlight. Ankur says rumors had circulated for weeks, so his first reaction was relief that the results were finally public and could be examined. He feels pulled in two directions: AI dramatically expands his ability to learn and connect ideas, but it destroys the familiar blueprint for advising graduate students and postdocs about which problems will make an impact. He says the result is excitement mixed with intellectual loneliness and anxiety that only spectacular achievements will stand out.

Lex cites Tao’s fear that AI releases casually disrupt years of patient work by students and postdocs: once a solution is known, the original exploratory path loses much of its value. Brunton is more hopeful because better mathematical tools could improve engineering, medicine, aircraft, and artificial hearts, though he fears humans may either be dwarfed by machine intelligence or see AI remain an extraordinary tool. Alex distinguishes “solving math” from “destroying math” and proposes a deliberately strong “solved axiom”: any problem solvable by a human interacting with AI could eventually be solved by AI alone. Even under that assumption, he argues pure mathematics remains valuable because understanding must live in human minds.

Alex says knowing that a solution exists can itself motivate humans: if another country invented teleportation, people would reverse-engineer it. Ankur argues that humans are computationally limited, so they often need general, abstract methods that explain neighboring problems rather than merely reaching one theorem. The panel stresses that mathematicians value not only truth but the ideas, definitions, and connections discovered along the way. Alex says the key question is whether AI can generate genuinely new definitions or merely combine existing moves. He often reduces a large goal to the simplest embarrassing open problem, because solving that small case may assemble tools for larger ones. They conclude that selecting meaningful questions and recognizing mathematical insight will become more important.

Ankur introduces the “doorman fallacy”: reducing a complex job to its most visible task and assuming automation replaces the whole role. A doorman does not merely open doors; he welcomes guests, recognizes regulars, handles luggage, hails taxis, provides security, discourages unwanted visitors, signals prestige, and makes people feel cared for. Ankur applies this to mathematicians, whose work includes deep thinking, building theories, forming conceptual connections, and creating new ways to view old problems—not merely proving famous conjectures. He says the field should not panic, but should reconsider its values and institutions. Alex adds that asking good questions requires mathematical insight, not just becoming a more polished online personality.

The panel debates whether AI’s results are interpolation or extrapolation. Alex explains a “convex hull” picture: known mathematical ideas form a balloon, and linear combinations remain inside it; the derogatory interpretation is that AI only did what humans could already have done. He prefers Tim Gowers’s “semigroup” intuition, comparing known ideas to tiles that can be repeatedly placed next to one another, potentially reaching much farther than humans have explored. In the quasi-Riemann-hypothesis result, AI used known techniques—especially a long, beautiful cascade involving cubic theta functions and cubic Gauss sums—but combined them in a way that changed the research landscape. Even elementary solutions can be field-altering when they reveal which problems are reachable.

Alex explains that mathematics is both engineering a proof and communicating its meaning. Lean is an interactive theorem prover that checks each logical step, while Mathlib is its carefully maintained library of formalized mathematics. AI can now generate Lean code that is checked down to axioms, avoiding the unreliability of natural-language proof. Alex had expected Fermat’s Last Theorem to require years of formalization, but Anthropic produced a 13-million-line, sorry-free Lean proof in days. He says only two or three humans may have been involved, mostly telling the systems to continue. The result is not yet elegant or Mathlib-quality—it is “jerry-rigged”—but it establishes that auto-formalizing existing human mathematics is essentially solved.

Alex describes primes as the fundamental particles of multiplication: every whole number decomposes into primes, but their distribution remains mysterious. Riemann discovered that a complex-valued zeta function encodes the primes, much as Fourier analysis decomposes Lex’s voice into frequencies and amplitudes and reconstructs it. The zeta function’s zeros act like frequencies in an orchestra playing the music of the primes. The Riemann hypothesis says all nontrivial zeros lie on the central line with real part one-half inside the critical strip. The quasi-Riemann result proves a substantial zero-free strip between real parts seven-eighths and one, far stronger than previous knowledge, but its methods cannot reach one-half; a genuinely new idea is needed for the full hypothesis.

Brunton explains Navier–Stokes as conservation of momentum for fluids, describing velocity, pressure, viscosity, and external forces across space and time. The difficulty comes from billions of interacting local volumes and nonlinear self-motion. The AI proof establishes a smooth external forcing in three dimensions that orchestrates a vortex whose radius shrinks indefinitely, producing finite-time blow-up and infinite vorticity while total energy remains finite. Two-dimensional fluids behave differently: weather supports a reverse cascade that helps form hurricanes, whereas three-dimensional turbulence cascades from large vortices to smaller ones. Brunton emphasizes that the result is mathematically important but highly nonphysical; day-to-day engineering still faces a trillion-dollar problem of predicting and controlling real turbulence efficiently.

Brunton says turbulence is difficult because every scale matters: unlike a gas, whose behavior can be summarized by temperature, pressure, and density, turbulent flow around a wing or inside a heart cannot yet be compressed into a few useful variables. He sees AI as promising for discovering a coordinate system that makes turbulent dynamics simple and generalizable, combining large simulation datasets with mathematical structure. Nature already demonstrates successful control: insects use strain-sensitive neurons embedded in their wings to react to gusts faster than signals could travel to the brain and back. Ankur adds that nature may be performing computation, as in slime molds or game-of-life-like systems, and that uncertainty-aware, robust models may be more realistic than assuming complete knowledge of initial and boundary conditions.

Ankur introduces satisfiability: variables are assigned true or false, clauses are ORs, and the whole formula is an AND; checking a proposed assignment is easy, while finding one may be hard. The PCP theorem showed that even probabilistically checking a purported proof can be robustly hard, founding hardness-of-approximation theory. The Unique Games Conjecture strengthens this framework and, if true, organizes optimal approximation thresholds for many problems. Human researchers had made major progress, including the Two-to-One Games Theorem, but AI’s release supplied a proof that the conjecture is true. Ankur was relieved to see the result public, yet worried that the community lost years of potential human discoveries and that researchers may become more secretive with crucial ideas.

The panel says mathematical publication is already strained by too many papers and exhausted referees; AI-generated results intensify the problem. Alex recalls how Perelman’s Poincaré-conjecture proof required years of seminars and line-by-line community verification, because a written proof is only the first step toward shared understanding. Ankur favors models resembling open-source software: public pull requests, maintainers, collaborative refactoring, and Mathlib-style formal specifications. Formalization prevents misuse because Lean will reject an improperly applied theorem, while natural-language exposition can focus on the story rather than every technical detail. Alex describes teaching real analysis by delaying formal construction of the real numbers until students encountered a theorem that required it, arguing that mathematical communication should tell the conceptual story alongside a machine-checked proof.

Ankur says AI’s mathematical progress does not yet explain why deep learning works. Worst-case optimization problems involved in fitting neural networks can be cryptographically hard, yet natural data makes these systems effective. He compares this with protein folding: folding is hard in the worst case, but evolution may have selected sequences that heuristics can fold reliably. “Nature abhors computation-hard problems” is his proposed intuition. For P versus NP, he remains skeptical that AI will solve it soon. SAT appears to require far more than reading the input in the worst case, even though practical SAT solvers work extremely well on structured instances such as compiler problems. The important future question is how to model real-world structure beyond worst-case analysis.

Ankur describes subliminal learning: one model prompted to love owls generates random-looking numbers, and fine-tuning another model on those numbers causes the second model to mention owls despite never seeing the prompt. He attributes this to many concepts being embedded as non-orthogonal vectors, so small coordinated effects can accumulate like a Ouija board. The panel warns that such hidden behaviors become dangerous when AI systems control infrastructure or high-stakes decisions. Brunton proposes a safety framework based on stakes and verifiability: AI-generated Lean proofs or aircraft designs are relatively safe because errors can be checked, while medical, legal, or critical infrastructure decisions require firewalls and established human protocols. They agree that intelligence does not imply morality or wisdom.

For students, Alex recommends learning as much difficult, abstract mathematics as possible and using AI to reach the “zone of proximal development,” while still learning fundamentals rather than outsourcing everything. Brunton emphasizes curiosity, reading, history, deep focus, and resisting the engagement economy; boredom can create room for independent thought. Ankur advises pursuing intrinsic reasons rather than external status, becoming comfortable with uncertainty, and pretending one has already won: what would one do after tenure or success? He says meaningful work combines importance, interest, and personal ability. The panel also stresses family, friendship, and community: as AI commodifies intelligence, human relationships may become more—not less—central to a fulfilling life.