Tao, Terence

mathematician, UCLA; Fields Medalist

Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

Lex Fridman Podcast · Jun 14, 2025

• Terence Tao
• mathematics
• Navier-Stokes
• twin prime conjecture
• Riemann hypothesis
• Collatz conjecture
• P vs NP
• Lean proof assistant
• AI and mathematics
• Poincaré conjecture

Core ideas

Tao structures the conversation around a handful of the "hardest problems," using each as a lens on how mathematicians actually think. He opens with the Kakeya needle problem (can a needle be rotated 180° using arbitrarily little area — Besicovitch showed yes, with zero-area constructions) and its 3D descendants, which connect to wave propagation and PDE blowup, before turning to the Navier-Stokes global regularity problem: his own 2016 result constructed finite-time blowup for an averaged version of the equations by exploiting supercriticality (the transport term dominating viscosity at small scales, unlike the critical 2D case solved in the 1960s). His proposed roadmap for the real 3D problem is a "liquid Turing machine" — a von-Neumann-style self-replicating machine built out of fluid, an idea he credits to thinking with his electrical-engineer wife — which he stresses is still a "pipe dream" missing key components, but draws a precise analogy to Conway's Game of Life, where engineered initial conditions (gliders, glider guns, self-replicating machines) produce structure that never emerges from typical/random configurations. This dichotomy between structure and randomness recurs as his single deepest theme, later resurfacing in his discussion of the primes: he argues mathematicians have very few tools to prove genuine randomness (as opposed to observing it statistically), and that essentially every famous open problem in number theory — twin primes, Goldbach, the Riemann hypothesis — is really a disguised version of "prove no conspiracy is hiding a pattern," blocked by what he calls the parity barrier, a specific, identified obstruction that current sieve methods cannot cross no matter how cleverly applied.

On mathematical practice, Tao describes himself as a "fox" rather than a "hedgehog" — moving between fields, willing to switch to a different problem rather than fixate, in explicit contrast to Grigori Perelman (who solved the Poincaré conjecture, via Ricci flow, by inventing new scale-invariant quantities that turned a supercritical problem critical, then classifying and surgically resolving all possible singularities — work Tao calls comparable in difficulty to his own wave-maps research) and Andrew Wiles (seven years of solitary work on Fermat's Last Theorem, which Tao witnessed firsthand as a Princeton graduate student). He offers a detailed account of his own problem-solving method — "cheating strategically" by turning off nine of ten difficulties in a problem to solve the tenth in isolation, then recombining, an approach he compares to how Hong Kong action films choreograph one-on-one fights — and describes an emotionally costly real example: a four-person collaboration that appeared solved after two months, only for one unestimated term in a lemma to block progress for two more years before an entirely different approach succeeded, a "false dawn" he says was necessary to keep the team motivated long enough to eventually succeed. He is notably unsentimental about failure, attributing this partly to never letting any single problem consume him the way "mathematical diseases" consume some colleagues, and describes his own atypical, accelerated mentored education as no template for what makes a good mathematician — arguing people find their "native math language" (visual, symbolic, competitive/game-based) through different repurposed brain systems, and that one-size-fits-all math education fails most students by not surfacing this.

The second half turns to AI and formalization. Tao explains Lean not as an alternative to writing proofs but as a system that attaches machine-checkable certificates to every step, making atomic-scale, trustless collaboration across dozens of strangers possible (he cites his Equational Theories Project — 22 million abstract-algebra implication questions, crowdsourced across ~50 contributors, all but two now resolved) and making proofs anti-fragile to later changes (updating one constant in a formalized theorem highlights exactly which downstream lines break, versus re-checking an entire pen-and-paper proof by hand). He is measured about current AI: large language models are useful as literature-search and "fancy autocomplete" tools inside Lean (citing Mathlib lemma search) and can already contribute partial results to real papers, but he stresses their central limitation is a lack of "mathematical smell" — the ability to sense, without proof, whether a proof strategy is heading toward or away from a solution — something AlphaZero-style systems developed for Go/chess but that mathematical LLMs conspicuously lack, making multi-step proofs fail combinatorially (a 10% per-step error rate compounds catastrophically over 20 steps). He predicts this decade will see an AI generate a genuinely novel, correct mathematical conjecture linking previously unrelated areas, considers a Fields-Medal-level AI-assisted proof plausible but distant, and diagnoses a deeper data problem: mathematics as a field only publishes successes, discarding the trial-and-error record of abandoned approaches that would teach a model how mathematicians recognize dead ends — "AI has to go through grad school," including office hours and mistakes, not just textbooks. He also discusses DeepMind's AlphaProof (silver-medal IMO performance, but requiring three days of server time per problem, not remotely scalable), the unsolved difficulty of translating natural language into formal statements, and physics parallels — general relativity's yet-unfound "Hamiltonian" for quantum gravity, and how unification (Maxwell, Newton) has always been physics' historical trajectory, giving him optimism a theory of everything is achievable in principle even though our two best theories are, ironically, victims of their own success (covering 99.9% of observable phenomena, leaving little experimental traction for a successor).

He closes on broader reflections: the danger of collapsing credit for large collaborative or centuries-long achievements onto one iconic name (Wiles, Perelman, Steve Jobs) as a necessary but distorting simplification; his respect for, but personal disagreement with, Perelman's rejection of the Fields Medal and Millennium Prize money as a matter of principle; his view that winning the Fields Medal shifted him from technical work toward being asked for opinions on the direction of the whole field, requiring him to "walk the walk" on AI/Lean rather than merely advocate for it; and a broadly optimistic view of collective human intelligence — math communities (MathOverflow, Polymath projects) regularly exceeding what any individual mathematician could produce alone — as reason for hope about civilization's trajectory generally.

Excerpts