Terence Tao on the Hardest Problems in Mathematics, Physics and the Future of AI — Notes

Four questions [Adler frame]

Q1 — What is it about? A long-form conversation tracing Tao’s mathematical practice across four domains: hard open problems (Kakeya, Navier-Stokes), the philosophy of mathematics versus physics, the structure–randomness dichotomy in combinatorics, and the emerging role of AI and formal proof systems in mathematics. The thread running through all of it is how mathematicians isolate difficulty, exploit analogy across fields, and build toward results that seem impossibly hard.

Q2 — How is it argued? Tao argues by concrete example and analogy throughout. Abstract concepts (supercriticality, universality, finitisation) are introduced through accessible metaphors — Maxwell’s Demon, Hong Kong action movies, monkeys and typewriters — before precise statements arrive. This mirrors his self-described “fox” style: cross-domain pattern recognition over narrow technical depth.

Q3 — Is it true? Technically accurate throughout. Tao is careful to distinguish established results (Szemerédi’s Theorem, Ladyzhenskaya’s 2D regularity, the liquid Turing machine as a programme for Navier-Stokes rather than a proof) from conjecture. The 2008 financial crisis discussion accurately identifies correlation breakdown as the failure point of Gaussian default models.

Q4 — What of it? The most transferable idea is strategic cheating: systematically turning off nine of ten difficulties, solving the simplified problem, then re-enabling difficulties one at a time. This is a general heuristic for any complex problem, not just mathematics. The liquid Turing machine section offers a striking argument that Navier-Stokes blowup is in principle constructible — a roadmap even if the engineering details are missing.

Glossary

Kakeya Problem — Originally: what is the minimum area to rotate a unit needle through all directions in the plane? Answer: arbitrarily small (Besicovitch). In higher dimensions: what is the minimum volume for a delta-thick needle to sweep all directions? The conjecture is that volume decreases only logarithmically slowly with thickness delta. Recently proved. Connected to wave propagation, number theory, and PDEs.

Navier-Stokes regularity problem — Millennium Prize Problem. Do smooth solutions to the 3D incompressible Navier-Stokes equations (governing water) always remain smooth, or can they develop singularities (velocities becoming infinite in finite time)? The latter is called finite time blowup.

Supercriticality — A qualitative property of PDEs. An equation is supercritical when nonlinear effects dominate linear (dissipative) effects at small scales. Navier-Stokes in 3D is supercritical; in 2D it is critical (Ladyzhenskaya proved global regularity in 2D in the 1960s). Supercriticality is why Navier-Stokes blowup cannot be ruled out by current techniques.

Finite time blowup — A scenario in which a PDE solution’s energy concentrates into a single point in a finite time interval, causing the solution to become undefined beyond that time.

Maxwell’s Demon — A thermodynamic thought experiment: a hypothetical demon that sorts gas molecules to create an entropy-decreasing configuration that is statistically impossible but not physically prohibited. Tao uses it as a metaphor for why global regularity of Navier-Stokes is so hard to prove: a fluid “demon” might concentrate energy into smaller and smaller scales.

Liquid Turing machine — Tao’s speculative construction: a configuration of fluid (Navier-Stokes) that computes — built from water-pulses acting as bits and fluid collisions acting as logic gates. A self-replicating, self-shrinking fluid computer would demonstrate finite time blowup by reducing scale at each iteration.

Structure vs randomness dichotomy — Tao’s framing of a fundamental split in combinatorics and analysis: any mathematical object is either essentially random (no discernible pattern) or essentially structured (explainable by a simpler, patterned object). Inverse theorems make this dichotomy precise.

Szemerédi’s Theorem — Any set of integers with positive density contains arithmetic progressions of arbitrarily long length. Proved by Szemerédi in 1975. Applies to both highly structured sets (odd numbers) and seemingly random ones. Proof involves the structure–randomness dichotomy.

Inverse theorem — A result that identifies when a function or set exhibits approximate structure. If something behaves almost additively, an inverse theorem says it must be close to a genuinely additive (structured) object. Key tool for making the structure–randomness dichotomy rigorous.

Universality — The phenomenon by which macroscopic behaviour (e.g. Gaussian distributions) emerges from many different micro-scale interactions. The central limit theorem is the archetypal universality result. Breaks down when inputs are systemically correlated — as in the 2008 financial crisis default models.

Finitisation — Converting an infinitary mathematical result into a quantitative finite statement. Slower to obtain but more intuitive: “how many monkeys, how long, to produce Hamlet?” rather than “almost surely, infinite monkeys will.”

Hamiltonian mechanics — A reformulation of Newtonian mechanics by Hamilton (19th century) where energy (the Hamiltonian) is the central object. Conservation laws follow from symmetries (Noether’s theorem). Crucially, the same Hamiltonian framework extends to quantum mechanics (the Schrödinger equation), enabling direct transfer of classical intuitions.

Noether’s theorem — Every continuous symmetry of a physical system corresponds to a conservation law: translational symmetry → momentum; rotational symmetry → angular momentum; time symmetry → energy. True in both classical and quantum mechanics via the Hamiltonian.

Wave maps equation — A PDE that describes a vector field constrained to lie on a sphere, evolving as a wave. One step below Einstein’s equations in difficulty. Tao proved global regularity for this equation at critical energy, using a gauge transformation discovered while lying on his aunt’s floor.

Lean — A formal proof verification language. Designed so that individual lines of code correspond to individual steps in a mathematical argument. Produces certificates (proofs with fully checked derivations). Backed by Mathlib, a library of tens of thousands of verified mathematical lemmas.

Mathlib — The community-built library of formalised mathematics for Lean. The current bottleneck in using Lean is lemma search: finding the right fact in tens of thousands of results. LLMs are beginning to assist with this search task.

Strategic cheating — Tao’s heuristic for attacking difficult problems: if there are ten sources of difficulty, turn off nine of them and solve the resulting simpler problem. Repeat for each difficulty individually, then begin combining them. Analogy: Hong Kong action choreography, where the hero always fights one opponent at a time.

The Kakeya Problem and wave concentration [§ First Hard Problem]

The Kakeya Problem illustrates Tao’s core method: a puzzle that looks recreational becomes a key node in a web of deep connections. The three-dimensional version asks how small a volume a delta-thick needle needs to sweep through all directions. The Kakeya conjecture says this volume shrinks only logarithmically as delta → 0. The connection to wave propagation: waves can be decomposed into wave packets (tubes in space-time), and the Kakeya geometry determines how tightly these tubes can be packed, which in turn governs whether certain waves can concentrate to a singularity (blowup).

Navier-Stokes and the liquid Turing machine [§ Navier-Stokes Singularity]

Tao’s 2016 paper constructs an averaged Navier-Stokes equation (with certain energy channels artificially blocked) that demonstrably blows up in finite time. This is not a proof that actual Navier-Stokes blows up, but an obstruction result: any proof of global regularity for the true equation must use some property that the averaged equation lacks.

The key structural obstacle in 3D (unlike 2D) is that naïve scale-to-scale energy transfer disperses too quickly; viscosity catches up. Tao’s fix: engineer an airlock mechanism — like a sequence of capacitors — that holds energy at one scale until all of it transfers before opening the next gate. This engineering mindset led to the liquid Turing machine idea: if a fluid could sustain computation (logic gates, clock, self-replication at decreasing scales), then each self-similar iteration would blow up in finite time. Analogous structures exist in Conway’s Game of Life (gliders, gates, self-replication), offering historical precedent.

The programme is currently a pipe dream (no water vortex ring has been shown to function as a logic gate), but Tao considers it physically consistent with the Navier-Stokes equations.

Structure vs randomness and Szemerédi’s Theorem [§ Infinity]

The structure–randomness dichotomy is Tao’s unifying lens across combinatorics, number theory, and analysis. Any set or function either has no pattern (behaves like random data) or is secretly structured (well-approximated by an explicit algebraic or geometric object). Inverse theorems make this actionable: approximate additivity implies proximity to exact additivity. Szemerédi’s Theorem — arithmetic progressions exist in any dense set — follows from this dichotomy: either the set is random (and random sets contain long progressions by the infinite monkey argument) or it is structured (and structured sets contain long progressions trivially).

Mathematics vs physics and the compression model [§ Math vs Physics]

Tao’s framing: science mediates between reality, observations, and models. Mathematics stays within the model, deriving consequences from axioms. Science collects observations and proposes models. A good theory is a compression of observations: fewer parameters than data points, high explanatory coverage. The dark matter model has ~14 parameters and fits petabytes of astronomical data. The most mysterious fact about the universe, per Tao, is that it is compressible at all (Wigner’s “unreasonable effectiveness”).

Universality — the central limit theorem as its most tractable case — provides a partial answer: many micro-scale processes produce the same macro-scale statistics, so the universe contains fewer effective degrees of freedom than it might.

Hamiltonian mechanics as conceptual bridge [§ Solving Difficult Problems]

Hamilton’s reformulation of Newtonian mechanics (replacing F=ma with the energy-centric Hamiltonian) was a change of language that turned out to be the correct language for quantum mechanics too. The Schrödinger equation is structurally identical to Hamiltonian evolution; Noether’s theorem applies in both regimes. This is Tao’s clearest example of the “right concept” revealing itself through notational convergence across disparate fields.

Lean and AI-assisted proof [§ AI-Assisted Theorem Proving]

Tao uses GitHub Copilot as a lemma-search tool within Lean: he types a description of what he needs (“fundamental theorem of calculus at this step”), and the LLM suggests a Mathlib lemma roughly 25% of the time. The bottleneck is not proof generation — formal derivation is handled by Lean’s type checker — but lemma retrieval: finding the right pre-existing result in a library of tens of thousands. AI for mathematics is currently an information retrieval problem, not a reasoning problem. Tao also uses LLMs for rapid exploratory coding (plotting functions, running simple iterations) that previously required two hours of Python debugging.

Strategic cheating as method [§ Solving Difficult Problems]

The most practically generalisable insight. When a problem has multiple interacting difficulties, never attack all simultaneously. Isolate each, solve in isolation, then recombine. The key discipline is knowing what not to try: obstruction results (like Tao’s averaged Navier-Stokes) rule out entire families of approaches, compressing the search space. Mathematics is as much about eliminating wrong paths as finding correct ones.