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

Speaker: Terence Tao Source: Lex Fridman Podcast URL: https://lexfridman.com/terence-tao-transcript

Key ideas

Strategic cheating solves hard problems. When a problem has ten sources of difficulty, turn off nine and solve the remaining one. Learn each difficulty in isolation, then recombine. Never try to fight all opponents simultaneously — a lesson equally applicable to engineering, coding, and research.
Supercriticality explains why Navier-Stokes is unsolved. In 3D, nonlinear effects dominate dissipation at small scales (supercritical). In 2D, they are balanced (critical) — hence Ladyzhenskaya proved 2D regularity in the 1960s. Supercriticality is also why weather is unpredictable beyond ~two weeks.
The liquid Turing machine is a roadmap for blowup. Tao’s 2016 paper showed an averaged Navier-Stokes equation blows up, ruling out whole classes of regularity proofs. The next step — constructing a self-replicating fluid computer — would prove actual Navier-Stokes blowup. Currently a pipe dream, but physically consistent.
Good theory is compression. A physical model’s quality is measured by how much observational data it compresses into how few parameters. The dark matter model: ~14 parameters, petabytes of astronomical data explained.
AI is now useful in formal proof via lemma search. Lean + Mathlib handles derivation; the bottleneck is finding the right pre-existing lemma. LLMs acting as information retrieval tools (like GitHub Copilot) close this gap roughly 25% of the time.

The Kakeya Problem

The Kakeya Problem asks: what is the minimum volume for a delta-thick needle in 3D space to sweep through every possible direction? The conjecture: the volume decreases only logarithmically as delta → 0. Recently proved after decades of work.

Importance: Kakeya geometry controls how wave packets (tubes in space-time) can concentrate. Tight packing of tubes in many directions is the geometric mechanism that could allow certain PDEs to produce singularities (wave blowup). Understanding Kakeya informs the wave concentration aspects of Navier-Stokes.

Navier-Stokes and finite time blowup

The Navier-Stokes Millennium Prize Problem asks whether smooth solutions to the 3D incompressible Navier-Stokes equations can develop singularities in finite time. Current consensus has shifted toward believing blowup is possible for specially crafted initial conditions, but no proof exists.

The obstacle: a “Maxwell’s Demon” of fluid dynamics — a hypothetical self-reinforcing energy cascade from large to small scales, faster than viscosity can damp it. In 3D, the equation is supercritical: at small scales, nonlinear transport overwhelms viscous dissipation.

Tao’s 2016 work: by surgically blocking certain energy transfer channels in an averaged Navier-Stokes equation, he forced energy to concentrate in a self-similar cascade and proved blowup for this modified system. This is an obstruction: any proof of regularity for the true equation must exploit features the averaged equation lacks. It rules out whole families of proof strategies without those strategies having to be tried.

The liquid Turing machine

Inspired by Conway’s Game of Life — where simple rules (three or four) support gliders, logic gates, self-replicating structures, and Turing-complete computation — Tao proposes the same for Navier-Stokes. If a fluid could sustain:

Logic gates (colliding vortex rings producing defined outputs)
A clock (regulating energy transfer scale by scale)
A self-replicating machine (a fluid robot that spawns a smaller version of itself, transfers all energy, then powers down)

…then each iteration would occur faster and at a smaller scale, converging to a singularity. The key engineering challenge in Tao’s 2016 construction was the airlock: energy must be fully committed to one scale before the gate to the next opens (direct cascade disperses too quickly and viscosity wins).

Structure vs randomness

The deepest organising principle in Tao’s work. Any mathematical object either has structure (it is well-approximated by a simpler, patterned object — arithmetic progressions, algebraic sets, low-rank functions) or behaves randomly (no discernible pattern). Inverse theorems make this rigorous: approximate additivity implies closeness to exact additivity.

Szemerédi’s Theorem: any set of integers with positive density contains arithmetic progressions of arbitrary length. Both structured and random sets satisfy this — for different reasons. The dichotomy framework unifies the proof.

Universality (the central limit theorem as the simplest case): many micro-scale processes produce the same macro-scale statistics (Gaussian). Breaks down when inputs are systemically correlated — the 2008 financial crisis was a failure of the independence assumption in mortgage default models.

Mathematics vs physics

Tao’s tripartite model: reality → observations → mental models. Mathematics works within models (axioms → consequences). Physics collects observations and proposes models. Engineering works backwards from desired consequences. Mathematics is unusual in being primarily antecedent-driven rather than consequent-driven.

A good theory compresses: fewer parameters than observations. The “unreasonable effectiveness of mathematics” is partly explained by universality — the universe has fewer effective degrees of freedom than its microscopic complexity suggests.

Hamiltonian mechanics and the right language

Hamilton reformulated Newtonian mechanics with energy (the Hamiltonian) as the central object rather than force (F=ma). This turned out to be the natural language for quantum mechanics too: the Schrödinger equation is Hamiltonian evolution on a Hilbert space, and Noether’s theorem (symmetry → conservation) holds in both regimes.

The implication: discovering the “right” concept in one framework transfers directly to another. Finding the analogous unifying object for a quantum gravity theory of everything is the open problem in fundamental physics.

Lean and AI-assisted proof

Lean is a formal proof language where each line of code corresponds to a step in a mathematical argument, and every object carries a type certificate. Lean’s compiler verifies proofs to 100% correctness (modulo compiler trust). Mathlib is the community library of tens of thousands of verified lemmas.

Current bottleneck: lemma search. Tao uses GitHub Copilot within Lean to retrieve relevant Mathlib lemmas by typing natural language descriptions. Success rate: ~25%. Separately, he uses LLMs for rapid exploratory coding (plotting, iteration) that previously took hours of Python setup.