Jun 14, 2025, 4:15 PM ETInterview

Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

Tao views difficult research problems as those situated on the boundary between solvable and hopeless, where existing techniques solve 90% of the problem but fail on the remaining 10%.
He adopts a "fox" persona, preferring to bridge disparate fields (e.g., number theory and geometry) rather than a "hedgehog" who focuses deeply on a single area.
His problem-solving strategy involves "strategic cheating": simplifying complex problems by removing up to 9 out of 10 difficulties to understand the core obstruction, then reintegrating them.
Tao emphasizes the "dichotomy between structure and randomness," arguing that most mathematical objects are either random or related to a structured model, with few exceptions.
He describes the process of formalizing proofs in Lean as creating a "supernatural" environment where one must act like a "pedantic colleague" to ensure absolute correctness.

The problem asks whether smooth initial conditions for incompressible fluid flow can lead to a "finite time blow-up" (infinite velocity) in the future.
Tao notes that while real-world fluids (like water) rarely exhibit blow-up due to viscosity, the mathematical possibility exists for supercritical equations where nonlinear transport dominates linear dissipation at small scales.
In 2016, Tao published a paper proving that an averaged version of the 3D Navier-Stokes equations can undergo finite-time blow-up.
This averaged equation was engineered by removing specific energy transfer channels to force energy to concentrate at smaller scales, acting as a counter-example to rules that might otherwise be proposed for the real equations.
The blow-up construction for the averaged equation involves a "fluid computer" capable of simulating a Turing machine, specifically designed to perform logic gates (AND/OR) using vortex rings and fluid dynamics.
This "fluid Turing machine" concept demonstrates that if fluid dynamics can support computation, it can theoretically support a self-replicating machine that shrinks in scale, leading to a singularity.
The actual Navier-Stokes equation is "supercritical," making it significantly harder to analyze than 2D equations (which are "critical" and proven to be regular) or "subcritical" equations.

Perelman solved the Poincaré conjecture using Ricci Flow, a partial differential equation that smooths out curved spaces to reveal their underlying topology.
Perelman's key innovation was introducing "reduced volume" and "entropy" to convert the problem from supercritical to critical, neutralizing the nonlinearities that cause singularities.
Tao highlights Perelman's isolation and refusal of the Fields Medal and the $1 million Millennium Prize as an outlier behavior, noting his belief that if the proof is correct, external recognition is irrelevant.
The conjecture posits that any simply connected, closed 3-dimensional manifold is topologically equivalent to a 3-sphere.

The Twin Prime Conjecture is considered exceptionally difficult because a "conspiracy" could theoretically eliminate twin primes while preserving other statistical properties of primes (like the Prime Number Theorem).
Tao and Ben Green proved the Green-Tao Theorem: the prime numbers contain arithmetic progressions of any arbitrary length.
Unlike twin primes, arithmetic progressions are "indestructible"; even if 99% of primes are removed arbitrarily, the remaining set still contains arbitrarily long progressions.
Current techniques are blocked from proving the Twin Prime Conjecture by the "parity barrier," which prevents proving a gap smaller than a certain threshold (currently 246) without new mathematical breakthroughs.
The Riemann Hypothesis remains a distant goal; Tao believes a proof will likely come from a "breakthrough in another area" rather than direct attack on the hypothesis itself.
Tao has made partial progress on the Collatz Conjecture, proving that 90% of starting numbers eventually decrease, though the existence of an infinite loop (outlier) remains unresolved.

Tao actively uses the Lean proof assistant, which translates mathematical arguments into code to verify correctness with 100% certainty, effectively creating a "trustless" mathematical environment.
Formalizing a proof in Lean currently takes roughly 10 times longer than writing it by hand, but it allows for modular collaboration and easy updates when parameters change.
A recent "Equation of Theories" project utilized Lean to settle 22 million questions in abstract algebra involving over 4,000 possible algebraic laws, involving ~50 collaborators.
Tao observes that while AI (like AlphaProof) can solve high-school level problems, it struggles with the exponential complexity of higher-level proofs where a single error at any step invalidates the result.
AI currently excels at generating "fancy autocomplete" for proofs but lacks "mathematical smell" to intuitively recognize when a proof strategy is doomed or when a step is trivial.
Tao predicts that within the next decade, AI-human collaborations will yield research-level published results, though a Fields Medal-winning proof generated primarily by AI is likely 20–50 years away.
He anticipates a "phase shift" similar to the adoption of LaTeX, where formalization becomes standard practice, allowing journals to accept papers with significantly less manual verification of correctness.

Tao suggests that human brains do not have a dedicated "math center" but rather repurpose visual, linguistic, or puzzle-solving regions for mathematical thinking.
He advocates for diverse teaching methods to accommodate different cognitive styles (visual vs. symbolic) rather than a one-size-fits-all curriculum.
Tao notes that the "establishment" role of a Fields Medalist can be constraining regarding time, but he views it as an opportunity to inspire and mentor younger generations.
He encourages students to embrace "structured procrastination" and to view failure as a necessary step in exploring the "space of proofs."
Tao believes the future of mathematics lies in the synergy between human intuition and AI's computational power, allowing for massive-scale collaboration on problems that are currently too complex for individuals.