Terence Tao — Fields Medal and Breakthrough Prize-winning mathematician, often called the 'Mozart of math,' known for groundbreaking work across an astonishing range of fields in mathematics and physics. He is a professor at UCLA and a celebrated collaborator.
The gist
Lex Fridman talks with mathematician Terence Tao about the deepest open problems in mathematics and physics and the nature of mathematical discovery. Tao explains the Navier-Stokes regularity problem, his idea of a fluid 'liquid computer' that could force a blowup, the Kakeya conjecture, and the dichotomy between structure and randomness underlying problems like the twin prime conjecture and the Riemann hypothesis. He discusses universality in physics, the difference between mathematics, physics, and engineering, and Perelman's proof of the Poincare conjecture. A major thread is the future of mathematics: the Lean proof assistant, massive crowdsourced formalization projects, and how AI is beginning to assist with proofs, computation, and literature review.
Big reveals
- Tao describes building a 'liquid computer' out of water that could function like a von Neumann self-replicating machine, providing a roadmap to force finite-time blowup in the Navier-Stokes equations.
- He explains his 2016 result constructing an averaged Navier-Stokes equation that blows up, by selectively turning off interactions to engineer a blowup and thereby rule out certain proof approaches.
- Tao introduces the Equational Theories Project, which generated about 22 million abstract-algebra implication problems and settled all but two using Lean and roughly 50 collaborators.
- He explains the 'parity barrier' that blocks proving the twin prime conjecture, Goldbach, and other problems because techniques can't push prime density inside almost-primes above 50%.
- Tao describes his Collatz result proving that statistically about 99% of inputs drift down to much smaller values, while the full conjecture remains out of reach due to possible outliers.
- Tao predicts that by 2026 there will be research-level math collaborations where AI generates part of the work, even if not Fields-Medal-level ideas.
- He explains how Perelman solved the Poincare conjecture by converting a supercritical problem into a critical one using new quantities (reduced volume and entropy) and then classifying all singularities of Ricci flow.
Things worth remembering
- The Kakeya needle problem originated from a 1918 puzzle by Japanese mathematician Soichi Kakeya about turning a needle around using as little area as possible.
- Of the seven Clay Millennium Prize Problems, each carrying a million-dollar prize, only one (the Poincare conjecture, by Perelman) has been solved.
- The infinite monkey theorem implies that an infinite random string eventually contains any finite pattern, including arithmetic progressions of any length.
- To model a gas of quintillions of particles, you only need about five or six parameters like temperature, pressure, and volume, an example of universality.
- Tao estimates that formalizing a proof in Lean currently takes about 10 times as long as writing it out by hand.
- DeepMind's AlphaProof achieved silver-medal-equivalent performance on IMO problems but took about three days of server time to solve a single high school math problem.
- 'Sexy primes' are primes that differ by six; cousin primes differ by four, and the current proven bound is infinitely many prime pairs differing by at most 246.
- As a Princeton graduate student, Tao witnessed Andrew Wiles announce his proof of Fermat's Last Theorem, recalling that Wiles's mailbox suddenly overflowed with mail.
- Tao got involved with Lean and AI after enthusiastically proposing a workshop on computer-assisted proofs, then feeling obligated to 'walk the walk' rather than just talk about it.
- In the Collatz process, the number 13 climbs to 40 before eventually descending to the repeating 4-2-1 cycle; such paths are called hailstone sequences.