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

What It Covers

Terence Tao, Fields Medal winner, explores hardest problems in mathematics including Navier-Stokes equations, Kakeya conjecture, wave concentration phenomena, connections between mathematical fields, proof formalization using Lean, and AI's emerging role in mathematical discovery and verification.

Key Questions Answered

• •Kakeya Problem Solution: The conjecture about minimum volume needed to rotate a needle in three dimensions connects to wave propagation and partial differential equations. Proving it required showing waves cannot concentrate efficiently at small scales, which has implications for understanding singularities in fluid dynamics and other nonlinear systems.
• •Navier-Stokes Blow-Up Strategy: Constructing a liquid computer using vortex rings as logic gates could theoretically create finite-time blow-up in fluid equations. The approach requires building self-replicating water configurations that transfer energy to progressively smaller scales faster than viscosity can dissipate it, mimicking Von Neumann machines in fluid form.
• •Mathematical Problem-Solving Method: Turn off nine of ten difficulties in a problem first, solve each separately, then combine solutions incrementally. This strategic cheating approach—changing dimensions, ignoring error terms, or simplifying nonlinearities—makes intractable problems manageable by isolating individual challenges before addressing their interactions.
• •Supercriticality in Equations: When nonlinear transport terms dominate dissipation at small scales, equations become supercritical and unpredictable. This explains why weather forecasting fails beyond two weeks while planetary motion predicts millennia ahead. The balance between competing forces at different scales determines whether systems remain stable or develop singularities.
• •Lean Proof Assistant Workflow: Formalizing mathematical proofs takes ten times longer than writing informal versions, but AI autocomplete now succeeds twenty-five percent of the time at suggesting correct proof steps. The bottleneck shifted from writing proofs to searching Mathlib's tens of thousands of lemmas, where large language models increasingly assist.