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

Tao is drawn to problems near the boundary of available technique, where existing methods do much of the work but leave a decisive gap. His example is the Kakeya problem: how little area or volume is needed to move a needle through all directions? A planar construction can use arbitrarily little area, while a thin three-dimensional tube raises a quantitative question as its thickness decreases. The geometric puzzle matters beyond its appearance. Wave packets trace tube-like regions in space-time, so understanding how differently directed tubes overlap can inform questions about concentration in partial differential equations. Tao describes the recent Kakeya resolution without treating it as a solution of every related wave problem.

Navier–Stokes regularity asks a stricter question than whether ordinary water usually behaves well: can some smooth initial velocity field develop an infinite velocity at a point? Tao stresses the gap between overwhelmingly typical behavior and a statement covering every case. Viscosity damps dispersed energy, but a hypothetical cascade could keep transferring energy into smaller regions quickly enough to evade dissipation. His Maxwell’s demon and digits-of-pi analogies illustrate an inability to exclude an exceptional conspiracy, not evidence that such a fluid configuration has been constructed. He explicitly calls Kakeya’s relevance to Navier–Stokes indirect, through a better understanding of concentration.

Tao engineered finite-time blowup for an averaged, modified Navier–Stokes equation by selecting which energy transfers remain active while preserving energy conservation. This does not establish blowup for the actual fluid equations. Its value is as an obstruction: a proof of regularity for the true equation must use a feature absent from the modified system. Conservation and viscosity alone cannot support every hoped-for general argument. The underlying distinction is criticality. In his account, supercritical scaling lets nonlinear transport become relatively stronger at fine scales, whereas critical or subcritical problems provide more established ways to control the competition with dissipation.

The construction needed timing, not simply the fastest possible cascade. Pushing at every scale simultaneously spreads energy and exposes it to viscosity. Tao instead designed delays resembling airlocks and circuit components, keeping energy localized before opening the next gate. This suggested a much more speculative route for actual water: logic gates, computation and a fluid machine that builds a smaller copy, transfers its energy and shuts down. He lists missing ingredients, including the gates, error control and a clean shutdown. A roadmap inspired by computation is therefore distinct from an implemented fluid computer or a proof of actual Navier–Stokes blowup.