Terence Tao is arguably the most celebrated living mathematician, and the Navier-Stokes equations are one of the seven Millennium Prize Problems — a set of questions the Clay Mathematics Institute considers so foundational that solving any one earns you a million dollars and a permanent place in the history of the field. When Tao writes publicly about Navier-Stokes, technical readers should pay attention, because he has spent more than a decade circling this problem and has produced some of the most consequential partial results in the area.

The Navier-Stokes equations describe how fluids move — everything from the airflow over an aircraft wing to blood in your arteries to the swirl of cream in coffee. Engineers use them constantly, plugging them into simulations and trusting that they behave. But mathematically, we still don't know whether smooth initial conditions in three dimensions always yield smooth solutions for all time, or whether the equations can develop singularities — points where velocity or pressure blows up to infinity. This is the "global regularity" question, and it's genuinely open.

Tao's most famous contribution here is his 2014 paper showing that an averaged version of Navier-Stokes does blow up in finite time. This was a bombshell because it demonstrated that any proof of global regularity for the real equations must exploit some fine-grained structural feature that the averaged version lacks — ruling out entire families of proof strategies. In effect, Tao told the community: "here is a whole class of arguments that cannot work; stop trying them."

A Mastodon post from Tao on this topic is likely one of three things worth reading:

• A new technical observation or partial result — Tao often uses his social presence to work through ideas semi-publicly, and even his casual posts contain insights that end up in later papers.
• An expository thread breaking down where the field stands, aimed at mathematically literate non-specialists.
• A pointer to a longer blog post or preprint, with Mastodon serving as the announcement channel.

For a technical audience — especially anyone who works with fluid simulation, numerical PDEs, or turbulence modeling — understanding why Navier-Stokes remains open shapes how much you should trust simulations at extreme parameter regimes. And for anyone who enjoys watching a first-rate mind think in public, Tao's feed is one of the best resources on the internet.