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Mathematics in the news
Two pieces of mathematics have reached the news, one about what a proof is and one about what a map is. OpenAI says its agents settled the Navier–Stokes problem, one of the seven Millennium Prize Problems, and mathematicians have neither seen nor accepted the proof. The United Nations General Assembly, meanwhile, has asked the world to stop drawing Mercator's map.
What the Navier–Stokes equations describe
Velocity
How fast the fluid moves, and where
Pressure
The push that drives and resists the flow
Viscosity
The fluid's own thickness, which smooths motion
Worked out from them, every day
- Weather. Air moving over the Earth
- Aircraft. Air over a wing
- Blood flow. Through arteries and veins
- Ships. Water around a hull
- They are Newton's second law written for a fluid, balancing pressure, viscosity and the fluid's own momentum.
- The difficulty sits in the term where the flow carries itself along, which is not proportional to velocity and lets small swirls feed large ones.
Kinds of flow
Laminar
Layers slide past each other without mixing
Transitional
The layers begin to wobble
Turbulent
Eddies of every size, mixing the fluid
One set of equations covers all three. Which one a fluid shows depends on its speed, its thickness and the size of what it flows around.
- Turbulence is where the equations resist analysis, which is why engineers solve them by computer rather than exactly.
The open question, and why the answer matters
A smooth start
In three dimensions, a fluid is set moving in a well behaved way.
The question
Must it stay smooth for all time, or can its velocity grow without bound in a finite time?
Blow-up
Such a point of infinite velocity is a singularity. Viscosity smooths motion, but scales and forces inside a fluid interact in ways nobody has pinned down.
Either way
The answer is theoretical. Weather forecasting, aircraft design and blood flow models already use the Navier–Stokes equations.
- A proof either way would not change a weather forecast, because engineers already solve the equations approximately and trust the results.
- A singularity would mean the equations stop describing a real fluid at that instant, marking the limit of the model rather than of the fluid.
- Smoothness for all time would show the model is self consistent, and methods built to prove such a thing usually outlast the result.
The seven problems
Navier–Stokes existence and smoothness
Open, and the subject of OpenAI's claim
Poincaré conjecture
Resolved by Grigori Perelman, who declined the prize
P versus NP
Open
Riemann hypothesis
Open
Yang–Mills existence and mass gap
Open
Hodge conjecture
Open
Birch and Swinnerton-Dyer conjecture
Open
- The Clay Mathematics Institute's own rules require a result to appear in a refereed journal and to stand unchallenged for two years before a prize is considered.
Why no flat map can be true to everything
Every flat map distorts at least one of these
AreaHow large a landmass is relative to others
ShapeThe true outline of a country
DistanceThe actual space between two points
DirectionThe correct compass bearing between places
Mercator (conformal)
- Keeps angles and local shapes
- Rhumb lines are straight, so it suits navigation
- Inflates areas more towards the poles
- Greenland looks as big as Africa, which is 14 times larger
Equal Earth (equal-area)
- Keeps relative areas of continents true
- Shapes stretch near the poles
- Suits thematic maps of population, climate and resources
Winkel Tripel (compromise)
- Keeps no property exact
- Spreads error across area, distance and direction
- A map projection keeps one property at the cost of the others, so the right choice depends on what the map is for.
What has happened
11 Sep 2026
Fields Medal winners warned that using mathematical problems as a benchmark pulls the subject away from understanding.
- Formal checking shows that a proof is correct, not how the model reached it.
- OpenAI's claimed 100 page proof is still neither independently verified nor accepted.
8 Sep 2026
OpenAI said that 10,000 of its artificial intelligence (AI) agents had cracked the Navier–Stokes problem in about 88 hours.
- Its paper builds a fluid whose velocity becomes infinite at a point in finite time while the flow's energy stays bounded, so it claims blow-up rather than smoothness.
- The proof was formalised in Lean, which checks each step of an argument mechanically.
- OpenAI cannot rule out that de-identified data from researchers using its products improved its models, which raises the question of epistemic ownership.
4 Sep 2026
The United Nations General Assembly adopted the Correct the Map resolution, which is not binding, naming the 2018 Equal Earth projection as more accurate than Mercator's of 1569.
- It passed 164 to 1, with the United States alone against.
- India voted for it but said the resolution authenticates no map, and that its territory, including Jammu and Kashmir and Ladakh, must be shown as in India's official map.
- The case made was that the projection a map uses shapes how important a region looks, which is a matter of image and soft power.