Every time I have coffee with a mathematician, I always ask which of the Seven Millennium Problems they think will be next. These are the most famous open questions in mathematics. Solve one and you’ll win a million-dollar prize, but that’s only happened once since the Clay Mathematics Institute announced the list in 2000.
Mathematicians often use Millennium Problems as a sort of yardstick, giving prestige to their own work by counting how many degrees separate it from a million-dollar payout. I see them as a way to feel movement in a discipline where advances often take decades to materialize.
And I recently started hearing a completely new answer to my question which one will fall on first. Lately, mathematicians have flagged one of seven problems that experts say have been beyond their reach for centuries. It’s about mathematicians’ attempts to understand something far more familiar than imaginary numbers or string theory: the puzzling movements of fluids.
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We have all been captivated by a crashing wave or cascading waterfall. These complex flows hide deep mathematical challenges, ones that prevent us from understanding exactly how planes fly or to perfectly predict next week’s weather. Mathematicians have struggled for centuries with the equations that govern these flows. In all that time, only one fundamental question about these equations has eluded them, despite the addition of a million-dollar bounty.
However, a continuing series of recent breakthroughs – some touted for their use of artificial intelligence – have now convinced some prominent mathematicians that victory is near. Meanwhile, others are still wondering how far AI can take us – and whether a deeper, more worldly understanding might be the most viable path.
A timeless trance
Let’s say you want to mathematically capture the flow of a river. To start, you would need a perfect snapshot of the river at any given time, down to the position and speed of each droplet. The well-known conservation laws – of energy, of momentum – would then govern what happens to the fluid next. Throw a rubber duck into this bubbling stream, and the laws should determine its every move, whether for the next 20 minutes or 20,000 years.
These laws, applied to water or any other “incompressible” fluid, take the form of four equations: the three-dimensional Navier-Stokes equations. Literally every possible way a fluid can swirl, from a calm sea to a roaring tsunami, constitutes its own distinct solution to these equations.
It is this unlimited menagerie of flows hidden in seemingly simple equations that baffles mathematicians. They want to be sure that the Navier-Stokes equations are mathematically valid, that they always make sense, and that they never fail to describe reality. They want to rule out rare monstrosities hiding within this vast menagerie.
Mathematicians call these hypothetical flows “explosions”: solutions to the Navier-Stokes equations where the fluid speed becomes infinite. If a whirlwind or streamlining can intensify beyond the mathematical breaking point (the equivalent of a small tornado suddenly swirling in your coffee), then the equations cannot be entirely reliable.
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The Millennium Problem asks whether the Navier-Stokes equations can “explode” in this way. If, from simple laws of physics, we can construct strange fluids that would tear reality apart. Today, after a quarter century of progress on the problem, some say the problem is on the verge of escalating into flooding.
New advancements start by removing friction from the equation. Friction is essential to fluids: it’s what gives them “viscosity,” which is, among other things, how fish are able to swim, using the friction of water to propel themselves forward. But viscosity – that feedback between motion and the medium – also makes the mathematics of a fluid particularly difficult to master.
So mathematicians hope that the easiest way to find an explosion in the Navier-Stokes equations is to first find it in their frictionless cousins, the Euler equations. “The natural path to Navier-Stokes would be to go through Euler,” says Javier Gómez-Serrano, a mathematician at Brown University.
In 2022, this approach worked for a variation of the millennium problem concerning compressible fluids, such as air – the feat earned mathematician Frank Merle this year illustrious breakthrough prize in mathematics.
But incompressible fluids like water cause additional complications. A person diving at one end of a pool will raise the level of a floating buoy at the other. Everything affects everything else, which makes the calculations much more difficult. Many theorists have turned to machines to seek an explosion in the vast landscape of possibilities.
Glimpses of infinity
Last September, a group of mathematicians including Gómez-Serrano reported in a preprint that they had seen glimpses of infinity on their computer screens. They simulated a frictionless fluid trapped in a cylinder, like coffee swirling in a cup. And the AI they were working with, built in collaboration with Google’s DeepMind team, located a point near the edge of the cup where the fluid’s speed seemed infinite.
“I want to discover an explosion. It doesn’t matter whether it’s with or without AI,” Gómez-Serrano said at a symposium at Columbia University in March. “It’s a tool that allowed me to go further, so I used it. »
It may take years for the team to prove mathematically that the supposed explosion actually obeys Euler’s equations. And Clay’s problem requires an explosion in a infinity fluid – less a cup of coffee, more a limitless sea. Still, the result suggests that computer-aided screening could one day decipher Euler’s equations – and perhaps Navier-Stokes’s.
The AI in question has little in common with the major linguistic models that are now shaking up virtually all sectors of society. But that hasn’t stopped some experts from citing the development as an omen that computers will come first to take over Navier-Stokes and then all other open mathematical problems, leaving an uncertain role at the domain’s frontier for the human mind.
In February, however, three mathematicians showed in another preprint that this AI revolution could actually be far from blowing up Navier-Stokes. The DeepMind computer, like other simulations, assumes that the fluid rotates around a central axis. Many mathematicians agree that an explosion can have this “axial symmetry,” and that’s what simplifies their simulations enough that modern computers can handle it. But the new proof showed that virtually any explosion of Euler’s equations exhibiting this symmetry won’t postponement to Navier-Stokes. Adding friction would make such an infinite finite. The viscosity of your coffee will prevent this little tornado from breaking out.
“It doesn’t look promising,” says Vlad Vicol, a mathematician at New York University who co-authored the preprint. “For axial symmetry, it would really take a miracle.” Merle agrees. “The document shows that the method, as it stands, does not work,” he says.
Still, if the DeepMind team finds an explosion of Euler’s equations for an infinite fluid, it would be “an incredible achievement,” Vicol emphasizes. “And I think that may be within the reach of this program,” he said. “Our paper basically says that just because you understand the Euler equations doesn’t mean you get the Navier-Stokes equations for free.”
But if the explosion is possible for the Navier-Stokes equations, the evidence suggests, it could result from the complex feedback between viscosity and flow. “There has to be some sort of interaction,” says Vicol. “That’s what we’re seeing.”
To achieve this interaction, mathematicians should avoid the simplifying tricks required by computers. In fact, conjuring up an explosion from the mathematical depths might require innate talent. feeling to know how the equations should come together, that is to say an understanding of fluids too fragile to be integrated into the vast assemblage of numbers of a current AI.
Ride the wave
Steve Shkoller has developed this ineffable sense of the sea since the age of five, growing up in San Diego.
“When you surf from a young age, the ocean gives you that sense of movement that equations alone don’t provide,” he says. Shkoller is now a mathematics professor at the University of California, Davis, but still spends at least two hours a day on the ocean near his home in Marin County. This is where he thinks best. “You have a sense of timing, geometry, position; you kind of feel like the wave is a living thing,” he says. “And you just get these ideas.”
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When a torn muscle last fall sidelined him for nearly a year, his twin loves — surfing and math — suddenly felt out of reach. One October day, recovering on his couch, he closed his eyes and tried to imagine himself back on the water. “I was trying to feel the energy. And then, simultaneously, I was thinking about math.”
As he mentally “surfed” a gargantuan wave, he began to imagine it as an infinite-speed feature – a changing, life-size image of an explosion. And seeing this, he decided that the AI people and others like them hadn’t gotten the point.
Many of Euler’s recent computer-driven advances describe the explosion as a somewhat static feature, what is called a “self-similar” shape. Imagine a freeze frame of a large wave in which, no matter how far you zoom in on its crest, you see the same curled tip shape.
There is good reason to assume that the explosions would be self-similar. Despite the chaotic mathematics of fluids, nearly identical whorls and gyres tend to emerge at almost every scale. Time and time again, mathematicians have discovered characteristics of fluids that exhibit this astonishing system. metric and exploited it to make mathematics manageable. So many of them thought the explosions would be no different. But in doing so, researchers abandoned what Shkoller saw as the essence of fluids: change.
From a distance, an ocean wave may look like a single, coherent shape. But no drop of water really ventures far from its starting point: the aqueous content of the wave turns over every moment. Maybe this change wasn’t an obstacle to ignore, Shkoller thought. Perhaps this was the key ingredient, the very origin of the explosion that everyone was looking for.
He grabbed his iPad and began turning those suspicions into math.
Over the next week, he lay on his back for 12 hours a day, the tablet held aloft, scribbling equations and sketches to construct a simplified picture of an explosion drawn from the depths of his intuition. “The first three days, I was so excited I couldn’t even sleep,” he says.
Its “wave,” the shape of its infinite-velocity fluid feature, was neither self-similar nor static. “You’re making a film rather than just one frame of the film,” Shkoller explains. Moreover, he drew his explosion from the constant renewal of the fluid entering and leaving the “wave”: “Imagine every frame of the film, you bring in a whole new cast. » He then proved that an explosion of Euler’s true equations could precisely follow his mathematical scheme.
In March, Shkoller published solid proof to the arXiv.org preprint server. It’s over 100 pages long and filled with dense math, so it will likely take several months for the community to verify. But the first shots are promising. “No one thought it would really be possible to prove this,” says Scott Armstrong, a mathematician at New York University. “Steve seems to have done it.
Although the proof does not rely on a boundary like DeepMind’s work does, it uses other shortcuts that the Millennium Problem does not allow. And even without them, Vicol’s warning still applies; adding friction to the mix is likely to kill Shkoller’s blast as well, so it doesn’t bounce back to Navier-Stokes.
But his key idea, Shkoller believes, will lives on because it draws on a deep truth that he felt throughout his life.
“You’re in an ever-changing environment. Every wave is different,” he says, much like the ideas that flood through him as he floats on his surfboard, looking out to sea. “They just kind of hit you: Why didn’t I think of this before? »
