Three mathematicians just proved a famous 30-year-old geometric conjecture, with only a tiny bit of help from AI. The conjecture says that even within huge, scattered, chaotic assemblages of points existing in innumerable dimensions, simple, ordered shapes will inevitably arise.
French mathematician Michel Talagrand presented this “convexity conjecture” in 1995 as a powerful and radical statement about the geometry of large shapes. He never thought he would live to see this proven.
“This is the most extraordinary result of my entire life,” says Talagrand, who won the 2024 Abel Prizeoften called the Nobel Prize in mathematics. “The appropriate word is ‘sensational.’
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In fact, until last week, when the new proof appeared onlineTalagrand didn’t even believe that his own conjecture was true.
It’s about building “convex” shapesthe kind that protrudes outward without any dimples or crevices. A pentagon is convex, just like a circle, but Pac-Man is not: connect two points above and below his mouth with a straight line, and that line will pass beyond his yellow perimeter. For a shape to be convex, any line between two points inside or on its perimeter must be entirely anchored to it.
Convex shapes also exist in higher dimensional space, such as the three-dimensional tetrahedron. Talagrand was interested in forms inhabiting hundreds, even billions of dimensions, or more.
This concept may seem obscure and niche, but many calculations rely on higher-dimensional mathematics, and the real world is full of data sets with countless parameters that each constitute a sort of “dimension.” “You unknowingly use it every time you Google something or ask ChatGPT a question,” says Assaf Noar, a mathematician at Princeton University who was not involved in the new work.
In 1995, Talagrand was thinking about how to construct these higher-dimensional shapes from a set of points.
Draw a few dots on a piece of paper. Now draw a convex shape that contains them all; lassoing them in a large circle would be enough. If you repeat this process in any dimension, there is a known way to construct a convex shape that always contains all the points. But as you would expect, the higher the dimension, the more difficult this procedure becomes because your shape will require more and more mathematical movements to draw.
But in 1995, Talagrand began to suspect that there was a much simpler way to construct a convex shape from high-dimensional points. In the most extreme case – a case he proposed but didn’t believe could be true – you might find a procedure of fixed complexity that doesn’t get more difficult as the dimension increases. Even in billions of dimensions, you can construct a remarkably simple shape that still manages to “encircle” many points.
To anyone familiar with high-dimensional geometry, this prospect would seem absurd. “I made this bold assumption without any basis, you know, it’s just a shot in the dark,” Talagrand admits. “When you say something like that, you feel like it can’t be true. It would be a real miracle.”
Talagrand viewed his conjecture as a challenge rather than a truth to be proven. He wanted to inspire someone to find a counterexample, a multidimensional set of points from which it was difficult to construct a convex shape. For years he wrote and lectured on the problem, even offering $2,000 to anyone who solved it and another related dilemma. No one collected the reward.
But last summer, Antoine Song, a mathematician at the California Institute of Technology, found a way to translate the question into the language of probability theory. Instead of talking about convex shapes, he transformed Talagrand’s conjecture into a statement about the selection of random points in space according to certain statistical rules.
After decades of mathematicians spinning their wheels, the problem suddenly seemed solved. “It was a total surprise and I thought it was a game-changer,” Noar says. When Song revealed his breakthrough at a conference in Princeton last December, Noar expected that full proof would soon follow. “There was a crack in the wall,” he said. “You haven’t made it to the other side, but you feel like it’s going to break.”
But Song couldn’t find the missing piece, which required manipulating a mathematical object he was unfamiliar with. So he and his student Dongming (Merrick) Hua turned to ChatGPT. With some encouragement, the Large Language Model (LLM) was able to fill the gap in understanding, providing a proof of the proposition they needed.
Then they heard from Stefan Tudose, a Princeton mathematician who had attended Song’s lecture in December. Tudose was familiar with the object in question and had spent the time developing his own proof.
Song and Hua decided that Tudose’s proof was more general and insightful than ChatGPT’s. In fact, they then found pre-existing posts with ideas very similar to those of the chatbot. Even so, they cannot penetrate the opacity inherent in the LLM “thought process” to know whether ChatGPT somehow drew inspiration from this existing but neglected material.
This proof is perhaps the most publicized mathematical result that explicitly cites the use of an LLM, but the artificial intelligence work was ultimately not used and its originality is impossible to determine. “From my point of view, AI hasn’t changed much,” says Tudose.
However, this shows that AI is becoming a pillar of the mathematician’s toolbox. “Historically, navigating unfamiliar mathematical literature required consulting specialists in the field,” says Song. “The advent of search engines has accelerated this process, and now AI tools make it even easier.”
As for mathematics itself, it’s too early to know the full ramifications of the proof, but its new unification of the geometric and probabilistic worlds could eventually lead to breakthroughs in how machines process high-dimensional data sets.
“I’m sure people will spin this evidence in all kinds of directions,” Talagrand says. “If I was 20 years younger, I would spend a year doing this to make sure I understand what’s behind it. »
Talagrand has since reorganized its various bets into one, recurring price which will be awarded for the first time in 2032 or the year following his death, whichever comes first. “The winner will be chosen by a jury on which I will not influence in any way,” specifies Talagrand. “But it seems clear that Song will be considered.”
