Three mathematicians just proved a famous 30-year-old conjecture in geometry, with only a tiny assist from AI. The conjecture says that even within enormous, scattered and chaotic assemblages of points existing across innumerable dimensions, simple, orderly shapes will inevitably crop up.
French mathematician Michel Talagrand posed this “convexity conjecture” in 1995 as a powerful, sweeping claim about the geometry of high-dimensional shapes. He never thought he would live to see it proved.
“This is the most extraordinary result of my entire life,” says Talagrand, who won the 2024 Abel Prize, which is often called the Nobel Prize of math. “The proper word is ‘sensational.’”
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In fact, up until last week, when the new proof appeared online, Talagrand didn’t believe his own conjecture was even true.
It’s about building “convex” shapes, the kind that bulge outward without any dimples or crevices. A pentagon is convex, and so is a circle, but Pac-Man isn’t: 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 of it or on its perimeter must be fully ensconced within it.
Convex shapes exist in higher-dimensional space, too, like the three-dimensional tetrahedron. Talagrand was interested in shapes inhabiting hundreds or billions of dimensions—or even more.
This concept may seem obscure and niche, but many computations hinge on higher-dimensional math, and the real world is full of datasets with innumerable parameters that each constitute a “dimension” of sorts. “You’re using it without knowing whenever 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 build these higher-dimensional shapes from a set of points.
Draw some dots on a sheet of paper. Now draw a convex shape that contains them all; lassoing them inside a big circle would suffice. If you repeat this process in any dimension, there’s a known way to construct a convex shape that always contains all the points. But as you might expect, the higher the dimension, the tougher this procedure gets because your shape will require more and more mathematical moves to draw.
But in 1995 Talagrand began to suspect that there was a much simpler way to build a convex shape from high-dimensional points. In the most extreme case—a case he proposed but didn’t believe could be true—you could find a procedure of fixed complexity that doesn’t get more difficult as the dimension grows. Even in billions of dimensions, you could construct a remarkably simple shape that still manages to “circle” many of the points.
To anyone familiar with high-dimensional geometry, the prospect would seem preposterous. “I made this bold conjecture really without any ground for it, you know—it’s just a shot in the dark,” Talagrand admits. “When you say something like that, you feel it cannot be possibly true. That would be a total miracle.”
Talagrand viewed his conjecture as a challenge rather than a truth to be proved. He wanted to entice someone to find a counterexample—a multidimensional set of points from which you couldn’t easily build a convex shape. For years he wrote and gave talks about the problem, even offering $2,000 to anyone who solved it and another related quandary. 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 turned Talagrand’s conjecture into a statement about picking random points in space according to some statistical rules.
After decades of mathematicians spinning their wheels, the problem suddenly seemed tractable. “It was a total surprise, and I thought it was a game-changer,” Noar says. When Song unveiled his breakthrough in a talk at Princeton last December, Noar expected a full proof to soon follow. “There was a crack in the wall,” he says. “You didn’t get to the other side, but you feel like it’s going to break.”
But Song couldn’t figure out the missing piece, which required manipulating a mathematical object he wasn’t familiar with. So he and his student Dongming (Merrick) Hua turned to ChatGPT. With some prodding, the large language model (LLM) was able to fill the gap in their understanding, providing a proof of the proposition they required.
Then they heard from Stefan Tudose, a Princeton mathematician who had attended Song’s December lecture. Tudose was familiar with the object in question and had spent the intervening time working out his own proof.
Song and Hua decided Tudose’s proof was more general and insightful than ChatGPT’s. In fact, they later found some preexisting publications with ideas very similar to the chatbot’s. Even so, they can’t pierce the inherent opacity of the LLM’s “thought process” to know whether ChatGPT somehow took inspiration from that extant-but-overlooked material.
This proof might be the highest-profile math result that explicitly cites the use of an LLM—but the artificial intelligence’s work ultimately wasn’t used, and it’s originality is impossible to determine. “From my perspective the AI didn’t change much,” Tudose says.
It does, however, show that AI is becoming a mainstay of the mathematician’s toolkit. “Historically, navigating unfamiliar mathematical literature required consulting specialists in the field,” Song says. “The advent of search engines accelerated this process, and now AI tools have made it even easier.”
As far as the math itself goes, it’s too early to know the proof’s full ramifications, but its new unification of the geometric and probabilistic worlds could conceivably lead to breakthroughs in how machines process high-dimensional datasets.
“I’m sure people will turn this proof in all kinds of directions,” Talagrand says. “If I were 20 years younger, I would spend a year doing this to make sure I understand what is behind it.”
Talagrand has since reorganized his various bets into a single, recurring prize that will first be awarded in 2032 or the year after his death, whichever comes first. “The winner will be chosen by a jury that I will not influence in any way,” Talagrand says. “But it seems obvious that Song will be considered.”