‘Sensational’ proof topples decades-old geometry problem | Scientific American
May 19, 2026<br>4 min read<br>Add Us On GoogleAdd SciAm<br>‘Sensational’ proof topples decades-old geometry problem
The sudden resolution of a well-known conjecture highlights the growing adoption of AI as an assistant in high-level mathematics
By Joseph Howlett edited by Lee Billings<br>yuanyuan yan/Getty Images
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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.<br>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.<br>“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.’”<br>On supporting science journalism<br>If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<br>In fact, up until last week, when the new proof appeared online, Talagrand didn’t believe his own conjecture was even true.<br>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.<br>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.<br>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.<br>In 1995 Talagrand was thinking about how to build these higher-dimensional shapes from a set of points.<br>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.<br>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.<br>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.”<br>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.<br>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...