It happened last week. Three mathematicians solved a geometry problem that has baffled the world since 1995. The solution was sensational, a word Michel Talagrand uses deliberately. He’s the guy who posed the riddle in the first place, the 2024 winner of the Abel Prize—which is basically the Nobel for math.
He didn’t expect to live long enough to see this. Honestly, up until the proof dropped online, he didn’t believe it was true. Not even for a second.
“This is the most extraordinary result of my entirely life. The proper word is sensational.”
The problem involves shapes. But not just any shapes. We’re talking high-dimensional spaces, hundreds or even billions of dimensions. In those vast, chaotic voids of scattered points, simple, convex shapes should inevitably appear.
Think about convexity. It’s that bulge-outward property. No dents, no crevices, no Pac-Man mouth gaps. If you connect two points inside the shape with a straight line, the whole line must stay inside the shape. A circle? Convex. A Pac-Man? Nope. Connect points above and below the open mouth and the line slices right out.
In lower dimensions, like our flat sheets of paper or three-dimensional rooms, this is manageable. But add another dimension? The math gets messy. It requires more and more complex steps. Or so we thought.
Talagrand suspected in 1995 that there was a shortcut. A simple way to build these convex containers from random points that didn’t get harder as you added dimensions. Even in a universe of billions of dimensions, the complexity of the shape could stay fixed. Simple. Clean.
To most experts, it sounded preposterous. A miracle, really.
“When you say something like that you feel it cannot be possibly true.”
Talagrand didn’t offer the conjecture as fact. It was a dare. A $2,000 challenge for anyone who could prove it—or better yet, disprove it with a counterexample. Years went by. Talks were given. Prizes went uncollected. Nobody could break it.
Then came Antoine Song.
He’s at Caltech, a mathematician who decided to rewrite the question in the language of probability. Instead of drawing lines and shapes, he started looking at picking random points in space. Statistical rules. Probabilistic outcomes.
Suddenly the wall had a crack.
Assaf Naor, at Princeton, called it a game changer. It felt like the structure was about to collapse. Song saw a path, but he couldn’t walk it. He hit a wall of mathematical objects he didn’t understand. So what did he do?
He asked ChatGPT.
Song and his student Dongming Hua turned to the AI when they were stuck. They needed help manipulating a specific mathematical concept, something unfamiliar territory for them. The LLM provided the missing link, offering a proof for the proposition they needed.
Did that mean AI solved it? No. Not exactly.
Enter Stefan Tudose, another Princeton mathematician. He’d heard the rumor. He knew the object. And while Song and Hua were chatting with a bot, Tudose was already working out a proof himself. One that was broader. More insightful.
Song and Hua checked it. Tudose was right. In fact, the AI’s solution mirrored some old, overlooked publications anyway. They couldn’t tell if ChatGPT was original or just regurgitating forgotten data. It’s a black box. Opacity remains.
But here’s the twist: they didn’t use the AI proof in the end. They used Tudose’s.
Is this a triumph for artificial intelligence? Or just another tool in the box?
“The advent of AI tools have made it even [easier]… Historically navigating unfamiliar mathematical literature required consulting [people]”
Song sees it as evolution. First search engines. Now AI. It accelerates the hunt through the literature. But the insight? The creativity? That still comes from us.
We don’t know where this goes next. Maybe it changes how machines process data. Maybe it stays a curious footnote.
Talagrand is just glad it’s done. Though he adds, with a shrug that sounds distinctly human, that if he were twenty years younger, he’d spend the whole next year just trying to understand the magic behind it all.
