Mathematicians Crack 30-Year Geometry Mystery With AI Help

A geometry problem that seemed impossible for 30 years just got solved, and even the mathematician who created it can't believe it's true.

French mathematician Michel Talagrand posed his "convexity conjecture" in 1995 as a challenge he never expected anyone to answer. The puzzle asks whether you can build simple, orderly shapes from chaotic collections of points scattered across billions of dimensions.

"This is the most extraordinary result of my entire life," says Talagrand, who won the 2024 Abel Prize, often called the Nobel Prize of math. "The proper word is 'sensational.'"

Until last week, Talagrand didn't even believe his own conjecture was true. He offered $2,000 to anyone who could prove it wrong, and for decades, no one could do either.

The breakthrough came when Antoine Song at the California Institute of Technology figured out how to translate the geometry problem into probability theory. Instead of talking about shapes, he turned it into a question about picking random points according to statistical rules.

When Song presented his work at Princeton last December, other mathematicians knew the wall was about to break. But Song hit a snag with a mathematical object he didn't fully understand.

That's where AI stepped in. Song and his student Dongming Hua asked ChatGPT for help, and the large language model filled the gap in their understanding with a working proof.

Meanwhile, Stefan Tudose at Princeton had been working on the same missing piece since Song's December talk. When they compared notes, Song and Hua decided Tudose's proof was more elegant and insightful than ChatGPT's version.

The three mathematicians published their complete proof last week, with AI playing a supporting role rather than the star. They later found that parts of the proof already existed in older publications, but no one had connected the dots until now.

Why This Inspires

This breakthrough matters beyond abstract math. High-dimensional geometry powers everyday technology from Google searches to ChatGPT responses.

More importantly, it shows how human creativity and AI assistance can work together on problems that seemed unsolvable. The AI didn't crack the case alone—it took human insight to frame the question differently and human judgment to evaluate the answers.

Talagrand spent nearly three decades believing his challenge would never be met, and now he's witnessed what he calls a miracle.

Sometimes the impossible just needs a fresh perspective and the right tools to become reality.