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.<\/p>\n
French mathematician Michel Talagrand posed this \u201cconvexity conjecture\u201d in 1995 as a powerful, sweeping claim about the geometry of high-dimensional shapes. He never thought he would live to see it proved.<\/p>\n
\u201cThis is the most extraordinary result of my entire life,\u201d says Talagrand, who won the 2024 Abel Prize<\/a>, which is often called the Nobel Prize of math. \u201cThe proper word is \u2018sensational.\u2019\u201d<\/p>\n On supporting science journalism<\/p>\n If you’re enjoying this article, consider supporting our award-winning journalism by subscribing<\/a>. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<\/p>\n In fact, up until last week, when the new proof appeared online<\/a>, Talagrand didn\u2019t believe his own conjecture was even true.<\/p>\n It\u2019s about building \u201cconvex\u201d shapes<\/a>, the kind that bulge outward without any dimples or crevices. A pentagon is convex, and so is a circle, but Pac-Man isn\u2019t: 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.<\/p>\n Convex shapes exist in higher-dimensional space, too, like the three-dimensional tetrahedron. Talagrand was interested in shapes inhabiting hundreds or billions of dimensions\u2014or even more.<\/p>\n 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 \u201cdimension\u201d of sorts. \u201cYou\u2019re using it without knowing whenever you Google something or ask ChatGPT a question,\u201d says Assaf Naor, a mathematician at Princeton University, who was not involved in the new work.<\/p>\n In 1995 Talagrand was thinking about how to build these higher-dimensional shapes from a set of points.<\/p>\n 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\u2019s 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.<\/p>\n 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\u2014a case he proposed but didn\u2019t believe could be true\u2014you could find a procedure of fixed complexity that doesn\u2019t get more difficult as the dimension grows. Even in billions of dimensions, you could construct a remarkably simple shape that still manages to \u201ccircle\u201d many of the points.<\/p>\n To anyone familiar with high-dimensional geometry, the prospect would seem preposterous. \u201cI made this bold conjecture really without any ground for it, you know\u2014it\u2019s just a shot in the dark,\u201d Talagrand admits. \u201cWhen you say something like that, you feel it cannot be possibly true. That would be a total miracle.\u201d<\/p>\n Talagrand viewed his conjecture as a challenge rather than a truth to be proved. He wanted to entice someone to find a counterexample\u2014a multidimensional set of points from which you couldn\u2019t 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.<\/p>\n 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\u2019s conjecture into a statement about picking random points in space according to some statistical rules.<\/p>\n After decades of mathematicians spinning their wheels, the problem suddenly seemed tractable. \u201cIt was a total surprise, and I thought it was a game-changer,\u201d Noar says. When Song unveiled his breakthrough in a talk at Princeton last December, Noar expected a full proof to soon follow. \u201cThere was a crack in the wall,\u201d he says. \u201cYou didn\u2019t get to the other side, but you feel like it\u2019s going to break.\u201d<\/p>\n But Song couldn\u2019t figure out the missing piece, which required manipulating a mathematical object he wasn\u2019t familiar with. So he and his student Dongming (Merrick) Hua turned to ChatGPT<\/a>. 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.<\/p>\n Then they heard from Stefan Tudose, a Princeton mathematician who had attended Song\u2019s December lecture. Tudose was familiar with the object in question and had spent the intervening time working out his own proof.<\/p>\n Song and Hua decided Tudose\u2019s proof was more general and insightful than ChatGPT\u2019s. In fact, they later found some preexisting publications with ideas very similar to the chatbot\u2019s. Even so, they can\u2019t pierce the inherent opacity of the LLM\u2019s \u201cthought process\u201d to know whether ChatGPT somehow took inspiration from that extant-but-overlooked material.<\/p>\n This proof might be the highest-profile math result that explicitly cites the use of an LLM\u2014but the artificial intelligence\u2019s work ultimately wasn\u2019t used, and its originality is impossible to determine. \u201cFrom my perspective the AI didn\u2019t change much,\u201d Tudose says.<\/p>\n