Transformation

Earlier this week, mathematicians at OpenAI announced that 10,000 autonomous AI agents under their direction had found a “singularity” in the Navier-Stokes equations — resolving one of the six remaining Millennium Prize problems. It potentially marks a major shift in what mathematics research will look like going forward. But it didn’t come without controversy. We wrote about the details of the result, the long research program that made it possible, and who deserves the credit. Read that story here.

A large language model has found an object that has eluded mathematicians since 1947, despite intense efforts to find it or prove it doesn’t exist.

In the mid-20th century, mathematicians sought to learn more about the nature of high-dimensional spheres by asking if it’s possible to create a “complex structure” on them. The idea is to divide a sphere into smaller pieces, and see if each piece can be mapped to a list of complex numbers (numbers of the form a + bi where i is the square root of –1) in a way that allows for consistent, well-behaved transitions between each piece.

Mathematicians have known since the 1850s that every point on the ordinary two-dimensional sphere can be identified with a single complex number. But can each point on the surface of a four-dimensional sphere be identified with a pair of complex numbers? Can each point on a six-dimensional sphere be identified with a triplet? And so on. (Since complex numbers each consist of two real numbers, it only makes sense to ask the question for even dimensions.)

When the German mathematician Heniz Hopf first posed these questions in 1947, he showed that four- and eight-dimensional spheres don’t have a complex structure. By 1951, researchers had shown the same for ten-dimensional spheres and above. But the six-dimensional sphere, S⁶, remained enigmatic. Was this phenomenon particular to two dimensions? If not, it might suggest something deeper about the relationship between geometry and complex variables. “In some sense, the six-sphere is the simplest manifold for which we did not know the answer,” said Mohammed Abouzaid, a geometer and topologist at Stanford University.

As numerous mathematicians have tried and failed to settle the question one way or the other, the stakes of the problem have grown.

On August 23, Levent Alpöge, a researcher at Anthropic, announced in a tweet that S⁶ has a complex structure. (A few days earlier, together with Tristan Buckmaster of New York University, he had found a proof of a singularity in the 3D Euler equations, which they announced earlier this week.) The tweet was accompanied by a nearly 100-page paper written using Claude, Anthropic’s AI model, which Abouzaid describes as “so poorly written that it will take a while to check.” (Alpöge agrees with the criticism, writing in a later tweet, “Yea clearly my bad for fucking up the writeup.”) He said he was too excited to take the time to explain the result clearly.

I spoke with several experts in the field who say that although it will take some time for a consensus to solidify about the correctness of Alpöge’s proof, early signs point to it being true. As Phillip Engel of the University of Illinois in Chicago, who has written an expository note streamlining and clarifying the arguments Alpöge first publicized, told me, “For myself, I did all the computations necessary to convince myself that it’s true.”

A version of the statement has also been formally proved by Boris Alexeev at OpenAI, but Abouzaid cautions that although the formalization “is making the community confident that the statement is correct, the effort to digest Alpöge’s proof, and its connection to the formalized proof, have just begun.”

Arguably one reason mathematicians struggled to find the object is that an influential 1996 paper asserted that it couldn’t possibly exist. (In the aftermath of the new proof, that paper is now thought to be wrong.) Engel expects the new construction will have some interesting implications, leading to a “little bit of a zoo” as mathematicians generalize it. “Maybe we all would have died not knowing the answer if the AI hadn’t discovered it,” he said.

“Exponential” is a widely misused term. It shows up as a synonym for “rapid,” but its precise meaning is far more remarkable. As many readers of Quanta will already know, it means to double and double and double again, the pace of change accelerating faster and faster. It’s hard to develop intuition for just how rapid this kind of growth is, not least because it rarely lasts: Changes begin to come so quickly that they run out of steam.

Computing power is an exception that has held for the better part of a century. Built atop that exponential growth, frontier artificial-intelligence models have likewise grown exponentially for the last decade and a half.

In the autumn of 2025, I began reporting a feature for Quanta about the impact artificial-intelligence algorithms were having on math research. In all candor, I expected that impact to be limited. I knew that AI had a tendency to hallucinate, and in my limited firsthand use of large language models, I’d found them unreliable. I’d been a journalist for two and a half decades, so tech industry hype was not new to me. I knew that some serious people were using AI on real problems — I expected to find that, like any other tool, it would have certain interesting niche uses. The possibility that it might transform the discipline seemed remote.

As I reported the story over the course of the last months of 2025 and the first of 2026, I was surprised by what I found. Fundamental change had not yet arrived, but it was impossible to avoid the conclusion that it was coming. My intuition for what exponential growth can do was — predictably — off.

Over this past summer, the scale of the change AI is bringing to math has become clearer. As Akshay Venkatesh of the Institute for Advanced Study wrote in a prescient essay: “Mechanical reasoning will change not only how we do mathematics, but what it is; this must be renegotiated amongst its practitioners and with society.”

These updates are intended to be a chronicle of that renegotiation, which is now underway at a rapid, sometimes frenetic, pace.

Our goals here are twofold.

First, as new results are established with the aid of artificial-intelligence algorithms, we will explain them in a timely fashion. We will put new results in mathematical context: How important do mathematicians think the latest counterexample is? How novel is a connection that AI discovered between different branches of math? What are the potential consequences of a new method of proof? Did AI reveal that a problem previously thought to be hard was simpler than it seemed, or did it solve a hard problem?

Second, we will cover the ways in which mathematicians are wrestling with the changes AI is bringing to their discipline. That response is, like the mathematical community itself, varied. Some welcomed these changes; others now call for resistance to AI or even “total opposition to the use of artificial intelligence in mathematics.” How effective is formalization as a tool for combating AI slop? Can an AI model be a co-author on a paper, or even its sole author? How do peer review and mathematical journals change when AI is used to do math?

It is my expectation that some aspects of what it is to be a mathematician will abide through the transformations brought by AI. Others will, in all likelihood, be wholly changed.

These are just some of the topics we will tackle. If you have an idea for something you think we should cover, we’d like to hear from you. Please write us at [email protected].

These are, of course, big questions. We will continue to cover them both here and in the rest of Quanta’s digital pages.

Welcome to Transformation.