AI Revolutionizes Math Research

The year 2026 will be remembered as the year that AI revolutionized mathematics research.

In August, AI made some real breakthroughs in some notable unsolved problems in mathematics, mostly involving disproof of conjectures by counterexample. But this week, Open AI published more than three hundred proofs of unsolved problems by various methods, that were significant although not quite as notable, across all major areas of mathematics on a single day (although a few of those proofs have been withdrawn or revised in response to comments).

Its single day results (rumored to be from an initial set of 400 problems), that probably took several months to put together and were just released all at once for drama. 

But its productivity on mathematical problems of this medium level of significance which are what the best human mathematics researchers might solve once every few years, rivals the productivity of the entire global mathematical research community over a time frame somewhere between many months and a few years.

The announcement this week leaves no doubt that AI will forever and profoundly change how mathematical research is done. We can expect a surge of mathematical breakthroughs for a few years as the backlog of AI solvable unsolved problems in mathematics are conquered and human mathematicians have to come up with new mathematics problems for humans and AI to solve together, in many cases probably building on the findings made in a new surge of proofs.

“If a human did this, it would be an instant Fields Medal, no questions asked.”

That’s how Alex Kontorovich, chair of the department of mathematics at Rutgers, responded to an A.I.-generated proof that OpenAI, the artificial intelligence giant, released on Tuesday.

And that single proof — worthy, in Dr. Kontorovich’s assessment on social media, of arguably the highest honor in the field of mathematics — was just one among more than 350 findings that the firm released that day, all at once.

Not all of them were quite so remarkable. But Martin Bridson, a mathematician at Oxford and the president of the Clay Mathematics Institute, which established the coveted Millennium Prize Problems, called OpenAI’s release “breathtaking.”

“Until very recently, it would have been impossible to imagine that the frontiers of mathematics could move so far in one day,” Dr. Bridson said.

If there had been any doubt that the leaps achieved by artificial intelligence this year would transform the discipline of mathematics, the tens of thousands of pages let loose on Tuesday have put that doubt to rest. The puzzles apparently conquered by OpenAI include crucial open questions in nearly every subfield of math, including algebra, number theory, theoretical computer science, mathematical logic and topology.

None of the proofs fully resolve any of the five remaining Millennium Prize Problems, which were designated at the turn of the century as a way of celebrating mathematics and its vast frontier of mysteries. But the trove of findings seemed to include results that make tangential or related progress on all of them. Perhaps the most significant result — the one Dr. Kontorovich was responding to — was a proof of the “quasi-Riemann hypothesis,” a conjecture related to the best known of the Millennium problems, the Riemann hypothesis, and a possible steppingstone of sorts to its resolution.

Ken Ono, a professor at the University of Virginia and the founding mathematician at Axiom Math, a math-focused artificial intelligence firm, wasn’t sure whether the quasi-Riemann solution would be a springboard for solving the original hypothesis. But he said that, as is expected with the original Riemann hypothesis, an encyclopedia of consequences would flow from this result, providing rich fodder for future breakthroughs. . . .

(As of Thursday, OpenAI had made a number of updates to the public online repository; withdrawing three papers and making fixes to several others. A spokeswoman said: “Where errors are identified, we will work to correct them promptly and withdraw papers if no fixes can be found. As with other research manuscripts, this is an iterative process.”) . . . .

For Bryna Kra, a mathematician at Northwestern University, problem No. 145 on the list was of great interest — “the Rokhlin problem on mixing implying higher order mixing,” which answers a question from the 1940s. “The writing, however, makes it impossible to understand,” said Dr. Kra, former president of the American Mathematical Society, who is among a group of mathematicians who are proactively working to chart a course for the field through this period of “uncertainty and seismic change,” as she described it. . . .

The French mathematician Jean-Pierre Serre, an emeritus professor at the Collège de France in Paris — who has won two of math’s top prizes, the Fields Medal and the Abel Prize, and celebrated his 100th birthday last month — was naturally attracted to finding No. 46 in OpenAI’s list, which solves positively a conjecture he made almost 70 years ago.

But Dr. Serre is conflicted about doing math with A.I. He uses the technology for things like references, and, with the help of friends, for chasing down hunches about errors. “Mathematicians take pleasure in doing maths in two different ways: learning and finding new things,” Dr. Serre said. “Hence a conflict: A.I. increases the first pleasure and lowers the second one. The problem is it may lower the pleasure too much; that is especially serious for young mathematicians.”

From the New York Times.

More analysis from Woit (a math professor at Columbia who blogs about and researches math, particle physics, and academic life) can be found here and here.