As they mentioned in their January 3, 2026 preprint, AlphaEvolve discovered that the Bruhat intervals of these particular permutation groups have a surprisingly special structure. When researchers studied the spacing, they found that they formed a high-dimensional cube called a hypercube. “I was very surprised to see what AlphaEvolve was thinking,” Libedinsky said. “If it were a human, it would be a very creative human.”
AlphaEvolve was answering questions we didn’t know we had. “We didn’t ask AlphaEvolve to find a big hypercube,” Ellenberg said. “We asked it to find something else, and when we thought about it, we realized it was a giant hypercube, which we didn’t expect to be there.”
“It’s a building that’s been sitting in front of us for 50 years, we just didn’t realize it,” Williamson said.
Previously, older machine learning techniques also enabled serendipitous mathematical discoveries like this, revealing patterns that no one had thought to look for. But Williamson said that previously it was “a real engineering effort. … You need to know how to write code, and you need to spend a lot of time looking into the details of training a neural network. For mathematicians without a strong background in machine learning, this was basically very difficult to do.”
With the LLM, “experiments that would have taken me two weeks two years ago suddenly take 20 minutes,” he said. Although “most of the time it doesn’t work,” AI can now be used in unprecedented ways “to discover a richer world beyond our imagination.”
around the sphere
Although bluehat intervals appear to be purely combinatorial objects, they also play an important role in a particularly abstract area of mathematics called algebraic geometry. Algebraic geometry is the specialty of Ravi Bakir, a mathematician at Stanford University and current president of the American Mathematical Society.
Algebraic geometry is the study of shapes defined by polynomials such as: ×3 +2×2y + xz =5. This includes the sum of variables raised to the power of an integer exponent. The order of the equation is the largest exponent the polynomial has, in this case 3.
Ravi Vakil and his colleagues recently came up with the idea for a new proof while chatting with a custom-built version of Gemini. “Who came up with that idea?” he asked. “Is it our fault? Is it the model’s fault?”
Vakil and his colleagues, Balázs Elek at the University of New South Wales and Jim Bryan at the University of British Columbia, were interested in studying how to embed spheres in specialized spaces called flag types. (Flag types also appear in the Bruhat team’s paper.) Each embedding (how each point on the sphere is related to a point within the flag type) can be defined by a polynomial.
There are many ways to embed the sphere. Mathematicians represent each embedding as a unique point in a separate high-dimensional space. We then study embeddings defined by polynomials of different degrees by analyzing the different spaces formed by the polynomials.
As the degree progresses, mathematicians want to understand how these spaces change. They knew that as the degree becomes arbitrarily large (as it approaches infinity), the space resembles the space of all consecutive embeddings, not just those defined by polynomials. But when will this similarity be realized?
Surprisingly, Vakil and his colleagues found examples that suggest it happens very quickly. “There was a consistency that wasn’t supposed to happen until we got to infinity, but it was already happening,” he said.
So, along with Freddie Manners and George Salafatinos, who were working at DeepMind at the time, we set out to prove it using two specialized modules built on top of Google Gemini. DeepThink is publicly available, and the other is a system called FullProof developed by Sarafatinos. They started with simpler cases. “The proof was very elegant, precise, and beautifully written. We were able to follow it line by line,” Vakil said. “We uncovered structures that weren’t obvious at the time. From there we learned how whole arguments and important generalizations might work.”
Vakil and his colleagues then went back to the AI model, sketched out proofs for common cases, and asked the AI model to fill in the details. It was successful, as they reported in a preprint on January 12, 2026. “For me, the real thing came first,” Vakil said. This is DeepMind’s proof for the simpler case. “The clarity of the argument gave us new ideas.” But he wondered, “To whom do we owe those ideas? To us? To the model?”
No matter how you look at it, Vakil said, “I think given enough time, I could have proven it.”
But then he hesitated. “I think so. I don’t know. I don’t know. Maybe I would have written it in a clunky way. This paper probably wouldn’t have happened without help.”
And finally, “We had to go back and forth. AI models can help us do the math by allowing us to do things we didn’t have time for before.”
This is perhaps the classic example of how AI can help today. A group of expert mathematicians, with the help of big tech companies, figure things out faster than they might otherwise. I’m also sure it’s correct because I can check it line by line.
everything you need to know
When we ask what AI is bringing to mathematics research, we shouldn’t just look at successes. Litt warned that the commons is being “massively polluted by AI-generated nonsense.” Joel David Hamkins of the University of Notre Dame said he is “desperate at this sea of slopes overwhelming our journal system.”
Mathematicians are pinning their hopes on formal proofs as a way to overcome this ocean of slope. They convert the proof into a language that computers can understand and use a computer program to verify that all the logic in the proof works. “AI without validation is too unreliable to be used in serious applications,” Tao said.
Currently, formalizing mathematical proofs in this way is a time-consuming and complex process that itself requires considerable mathematical knowledge and is a bit of an art. Mathematicians are therefore increasingly turning to “automatic formalization,” where AI models transform mathematical statements into formal, logical statements and prove them. “For the first time, it feels like we can formalize a significant portion of mathematics through AI,” Tao said.
Another big challenge that many mathematicians see as a result of AI’s improved math capabilities is how it will impact the way students learn. Even the most ardent supporters of AI are concerned. Ken Ono, a professor at the University of Virginia who recently took a leave of absence to become Axiom’s “founding mathematician,” said, “While I see a rosy picture of how AI will help mathematics research, I am very concerned about the role of AI in the future of work and training at all levels.”
Professor Tao said, “Many of the problems we pose can be solved instantly by AI. This may prevent many students from developing their mental strength.”
Hamkins agreed. “I used to give a lot of homework, but I can’t do it anymore,” he said. A significant portion of the assignments submitted by students are written by AI. “I don’t want to read. I don’t want to be an AI cop.” Homework used to be of great educational value, but now “everything has to be quizzed and worked on in class. This is an issue across the academic profession.”
Another mathematician at a major research university told me: “There is a serious risk that AI will hinder the growth of mathematics researchers at the same time that it accelerates the progress of serious mathematics researchers.”
Despite rapid changes over the past year, none of the mathematicians I spoke to in reporting this paper feared the subject would become obsolete. Tao compared mathematicians to trying to climb “a big mountain range with a lot of high mountains and a lot of hilly areas.” Humans can only climb one step at a time, but they can plan a route to the top of a mountain like Mount Everest. On the other hand, Tao said that current AI is like a robot that jumps. They can sometimes “parkour to the top of six-foot walls” that humans cannot climb. But you can’t make long-term strategic plans. Tao imagines that 6 feet could become 10 feet or even 100 feet. But “a little robot that jumps is a long way from the Everest of mathematics.”
Mr. Park said that the particular Everest, the total π + e It can be written as a fraction, but it will remain unresolved for centuries. “I really doubt that AI can have any impact there,” he says. “This is not something that AI can do, but I am confident that if humanity survives, we will eventually be able to solve it.”
Of course, much depends on how the capabilities of AI algorithms change and improve over the next few years. Even the most astute and careful observer cannot say with certainty how a model will develop. Few companies are showing any signs of stagnation. “Things are moving very fast. There are no signs of slowing down,” Litt said. The first few months of 2026 have seen a flurry of new work announced by big companies like Google and OpenAI, smaller companies like Axiom, and even academics and hobbyists.
“My prediction is that in 20 years, we will definitely have AI tools that produce mathematics that is better than any human mathematician in many measurable ways,” Litt said. “I’d be shocked if that didn’t happen.”
But, as Venkatesh told me, “At the end of the day, there are an infinite number of ways to formulate any mathematics.” The choices we make, he said, are driven by human values and shaped by the fact that mathematics is not only a science but also an art.
The balance between science and art is a big part of what gives mathematics its beauty, and is one of the “precious things in our culture” that Venkatesh wants to preserve. If AI moves mathematics away from its artistic heritage, the discipline will decline, even if more theorems are proven every month. After all, no poet seriously talks about doing statistical regression on sonnets to find the best one.
The biggest hope for AI is that it will help mathematicians discover and prove things that would otherwise remain mysteries. Most mathematicians agree that that’s what computers have been doing for the past 80 years. However, many are concerned about the scale of the changes currently underway.
The world’s largest annual mathematics conference is held in early January each year. Washington DC in 2026 was full of nervous jokes about AI making us obsolete, even if everyone on record claimed that AI would help human mathematicians. Williamson, who has been working on AI for years and is very excited about it, was chosen to deliver a series of prestigious talks on AI and mathematics to the entire conference. He told the audience that it would be a mistake to react to advances in AI with ignorance or fear.
But he says he understands where that fear comes from. He sees mathematics as “a craft to which people have spent their lives, to which they have dedicated their lives. Its value may be greatly diminished in the future.”
