GoKawiilAs they related in a preprint on January 3, 2026, AlphaEvolve had found that the Bruhat intervals in these particular permutation groups had a surprisingly special structure. When the researchers studied the intervals, they found that they formed higher-dimensional cubes called hypercubes. “If you look at what AlphaEvolve was thinking, I was super surprised,” Libedinsky said. “If it was a human, it would be an extremely creative human.”
AlphaEvolve had answered a question they didn’t know they had. “We didn’t ask AlphaEvolve to find big hypercubes,” Ellenberg said. “We asked it to find something else, and we thought about it and realized it was a gigantic hypercube which we had not anticipated was there.”
Excerpt from a prompt that mathematicians gave to AlphaEvolve, in which they asked it to construct an object called a Kakeya set. Mathematicians have found that AI performs better with encouragement.
As Williamson put it, “It’s a structure that’s been sitting there for 50 years in front of our nose. We just hadn’t noticed it.”
Older machine learning methods had previously enabled such serendipitous mathematical discoveries, too — uncovering patterns no one had thought to look for. But in the past, Williamson said, it was a “real engineering effort. … You need to know how to code, spend a lot of time looking at details of neural network training. It was basically extremely difficult for a mathematician with no significant machine learning background to do this.”
With LLMs, “I can suddenly do an experiment in 20 minutes that two years ago would have taken me two weeks,” he said. Though “most of the time it doesn’t work,” AI can now be used like never before “to discover the world that has riches beyond our imagination.”
Around Sphere
Though Bruhat intervals seem like purely combinatorial objects, they also play an important role in a particularly abstract area of math called algebraic geometry, which Ravi Vakil, a mathematician at Stanford University and the current president of the American Mathematical Society, specializes in.
Algebraic geometry is the study of shapes defined by polynomial equations like x3 + 2x2y + xz = 5, which involve a sum of variables raised to whole-number exponents. The degree of the equation is the highest exponent the polynomial has, in this case 3.
Ravi Vakil and his colleagues recently came up with a novel proof idea while chatting with a bespoke version of Gemini. “Who is that idea due to?” he asked. “Is it due to us? Is it due to the model?” Rod Searcey
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