Mathematicians use AI to find mysterious symmetries, solving decades-old problem
This past May mathematicians headed into a room near the top of the California Institute of Technology’s tallest building to prep for a sprint. The American Institute of Mathematics was soliciting questions of a particular kind: ones ripe for harnessing artificial intelligence’s unique ability to sift through larger-than-human-scale possibilities. Rachel Pries, a mathematician at Colorado State University, took to the podium to present a particularly strong candidate. “This problem had been open for a very, very long time,” she said.
Her challenge focused on the deep connections between polynomial equations such as 4x2 – 1 = 0 and the symmetries buried in their solutions. Within a few short months of the conference, by intertwining human insight with AI scale, the decades-old mystery would collapse.
Pries’s suggestion, known as the inverse Galois problem, was inspired by the work of the 19th-century French mathematician Évariste Galois, who perished after he was wounded in a gun duel at age 20. He had been studying patterns of symmetries that arise in polynomial equations with rational coefficients (numbers that can be written as fractions, such as 3 or 1⁄2). Mathematicians have long known that some polynomial equations have only rational solutions, while others have irrational solutions (such as ). But Galois’s work classified polynomials far more finely. By studying their solutions, he isolated symmetries that were peculiar to each equation, defining what eventually came to be known as “Galois groups.”
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Consider the equation x2 + 1 = 0, which has imaginary solutions—called roots—of +i and –i, where i = .
Imagine sneaking into a math library and combing through every mathematical text ever written. You secretly replace every i with –i and every –i with i. In the end your prank fails because the swap is a genuine symmetry, so all the equations in each book are still true. Pair that with the “do nothing” identity symmetry, and you have the two elements of the x2 + 1’s Galois group.
The more complicated a polynomial is, the more ways there are to shuffle its roots without anyone noticing the difference, leading to a richness of symmetries.
For any polynomial, known algorithms tease out its Galois group. But the reverse direction is opaque: Given a particular Galois group of symmetries, can you always find an associated polynomial? The quandary became known as the inverse Galois problem.
Most Galois groups come in orderly families. Shuffle three polynomial roots, and you get one group, shuffle four, and you get the next group; and so on up a ladder. But 26 different groups refuse to fall in line. The “sporadic” groups belong to no family and follow no pattern, with no simple explanation for why they exist at all. The first five sporadic groups to be discovered were the Mathieu groups: M11, M12, M22, M23 and M24.
In the 1980s teams of mathematicians around the world unearthed the polynomials corresponding to 25 of the 26 sporadic groups—but not M23. “There’s this one last holdout,” says Bjorn Poonen, a mathematician at the Massachusetts Institute of Technology. “I think some people were even wondering whether there might be no polynomial giving M23.”
After the Caltech talks concluded, the organizers asked participants to bid on problems they wanted to attack. The inverse Galois problem quickly racked up bids. In the end, six participants who had never worked together soon found themselves part of a mathematical dream team. Pries and Poonen were joined by Xiaoyu Huang of Temple University, Blake Jackson of the Institute for Computer-Aided Reasoning in Mathematics, Kyu-Hwan Lee of the University of Connecticut and Shaowu Zhang, a Ph.D. student at Caltech.
And before three months had passed, M23’s mysteries fell apart.
“We could do it very efficiently,” Kyu-Hwan Lee says. “That wasn’t really possible five years ago.”
The team used AI to comb through M23 for combinations of symmetries. Of the results, they took the smallest collection—a set of seven surfaces—and asked the AI to numerically approximate their equations. Perhaps one could be pinned to exact numbers? But the decimals didn’t resolve into anything recognizable. The AI kept suggesting more precision, but by the time they hit 90 digits, they had maxed out the computer’s memory. The project halted. Huang posted a suspicion in the team’s chat that the choice of coordinates was to blame. “I think Bjorn pointed out that we could try some other way,” she said.
The group set a raft of AI agents to work on new, more harmonious coordinates. The group described what happened next as “miraculous” in a paper posted on the preprint server arXiv.org. Huang says most of the agents failed, but one came back excited about one of the seven surfaces. “This might work!” the agent wrote.
With just a few more steps, the group turned the approximation of the surface into an explicit equation and, ultimately, a family of polynomials with degree-23 (meaning the variable climbs to x23 and there are 23 roots to shuffle around). Here’s one whose symmetries form M23:
x23 – 184x21 – 1150x20 + 26151x19 + 18400x18 – 1808490x17 + 1545462x16 + 67672923x15 – 42732528x14 – 1333395744x13 + 290615166x12 + 10550424369x11 + 3700476348x10 + 35123826654x9 – 194398310718x8 – 1023887308293x7 + 3961650395556x6 + 1949980486716x5 – 28142323927002x4 + 53599151839311x3 – 46185312415788x2 + 19169943578802x – 3150159884154
The M23 solution is one small-but-mighty piece of an enormous puzzle. Mathematicians suspect there is a master group, the “absolute Galois group,” that describes the symmetries of all possible polynomials at once. Because groups and polynomials turn up everywhere in number theory, mapping this all-encompassing group is one of the field’s biggest ambitions. Each new group arising from a polynomial inches mathematicians closer to the absolute Galois group’s structure.
In an adjacent development, a crowdsourced competition—open to amateurs and professional mathematicians alike—launched with the goal of finding polynomials for every group acting on 24 roots; the largest such group is the symmetric group S24 containing all permutations. While M23 had been a mystery, examples of S24 mapping to polynomials were already plentiful, so the competition was more about constructing unknown, elegant solutions. Each such pattern illuminates a fragment of the absolute Galois group’s structure. The competition’s host, the Foundation for Science and AI Research (SAIR), was founded by top scientists, including Fields Medalist Terence Tao, an advocate for ethical AI use in math.
“If you just pick one [polynomial] arbitrarily,” says Jen Paulhus, a mathematician at Mount Holyoke College and one of the SAIR competition’s organizers, “its [Galois group is] likely going to be part of the big symmetric group. The interesting groups are the small, rigid ones, and random searches almost never land on them.”
Participants would need a strategy.
Submissions soon began pouring in from around the world, including everyone from career number theorists to hobbyists who may have had to look up what a Galois group was, with work powered by everything from AI to stubborn human brainpower. The first phase of the competition concluded in late August, with all 25,000 relationships between polynomials and symmetry groups realized. “I was a little surprised that we got all the groups that we were looking for in the first pass,” Paulhus says.
Emerging as the final winners were a pair of German mathematicians. And despite the pervasive use of AI as a tool throughout the competition, its only contribution to the winning team was to write an upload script. “It was open to AI, and it was still these folks who did the best,” Paulhus says. “As a working mathematician, that heartened me.”