Science1 publisher3 min readPublished
The M23 holdout fell three months after mathematicians bid to work on it at Caltech
The American Institute of Mathematics asked for problems where AI could search further than a person can, and Rachel Pries offered the one sporadic group with no known polynomial. Six mathematicians closed it in under three months.
The Scientist · Science desk
What happened
- Rachel Pries of Colorado State University put forward the inverse Galois problem, which asks whether a given group of symmetries can always be matched to a polynomial that produces it.
- Organizers had participants bid on the problems they wanted after the talks, and the inverse Galois problem drew enough bids to put together a team of six who had never collaborated.
- Less than three months later the M23 case came apart, and Kyu-Hwan Lee of the University of Connecticut said the work could not have been done as efficiently five years ago.
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Why it matters
- capability Questions abandoned in the 1980s because nobody could construct an example are worth re-opening when the obstruction is the size of the search space.
- precedent A solicitation plus a bidding round is cheap enough for any department to copy, and it produces something a grant panel can use: a ranked list of open problems whose obstruction is enumeration.
- constraint The filter that made this work also bounds what it demonstrates, because the workshop selected for search scale, so a problem needing a genuinely new concept was never in the sample.
- decision Anyone funding AI-for-mathematics should ask which step the model performs; here it enumerated symmetry combinations and approximated equations numerically, and the exact result came from the six mathematicians.
One direction of this problem has been routine for a long time. Given a polynomial, known algorithms tease out its Galois group, the shuffles of its roots that leave every true equation true [5][6]. Going the other way, from a group of symmetries to a polynomial that produces it, has no such recipe [5]. Most Galois groups sit in tidy families: shuffle three roots and you get one group, shuffle four and you get the next, on up a ladder [7]. The 26 sporadic groups belong to no family and follow no pattern [8]. Without a ladder to climb, finding a polynomial for one of them means combing through combinations.
In the 1980s, teams around the world produced polynomials for 25 of the 26 sporadic groups [10]. Subtract, and one case was left [1]. M23 was among the first five sporadic groups ever found, the Mathieu groups, so the holdout was not a late or exotic discovery [9][2]. "There's this one last holdout," Bjorn Poonen of MIT said [11]. "I think some people were even wondering whether there might be no polynomial giving M23," he said [12].
The American Institute of Mathematics was soliciting that shape of question, ones ripe for AI's ability to sift through larger-than-human-scale possibilities [2]. Pries, of Colorado State University, took the podium at Caltech in May and presented it [1][3]. "This problem had been open for a very, very long time," she said [4]. After the talks, organizers asked participants to bid on the problems they wanted to attack, and the inverse Galois problem quickly racked up bids [13]. Six people who had never worked together ended up on it: Pries, Poonen, 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 Caltech doctoral student [14].
In the work that followed, the machine's jobs were enumeration and approximation. The team used AI to comb through M23 for combinations of symmetries, took the smallest collection that came back, a set of seven surfaces, and asked the AI to numerically approximate their equations [17][18]. The decimals did not resolve into anything recognizable [18].
Before three months had passed, according to Scientific American, M23's mysteries fell apart [15][19]. "We could do it very efficiently," Lee said. "That wasn't really possible five years ago" [16]. Scientific American reports no benchmark figures; the only figure in the account is elapsed time.
The choice that carried this result was made before any computation, when a solicitation and a bidding round sorted one question out of many as the kind whose obstruction was the size of a search [2][13]. One problem is not a sample, and the workshop had pre-filtered for search scale, so nothing here tests what the same tools do on a problem that needs a new idea. If the format transfers, it transfers through the selection step, which cost a session of talks and a round of bids [13].
What to watch
- Publication and independent verification of the M23 polynomial, which would show how much of the final step from decimals to exact equations was human.
- Whether AIM repeats the solicitation-and-bidding format, and whether it reports on the problems chosen at Caltech that did not close.
- Whether the same enumeration approach reaches inverse Galois cases outside the 26 sporadic groups.