Post by @tao@mathstodon.xyz

2026-08-10 05:29 UTC

The first phase of the #SAIR / #LFMDB inverse Galois challenge https://competition.sair.foundation/competitions/igp24/overview will close in a week. One notable milestone of the competition so far: the inverse Galois conjecture has now been verified for all 25,000 of the transitive groups acting on 24 elements, by explicitly matching each such group with a degree 24 polynomial with that group as its Galois group! Prior to the competition, only 79 of these groups had been so matched. In some ways, our coverage of the degree 24 case is now slightly better than the degrees less than 24: of the 5512 eligible transitive groups in that case, all but one is known to be matched to a Galois group (the exception being the Matthieu group M_23). This does not mean that Stage 1 has been solved prematurely! We are not just tracking the Galois group of a polynomial, but also the number of real roots, increasing the number of signatures from 25,000 to 160,164. Currently, all but 5,672 of these signatures are matched to a polynomial. We are also tracking the discriminants of the polynomials submitted, and there is scope to improve them further. In Stage 2, we will reveal all the polynomials submitted, and switch to a collaborative mode in which contributors post their polynomial-finding techniques. We will then use AI to create a "superteam" who will be tasked with beating the best Stage 1 polynomials using all the ideas contributed. Stay tuned for more details!

Replies (4)

@highergeometer@mathstodon.xyz 2026-08-10 06:55

@tao@mathstodon.xyz I guess that one aim is to get the "superteam" to attack the case of M_23?

Open ##5119824
@wikiemol@mathstodon.xyz 2026-08-10 18:52

@tao@mathstodon.xyz In the future, do you see mathematics as having any differentiating features from empirical science and engineering? If so what features differentiate it from these fields? Previously, I think mathematics was very different from these fields. But these sorts of projects look way closer to science and engineering than mathematics (a comparison you have made before, I think). I’m not saying that’s a problem from a utilitarian perspective, but will this be the norm? whether or not this is the norm plays a role in whether I want to be a mathematician in the future or not. Or is it just the modern form of calculating a large table of values of e.g trig functions, as was done in the past before calculators?

Open ##5119827
@theking@mathstodon.xyz 2026-08-11 02:06

@tao@mathstodon.xyz FYI: inverse Galois problem for n=23 has recently been proved! https://math.mit.edu/~poonen/papers/M23.pdf

Open ##5119829
@gotanegra@mathstodon.xyz 2026-08-11 19:52

@tao@mathstodon.xyz I followed the extension structure of 24T10478 a bit further. The minimal obstruction to splitting occurs over Q₈ < GL(2,3). Surprisingly, its exceptional defect geometry has a clean projective description: the 64 local complements partition canonically into four 16-element blocks, equivariantly indexed by the four Sylow-3 subgroups of GL(2,3), i.e. P¹(F₃). More explicitly, an ordered generating pair (a,b) of Q₈ selects the unique order-3 generator p satisfying aᵖ = b, bᵖ = ab. The 24 pairs map equivariantly onto the four projective points, with fibres of size 6. GAP certificate available.

Open ##5119830