13 problems

Mináč–Tân's conjecture. For every field $F$, every prime $p$, and every positive integer $n$, $G_F$ satisfies $n$-Massey vanishing with respect to $p$.

Let … be the meta-profinite fundamental group of the Teichmüller tower, and let … be its Grothendieck–Teichmüller automorphism group. The absolute Galois group is denoted by…

Let $K$ be a field containing an algebraically closed field, let $G_K$ denote its absolute Galois group, and let $p$ be a prime. Write $[G_K,G_K]$ for the commutator subgroup of…

Hilbertian-projective implies pro-free conjecture. If $K$ is Hilbertian and projective, then $G_K$ is profree.

Let $m$ be a positive integer, and consider subsets of $(\operatorname{Aut}(\overline{\mathbb{Q}}))^m$ defined by existential, universal, or $\Sigma^0_2$-separated formulas, with p…

Let $K\subset\overline{\mathbb Q}$ and suppose that the absolute Galois group $G_K$ is a projective profinite group. The group $G_K$ is profree if it is a free profinite group. Hil…

Let $F$ be an infinite field with finitely generated absolute Galois group. Koenigsmann's conjecture. Then $F$ is ample. This conjecture concerns which fields are ample and is pres…

Let $K$ and $L$ be finitely generated infinite fields, let $\bar K$ and $\bar L$ be algebraic closures, and write $K^{\rm perf}$ and $L^{\rm perf}$ for their perfect closures. Let…

Let $k_0$ be a field, let $\mathbf{Var}_{k_0}$ be the category of varieties over $k_0$, and let $\mathcal{V}$ be a subcategory of $\mathbf{Var}_{k_0}$. The absolute I/OM asks wheth…

Let $F$ be a field containing a primitive $l$-th root of unity, and let $G_F$ be its absolute Galois group. Consider tuple Massey products of degree-one elements in the cohomology…

Let $F$ be a field containing a primitive $l$-th root of unity, and let $G_F$ be its absolute Galois group. Write $H^*(G_F,\mathbb Z/l)$ for the cohomology algebra of $G_F$ with co…

Let $F$ be a field, let $F_s$ be a separable closure, and let $G_F={\rm Gal}(F_s/F)$ be its absolute Galois group. Let $p$ be a prime number and $n\geq 3$ an integer. The vanishing…

Let $K$ be a fully Hilbertian field, and let $G_K$ denote its absolute Galois group. A profinite group of infinite rank $m$ is semi-free if every finite split embedding problem for…