38 problems

Let ${\mathcal T}_{\infty}$ be the theory under consideration, and let $\sim$ be a definable equivalence relation over ${\mathcal T}_{\infty}$. The relation is finite if its equiva…

Let $K$ be an ultraproduct of the fields $\boldsymbol{Q}_{p_i}$ or $\boldsymbol{F}_{p_i}((t))$ over a non-principal ultrafilter, and let $\varphi(x,y)$ be a partitioned formula with…

Let $(K,v)$ be a henselian valued field of positive characteristic. A valued field is tame if it satisfies the standard tameness conditions for henselian valued fields. NTP2 transf…

Let $\mathcal{K}$ be a complete discrete valued field of characteristic zero with residual characteristic $p$, and let $\kappa$ be its residue field. For a field $E$ of characteris…

Axiomatizability conjecture. There exists a theory in the language of globally valued fields whose models are exactly the existentially closed globally valued fields of archimedean…

Let $(K,v)$ be an algebraically maximal valued field with divisible value group and perfect residue field $k$. A field is called geometrically $C_1$ if every smooth projective sepa…

Let $(K,v)$ be a valued field, and let an Artin–Schreier (AS) tower mean a tower $M/K$ formed through Artin–Schreier extensions. Assume that $(K,v)$ is Henselian, perfect, and has…

Definable compact quotient conjecture. There is a $\mathbb{C}((t))$-definable dfg subgroup $H$ of $G$ such that $G/H$ is definable, rather than merely interpretable, and definably…

Split-solvability conjecture. The group $G$ is virtually a $\mathbb{C}((t))$-split solvable algebraic group.

Connected-component conjecture. One has

Conjecture on definably compact groups. The group $G$ is generically stable, and

Let $K$ be a henselian defectless valued field with divisible value group and residue field $k$. A field has the separably rationally connected rational-point property if every sep…

Let $K$ be a tame valued field with divisible value group and residue field $k$. Let $V$ be a separably rationally connected variety over $K$, and set $F=K(V)$. Valuation conjectur…

Let $(K,v)$ be an algebraically maximal valued field with divisible value group and perfect residue field $k$. A field is geometrically $C_1$ if every smooth projective separably r…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue field $k$, and let $A$ be a $p$-algebra over $K$. A maximal subfield of $A$ is totally ramified cycli…

Generalised Albert form conjecture. Every such form $\varphi$ is a generalised Albert form.

Let $K$ be a complete discrete valued field of characteristic $p>0$ with residue field $k$, and let $A$ be a $p$-algebra over $K$, meaning a central simple algebra whose center has…

Let $K$ be a field whose pure field theory is NIP$_n$, and let $v$ be a henselian valuation on $K$. Write $(K,v)$ for the resulting valued field. Jahnke's NIP$_n$ henselian expansi…

Let ${\mathcal O}_\mathbb{K}$ be the valuation ring of a field with residue map $\operatorname{res}$, and let $G\subseteq {\mathcal O}_\mathbb{K}^n$ be a $\emptyset$-definable grou…

For each $d\in\mathbb{N}_{\geq 1}$, let $c,s>0$ be constants. Second selection lemma over valued fields. For every valued field $K$, every $\alpha\in(0,1]$, every $n\in\mathbb{N}$,…

Sierksma's Dutch cheese conjecture. There are at least $((r-1)!)^d$ partitions of $X$ into parts whose convex hulls intersect. The conjecture extends the Dutch cheese conjecture fr…

Let $K$ be a field. A structure is distal if it has the model-theoretic distality property. Henselian residue-field conjecture. Suppose $K$ has a distal expansion; then $K$ has a h…

Let $K$ be a perfect field of positive characteristic, and let $\tau$ be a $W_n$-topology on $K$. Positive-characteristic decomposition conjecture. The topology $\tau$ is generated…

Valuation conjecture. If $K$ is an unstable dp-finite field, then $K$ is valuation type.

Valuation conjecture. The group $I_K$ is an ideal in a valuation ring on $\mathbb{K}$; equivalently, $\operatorname{Stab}(I_K)$ is a valuation ring.