58 problems

A NIP field is a field whose first-order theory has the non-independence property. Shelah's conjecture. Any NIP field is either finite, separably closed, real closed, or admits a n…

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

Let $(K,v)$ be a Henselian valued field, and let $L/K$ be a finite simple field extension such that $(L/K,v)$ is defectless. Let $r$ and $s$ be the minimal numbers of generators of…

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…

An NIP field is an infinite field whose first-order theory does not have the independence property. NIP fields conjecture. Every infinite NIP field is either separably closed, real…

Let $(K,v)$ be an NIP valued field. Henselianity conjecture. Every NIP valued field $(K,v)$ is henselian. This is one of the main conjectures on NIP fields and would constrain the…

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…

NIP real field classification conjecture. Every NIP real field is real closed, or admits a non-trivial definable henselian valuation.

Folklore conjecture. The Puiseux series field $\bigcup_{n=1}^{\infty} k(\!(t^{1/n})\!)$ is also $C_i$. This is related to the question of whether every $C_i$ field is stably $C_i$;…

Kuhlmann–Rzepka's characterization conjecture. A henselian field is a roughly deeply ramified field if and only if all of its algebraic extensions are independent defect fields.

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…

Connected-component conjecture. One has

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

NIP field conjecture. Every infinite NIP field is algebraically closed, real closed, or admits a non-trivial definable henselian valuation.

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…