26 problems

Let $n$ be a positive integer and let $K$ be an $n$-dependent field, meaning that its first-order theory is $n$-dependent. A field is simple if its theory is simple, and separably…

Supersimple pure-field conjecture. The only supersimple pure fields are bounded perfect PAC fields. Consequently, the only geometric simple pure fields are bounded perfect PAC fiel…

An infinite field is a field with infinitely many elements, and an $\mathrm{NIP}$ field is a field whose complete first-order theory has the non-independence property; an $\infty$-…

Let $K$ be a field in the ring language, and let $n$ be a positive integer. A theory is n-distal when it satisfies the corresponding $n$-ary distality condition. n-distal field con…

Let $K$ be a field considered in the ring language. A theory is n-dependent when it has no independence pattern of order $n$; 1-dependent is the corresponding usual NIP condition.…

O-minimal field conjecture. Any infinite field trace definable in an o-minimal structure is real closed or algebraically closed of characteristic zero.

2-IP conjecture. Every IP field is necessarily $2$-IP.

Order-or-valuation conjecture. Every unstable NIP field admits either a definable field order or a definable non-trivial valuation.

Chernikov3Hempel conjecture. Every IP field is $k$-IP for all $k$; equivalently, every IP field is locally trace maximal.

A field is viewed as a structure in the pure ring language, and $n$-dependence is understood in the model-theoretic sense. Chernikov's field conjecture. If a field is $n$-dependent…

The paper considers motivic cohomology groups $\operatorname H_{\mathrm M}^{n,i}(K)$ of a field $K$, with $\delta(K)$ denoting its Kronecker dimension. Growth-control conjecture. F…

Let $n\geqslant 2$. A pure field is a field considered only in the language of rings, and a theory is strictly NIP$_n$ when it is NIP$_n$ but has IP$_{n-1}$. The NIP$_n$ Fields Con…

Order-or-valuation conjecture. Every unstable NIP field admits a definable field order or a definable non-trivial valuation.

Chernikov–Hempel conjecture. If a field is $k$-independent for some $k\ge 2$, then it is $k$-independent for all $k$.

No strictly $n$-dependent fields conjecture. There are no strictly $n$-dependent fields for $n\geq 2$.

Field-dependence conjecture. Every $n$-dependent field is dependent.

Let ${\mathscr{F}}$ be an expansion of an infinite field $F$. A family ${\mathcal X}$ of subsets of $F$ is directed when any two members have a common member containing their union…

Let $\oplus$ and $\otimes$ be binary operations on $\mathbb{N}$ such that $(\mathbb{N},\oplus,\otimes)$ is a field. Ordered-field IP conjecture. The expansion…

Let ${\mathscr{R}}$ be an NIP expansion of $(\mathbb{R},<,+)$ by closed subsets of Euclidean space. A definable set has dimension zero when its dimension in the paper's dimension t…

Let ${\mathscr{R}}$ be an NIP expansion of $(\mathbb{R},<,+)$ by closed subsets of Euclidean space. Call ${\mathscr{R}}$ field-type when it satisfies condition (1): there is a none…

Nonexistence conjecture. There are no strictly $k$-dependent fields, for any $k\geq 2$, whose theory is simple.

Let $G$ be a group. The theory $G$-$\operatorname{TCF}$ is the model companion theory for fields equipped with an action of $G$. A group is locally virtually free if every finitely…

Let $G$ be a finitely generated group. The theory $G$-$\operatorname{TCF}$ is the model companion theory for fields equipped with an action of $G$. A group is virtually free if it…

Let $T$ be the complete theory of a field $F$ in the language of fields. VC-minimal fields conjecture. $T$ is VC-minimal if and only if $F$ is real closed or algebraically closed.…

PRC and PpC NTP2 conjecture. A PRC field is $\operatorname{NTP}_{2}$ if and only if it is bounded. Similarly, a PpC field is $\operatorname{NTP}_{2}$ if and only if it is bounded.