415 problems

For every entire function $f:\mathbb{C}\to\mathbb{C}$, is every subset of $\mathbb{C}$ definable, with parameters, in the structure $(\mathbb{C};+,\cdot,0,1,f)$ either countable or…

Every equality between convergent periods follows from the standard formal relations for periods: linearity of integration, additivity with respect to domains, and algebraic change…

For all positive integers $n$ and $k$, if $M$ is an $n$-dependent structure, $R(y_1,\ldots,y_m)$ is a relation definable in $M$, and $f_i:X_i^{r_i}\to M$ are arbitrary functions wi…

Let $n>0$, and let a connected group $G$ of finite Morley rank act faithfully and transitively on a definable set $X$ of Morley rank $n$. If the action is generically $(n+2)$-trans…

For every coherent theory $\mathcal{T}$, every topological space $X$, and every continuous assignment of model-theoretic types $p:X\to S(\mathcal{T})$, does there exist a sheaf mod…

For all integers $\alpha,\beta>1$ with $\gcd(\alpha,\beta)=1$, determine whether the existential fragment of $\mathsf{FO}(\mathbb{Z};<,+,V_\alpha,V_\beta)$ is decidable; that is, w…

Let $T$ be a countable complete theory and let $P$ be a distinguished unary predicate in its vocabulary. Assume that $T$ fails the Gaifman property; that is, assume that there exis…

For every $Z$-group $G$ and every normal subgroup $N\trianglelefteq G$, is $N$ again a $Z$-group? Equivalently, is the class of all $Z$-groups closed under taking normal subgroups?

Every locally nilpotent countably categorical group is nilpotent; equivalently, for every group $G$, if $G$ is locally nilpotent and countably categorical, then $G$ is nilpotent.

Let $(k,\delta)$ be a differential field of characteristic zero with field of constants $C$. Consider a first-order differential equation over $k$, and, respectively, an autonomous…

The conjecture asks for a Vaught-type classification of the countable models of every countable complete $\omega$-stable theory $T$, based on the model-theoretic Martin conjecture…

Shelah's conjecture for NIP real fields. Any NIP real field is almost real closed.

Let $\mathfrak{g}$ be an infinite simple Lie ring of finite Morley rank, and suppose that its characteristic is sufficiently large. Cherlin–Zilber conjecture for Lie rings. Then…

Let $K$ be an infinite field. It is model-theoretically minimal if every definable subset of $K$ is finite or cofinite. Podewski's conjecture. If $K$ is model-theoretically minimal…

Generalized henselianity conjecture. The ring $R$ is a henselian local ring.

Bodirsky–Pinsker conjecture. If $\str M$ is a first-order reduct of a finitely bounded homogeneous structure in a finite relational language, then either…

Zilber's trichotomy principle relative to $T$. If $\mathcal P$ is strongly minimal and non-locally modular, then there is an infinite field interpretable in $\mathcal P$.

For $r\in(1,\infty]$, let $L(\mathbb F_r)$ be the interpolated free group factor, and let the first-order fundamental group of $L(\mathbb F_r)$ be the group of positive scalars $t$…

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

Zilber's quasiminimality conjecture. Every such invariant subset of $\mathbb{C}$ is countable or cocountable; equivalently, $\mathbb{C}_{\operatorname{exp}}$ is quasiminimal.

Let $F$ be a strongly NIP ordered field. Such a field is almost real closed if it admits a henselian valuation whose residue field is real closed. Shelah–Hasson conjecture for orde…

Let $\Gamma$ be a homogeneous structure with a finite relational signature. An expansion of $\Gamma$ is a structure obtained by adding relations to its signature; its age is the cl…

Shelah's three cardinal and model-existence claims. The following assertions hold:

Let $G=G(\mathfrak{C})$ be a definable group, and let $\mu\in\mathfrak{M}^{\operatorname{def}}_G(\mathfrak{C},M)$ be fim over $M$. Its right stabilizer, denoted…

Let $\Lambda$ be a partial commutative monoid generated by a tuple $\bar\delta$; equivalently, let $\Lambda$ be a trace monoid (Cartier–Foata monoid). Let $T^\Lambda$ denote the co…