793 problems

An Abstract Elementary Class is a class of structures equipped with a strong substructure relation; $\text{LS}(d53e)$ denotes its Löwenheim–Skolem number, and $d53e$ is categorical…

Let $T$ be a first-order theory, and consider its countable models up to isomorphism. Vaught's conjecture. The number of countable isomorphism types of models of $T$ is either coun…

Let a strongly minimal structure be a structure in which every definable subset of its universe is finite or cofinite, and let a structure be non-locally modular when it is not loc…

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…

Let $\mathbb{C}$ be equipped with its standard exponential map. Boris Zilber's axiomatization defines a unique exponential field satisfying the stated axioms and the Schanuel conje…

Newelski's conjecture. In good circumstances, for example when $T$ is an NIP theory, the canonical epimorphism

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 $\mathfrak{A}$ be a countable homogeneous structure in a finite relational signature. A reduct of $\mathfrak{A}$ is a structure obtained by restricting its definable relations,…

A definable simple Lie ring of finite Morley rank is a Lie ring $\mathfrak{g}$ definable in a structure of finite Morley rank. Its characteristic is the characteristic associated w…

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…

Let $\lambda$ be a cardinal. For an abstract elementary class (AEC) $\boldsymbol{K}$, write $\operatorname{LS}(\boldsymbol{K})$ for its Löwenheim–Skolem number, and say that…

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 $t$…

Let $G$ be a permutation group on a set $\Omega$. For a positive integer $n$ and tuples $I=(\omega_1,\ldots,\omega_n)$ and $J$ in $\Omega^n$, write…

A structure is monadically $IP$ if every expansion by unary predicates is NIP. Monadic NIP trace-definability conjecture. Monadically NIP structures cannot trace define infinite gr…

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

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…

Kaplan–Simon conjecture. Every $c9$-categorical structure admits an $c9$-categorical expansion with a CIR.

Let $T$ be a complete theory with quantifier elimination in a relational language $L$, and let $P$ be a distinguished unary predicate symbol in $L$. For a model $M\models T$, write…

A first-order sentence is a sentence in the language of groups, and a few-relator random quotient is a random quotient of a free group by a small number of relators. Knight's conje…

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

Wilson's conjecture. Every locally nilpotent omega-categorical $p$-group is nilpotent.

Model-companion conjecture. The theory $\operatorname{GVF}_e$ has a model companion for every Archimedean error $e\geq 0$.

Let $G$ be a definable group in an $o$-minimal theory. Assume that $G$ is definably compact and definably connected, so that $G=G^{0}$. Pillay's conjecture. The quotient $G/G^{00}$…

Stable-field conjecture. Every infinite stable field has no separable extensions; equivalently, every infinite stable field is separably closed.

Let $\mathcal{L}_{\omega_1,\omega}$ be the infinitary logic allowing countable conjunctions and disjunctions but only finite strings of quantifiers. Write $\aleph_{\omega_1}$ for t…