18 problems

Let $(\Gamma,\psi)$ be the asymptotic couple under consideration, and let $\chi\colon\Gamma\setminus\{0\}\to\Gamma$ be the contraction map defined by … A non-definability conjectur…

Let $n\geq 2$. A property of complex algebraic varieties in ${\Bbb C}^n$ is definable if there is a first-order sentence in the corresponding language with a predicate for membersh…

A topological model consists of a topological space together with the valuation of proposition letters and nominals; let $T_2$ denote the class of Hausdorff spaces, and let…

A representable construction is a construction in a model of set theory that is representable there; it is uniformisable when its associated choices can be made uniformly, and it i…

'Ez-field conjecture. All known logically tame perfect large fields, and in particular all logically tame perfect fields known before the cited work, are 'ez.

Definable-basis conjecture. If $K$ admits a gt-henselian topology with a basis consisting of definable sets, then the $\mathscr{E}_K$-topology is a field topology.

Let $\mathbb{R}_{S_n}$ be the reals at definability level $S_n$, let $\mathbb{R}_{S_\omega}=\bigcup_{n\in\mathbb{N}}\mathbb{R}_{S_n}$, and let $\mathbb{A}_{S_n}$ and…

Let $K$ be a non-big field, meaning an algebraic extension of $\mathbb{Q}$ that is not big, and let $R_K$ denote the ring constructed from the unit group of $K$. Triviality conject…

Let $\mathbb{Q}^{\operatorname{ab}}$ be the maximal abelian extension of $\mathbb{Q}$, and let $\mathbb{Q}^{\operatorname{ab},+}$ be its maximal totally real subfield. Write…

Let $\mathcal{F}$ be a definable family of sets in an NIP structure with the $(p,q)$-property: for every $p$ sets in $\mathcal{F}$, some $q$ intersect, where $p\geq q$ and $q$ is s…

Let $A$ be a simple abelian variety, let $[0,+\infty]$ be the extended nonnegative real interval, and let … be the path under consideration. Uniqueness conjecture. The germ of $q$…

Recursive definability conjecture. There are formulas $\phi_{n,g}(\vec{x})$, $\psi_{n,g}(\vec{x},\vec{y})$, and $\rho_{n,g}(\vec{x},\vec{z})$, depending recursively on $n$ and $g$,…

Let $O$ be a ring of algebraic integers, that is, the integral closure of $\mathbb{Z}$ in a number field. A subset is e-definable if it is definable by an existential system of pol…

Suppose $\mathcal{M}$ is a model of $\mathrm{ZFC}+\mathrm{V}=\mathrm{L}$. Let $\mathcal{D}_{\mathcal{M}}$ be the collection of $\mathcal{M}$-definable subsets of its domain, and ca…

Non-definability conjecture. In general, $M$ need not be definable in its set-forcing extension $M[G]$.

Let $\mathbb{Q}^{ab}$ be the maximal abelian extension of $\mathbb{Q}$, let $\mathbb{Q}^{rab}=\mathbb{Q}^{ab}\cap\mathbb{R}$, and let $\mathbb{Z}^{ab}$ and $\mathbb{Z}^{rab}$ denot…

Finite-pattern definability conjecture. There is a finite set $\Delta'$ of formulas such that, for every $\phi(x;y)\in p$ and $b,b'\in\mathcal U$, if

Bi-Interpretability Conjecture. The Turing definable relations are exactly those whose information content is describable in second-order arithmetic. Moreover, every description of…