14 problems

Let $K$ be a $p$-adically closed field and let $A$ be an infinite definable subset of $K^n$ that is open in its closure. Let $(B_k)_{k\leq q}$ be a finite collection of closed defi…

Let $\mathcal{R}$ be an expansion of a divisible ordered Abelian group of dp-rank $2$. Consider infinite definable sets that are discrete, dense and codense in an interval, or Cant…

Transfer conjecture. Many assertions that hold for o-minimal expansions of ordered fields should also hold for definably complete locally o-minimal expansions of ordered groups hav…

Let $X_t$ be the sections of a definable set, and let $X_0$ be the reference section. Assume that $X_t$ has a limit as $t$ tends to $0$, not necessarily equal to $X_0$ but containe…

Upper-limit medial-axis conjecture. One should have

Combinatorial characterization conjecture. The trees of balls $T(X)$ of definable sets $X\subset \mathbb{Z}_p^n$ are characterised by a simple combinatorial condition.

Let $K \subset {\mathbb R}^n$ be a compact definable set in an o-minimal structure over $\mathbb R$, and let $\{S_\delta\}_{\delta>0}$ be a monotone family of compact subsets of…

Let $K$ be a $p$-adically closed field, and let the relevant structure on $K$ be one for which Lemma 2-dimensional sets applies. Converse of Lemma 2-dimensional sets. The converse…

Let $\mathcal M$ be an o-minimal structure and let $D$ be o-finitistic, meaning the property denoted by the source macro $o-finitistic$. Consider the o-minimalistic expansion…

Let $D$ be the set obtained in Example, and let power dominant mean that every definable discrete set in the relevant o-minimalistic structure is definably embeddable into some Car…

Gabrielov–Vorobjov homotopy equivalence conjecture. The approximating compact sets should remain homotopy equivalent to $S$ even without the separability hypothesis.

Let $Y \subset \operatorname{T}(\mathbb{Z}_p^n)$ be definable, where $\operatorname{T}(\mathbb{Z}_p^n)$ is the tree of residue classes arising from $\mathbb{Z}_p^n$. A skeletal cel…

Let $X \subset \mathbb{Q}_p^n$ be a definable set in the language of fields. For each $\lambda \geq 0$, let $X_\lambda$ be the image of $X \cap \mathbb{Z}_p^n$ in…

Definable-set lifting conjecture. If $T'$ does not have the Schröder–Bernstein property, then $T$ does not have the Schröder–Bernstein property.