9 problems

Let $\mathcal{Z}$ be a dp-minimal proper expansion of $(\mathbb{Z},+,0,1)$. A classification conjecture for dp-minimal expansions of the integers. Exactly one of the following hold…

Ordered-group decomposition conjecture. There is an o-minimal expansion ${\mathscr R^{\square}}$ of an ordered abelian group such that

Characterization conjecture. Let $(\mathbb{Z},+,0,1,v)$ be distal. Then the following are equivalent:

$p$-adic completion conjecture. The structure induced on $\mathbb{Z}$ by ${\mathscr{Z}}^{\square}$ is interdefinable with ${\mathscr{Z}}^{\mathrm{Sh}}$, and every…

Generic smooth expansion conjecture. There is a comeager subset $\Lambda\subseteq C^\infty(I)$, possibly depending on $\alpha$, such that for every $f\in\Lambda$: 1.…

Topology conjecture. Any proper dp-minimal expansion of $(\mathbb{Z},+)$ which eliminates $\exists^\infty$ defines a non-discrete group topology on $(\mathbb{Z},+)$.

Assume that $T$ is dp-minimal. If the formula $\phi(x;b)$ does not fork over $M$, dp-minimal definability conjecture. it extends to an $M$-definable type. The paper states this as…

Countable coheir detection conjecture. There is a countable subset $Q\subseteq S_y(\mathcal U)$ of coheirs of $q$ such that, for every invariant type $p$, if $p$ commutes over $M$…

Definable extension conjecture. The formula $\phi(x;b)$ extends to an $M$-definable type.