25 problems

Petrykowski's conjecture. If $G$ has a bounded orbit, then $G$ is definably amenable.

Let $G$ be a definably compact group, let $M_0$ be the parameter structure over which $G$ is defined, and let $G^{00}$ denote the infinitesimal subgroup. Contractibility conjecture…

Assume $2\leq k\in\omega$, let $T$ be a $k$-distal NIP theory, and let $G$ be a definable group. Let $\mu\in\mathfrak{M}_{G}(\mathbb{M})$ be a generically stable, $G$-invariant mea…

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…

Definable compact quotient conjecture. There is a $\mathbb{C}((t))$-definable dfg subgroup $H$ of $G$ such that $G/H$ is definable, rather than merely interpretable, and definably…

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

Connected-component conjecture. One has

Conjecture on definably compact groups. The group $G$ is generically stable, and

Let $T$ be a $k$-dependent theory, let $A$ be a small parameter set, and let $G$ be a type-definable group over $A$. Let $\mathcal{M} \supseteq A$ be a small model and let…

Let $M$ be a highly saturated $P$-minimal structure over a $p$-adic field $K$, let $G$ be a definable group, and set $n=\dim(G)$. Let $M_0\prec M$ be a small model defining $G$. Re…

Let $M$ be a highly saturated $P$-minimal structure over a $p$-adic field $K$, and let $G$ be an $n$-dimensional definable group. Recall that $G^{00}$ is the smallest type-definabl…

Let $G$ be a definably amenable NIP group. An fsg group is a group with finitely satisfiable generics, and a dfg group is a group with definable f-generics. Pillay–Yao's conjecture…

Let ${\mathcal O}_\mathbb{K}$ be the valuation ring of a field with residue map $\operatorname{res}$, and let $G\subseteq {\mathcal O}_\mathbb{K}^n$ be a $\emptyset$-definable grou…

Let $H$ be a group definable in an $o$-minimal structure or a $p$-adically closed field, and assume that $H$ is a $\mathrm{dfg}$ group. Say that $H$ is bad when it has the badness…

Let $G$ be a definable, definably amenable group defined over either an $o$-minimal structure or a $p$-adically closed field. Say that $G$ is stationary when it has the stationarit…

The dfg conjecture. If $G$ is a $\mathbb{Q}_p$-interpretable zero-dimensional, definably amenable group, then $G$ has dfg.

Let $G$ be a group definable in a NIP structure. Say that $G$ is dfg if it has a definable global f-generic type, and call a global f-generic type almost periodic when its orbit un…

Let $\mathcal{M}$ be an o-minimal structure, and let $G$ be an abelian, two-dimensional, torsion-free group definable in $\mathcal{M}$. Splitting conjecture. The group $G$ is the p…

Let $M$ be a first-order structure and let $G$ be an $M$-definable group. The space $S_{ext,G}(M)$ is the universal definable flow of $G$ over $M$, and $G^{00}$ is the smallest typ…

Let $G$ be a definably simple group definable in an o-minimal structure $\mathcal{M}$. A subset $X\subseteq G$ is syndetic if there is a finite set $E\subseteq G$ such that $XE=G$.…

Weak definable amenability conjecture. If $G$ is weakly definably amenable, then

Definable topological amenability conjecture. If $G$ is definably topologically amenable, then

Let $\langle G,*\rangle$ be a definable group. A group is small if it is a definable set of the corresponding smallness notion in the dense expansion, and a group is $\bigvee$-defi…

Let $K$ be a $p$-adically closed field and let $H$ be a definable subgroup of $SL_2(K)$. Assume that $H$ is bounded and nonsolvable, and that $w$ does not normalize $H$. Let…

Solvable-by-finite conjecture. Then $G$ is solvable-by-finite.