47 problems

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

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 $G/G^{00}$…

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…

Let $G$ be a group definable over $M$. A definable action of $G(M)$ on a compact space is an action for which each orbit map is definable. The group is weakly definably amenable wh…

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

Let $G$ be a definably compact group in an $o$-minimal structure. The group $G$ is compactly dominated by the quotient $G/G^{00}$, where $G^{00}$ is the smallest type-definable sub…

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…

Let $G$ be a definably compact group, let $G^{00}$ be its type-definable connected component, and let $\pi\colon G\to G/G^{00}$ be the quotient map. Write $m$ for Haar measure on t…

Let $G$ be a definably compact group, and let $\mu\colon \operatorname{Def}(G)\to [0,1]$ be a finitely additive measure on the definable subsets of $G$. Uniqueness conjecture. Ther…

Let $G$ be a definably compact, definable abelian group, and let $\pi_1(G)$ denote its o-minimal fundamental group. The group $\pi_1(G)$ is a torsion-free finitely generated abelia…

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 $M$ be a small model, and let $p_x$ and $q_y$ be invariant types such that $p$ is definable. Let $F_M$ be the retraction from invariant types to finitely satisfiable types. Rev…

Let $T$ be an NIP theory, let $G$ be a definably amenable group, and let $M$ be a small model. Write $F_M$ for the retraction from invariant types…

Let $T$ be an NIP theory, let $G$ be a definably amenable group, and let $F_M$ denote the retraction from invariant types to finitely satisfiable types, with the associated analysi…

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…

Badness–non-stationarity conjecture. $H$ is bad if and only if $H$ is non-stationary.

Non-semisimplicity conjecture. There exists an infinite, normal, abelian subgroup $N\trianglelefteq G$ such that

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…