39 problems

Let $\mathcal{K}=(K,+,-,\cdot,0,1,<,E)$ be an ordered exponential field, meaning that $E$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the orde…

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

Let $M$ be a first-order topological structure, and let $\operatorname{Spec}(A)$ denote the Stone spectrum of a definable set $A\subseteq M^k$. Say that $M$ is definably $T_5$ when…

Let $M\preccurlyeq N$, let $X=\operatorname{Spec}(A)$ be the o-minimal spectrum under consideration, let $Z\subseteq X$ be definably closed, and let $W:=\theta^{-1}(Z)$, where…

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…

O-minimal field conjecture. Any infinite field trace definable in an o-minimal structure is real closed or algebraically closed of characteristic zero.

Let $T$ be an o-minimal theory and let $\mathfrak U\models T$ be a monster model. A type over $U$ is invariant over a small set if it is invariant over some small subset of the mon…

Chernikov's VC-density conjecture. The following assertions hold:

Peterzil's conjecture. Every strongly minimal ${\mathcal R}$-relic is either locally modular or interprets an algebraically closed field.

Berarducci–Peterzil–Pillay's conjecture. The natural inclusion

O-minimal istem conjecture. All o-minimal structures are istem.

Let $\mathcal R$ be an o-minimal expansion of a real closed field, and let $\mathcal M$ be a non-locally modular strongly minimal $\mathcal R$-relic, meaning a strongly minimal str…

Let $\mathbb R_{\mathrm{LN},\mathrm{PF}}$ be the structure generated by the larger class of Noetherian and Pfaffian functions considered in the paper. Noetherian–Pfaffian counting…

Let $(X,\tau)$ be a definable topological space in an o-minimal expansion of an ordered field. Call $(X,\tau)$ affine if it is definably homeomorphic to a set with the o-minimal Eu…

Let $\mathbb{R}_{\mathrm{an},\exp}$ denote the real field expanded by restricted analytic functions and the exponential function. Let $f:\mathbb{R}\to\mathbb{R}$ be definable in…

Let the self-dual flux-vacuum locus be the locus of flux vacua satisfying the self-duality conditions, and let $(F,D)$ denote its sharp o-minimal format and degree. Sharp o-minimal…

Let $G$ be a definable group in an o-minimal structure, let $S\subset G$ be a definable subset avoiding unboundedly large arithmetic progressions, and let $k=\dim(S)$. Consider the…

Let $C<G$ be a connected quasi Frobenius group definable in an o-minimal structure, with $C$ definable and connected. Assume that $G$ contains involutions, and that … contains all…

Let $G$ be a definably connected group in an arbitrary o-minimal structure. A Levi decomposition conjecture. $G$ has a Levi decomposition in the sense of the definition given in th…

Optimal point-concentration conjecture. In the setting of Theorem baby-forwards, one has $\Pr(X\in S)\le C_S/\sqrt n$ for some constant $C_S$ depending only on $S$.

Wilkie's conjecture. There exist constants $c=c(X)$ and $d=d(X)$ such that, for every $H\in\mathbb{N}$ with $H>e$,

Conjecture on optimal o-minimal bounds. The same bounds should hold in any $o$-minimal structure, potentially via Davenport–Schinzel sequences. Moreover, every $\Phi(x;y)$ in an…

Let $M$ be a definably complete uniformly locally o-minimal structure of the second kind, abbreviated DCULOAS, and suppose that $M$ enjoys the definable Tietze extension property.…

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.…

Let ${\mathscr{R}}$ be an expansion of $(\mathbb{R},<,+)$, and let ${\mathscr{R}}^\circ$ denote its open core. The structure $({\mathcal P}(\mathbb{N}),\mathbb{N},\in,s)$ is the tw…