100 problems

Let a strongly minimal structure be a structure in which every definable subset of its universe is finite or cofinite, and let a structure be non-locally modular when it is not loc…

Let $X$ be a definable set in the structure $\mathbb R_{\exp}$, and consider the Pila–Wilkie counting function for arithmetic points of bounded height after removing the algebraic…

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 $\mathbb{R}_{\text{an}}$ be the expansion of the real field by restricted analytic functions, let $\mathbb{R}_{\text{an}}^\mathbb{R}$ be its expansion by all real power functio…

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 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 $G^{00}$ denote its type-definable connected component, and let $\widetilde{G^{00}}$ be the corresponding closed subspace of the spectral…

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…

van den Dries–Miller conjecture. There are no structures on $(\mathbb R,+,\cdot)$ lying strictly between $\mathfrak S^{an}$ and $\mathfrak S^{an,\, exp}$.

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…

Miller–Starchenko conjecture. Every $\mathcal I$-definable one-dimensional torsion-free ordered group is $\mathcal I$-definably isomorphic to either $(I,0,+,<)$ or…

Countability necessity conjecture. Neither the countability of $I$ nor the countability of the languages can be dispensed with without additional hypotheses.

Let $X\subseteq\mathbb R^n$ be definable in a sharply o-minimal structure with sharp derivatives, and let $k=0,\ldots,n-\dim X-1$. For $g,h\in\mathbb N$, let…

Let $X$ be a set definable in a sharply o-minimal structure, and let $V$ range over algebraic varieties over $mathbb Q$ of dimension $k<\operatorname{codim} X$. Set…

Let $\cM_{\rm cs}$ be the complex-structure moduli space of a Calabi–Yau manifold, and let its period integrals be the functions obtained by integrating the relevant Calabi–Yau for…

Fix a spacetime dimension $d$ and a finite energy cutoff $\Lambda>0$ in $d$-dimensional Planck units. For each EFT, let $(\cF_{\rm EFT},\cD_{\rm EFT})$ denote the format and degree…

Let an EFT be an effective field theory compatible with quantum gravity. Its tame complexity is a pair of integers $(\cF_{\rm EFT},\cD_{\rm EFT})$ measuring the format and degree o…

Beyond-semi-algebraicity conjecture. A variant of the learning-rate gradient-stability theorem should hold without the semi-algebraic assumption, using Baire category arguments.

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.