41 problems

Let $\mathcal W$ be a web, and let its abelian relations mean the relations among the defining foliations obtained by summing closed one-forms along the leaves. A Liouvillian exten…

Let $(K,\partial)$ be a differential field of characteristic zero. Let $A$ be a differentially finitely generated differential $K$-domain, let $f\in A$ be a non-unit, and let $P$ b…

Dimension Conjecture. If $m<n$, then every irreducible component $\Sigma_1$ of $\Sigma$ satisfies

Jacobi Bound Conjecture. The absolute dimension of $\Sigma_1$ satisfies

Let $\operatorname{Av}(4231,4123,4312)$ be the permutation class avoiding $4231$, $4123$, and $4312$, and let $F(x)=\sum_{n\geq 0}a_nx^n$ be its ordinary generating function. Non-di…

Let $(C_n)_{n\in\mathbb{N}}$ be the Catalan numbers, and let $d\in\mathbb{N}\!\setminus\!\{0\}$. A sequence is D-algebraic of order $2$ and degree $2d$ if it satisfies a polynomial…

Let $\Lambda$ be a partial commutative monoid generated by a tuple $\bar\delta$; equivalently, let $\Lambda$ be a trace monoid (Cartier–Foata monoid). Let $T^\Lambda$ denote the co…

Let $\mathfrak C$ carry a topology $\tau$ satisfying suitable conditions, and let $\tau_{\bar \delta}$ be the topology induced by $x\mapsto\operatorname{Jet}(x)$ into…

Let $\langle \mathfrak C,\bar \delta\rangle$ be a monster model of $T^{\bar \delta}_g$. For an $L^{\bar \delta}$-definable set $X$, let $\omega_X$ be the supremum of the relevant K…

Let $T$ be an algebraically bounded theory, and let $T^{\bar \delta,?}_g$ be its theory of generic derivations. Elimination of imaginaries modulo $T^{eq}$ means that imaginaries in…

The $\omega$-categoricity conjecture. The set $\mathcal{X}$ is $\omega$-categorical if and only if there is a positive integer $k$ such that, for every such tuple $x_1,\ldots,x_n$,…

Let $X$ be a symmetrically self-similar graph with bounded geometry, and let its Green's function be the generating function for walks on $X$. A star is a graph consisting of finit…

Maximal Hardy-field conjecture. If $H$ is maximal, then there is no $y\in\mathcal{C}^1$ such that

Jin's conjecture. The following are equivalent:

Let $\Delta=\{\delta_1,\dots,\delta_m\}$ be a set of $m$ commuting derivations in characteristic zero, and let $G$ be a strongly connected differential-algebraic group of $\Delta$-…

General type characterization conjecture. A first order differential equation over $k$ (respectively, an autonomous differential equation over $C$) is not of general type if and on…

Let $n\geq 1$, let $\mathrm{Y}_2(1)$ be the target variety for the tuple-valued function $J$, and let $\mathrm{E}_J^n$ be its graph over $\mathbb{H}^n$. A variety…

Let $f(x)$ be a generic differential equation of fixed order $h>1$ and fixed degree $d>1$, and let $a$ and $b$ be two solutions of $f$. Geometric triviality conjecture. Any two sol…

Let $f(x)$ be a generic differential polynomial of fixed order $h>1$ and fixed degree $d>1$, meaning that its coefficients are independent differentially transcendental elements an…

Let $h$ be an entire function, let $\Gamma$ denote the Euler gamma function, and let $\mathcal{M}_0$ be the field of meromorphic functions $y$ satisfying $T(r,y)=o(r)$ as…

Let $J:\operatorname{\mathbb{H}}\to\operatorname{\mathbb{C}}^3$ be defined by … and extend it coordinatewise to $\operatorname{\mathbb{H}}^n$, with graph…

Let $K$ be a field of characteristic zero and let $K[x]=K[x_1,\ldots,x_n]$. Consider the derivation … Let $\operatorname{Aut}(K[x])_D$ denote the subgroup of polynomial automorphis…

The converse conjecture. The derivation $D$ is simple if and only if

The trivial-centralizer conjecture. If $D$ is a simple Shamsuddin derivation of $K[x]$, then

Let $C$ be an algebraically closed field with $\operatorname{td}(C/\mathbb{Q})=\aleph_0$, let $\mathfrak{C}$ be the class of $E_j$-fields with constant field $C$, and let $U$ be it…