85 problems

Let $Q(x,y,t)$ be the generating series of a lattice walk in the quarter plane with small step set, and associate to the walk the group of birational automorphisms of…

Let $(K,\partial)$ be a differential field of characteristic zero, and let $u_1,\ldots,u_n\in K\lbrace x_1,\ldots,x_n\rbrace$. Let … Let $\Sigma_1$ be an irreducible component of…

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 $\langle \mathbb K,\bar \delta\rangle$ be a model of $T^{\bar \delta,?}_g$. For $p\in S_{L^{\bar \delta}}^n(\mathbb K)$, choose a realization $\bar a$ and let $\tilde p$ be the…

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…

Let $P$ be an algebraic differential expression in $\zeta$ whose coefficients are polynomials in the Euler gamma function $\Gamma$ and its derivatives; equivalently, $P$ is a polyn…

Let $\mathbb{H}$ be the complex upper half-plane and let $J=(j,j',j”):\mathbb{H}\to\mathbb{C}^3$, extended coordinatewise to $\mathbb{H}^n$. A subset $U\subseteq\mathbb{H}^n$ is…

Let $y1, ldots,yn$ be solutions of a fixed Painlevé equation whose complex parameters are generic. Write $operatorname{td}(L/K)$ for the transcendence degree of a field extension…

Algebraic-to-Ritt conjecture. If one can solve the problem of computing a characteristic decomposition of a radical differential ideal from generators completely algebraically—by f…

Let $\zeta(s)$ denote the Riemann zeta function. An algebraic differential equation with rational coefficients is an equation of the form … where $n$ is a nonnegative integer and…

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 ${\rm P}_{\rm II}^{(n)}$ denote the $n$th equation in the second Painlevé hierarchy, with parameter $\alpha_n$. Irreducibility means that, for generic values of $\alpha_n$, no…

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…

Let $(K,\partial)$ be a differential field of characteristic zero. Let $u_1,\ldots,u_i\in K\lbrace x_1,\ldots,x_n\rbrace$, and suppose that the differential ideal…

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

The limiting ratio conjecture. Except for countably many values of $a$,

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 $\mathfrak C$ 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 $\mathbb{Q}\subseteq\mathcal{C}$ and let $\mathcal{R}_\mathrm{T}$ be freely generated by $\{a_i\mid i\in I\}\subseteq\mathcal{R}_\mathrm{T}$ as a $\mathcal{C}$-module. Let…

Let $\mathbb{Q}\subseteq\mathcal{C}$, let $\mathcal{R}_\mathrm{T}$ be generated as a $\mathcal{C}$-module by $\{a_i\mid i\in I\}\subseteq\mathcal{R}_\mathrm{T}$, and let $\le$ be a…

For $n,m,l\ge 1$, let $c_{n,m}$ denote the constants arising from the cyclic case, and let $G_d$ be the set of relations defined from the displayed relations for…

Let $(\mathcal{R},{\partial})$ be a commutative differential ring with constants $\mathcal{C}$ and quasi-integration ${{\mathrm{Q}}}$, and suppose that $\mathcal{R}_{{\mathrm{T}}}$…

Irreducibility conjecture. The operator … Sigma($Gamma_{P,Q})$, meaning that it is irreducible for generic values of … mu…