27 problems

Matzat's conjecture. The absolute differential Galois group of $F$ is the free proalgebraic group on a set of cardinality $|F|$.

Differential Galois integrability conjecture. The identity component of the differential Galois group of the $k$-th order variational equations is Abelian for every…

Grothendieck–Katz and Bombieri–Dwork conjectures. The Lie algebra of the differential Galois group of $D$ is minimal with the property that, for almost all primes $\mathfrak p$ of…

Non-integrability conjecture. The system is non-integrable for the case $2\xi=3(1-C)$; in particular, it does not admit an additional real meromorphic first integral functionally i…

The paper considers gradient-flow dynamics for general neural networks and the differential equations governing those dynamics, whose differential Galois group measures the algebra…

A generalised Frenkel–Gross $G$-connection is a connection on the multiplicative group with an irregular singularity at infinity of slope $1/h$ and a regular singularity at $0$. Th…

Let $T$ be a minimal-order telescoper for an integral … where $a$ is algebraic in $x$ and $h$. Finiteness conjecture. If the Galois group of $T$ is virtually solvable, then it is f…

The trivial-Galois-group conjecture. However large $\alpha$, $\beta$, and $r$ are, the Galois group of the associated third-order homogeneous equation is trivial in the category of…

Let $\alpha$, $\beta$, and $r$ be parameters satisfying … and suppose also that the explicitly computed range in the paper satisfies $\alpha<30$, $\beta<30$, and $r<15$. The nice-s…

Let $\lambda$ and $E$ be the parameters of the axisymmetric Bianchi IX cosmological model, and let $\gamma_1,\gamma_2\in\mathbb{C}$ be the constants appearing in the differential G…

Let a complex analytic Hamiltonian system be meromorphically completely integrable in a neighborhood of an integral curve $\Gamma$, and let its variational equations be expanded to…

Let $k$ be an algebraically closed field of characteristic zero, set $K=k(x)$ with derivation $d/dx$, and let $F_X$ be the abstract free group on a set $X$ of cardinality $|k|$. A…

Kovacic first-case integrability conjecture. If the corresponding partial differential equation is integrable via a Laplace transform and its transformed linear ordinary differenti…

The differential Galois group algorithm takes as input a linear differential equation over $C(t)$ and computes its differential Galois group using an algebraic-group approximation…

Conjecture. Under assumptions (H1) and (Hk), for some $k\geq 2$, the polynomial field has no elementary first integral.

The three-dimensional double pendulum is a dynamical system whose first integrals include its known integrals of motion. Non-integrability conjecture. The three-dimensional double…

Consider the higher-order linear differential operators in the paper's family, decomposed into factors $U_n$ in the displayed canonical forms. Here each $U_n$ is a self-adjoint dif…

A linear differential operator is an operator annihilating a diagonal of a rational function, and an irreducible factor is an irreducible differential-operator factor of such an op…

Let $\chi^{(n)}$ denote the $n$th Ising susceptibility, and consider the differential operators whose solutions include the multifold integrals defining these susceptibilities. The…

Adjoint-homomorphism conjecture. Every irreducible factor of $L$ should be homomorphic to its adjoint, possibly after passing to an algebraic extension.

Let $\mathrm{LVE}^m_{\phi_0}$ denote the linearized variational equation of order $m$ along the solution $\phi_0$, and let $\tilde{A}_{m+1,R}$ be the output of the reduction proced…

Katz's Lie-algebraic $p$-curvature conjecture. With this notation, $\mathfrak g$ is the smallest algebraic Lie subalgebra of $\operatorname{End}_{\mathcal O_X}V\otimes K(X)$ such t…

Let $n\in\mathbb N$ and $m\ge3$, and consider the algebraic Lamé equation … where $f(z)=4z^3-g_2z-g_3$. For prescribed $n$ and $m$, consider the conditions on $B$, $g_2$, and $g_3$…

Let … be the Hamiltonian of the minimally coupled scalar-field system. Assume $\Lambda\ne0$, let $n$ be an integer satisfying … and consider a generic energy level $E\ne0$. Integra…

Consider the Brioschi case under the assumptions of the subsection, with integer parameter $n$, parameter $k$, and energy level. Brioschi integrability conjecture. If the system is…