234 problems

For mean-zero solutions of advection–diffusion equations on the two-dimensional torus, derive explicit constructive lower bounds that rule out excessively fast mixing in three regi…

For the stochastic heat equation $\partial_tu=\tfrac12\partial_x^2u+\delta_0(u)+\dot W$ on $[0,1]$ with Dirichlet boundary conditions, does its Markov semigroup have at most one in…

Under the paper’s smoothness, positivity, Schwartz-decay, and score assumptions, are all steady solutions of the Coulomb Vlasov–Maxwell–Landau system necessarily spatially uniform…

Bombieri–Dwork conjecture. The minimal differential equation of a $G$-function should come from geometry.

Ablowitz–Ramani–Segur conjecture. There exists a connection between nonlinear integrable PDEs and nonlinear ODEs with the Painlevé property such that any ODE emerging from the redu…

Let the linear multistep neural network loss have infinitely many global minimizers because the governing-function components $f_n$ are underdetermined by the training equations, a…

A differential equation is globally nilpotent if its $p$-curvature is nilpotent for almost all primes $p$. A Gaussian hypergeometric equation is a differential equation associated…

Let a bicycle segment have unit length, and let its front track be a closed convex curve bounding an area greater than $\pi$. The associated monodromy is the Möbius transformation…

Bank–Laine conjecture. If $\rho(A)\notin\mathbb{N}$, then

Vershik's conjecture. Despite examples in which such tensor fields have been defined and computed, there is probably no general definition of a curvature tensor for non-holonomic d…

Let $\alpha$ be a nonnegative integer, and let $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ be the generalized Laguerre polynomials with parameters $M$ and $N$. Differential-equation c…

Let $\mathfrak g$ be a Lie algebra with Cartan subalgebra $\mathfrak h$, and consider the De Concini–Procesi connection on the set of regular elements of $\mathfrak h$. Its horizon…

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…

Henn's conjecture. There always exists a rotation into a basis $\boldsymbol{J}(\boldsymbol{x},\varepsilon)$ in which the master integrals satisfy an $\varepsilon$-factorized differ…

Let $V$ be a $d$-dimensional vector space of real analytic functions on an interval $I\subseteq\mathbb{R}$, and let $\operatorname{Wr}(V)$ denote its Wronskian. Say that $V$ is dis…

Consider the simplest equation … obtained from Hayman's equation by setting $a_2=a_3=0$, where $a_1$, $b_1$, $b_2$, $b_3$, and $b_4$ are polynomials in $z$ and $a_1\not\equiv0$. Ha…

Let $D_M$ be the differential operator defining the homogeneous Heun equation for $\Pi_M$, and let $d$ be the parameter specified by the relation referenced in the source. Polynomi…

Let a linear differential or $q$-difference system be given, with its Malgrange groupoid acting on the fibers of the associated system. Casale–Roques conjecture. For linear $(q)$-d…

Let $\mathfrak d(z)$ be a degenerate Lamé operator, meaning that its leading coefficient satisfies $\deg Q_k<k+r$, and let $N_0$ be any positive integer. Unbounded-root conjecture.…

Right-divisibility conjecture. This property holds for all values of $n$. The conjecture proposes that the observed nested factorization of the differential operators annihilating…

The Heun differential equation is a Fuchsian differential equation, and a contour integral means an integral representation of its solution in terms of a simpler elementary functio…

The paper considers nonlinear differential equations, infinite-order linear differential equations, finite-difference equations, and q-differential equations, together with their q…

A generalized Briot–Bouquet equation is an equation … where $F$ is a polynomial in two variables and $k$ is an even or odd integer; $W$ denotes the class of functions under conside…

Gran's conjecture. The polynomials $\{P_n^{\alpha,\alpha,M,M}(x)\}_{n=0}^{\infty}$ satisfy a differential equation of the form with coefficients $\{c_i(x)\}_{i=0}^{\infty}$ given b…

Consider a classical confluenting family of hypergeometric equations, with a nonperturbed equation having an irregular singularity. Let the perturbed equation have appropriately no…