70 problems

For every $n\ge 1$, real coefficients $c_1,\ldots,c_n$, and polynomials $P_1,\ldots,P_n\in\mathbb{R}[x]$ satisfying $\deg P_k\le 2$ and $P_k(x)>0$ for all $x\in\mathbb{R}$, if…

Let $R(t,w)\in\mathbb{C}(t,w)$ be rational in $t$ and $w$, and assume that $R$ is not a polynomial in $w$ of degree at most $2$. If a function $f$ meromorphic on $\mathbb{C}$ satis…

Let $(k,\delta)$ be a differential field of characteristic zero with field of constants $C$. Consider a first-order differential equation over $k$, and, respectively, an autonomous…

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…

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…

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…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and set $p_0(x)=x$. Consider the formal expansion with recursively defined polynomial coefficients $p_k$, and l…

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.…

Let $N\geq 3$, let $A_j(t)$, for $j=2,3,\ldots,N-1$, be polynomial functions, and consider the differential equation … A solution is a center at the origin if nearby solutions have…

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…

Darboux characterization conjecture. Any polynomial solutions to the bilinear hypergeometric equations of propositions and are given by … , respectively.

WKB Stokes-period conjecture. The support $\Gamma_{Q_k}$ is

Mother-body conjecture. The root-counting measures of $Sp_n$ converge weakly to a positive measure $\mu_{Q_k}$ depending only on $Q_k$. Its support is a finite planar tree…

Spectral-tree and mother-body conjecture. The measures $\mu_n$ converge weakly to a probability measure $\mu_Q$ depending only on $Q_k$. Its support is a finite planar tree contain…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and let $p_0(x)=x$. Assume that the formal expansion has polynomial coefficients $p_k$ as in the polynomial sol…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ be a polynomial satisfying … and set $p_0(x)=x$. Let $p_k$ denote polynomials whose coefficients are determined recursively by the formal expansion…

Yang–Li's conjecture. The equation has no transcendental meromorphic solution when $P_1/P_3$ is a non-zero constant and $P_2/P_3$ is not the square of any rational function.

Let $P(t)$ be a non-degenerate quadratic polynomial, and consider the differential equation … An integral solution is a solution at $t=0$ of the form … with $y_n\in\mathbb{Z}$ for…

Let $k$ be a positive integer, let $a,b\in\mathbb{C}$ with $a\neq b$, let $a_i\in\mathbb{C}$ for $i=0,1,\ldots,k$ with $a_k=1$, and let $\varphi$ be an entire function. For a funct…

Gao's conjecture. This equation has no entire solution.

Gao's conjecture. If $9ab^2=-4$, then every entire solution is either

Let $\mathcal{C}_n$ be the vertex algebra under consideration, and let … be its character. An MLDE is a modular linear differential equation of the form described using Serre deriv…

Let $X/\mathbb{C}$ be a smooth projective curve, let $D\subset X$ be a reduced effective divisor, and let $(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega_X^1(\log D))$…

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $X$ be a smooth $R$-scheme, let $(\mathscr{E},\nabla)$ be a flat vector bundle on $X/R$, let…

Let $g\in \mathbb{Q}(z,y_0,\ldots,y_{n-1})$ be a rational function, let $t_0,\ldots,t_{n-1}\in\mathbb{Q}$ be initial values for which $g(0,t_0,\ldots,t_{n-1})$ is defined, and let…