75 problems

Let $m\in\mathbb N_+$, let ${\mathbb C}^{[m]}={\mathbb C}[X_1,\ldots,X_m]$, and let … be the Laplace operator. Let $f\in{\mathbb C}^{[m]}$ be homogeneous of degree $4$. Vanishing C…

Let $\mathbb{C}[z]$ be a polynomial ring and let $\mathbb{D}[z]$ be the set of differential operators $\Phi$ of $\mathbb{C}[z]$ of the form … for some $h(z)\in\mathbb{C}[z]$ and…

Let $X$ and $Y$ be independent non-centered normal random variables, and let $W=XY$. A second-order Stein operator for $W$ is a differential operator acting on smooth functions wit…

Monotonicity conjecture. If $T$ is not monotone, then $T$ is not hyperbolicity preserving.

Let $A={\mathbb C}[\xi,z]$, where $\xi=(\xi_1,\ldots,\xi_m)$ and $z=(z_1,\ldots,z_m)$ are two sets of commuting indeterminates. For each $i$, define…

Let $u_r$ be the symmetric homogeneous polynomial coordinates in $N$ variables constructed in the main theorem. For a partition $\sigma\vdash r$ and a differential operator of orde…

Let $z=(z_1,z_2,\dots,z_n)$, let $D_i=\frac{\partial}{\partial z_i}$, and let $\Delta_n=\sum_{i=1}^n D_i^2$ be the Laplace operator on the polynomial algebra…

Let $\mathrm{Dif}_1$ denote the Lie algebra of differential operators of order at most one on the circle, and let $\Psi_3,\widetilde\Psi_5,\widetilde\Psi_7,\widetilde\Psi_9,\dots$…

Let $G$ be an abstract completely continuous operator, let $\Lambda:=\left\{\lambda_j\right\}$ be the set of its eigenvalues, and fix $\delta>0$. For each $\lambda$, define the circle…

Let $M$ be a compact complex manifold, let $obreak\lambda$ be a line bundle on $M$, and let $\boldsymbol1_\lambda$ denote its identity differential operator. Write…

Let $S$ be an affine semigroup with standard expression, and let $S_2$ be the semigroup appearing in the standard expressions of $S$ and $S_2$ referenced in the source. Write…

The weighted derivative-substitution conjecture. This mapping determines a map

The geometric Langlands correspondence. There exists a canonical equivalence of categories

Let a boundary-value problem be subject to the estimate referenced in the source as … . The conjecture proposes a general regularity consequence of polynomial resolvent growth, ext…

Andersen's scalar-operator conjecture. If

Let $V$ be an $n$-dimensional vector space over $\Bbbk$, let $G=\operatorname{GL}(V)$ with Lie algebra $\mathfrak g=\mathfrak{gl}(V)$, and let $P\subset G$ be the parabolic subgrou…

Let $\mathrm{Vect}(\mathbb R^n)$ be the Lie algebra of vector fields on $\mathbb R^n$, let $\Omega(\mathbb R^n)$ denote the space of differential forms, and let…

Equivalence conjecture. Any stack of $\mathbb{C}_X$-twisted $\mathcal{D}_X$-modules is $\mathbb{C}$-equivalent to a stack of the form

Common-orbit conjecture. If $P_{1},P_{2}\in\mathcal{R}_{n}$, then there exist differential operators $\Lambda_{1},\Lambda_{2}\in\mathcal{S}$ such that

Let $k$ be any field, and let $R\hookrightarrow S$ be an inclusion of $k$-algebras that splits as a map of $R$-modules. Simplicity descent conjecture. If $D_k(S)$ is a simple ring,…

Let $k$ be a field, let $W$ be a finite-dimensional $k$-representation of a linearly reductive group $G$, and let $R=S(W)^G$ be the ring of invariants in the symmetric algebra…

Local regularity conjecture. Under these assumptions,

Let $\mathrm{osp}(1|2)$ act on the space ${\cal D}_{\lambda,\mu}$ of differential operators from tensor densities of weight $\lambda$ to tensor densities of weight $\mu$, and let…

Eventual real-rootedness conjecture. For all sufficiently large $n$, every zero of $q_n^{u,v}$ is real.

Weak Nakai conjecture. If $der_k^q(A)=Der_k^q(A)$ for each integer $q\geq 1$, then $A$ is regular.