168 problems

Let $\mathcal D(W)$ be the algebra of differential operators associated with the weight matrix $W$, and let $I$, $L_1$, and $L_3$ be the operators used in the paper. Full-algebra g…

Let $A$ be a $mathsf k$-algebra, and let $\mathcal{D}(A)$ be its ring of differential operators. In characteristic zero, for an affine regular domain, let $\Delta(A)$ denote the su…

Nakai conjecture. If

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 $H$ be the Fuchsian operator … where $m_1,\ldots,m_M$ are positive integers, $\alpha+\beta-\gamma-\delta+\sum_{k=1}^M m_k+1=0$, and $a_1,\ldots,a_M$ are apparent singularities…

Let $K$ be a field of characteristic zero, let a reductive group act linearly on a polynomial ring over $K$, and let $R$ be the corresponding ring of invariants. The ring of $K$-li…

Zhao's Image Conjecture. If $f\in\mathbb C[\zeta,x]$ satisfies $f^m\in\operatorname{Im}D$ for every positive integer $m$, then for each $g\in\mathbb C[\zeta,x]$,

Let $\Lambda\in\mathbb C[\partial_1,\ldots,\partial_n]$ be a differential operator with constant coefficients, and let $P,Q\in\mathbb C[x]$. Zhao's Generalized Vanishing Conjecture…

A Hertz potential is a solution of a higher-spin field equation obtained by applying linear differential operators to a scalar Debye potential, which solves a suitable scalar wave-…

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…

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 $g$ be the genus of a hyperelliptic sigma-function, let $\mathscr{P}=\mathbb{Q}[\Lambda]$, and let $\mathscr{Q}=\mathscr{P}[D_1,\dots,D_g]$ be the graded ring of linear differe…

Let $K$ be a field of characteristic zero, let ${\mathcal A}$ be a generic hyperplane arrangement in $K^n$ composed of $r$ hyperplanes, and let $D^{(m)}({\mathcal A})$ be the $S$-m…

Feigin–Losev–Shoikhet's integral conjecture. The FLS-functional satisfies

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

The paper considers the construction of the operator ${\bf S}$ and the associated quantization theorem for a complex simple Lie algebra ${\mathfrak g}$, with the preceding treatmen…

The paper considers bispectral manifolds of polynomial Darboux transformations of Bessel and Airy planes, together with the involutions $a$ and $b$; in the Airy case, the relevant…

Bispectral operator conjecture. All bispectral scalar differential operators can be obtained by the constructions described in this paper, namely through the indicated highest weig…

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…

Lifting cocycle decomposition conjecture. The two summands are noncohomologous cocycles, and every nontrivial linear combination of them is not cohomologous to zero. The claim exte…

Feigin's degeneration conjecture. This spectral sequence degenerates at $E_2$. The conjecture is part of Feigin's proposed description of the cohomology of $\mathrm{Dif}_1$; the so…

Let $M$ be a compact complex manifold, let $obreaklambda$ be a line bundle on $M$, and let $boldsymbol1lambda$ denote its identity differential operator. Write…

Let $\operatorname{Dif}_n$ denote the algebra of differential operators in $n$ variables, and let ${\mathfrak{gl}}_\infty^\mathrm{fin}(\operatorname{Dif}_n)$ be the stable Lie alge…

Let $D$ and $E$ be the commuting differential operators acting on the sequence of matrix-valued orthogonal polynomials $\{P_w\}$ constructed in Theorem, with each $P_w$ a simultane…

Let $T$ be a degenerate exactly-solvable differential operator of order $k$, and let $p_n$ be its degree-$n$ eigenpolynomial. Write $r_n$ for the largest modulus of the roots of…