17 problems

Let the $d$-shift module be the Hilbert module over the polynomial ring on the complex unit ball, and let a submodule be homogeneous when it is generated by homogeneous elements. A…

Geometric Arveson–Douglas conjecture. The quotient module $Q$ is $p$-essentially normal for every $p>\dim M$.

Let $I\subseteq\mathbb{C}[\mathbf{z}]$ be an ideal, let $\mathcal{Z}$ be its zero variety, and let $\mathcal{S}$ be the submodule obtained by taking the closure of $I$ in…

Let $\mathcal{M}$ be a finite-rank, quasi-free, $p$-reductive Hilbert module over $A(\Omega)$, and let $\mathcal{S}$ be a submodule such that $\mathcal{S}^{\bot}$ is spanned by gen…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$, and consider submodules of $B^2(\Omega)\otimes\mathbb{C}^k$. Finite-multiplicity p-reductivity conjecture…

Let $\Omega$ be a bounded, strongly pseudoconvex domain in $\mathbb{C}^m$ with smooth boundary, let $B^2(\Omega)$ be its Bergman space, and let $\mathcal{S}\subseteq B^2(\Omega)$ b…

Conjecture A. The quotient $H_V$ is essentially normal.

Let ${\mathcal{S}}$ be an essentially reductive submodule of ${\mathcal{M}}_\Lambda\otimes {\mathbb{C}}^k$, and let $[{\mathcal{S}}^\bot]$ denote the associated element of…

Direct-summand conjecture. The modules $\mathcal{R}_1$ and $\mathcal{R}_2$ are quasi-free of ranks $k_1$ and $k_2$, respectively, and

Let $V$ be a one-dimensional smooth holomorphic zero variety of an open neighborhood of the closed unit ball $\overline{\mathbb B}$, transversal to the boundary…

Let $\mathcal{H}$ be a left-Hilbert module that is finite-dimensional as a vector space. Let $X$ be a measure space, let $\mathbb{F}$ be the underlying scalar field, and let…

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of distinct Clifford numbers in $B^{m+1}$, and let $T_{a_1,\ldots,a_n}$ be the functions obtained by the Gram–Schmidt orthogonalization p…

Let $K$ be a finite dimensional Hilbert space, let $H^2_d$ denote Drury–Arveson space, and let $L\subseteq H^2_d\otimes K$ be a graded submodule. Write $\dim(L)$ for the dimension…

Let $T=[T_1,dots,T_d]$ be a pure graded finite-rank $d$-contraction, and let ${\mathfrak S}_p$ denote the Schatten $p$-class of compact operators whose singular numbers lie in…

Let $\mathcal M$ be a Hilbert module of holomorphic functions on a bounded open connected subset $\Omega$ of $\mathbb C^m$, and let $\mathcal I\subseteq\mathbb C[\underline{z}]$ be…

Douglas's refinement of Arveson's conjecture. The quotient module is $p$-essentially reductive for

Finite-codimensional submodule conjecture. Such submodules can occur only for $\vphantom{S}\text{Šilov}$ modules.