14 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 $\boldsymbol z=(z_1,ots,z_d)$ and let $mathcal H^{(lambda)}(mathbb B_d)$ be the weighted Bergman space on the Euclidean ball. For a homogeneous ideal…

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

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 $\mathcal{M}$ be an infinitely generated submodule of $H^2(\mathbb D^2)$, and let $\mathcal{M}^\perp$ be its associated quotient module. A quotient module is essentially normal…

Arveson–Douglas conjecture. The quotient submodule $M_V^\bot$ is $p$-essentially normal for every $p>\dim V$.

Let $V\subset\mathbb{B}^n$ be an affine homogeneous algebraic variety, meaning that there exist finitely many holomorphic homogeneous polynomials $q_1,\ldots,q_m$ such that … Let…

Let $\mathbb{B}^n$ be the unit ball and let $M$ be a graded submodule of the Drury–Arveson module. A Hilbert submodule is $p$-essentially normal if all cross-commutators belong to…

Let $I \subset \mathbb{C}[z_1,\ldots,z_d]$ be a homogeneous ideal of infinite co-dimension, let … and set $I^\bot=H^2_d\ominus I$. Define $S^I=(S_1,\ldots,S_d)$ on $I^\bot$ by … Th…

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 $p\in\mathbb{C}[z_1,\ldots,z_n]$, let $Z(p)$ be its zero set, and let $\mathscr{Q}_p$ be the associated quotient module with operators $Q_f$. Consider the commutators of these…

Let $p\in\mathbb{C}[z_1,\ldots,z_n]$, let $\mathscr{Q}_p$ be the relevant quotient Hilbert module, and let $Q_f$ denote the operator induced by a polynomial or multiplier $f$ on…

Let $p\in\mathbb{C}[z_1,\ldots,z_n]$ be a polynomial, let $X_p$ be the associated compact subset of $\partial\mathbb{B}_n$, and let $[p]\in K_1(X_p)$ be the odd K-homology class de…