8 problems

Factorization conjecture. If $\theta$ does not factor in this way, then $W(S_\theta^1)$ and $W(S_\theta^2)$ are open. If $\theta$ does factor in this way, then $W(S_\theta^1)$ and…

Let $\mathbb{B}^n$ be the unit ball in $\mathbb{C}^n$, and let an inner function on $\mathbb{B}^n$ mean a bounded holomorphic function whose radial boundary values have modulus one…

Let $\Omega\subset\mathbb R^n$ be an $(\epsilon,\delta)$ domain, meaning a domain satisfying the quantitative curve-connectivity condition introduced by Jones. Such domains are Jon…

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, let $\mathbb S_n$ be its boundary sphere, and let $\mathcal{Z}(p)$ denote the zero set of a polynomial $p$. A polynomial is cyc…

Bessmertnyĭ's representation conjecture. An appropriate weakening of the metric conditions for the Bessmertnyĭ operator pencil should lead to a representation for the most general…

Minimality conjecture. The Agler decomposition above is minimal in the sense of the cited definitions. This would identify the canonical Agler decomposition arising for matrix-valu…

Ahern–Sarason conjecture. Every positive log-integrable function can be written as $|f|^2$ for some $f\in A^2(\omega)$ such that

Let $\mathbb B^d$ be the unit ball in $\mathbb C^d$, and let $O^+$ denote the class of functions with positive real part on $\mathbb B^d$. An inner function is a function whose bou…