9 problems

Compact-space conjecture. If $X$ contains neither a copy of $\beta\mathbb N$ nor a Cantor set, then $X$ has the CSWP.

Automorphism conjecture. Every algebra automorphism of $\mathcal{A}$ must be a composition operator induced by some disc automorphism.

Polynomial cluster value conjecture. A polynomial cluster value theorem holds for $\ell_1$ and for $\ell_1^n$ with $n\in\mathbb{N}$.

Let $\Omega\subset\mathbb{C}^2$ be a domain and let $z\in\partial\Omega$. The point $z$ is a local MDM point for $\Omega$ if there exists a ball $B$ centered at $z$ such that, for…

Let $X$ be a connected Riemann surface such that the Banach algebra $H^\infty(X)$ of bounded holomorphic functions on $X$ separates the points of $X$. Let ${\rm cl}\,X$ be the clos…

Let $H^\infty$ be the algebra of bounded holomorphic functions on the unit disk, and let $L^\infty$ denote the algebra of essentially bounded functions on the boundary. A Douglas a…

Let $M$ and $N$ be manifolds possessing a $C^k$-differentiable structure, and let $C^k(M)$ and $C^k(N)$ denote their algebras of $C^k$ functions. The Banach–Stone conjecture. $M$ a…

Quantitative stable-rank conjecture. There exist $h_1,h_2,g_1,g_2\in H^\infty_\mathbb{R}(\mathbb{D})$ such that

Let $H^\infty_\mathbb{R}(\mathbb{D})$ denote the algebra of bounded holomorphic functions on the unit disk that are real on the real interval. The Bass stable rank conjecture. The…