8 problems

A circle domain $\Omega\subset\widehat{\mathbb C}$ is a domain whose boundary components are points or circles. It is conformally rigid if every conformal map of…

Let a Jordan curve have a conformal welding homeomorphism, and let a curve be conformally removable when every homeomorphism of the Riemann sphere conformal off it is a Möbius tran…

Let $\textrm{CH}(\gamma)$ be the set of homeomorphisms of the Riemann sphere that are conformal off a curve $c(cgamma)$. A curve is conformally removable if every element of…

For each $\underline{\kappa}\in(0,4)$, let $\rho_c=\rho_c(\underline{\kappa})$ lie in $(-2-\underline{\kappa}/2,-2)\cap[\underline{\kappa}/2-4,-2)$, and let $\eta$ be an SLE…

For $\kappa \in (4,8)$, let $\eta$ be an SLE$_\kappa$ curve from $0$ to $\infty$ in $\mathbb{H}$. The adjacency graph of the connected components of $\mathbb{H} \setminus \eta$ has…

Let $C$ be the middle-thirds Cantor set. A planar compact set $K$ is said to contain a homeomorphic copy of $C\times[0,1]$ if there is a subset of $K$ homeomorphic to…

He–Schramm's removability conjecture. If every Cantor set contained in $\partial\Omega$ is conformally removable, then $\partial\Omega$ is conformally removable.

Local removability conjecture. A compact set is conformally removable if and only if it is locally conformally removable.