12 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…

A set is conformally removable if every homeomorphism of the sphere conformal off the set is Möbius. A set is CNED (countably negligible for extremal distance) if it satisfies the…

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…

Let $\rho_c:(0,4)\to\mathbb{R}$ be the critical threshold from the source's SLE$_{\underline{\kappa}}(\rho)$ phase-transition conjecture. Critical-threshold behavior conjecture. Th…

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…

Let $\kappa_c\in(4,8)$ be the critical parameter from the source's SLE conjecture on conformal removability and bubble connectivity, and let scenario denote the second scenario in…

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 $\Gamma$ be the graph of one-dimensional Brownian motion. A compact set $K\subset\mathbb{C}$ is conformally removable if every homeomorphism of $\mathbb{C}$ that is conformal o…

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.