Question 1. In conformal geometry, there is a notion of “conformally removable” set meaning that the set is negligible concerning analytic continuation of conformal maps. Consider the image of a curve. This analytical property is related to the regularity of the curve. A countably infinite number of unions of conformally removable curves is not necessarily conformally removable. An interesting classes of curves to consider are the sample curves of an SLE. Using the Jones and Smirnov Criterion for simple curves, the SLE curves of parameter kappa are shown to be almost surely conformally removable for Kappa in (0,4), as shown in an article of Sheffield and Schramm. For a subset of the kappa, in the self-intersecting non-space filling regime (4,8), this question is partially answered. See recent article (arxiv:2302.10857) by Kavvadias and Miller and also that by Gaynne and Pfeffer.
I learnt this series of work from Lukas Schoug’s talk at EPFL and posed him a question that I now share. It is on the universality aspect of the conformal removability of SLE curves.
Conjecture: Let me denote the collection of $\kappa\in (0, 8)$ for which the sample paths of the SLE curves are conformally removable. Let us take a representation of the SLE curves with the stochastic Loewner equation.: $$\partial_t g_t = 2/( g_t -\kappa W(t))$$, $g_0(z)=z$ where $W(t)$ is a Brownian motion. Then there exists a set of full Brownian paths, for which the SLE curves are conformally removable for all $\kappa \in I$.