17 problems

Let $E$ be a positive-area set with no interior, and let $\operatorname{CH}(E)$ denote the homeomorphisms of the Riemann sphere that are conformal off $E$. The zero-area image conj…

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…

Vanishing energy-gap conjecture. If

Let $\alpha\in\mathbb{T}$ be a parameter, and let $GCS_n(\alpha)$ denote the generalized cylinder set associated with the first $n+1$ letters of the itinerary $I(\alpha)$. Say that…

Beltrami-welding conjecture. The above Beltrami welding yields an SLE loop measure and thus an independent second construction of the SLE loop in Theorem 1.3 of Ang (2021).

For $\ell>0$, let $\overline{\mathrm{MD}_{0,2}^n}(\cdot,[\ell n^{1/2}])^\#$ and $\underline{\mathrm{MD}_{0,1}^n}([\ell n^{1/2}])^\#$ be the probability measures on the two types of…

Fix $W_1\geq\frac{\gamma^2}{2}$ and $W>2$. Let $\mathrm{Bubble}_{\mathbb H}(p)$ denote the space of bubbles in $\mathbb H$ rooted at $p$, and let…

Consider an annulus represented as a cylinder of length $\pi\tau$ and radius $1$. Set $q=e^{-2\pi\tau}$, and let $\mathcal{Z}_{\mathrm{Annulus}}$ denote the no-insertion-point part…

Fix a permutation $p$ of the natural numbers, and let $\sigma_N$ be the associated finite factorization maps with limit $\sigma(p,w)$, where $w=(w_n)$ belongs to the product of uni…

Let $\mu_c$ be the conjectural family of measures with values in $\operatorname{Det}^c$, let $\nu_c$ be its induced measure on welding homeomorphisms, and let $a$ be the diagonal p…

Let $\mu_0$ be Werner's measure on self-avoiding loops, let $\nu_0$ be its induced measure on welding homeomorphisms $\sigma$, and let $W$ be the welding map. A quasicircle is a Jo…

Let $\nu_0$ be the probability measure induced by Werner's loop measure on the triangular factorization parameter $a$ of a welding homeomorphism, and let $\beta_0<\frac{5\pi^2}{4}$…

Atomicity conjecture. A corresponding limit of the normalized measures is nontrivial for $\beta\geq\beta_c$ and is atomic for $\beta>\beta_c$, thereby producing a discontinuous map…

Local SLE conjecture. The curves $\gamma$ are locally related to, or locally look like, $\operatorname{SLE}(\kappa)$ for $\kappa<4$; in the more specific formulation, they locally…

Let $\tau_\varepsilon$ be the regularized random measure used to construct the circle homeomorphism, and define … Extend $h_\varepsilon$ periodically over $\mathbb{R}$ and use it t…

The construction produces random closed curves by conformally welding random homeomorphisms of the unit circle. For $\beta<\beta_c$ (in the paper's high-temperature regime,…