22 problems

Uniqueness conjecture for simply connected quadrature domains. Simply connected quadrature domains are uniquely determined by their quadrature function, up to conformal radius.

Let $\Omega$ be a simply connected domain, let $F:\Omega\rightarrow\mathbb{D}$ be a biholomorphic map, and let $p\in\mathbb{R}$. Brennan conjectured that … for every…

Let $\Omega\subset\mathbb{C}$ be an extremal domain for analytic content, meaning that it is a bounded domain with real analytic boundary and … An annulus is a domain conformally e…

Let $K\subset\mathbb{C}$ be compact, let $R(K)$ be the uniform closure on $K$ of rational functions with poles outside $K$, and define its analytic content by … For a bounded domai…

Let $S_i$ and $S_e$ denote the interior and exterior shape factors, respectively, and let assumption A be the condition that each boundary section possessing a given boundary condi…

Let a domain be a connected open subset of the Riemann sphere, and let a circle domain be a domain whose boundary components are points or circles. Existing proofs of Koebe's conje…

The stationary Hastings–Levitov process with parameter $\alpha$, denoted $\operatorname{SHL}(\alpha)$, is the stationary version of the Hastings–Levitov model on the upper half-pla…

Let $L_0$ be a longest member of a finite collection of pairwise disjoint open arcs on the unit circle, each of Euclidean length less than $\pi$. In the Poincaré disk model, remove…

Let $M>0$ be a positive real number. Consider exterior quadrilaterals with prescribed angles and vertices $A_1=1$, $A_2=0$, $A_3$, and $A_4$, and let their conformal modulus be $M$…

Let $\Omega\subset\mathbb{C}$ be a finitely connected domain with disjoint boundary components $\gamma_0,\gamma_1,\ldots,\gamma_n$, where $\gamma_0=\partial\mathbb{D}$ and the rema…

Let $Z_t=X_t+iY_t$, $t\geq 0$, be standard planar Brownian motion starting at the origin, and let $T^D$ denote its first exit time from a domain $D$ containing $0$. Let…

Boundary-oscillation conjecture. The number $N$ increases at most linearly with $n$; the specific proposed bound is

Suppose $F:\mathbb D\to\Omega$ is a conformal map of the unit disk onto a Jordan–Lipschitz domain $\Omega$, and let $L(\Omega)$ denote the perimeter of $\Omega$. For $\alpha>0$, th…

Exactness conjecture. These lower estimates are exact: for sufficiently large and sufficiently small moduli, the number of spherical quadrilaterals with the prescribed angles and m…

Let $delta$ and $sigma$ be any pair of parameters, let $Dc$ be the domain under consideration, let $K\textsubscript{}$ be a hull contained in $Dc$, and let…

Let $t_1,t_2$ satisfy … For a real parameter $\lambda$, let $f$ be the solution of … normalized by $J_f(0)=(0,1,0)$, and call $f$ a gear mapping when its image is a gear-shaped dom…

Two-gear conjecture. When $\beta$ is fixed, there is a threshold $t_\beta\in(0,\pi/2)$ such that, for each $t\in(0,t_\beta)$, exactly two values of $\gamma$ satisfy

Let $\mathbb{D}$ be the unit disk, let $\mathcal{RU}_n$ denote the rational functions of degree at most $n$ with no poles in the closed disk and univalent in $\mathbb{D}$, and let…

Square-lattice extremality conjecture. The square lattice should be extremal, and for the extremal function, $f(0)$ should be a center of the complement.

Let $\Omega$ and $\Omega^*$ be bounded doubly connected domains in $\mathbb{C}$, and suppose that there is an energy-minimal diffeomorphism … Write $\operatorname{Mod}\Omega$ and…

Let $\Omega^*$ be a bounded doubly connected domain in $\mathbb{C}$, and let $\mathsf{E}(\tau,\Omega^*)$ denote the infimum of the Dirichlet energy over deformations from a doubly…

Bañuelos–Carroll conjecture. The extremal domain is the same for all the three problems.