17 problems

Let an annulus have analytically parametrized boundary components. Cutting a hole in it, analytically parametrizing the new boundary component, and sewing an analytically parametri…

Subtraction conjecture. The subtractions in the corresponding evolution equation take this loss into account.

The chordal SLE trace from $0$ to $\infty$ in the upper half-plane $\mathbb{H}$ is denoted by $\gamma$. The inversion $z\mapsto -1/z$ maps $\mathbb{H}$ to itself. SLE reversibility…

Let $f$ and $g$ be continuous functions on $[a,b]$, with $-\infty<a<b<+\infty$ and $f(x)<g(x)$ for all $x\transport?$. Let $\text{\boldmath Q}=(Q;z_1,z_2,z_3,z_4)$ be the quadrilat…

Let $f_t$ be the deterministic conformal map associated with the Loewner--Kufarev evolution at fixed time $t$, and let $z$ be a point in its domain. The cluster has half-plane capa…

Strong discrete uniformization convergence conjecture. The discrete uniformization maps $f_n$ associated to these data converge uniformly to the Riemann mapping associated to…

Uniqueness conjecture. Energy-minimizing parametrizations are unique up to a conformal diffeomorphism provided that every energy-minimizing parametrization is a homeomorphism.

Let $\operatorname{SHL}(\alpha)$ denote the stationary Hastings–Levitov process obtained by normalizing slit sizes according to the conformal radius, for a parameter $\alpha\neq 0$…

Let $n\ge 4$ equal-strength cavities be represented by a conformal map $\omega$ from a slit domain to the exterior elastic domain $D^e$. Let $\Psi(z)$ be the Kolosov–Muskhelishvili…

Harmelin–Minda conjecture. The constant $A$ can be reduced to $2$.

Positivity conjecture for Faber-polynomial error norms. If $L$ has at least one corner with exterior angle different from $0$, $\pi$, and $2\pi$, then

Ebenfelt–Khavinson–Shapiro conjecture. The fingerprints of rational lemniscates of degree $n$ are precisely the $C^{\infty}$ diffeomorphisms $k:\mathbb{T}\to\mathbb{T}$ arising fro…

Let $K\geq 1$ parameterize the affine Riemann surface obtained by gluing a rectangle of side-length ratio $K$ to the complement of a square in ${\mathbb C}$ along their boundaries…

Let a compact bordered Riemann surface have gonality equal to the least degree of a conformal map to the disc, and let $f_{a,b}$ denote an Ahlfors extremal function associated with…

Let $D$ be a simply connected Jordan domain, let $x,y\in\partial D$ be distinct boundary points, and let $\kappa>0$. For parameters $\rho_1,\rho_2>-2$, an…

Let $D$ be a contractible domain, and consider the extremal maps defining its squeezing function. Carathéodory-map conjecture. The extremal maps of $D$ are given by Carathéodory ma…

Extremal entire-function conjecture. The function $F$ has the displayed form, where $\phi$ maps the upper half-plane onto the region in the upper half-plane above $\gamma$, satisfi…