38 problems

Let $k=\|\mu\|_\infty<1$ and let $K$ be the distortion parameter inferred from $k$ through the relation specified in the source. Consider the area-distortion theorem stated immedia…

Astala's conjecture. The optimal upper bound for the Hausdorff dimension of a $K$-quasicircle is

Ehrenpreis–Siegel conjecture. One can choose finite unbranched coverings $\pi_1$ and $\pi_2$ such that $\widetilde X_1$ and $\widetilde X_2$ have the same genus and there is a…

Let $f\in\Sigma(0)$ be a univalent function with a quasiconformal extension, let $k(f)$ denote its Teichmüller norm, and let $\varkappa_p(f)$ denote its $p$-th truncated Grunsky no…

Let $M$ be a rank one symmetric space, and let a quasiconformal self-homeomorphism of the boundary at infinity of $M$ be a quasiconformal homeomorphism from that boundary to itself…

Let $f:S^1\to S^1$ be a quasi-symmetric selfmapping of the unit circle. A map of the unit disk is hyperbolic-harmonic if it is harmonic for the hyperbolic metric, and it is quasico…

Let $S$ and $R$ be Riemann surfaces, let $f_1:S\to R$ be a quasiconformal map, and let $\mathcal{CM}(R)$ be the space of conformal metrics on $R$ with unit area and at most conical…

A tame quasiconformal manifold is a topological manifold equipped with a quasiconformal structure whose chart transition functions are tame quasiconformal mappings. Tame quasiconfo…

Let $E\subset\mathbb{C}$ be a discrete set. Call $E$ well distributed if there exists $c>0$ such that $E$ intersects every disc of Euclidean radius $c$ in the complex plane. Markov…

Let $h$ be a quasisymmetric homeomorphism of $\widehat{\mathbb R}$, let $f_0$ be an extremal quasiconformal extension of $h$, and let $Q=U(z_1,z_2,z_3,z_4)$ range over quadrilatera…

Ahlfors' conjecture. The characterization of such conformal maps should be expressible in analytic properties of the logarithmic derivative $\log f'=f”/f'$ and related analytic inv…

Let $F:\mathbb{S}^1\to\mathbb{S}^1$ be a critical circle mapping with rotation number $\theta$. Let $H:\mathbb{S}^1\to\mathbb{S}^1$ be the normalized linearizing map satisfying … a…

Let $f$ be a real analytic critical circle map with irrational rotation number $\rho(f)=[a_0,a_1,a_2,\dots]$, and let $h:\mathbb{S}^1\to\mathbb{S}^1$ be the normalized linearizing…

The Brazilian Dream Problem. Then

Weak quasiconformal uniformization conjecture. Every metric sphere admits a weakly quasiconformal parametrization from $\hat{\mathbb{C}}$.

A Julia set is the Julia set of a rational map, and a Kleinian limit set is the limit set of a Kleinian group. Two subsets of the sphere are quasiconformally homeomorphic when ther…

Martio's conjecture. If $n\geq3$ and $K_I(f)<2$, then

Let $\Omega\subset\mathbb{C}$ be a simply connected domain. Define $K(\Omega)$ as the infimum of the maximal dilatations of quasiconformal maps in $\mathcal{F}$, and…

Let $\Pi_0^-$ be the left part of the symmetric comb domain, let $\Theta_0$ be the associated conformal map, and let $\hat\Pi$ be the comb domain with straight slits whose bases ar…

A connected open set $\Omega$ in the Riemann sphere $\widehat{\mathbb C}$ is a circle domain if each boundary component of $\Omega$ is either a circle or a point. It is rigid if ev…

Let $\mathcal G_s(K)$ be the class defined in the paper by, and let $\mathcal F_s(g)$ be the class defined by. For a map $f$, write $K_f$ and $K_{f^{-1}}$ for the distortions of…

Let $\Sigma^0$ denote the class of normalized univalent functions on the exterior disk $\mathbb{D}^*$, and let $\varkappa(F)$ and $k(F)$ denote respectively the Grunsky norm and th…

Sierpiński carpet non-removability conjecture. Every Sierpiński carpet is non-removable for quasiconformal maps and for $W^{1,p}$ functions, for

Let ${\cal F}$ be the exponential family or a family of unicritical polynomials. Two maps in ${\cal F}$ are combinatorially equivalent when they have the same combinatorial data, s…

Let ${\cal F}$ be a family of maps, and let $f\in{\cal F}$ be non-hyperbolic. The map $f$ is quasiconformally rigid when it is not quasiconformally conjugate to any other map in a…