17 problems

Let $S_1$ and $S_2$ be closed Riemann surfaces of the same genus, and let $\tilde S_i$ denote covers of $S_i$ for $i=1,2$. A cover is conformal when the covering map is conformal,…

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…

The Brazilian Dream Problem. Then

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 $T$ be the Ahlfors–Beurling operator on $L^p(\mathbb C)$, let $p^*=\max\{p,p/(p-1)\}$, and write $\|T\|_p$ for its operator norm. Iwaniec's conjecture. For every…

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…

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…

The conjectured value of $L(a)$. For $n=2$, numerical experiments suggest that

Schoen's conjecture. There exists a unique harmonic and quasi-isometric map

Higher regularity conjecture. The map $f$ is conformal at every point $p\in S$ and belongs to $C^{\infty}(S)$; that is, derivatives of all orders exist on $S$ in the sense of the d…

Let $T=\triangle(,0,)$ be a triangle and denote its angle at $0$ by $\theta=\measuredangle(\xi,0,\eta)$. Let $T^q$ be the $q$th level of the regular 1-4 subdivision of $T$. For…

Let $f\colon \mathbb{C}\to\mathbb{C}$ be a reduced quasiconformal map, and consider the image $f(\mathbb{R})$. A quasisymmetric graph is the image of $\mathbb{R}$ under a suitable…

Let $0<k<1$, and let a $k$-quasicircle be a quasicircle with parameter $k$. Astala's conjecture. There exists a $k$-quasicircle $\Gamma$ with Hausdorff dimension … Astala's upper e…

Let $QC_K$ be the family of all $K$-quasiconformal self-maps of the unit disk $\mathbf{B}^2$ that fix the origin, and let $M(2,K)$ be the least constant such that … for every…