17 problems

Let $d\in(0,2]$, let $E\subset\mathbb C$ be compact, and let $h$ be a principal $K$-quasiconformal mapping conformal on $\mathbb C\setminus E$. Here $\mathcal M^d$ denotes $d$-dime…

Let $f=h+\overline{g}\in\mathcal{S}^0_H(K)$, with $h$ and $g$ having the series representations specified in the source. Let $A(n,k)$ and $B(n,k)$ be the quantities defined in the…

Covering-radius conjecture. Every function $f\in\mathcal{S}^0_H(K)$ contains the disk

Covering-radius conjecture. The radius satisfies

Let $\mathcal{S}_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Harmonic K-quasiconformal Hardy-order conjecture. On…

Let $\mathcal{S}^0_\mathcal{H}(K)$ be the class of normalized harmonic $K$-quasiconformal mappings, and write $f=h+\overline{g}$. Second-coefficient conjecture. One has … The bound…

Let $0\le \theta<1$ and let $E\subset\mathbb{R}^n$. Write $\dim_A^\theta(E)$ for the Assouad spectrum with parameter $\theta$, and let the global quasiconformal Assouad spectrum di…

Let $G$ be a Carnot group other than $\mathbb{R}^n$ or the Heisenberg group $\mathbb{H}_n$ for any $n$, and let $U,U'$ be open subsets of $G$. Let $f:U\to U'$ be a quasiconformal h…

Let $G$supset U$and$U' ormal?" Wait. Cannot introduce invalid notation. Let $G$ be a rigid Carnot group, and let $U,U' e othing$ be open subsets of $G$. A quasiconformal homeomor…

Uniqueness conjecture. If $\hat{\Phi}$ is holomorphic, then $\hat{h}=\operatorname{identity}$.

Let $G$ be a Carnot group, and let $U,U'\subset G$ be open sets. Xie's conjecture. If $G$ is neither $\mathbb{R}^n$ nor a Heisenberg group $\mathbb{H}_n$, then every quasiconformal…

Let $G$ be a Carnot group. A group is rigid if the space of smooth contact embeddings $G\supset U\to G$ is finite-dimensional for every connected open subset $U\subset G$. Regulari…

Mocanu's conjecture. Every mapping in $\mathcal{M}$ belongs to $\mathcal{S}_{\mathcal{H}}^0$; equivalently, $\mathcal{M}\subset\mathcal{S}_{\mathcal{H}}^0$. The conjecture concerns…

Alternative p-adic condition conjecture. In the p-adic case, the renormalized average series condition in the definition of an adelic Beltrami differential can be replaced by the c…

Let $K>1$, $n>2$, and $r\in(0,1)$. Define $\alpha=K^{1/(1-n)}$ and let $\varphi_{K,n}$ be the quasiconformal distortion function. Hyperbolic distortion conjecture. … The conjecture…

Let $U$ be the Burkholder function appearing in the source, let $\partial$ and $\overline{\partial}$ denote the complex derivatives, and let $f\in C_0^{\infty}(\mathbb{C})$. Burkho…

Let $S$ be the Donaldson–Sullivan signature operator on differential forms over $\mathbb{R}^n$, defined by … For $1<p<\infty$, let $p^*=\max\{p,p/(p-1)\}$. Iwaniec–Martin conjectur…