6 problems

Let $f:S^1\to\mathbb{CP}^1$ be a CaTherine wheel arising from an expanding Thurston map. Uniform quasiconformality conjecture. There exists a finite constant $K$ such that $f$ is a…

Let $R>r>0$, let $G$ be a Kleinian group, and let $f$ be a CaTherine wheel invariant under $G$. Assume that the injectivity radius of $\bbH^3/G$ lies everywhere in the interval…

Let $f:\mathcal{H}_q\to\mathcal{H}_q$ be a critical quasicircle map with irrational rotation number, and let $\mathcal{N}$ be the Banach neighborhood of $f$ consisting of nearby un…

Let $h_{\mathbf{H}^2}$ be the Hitchin map and let $\sigma_{\mathbf{H}^2}$ be the Hitchin–Li–Mochizuki section, whose image parametrizes quasisymmetric maps associated with the corr…

Let $\mu_0$ be Werner's measure on self-avoiding loops, let $\nu_0$ be its induced measure on welding homeomorphisms $\sigma$, and let $W$ be the welding map. A quasicircle is a Jo…

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…