8 problems

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

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 $d4a2_1$ and $d4a2_2$ be the complementary regions of a Riemann surface separated by a curve, and let $T(d4a2_1,d4a2_2)$ denote the transmission operator between the associated…

Let a Riemann surface be separated by a curve, and consider the jump isomorphism from the harmonic Dirichlet space on either side to the direct sum of the corresponding holomorphic…

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…