15 problems

Let $S\subset \mathbb{R}^3$ be a closed, $C^2$ convex surface with principal curvatures $\kappa_1,\kappa_2>0$. Suppose that, at every point, for some constant $c>0$, … and equality…

Let $G$ be a geometrically finite Bers boundary group, and let $\Lambda(G)$ be its limit set. Form the contact tree whose vertices are the components of the domain of discontinuity…

Let $K_i$, for $i\in\{1,2\}$, be limit sets of hyperbolic conformal dynamical systems, such as limit sets of convex co-compact Kleinian groups or Julia sets of hyperbolic rational…

Let $\operatorname{Mol}$ denote the main molecule in the quadratic family $z^2+cz$, and let $Q$ be the quadratic map used in the source for the standard Basilica Julia set. Molecul…

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems. A map is weakly type-preserving if it sends parabolic points to parabolic…

Let $K_i$, for $i\in\{1,2\}$, be Basilica limit sets of geometrically finite conformal dynamical systems, such as limit sets of geometrically finite Kleinian groups or Julia sets o…

Let $J$ be a Julia set and let $\Lambda$ be a limit set. Suppose that both $J$ and $\Lambda$ are connected and that neither is homeomorphic to the circle or the 2-sphere. The LLMM…

Let $\mathcal{M}$ be the Mandelbrot set, let $\mathcal{H}\subset\mathcal{M}$ be a hyperbolic component, and let $\mathcal{M}_{\mathcal{H}}$ be the associated small copy of…

Let $\mathcal J$ be the Julia set of a rational map and let $\Lambda$ be the limit set of a Kleinian group. Suppose that $\mathcal J$ and $\Lambda$ are connected and neither is hom…

Let $J$ be the Julia set of a rational map and let $\Lambda$ be the limit set of a Kleinian group. Suppose that $J$ and $\Lambda$ are connected, and neither is homeomorphic to a ci…

Let $M(\omega)$ be the moduli space associated with a sequence $\omega\in(0,1)^{\mathbb N}$, and let $m_{\infty}$ denote the standard product probability measure on…

Quasiconformal Reeb-field conjecture. Every quasiconformal Reeb field must be associated with a tight contact structure.

Rajala–Romney conjecture. The optimal constant in this lower bound should be

Let $G$ be a locally compact group acting effectively and quasiconformally on a Riemannian manifold. Martin's conjecture. The group $G$ is a Lie group. This is a particular case of…

Conjecture I. If there exists an automorphism of infinite order $h\in\operatorname{Aut}(\mathbb{C}\setminus A)$ such that the quotient space…