16 problems

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…

The Mandelbrot set $\mathcal{M}$ is the connectedness locus of the quadratic family $p_c(z)=z^2+c$. MLC conjecture. The Mandelbrot set $\mathcal{M}$ is locally connected. Local con…

Lipa's pseudo-monodromy marker conjecture. The monodromy action of $\rho(\gamma)$ is described by compositions of the markers

Density of hyperbolicity conjecture. The set $\mathcal{H}$ is dense in $\mathcal{M}$.

Let $m=2^n m_0$, where $m_0$ is odd, and let $u= u2$ denote the 2-adic valuation. Let $am$ be the Taylor coefficients of the reciprocal Riemann mapping function. **Zagier-type con…

Let $a_m$ be the Taylor coefficients of $\Theta(z)=1/\Psi(1/z)$, let $m=2^n m_0$ with $m_0$ odd, and fix $n=\overline{n}$. Write $u= u2$ for the 2-adic valuation. **Asymptotic slo…

Let $m=2^n m_0$, where $m_0$ is odd, and let $u= u2$ denote the 2-adic valuation. Define $u((k)!)$ by the 2-adic valuation of the factorial. Zagier's conjecture. For the Laur…

The principal-vein Master Teapot conjecture. The set $\Upsilon_{p/q}$ is given by the closure of the pairs consisting of $e^{h(\theta)}$ and its Galois conjugates, as displayed abo…

Let $\mathbf M$ be the Mandelbrot set of quadratic polynomials, and let $\mathbf M_1$ be the connectedness locus in the family of rational maps with multiplier $1$ at a fixed point…

Let $\mathcal{H}$ be the main cardioid of the Mandelbrot set, and let $\delta: \mathcal{H} \to [0,2)$ be the Hausdorff dimension function. Noncriticality conjecture. The function…

Mandelbrot homeomorphism conjecture. The Mandelbrot set $M$ is homeomorphic to $M_{\Gamma}$. The family $\mathcal{F}_a$ has been shown to mate parabolic quadratic maps with…

Let $c\in\partial\boldsymbol{\Delta}$ be a non-parabolic parameter, let $f_c$ be the corresponding quadratic polynomial, and let $H_c$ denote its mother hedgehog: if the closed Sie…

Douady–Hubbard's conjecture. The points $c_1$ and $c_2$ do not lie in conjugate limbs of the Mandelbrot set if and only if

Let $\theta_1<\theta_2$ be external angles whose parameter rays $R_M(\theta_1)$ and $R_M(\theta_2)$ land at the same parameter on the boundary of the Mandelbrot set. Let…

Trivial-fibers conjecture. In the parameter spaces of quadratic polynomials and exponential maps, all fibers are trivial.

Douady–Hubbard's local connectivity conjecture. The set $\mathcal{B}$ is locally connected.