11 problems

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…

For $\kappa\in[0,8]$, let $\mathrm{SLE}_\kappa$ denote Schramm–Loewner evolution with parameter $\kappa$, and let its trace be the associated random curve. SLE trace minimality con…

Let $B$ be $n$-dimensional Brownian motion. High-dimensional Brownian motion conjecture. For all sufficiently large $n$, $B$ is not minimal almost surely; moreover, for all suffici…

Suppose $f\colon (S^2,A)\righttoleftarrow$ is a post-critically finite hyperbolic-type branched covering without a Levy cycle. Let $\Gamma$ be a completely $f$-invariant multicurve…

Let $f\colon (S^2,A)\righttoleftarrow$ be a post-critically finite hyperbolic-type branched covering, and let $\overline{E}^p(f)$ denote its asymptotic $p$-conformal energy for…

Let $(\pi,\phi)$ be a recurrent virtual graph endomorphism, and for $q\in[1,\infty]$ let $\overline{E}{}^q[\pi,\phi]$ denote its asymptotic $q$-conformal graph energy. Strict-decre…

Conformal-dimension characterization conjecture.

Bonk–Kleiner's conjecture. The conformal dimension of $\partial\Gamma$ is attained by a metric in its conformal gauge.

Fractal-rug conjecture. For any $c\geq 1$ there is a Jordan arc $V_{c-1}$ such that

Range conjecture. For every $d\in[2,\infty]$ there is a metric space $X$ with

For every $d in[2,\infty]$, let $X$ be a metric space and denote its topological conformal dimension by $\dim{tC}X$. Realization conjecture. For every $din[2,\infty]$ there is a m…