19 problems

Kleiner's conjecture. Approximately self-similar and combinatorially Loewner metric spaces attain their conformal dimension.

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

For $\gamma\in(0,2)$, consider the $\gamma$-Liouville quantum gravity ($\gamma$-LQG) sphere equipped with its associated metric, whose construction is known in the generality descr…

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 $f\colon [0,1]\to\mathbb{R}$ be continuous, and let … be its graph. Suppose there is a set $A\subset\mathbb{R}$ of positive Lebesgue measure such that, for every $a\in A$, the…

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…

Let $B([0,1])\subset\mathbb{R}^2$ denote the trace of one-dimensional Brownian motion on a time interval, viewed as a metric subset with its induced structure. Planar Brownian trac…

Let $\Gamma$ be the graph of a continuous function $f:[0,1]\to\mathbb{R}$, and let $Z_a=(-\infty,\infty)\times\{a\}$ be the horizontal line at height $a$. Suppose there is a set…

Let $(K,d,m)$ be the Sierpiński carpet, let $d_{\mathrm{ARC}}(K,d)$ be its Ahlfors regular conformal dimension, and suppose this dimension is attained. Let $\mathcal{V}_0$ be the b…

Conformal-dimension conjecture. The conformal dimension of $K$ is

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 $(X,d,\mu)$ be the metric measure space under consideration, let $\alpha_p$ be its critical Besov exponent for $p\geq 1$, and define … where $d_h$ is the Hausdorff dimension. A…

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.

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…