51 problems

For $n\geq 3$, let $b_n>0$ be a universal constant such that every domain $\Omega\subset\mathbb{R}^n$ satisfies $\dim w_\Omega\leq n-b_n$, where $w_\Omega$ denotes harmonic measure…

Let $n\geq 2$. A two-sided NTA domain is a domain in $\mathbb{R}^{n+1}$ whose interior and exterior both satisfy the conditions of a non-tangentially accessible domain, and whose i…

Let $\Omega$ be a Jordan domain in $\bold C$, and let $\mmp P$ and $\mmp Q$ be univalent circle packings for the same tangency graph $\mcompk$, with $(\compk,p_{\p Q})$ the random…

Euclidean harmonic-measure dimension conjecture. The dimension of harmonic measure in $\mathbb{R}^n$ is conjectured not to exceed

Oksendal's conjecture. The dimension of harmonic measure in $\mathbb{C}$ should never exceed $1$, although the Hausdorff dimension of its support can be as large as $2$.

Bishop–Jones conjecture. The optimal constant satisfies

Let $\Omega\subset\mathbb R^n$ be a domain, and let $\omega_\Omega$ denote harmonic measure on its boundary. Write … where $\dim_H$ denotes Hausdorff dimension. Harmonic measure di…

Let a domain have separated connected components with harmonic measures defined on their common boundary. Bishop's two-phase conjecture. Mutual absolute continuity of the harmonic…

Let a domain have a boundary, with harmonic measure and surface measure defined on that boundary. Bishop's conjecture. Absolute continuity of harmonic measure with respect to surfa…

Let $G$ be the fundamental group of a compact manifold with constant negative curvature, and let $\rho$ be the distance on $G$ induced by the Riemannian metric of its universal cov…

For $d\geq3$, integers $n\geq2$ and $m\geq3$, define … where $\|x_1-x_2\|$ is graph distance. Separation-distance conjecture. (1) There exist $C(d,m),c(d,m)>0$ such that, for all…

For $d\geq3$ and $n\geq2$, let $\mathfs{M}_n(\mathbb{Z}^d)$ be the least positive harmonic measure among subsets of $\mathbb{Z}^d$ of cardinality $n$. Two lattice vertices are $*$-…

Let $\psi(\mathbb{Z}^2)$ denote the reciprocal extremal factor governing the exponential lower bound for positive harmonic measures on finite subsets of $\mathbb{Z}^2$. The propose…

For $n\geq2$, let $\mathfs{M}_n(\mathbb{Z}^2)$ denote the least positive harmonic measure among subsets of $\mathbb{Z}^2$ of cardinality $n$. Calvert, Ganguly and Hammond construct…

Let $A\subset\mathbb{Z}^2$ be finite and let $y\in A$ have positive harmonic measure $\mathbb{H}_A(y)$. The least positive harmonic measure among sets of cardinality $n$ is denoted…

Let $K$ be one of the sets under consideration, with complement $\Omega=\mathbb{R}^2\backslash K$, and let $G_1$ and $G_2$ be two Green's functions associated with the relevant dom…

Let $K\subset\mathbb{R}^2$ be a compact set, let $\Omega=\mathbb{R}^2\backslash K$, and suppose that the Green function $G^x$ of $\Omega$ with pole at $x\in\Omega$ satisfies … for…

Let $G$ be the group of isometries of the hyperbolic plane $\mathbb{H}^2$. Let $\mu$ be a finitely supported probability on $G$ whose support generates, as a semigroup, a discrete…

Let $L$ be an arbitrary Lévy operator satisfying the assumption of Theorem, and let $\check{H}_D$ denote the corresponding harmonic measure for a bounded open set $D$. A support is…

Assume $\Omega=\mathbb{H}^N$ is real hyperbolic space of dimension $N\geq 2$, and let $\Gamma$ be convex cocompact. Let $\mu$ be a measure on $\Gamma$, and let $\nu$ be its harmoni…

Let $J$ be a conformal self-similar set of dimension $\delta<1$, and let $\omega_{\mathbb C\setminus J}$ denote harmonic measure for its complement. Dimension-drop conjecture. The…

Let $\Omega\subset\mathbb R^n$ be a one-sided NTA domain whose Green function with some pole $p$ satisfies, for some $\delta>0$ and constants $c_1,c_2>0$, bounds of the form … for…

Let $L_0$ be a longest member of a finite collection of pairwise disjoint open arcs on the unit circle, each of Euclidean length less than $\pi$. In the Poincaré disk model, remove…

Square-spiral conjecture. Asymptotically, the square spiral realizes the least positive value of harmonic measure, in the sense that

Exponent-gap conjecture. The constants satisfy