102 problems

Let $p>2$, let $\Omega\subset\mathbb{R}^{n}\times\mathbb{R}$ be a space-time domain, and let $z_{0}\in\partial\Omega$. For the $p$-parabolic equation…

For a compact set $K\subset\mathbb{R}$, define the integer Chebyshev constant by…

For every compact set $K\subset\mathbb{R}^{2}$, let $\operatorname{cap}_{\log}(K)$ denote its logarithmic capacity and let $\operatorname{Cap}_{N}(K)$ denote the Newtonian (electro…

Let $K$ be a complete non-archimedean field, let $X$ be a normal projective variety over $K$, and let $L$ be a line bundle on $X$. For every continuous metric $\varphi$ on…

Let $C_r(K)$ denote the Riesz $r$-capacity of a compact set $K\subset\mathbb{R}^n$. The Clark–Laugesen conjectures ask for sharp extremal inequalities for the ratio $C_p(K)/C_q(K)$…

Let $n\ge 3$. For every open set $\Omega\subset\mathbb{R}^n$, set $F:=\mathbb{R}^n\setminus\Omega$ and, for $0<\alpha\le 1$ and $x\in\Omega$, define…

For the three-dimensional axially symmetric steady incompressible cavity-flow free-boundary problem, determine whether the cavity solutions whose existence was established by Garab…

Conjecture 1.3 (Maxwell; [11], see also Section 4). The total number of points of equilibrium, all assumed nondegenerate, of any configuration with $l$ charges in $\mathbb{R}^{3}$…

Let $(X_n)_{n\ge 0}$ be a centrally excited random walk on $\mathbb{Z}^d$, $d\ge 2$, which otherwise moves as simple random walk, and let $V_n=\{X_0,X_1,\ldots,X_n\}$. Kozma conjec…

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_{\Omega}$. For $z_0\in\Omega$, let $c_{\beta}(z_0)$ denote its logarithmic capacity and let…

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_\Omega$, let $u$ be harmonic on $\Omega$, and set $\rho=e^{-2u}$. Define $B_{\Omega,\rho}$ from an…

Let $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of po…

Support-containment conjecture. If $A\subset B$, then

Let $h^{\infty}$ denote the space of bounded harmonic functions on the unit ball $\mathbb{B}^n:=\{x\in\mathbb{R}^n:|x|<1\}$. For $x\in\mathbb{B}^n$ and $l\in\mathbb{S}^{n-1}$, let…

Let the Laplacian operator be the operator defined by the jump conditions in --, and let $h_1$ denote the function in Theorem. Uniqueness conjecture. The unique, up to multiplicati…

Let $K\subset\mathbb{C}$ be compact and connected with more than one point. Let $M_K$ be the smallest number such that, whenever $p$ has degree $n\geq1$ and $p=p_1\cdots p_m$, … He…

Let $\Sigma_0$ be the class of univalent meromorphic functions $F$ in the exterior of the unit disk, normalized by $F(z)=z+O(z^{-1})$ as $z\to\infty$. Let $K=F(\mathbb{B}^c)^c$ con…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $\mu_K$ be the equilibri…

Let $K\subset\mathbb{C}$ be compact and connected, with logarithmic capacity $\operatorname{cap}(K)=1$, $0\in K$, and conformal centroid at the origin. Let $g$ be the Green's funct…

Complex-charge equilibrium zero conjecture. The meromorphic function

Let $\{a_i\}_{i\in\mathbb{N}}$ be positive with finite sum, and let $\{z_i\}_{i\in\mathbb{N}}$ be distinct complex numbers such that … for some $\rho>0$. Borel-series zero conjectu…

Let $\Gamma$ be a metrized graph, with total length $\ell(\Gamma)$ and tau constant $\tau(\Gamma)$. The ratio $\tau(\Gamma)/\ell(\Gamma)$ is invariant under scaling all edge length…

One-dimensional Maxwell bound. For any values of the charges, the function $V_{\alpha}^*(x)$ has at most $2l-1$ real critical points. This is a stronger one-dimensional version of…

Mother-body conjecture. The root-counting measures of $Sp_n$ converge weakly to a positive measure $\mu_{Q_k}$ depending only on $Q_k$. Its support is a finite planar tree…

Spectral-tree and mother-body conjecture. The measures $\mu_n$ converge weakly to a probability measure $\mu_Q$ depending only on $Q_k$. Its support is a finite planar tree contain…