29 problems

General convergence conjecture. The process $\Pi^\epsilon \circ Z^\epsilon$ converges weakly to $\bar{Z}$ in $C([0,\infty);\Gamma)$, uniformly with respect to the initial condition…

Let $G$ be a graph, and let $\Gamma$ be the associated $\mathbb Q$-graph. For an integer $r\geq 1$, let $D(G,r)$ and $D(\Gamma,r)$ be the minimal degrees of a $g^r_d$ on $G$ and…

Inradius upper-bound conjecture. There is no universal constant $C>0$ such that

Mean-distance upper-bound conjecture. There is no universal constant $C>0$ such that

Let $G=(V,E,l)$ be a connected metric graph endowed with an interval exchange transformation $T$ partitioning $G$ into $n$ intervals with lengths $\ell_1, \ell_2, \ldots, \ell_n$ l…

Let $\Gamma$ be a metric graph of genus $g$, and let $d,r$ be integers satisfying $0<2r\leq d<g$. The Brill–Noether rank $w_d^r(\Gamma)$ is the tropical Brill–Noether rank. Jensen–…

For each rank $r\geq 3$, let $C_r$ be the constant from the theorem asserting that every rank-$r$ metric graph contains a proper subgraph whose entropy is at least $C_r$. Conjectur…

Let $p\in(2,6)$ and let $\nu>0$ be a mass. An energy ground state is a positive solution of the NLS variational problem on the $\mathcal{T}$-metric graph that minimizes the energy…

Let $\Gamma$ be a compact metric graph with edge set $\mathcal{E}$. A Gaussian Markov random field of order $p$ is assumed to satisfy some mild regularity conditions. On each edge…

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a domain, possibly equal to $\mathbb{R}^d$, and let $V:\Omega \to \mathbb{R}$ be a sufficiently smooth potential bounded from belo…

Fix a positive integer $r$. For a metric graph $\Gamma=(G,l)$ with rational edge lengths, the $r^{th}$ Metric Divisorial Gonality problem asks whether…

Full support eigenfunctions conjecture. For any metric graph $(\Gamma,\boldsymbol{\ell})$, and any choice of a complete orthonormal sequence of eigenfunctions, there are infinitely…

$\mathbb{Q}$-independent spectrum conjecture. For every graph $\Gamma$, possibly except for some pathological cases, there is a generic set $G\subset\mathbb{R}_{+}^{N}$ such that,…

Let $X$ be a planar metric graph, meaning a geodesic space homeomorphic to a finite $1$-dimensional simplicial complex that admits a planar embedding. Let $G$ be a coefficient grou…

Let $G$ be a connected loopless multigraph, let $\Gamma(G)$ be the corresponding metric graph with unit edge lengths, and let $r \geq 1$. For each integer $k \geq 1$, let…

Let $\left\{\Gamma^{(\beta)}\right\}_{\beta\nearrow\infty}$ be any sequence of standard graphs, labeled by their first Betti numbers. For each $\Gamma^{(\beta)}$, choose arbitrary…

Let $r\geq 3$. A metric graph of rank $r$ has unit entropy when its entropy is normalized to equal $1$, and a subgraph is proper when it is a strict subgraph of the graph. Uniform…

Loose-versus-non-exhaustive partition conjecture. The infimum over non-exhaustive rigid $k$-partitions satisfies

Let $\mathcal{G}$ be a metric graph and let $k\geq1$. A rigid $k$-partition is a partition of $\mathcal{G}$ of the type considered in the paper, and $\mathcal{P}$ is called interna…

Let $\mathcal{G}$ be a finite, compact, connected metric graph. An exhaustive rigid $2$-partition of $\mathcal{G}$ is a partition ${\mathcal P}_2=\{\mathcal{G}_1,\mathcal{G}_2\}$ w…

Let $\Gamma$ be a metric graph whose skeleton is the complete graph $K_4$, and let $D=K$ be its canonical divisor. Write $|D|$ for the associated linear system, and call the cells…

Let $\Gamma$ be a metric graph of genus $g$. For integers $d$ and $r$ satisfying … let $w^r_d(\Gamma)$ denote the Brill–Noether rank, namely the largest integer $\rho$ such that ev…

Let $\Delta\subset\mathbb{R}^2$ be a two-dimensional lattice polygon, and let $\Delta^{(1)}$ denote the convex hull of the interior lattice points of $\Delta$. Write…

Let $\Gamma$ be a metric graph of genus $g$. A minimal rank-determining set is a rank-determining set of $\Gamma$ that has no proper subset that is rank-determining. Cardinality bo…

Geodetic approximation conjecture. For every $c\in\widehat{H'_1}$, there is a Cauchy sequence $(\chi_i)_{i\in\mathbb N}\in c$ such that every $\chi_i$ is the homology class of a fi…