21 problems

Linear growth conjecture. The length $|\ell(R)|$ is linear in $R$ as $R\to\infty$.

Let ISE denote Integrated SuperBrownian Excursion, and let a Continuum Random Map (CRM) be a continuum analogue of the labelled-tree construction that encodes distances. For scaled…

Let $r_n$ be the radius of a random quadrangulation with $n$ faces. Schaeffer's radius scaling conjecture. There exists a constant $c$ such that … The paper states this conjecture…

Let $(\mu,k)$ be a spherical profile, meaning profile data for square-tiled surfaces on the sphere. Let $\operatorname{ST}(\mu,k)$ denote the corresponding set of square-tiled surf…

An $(n,t)$-quadrangulation is a quadrangular embedding of a simple graph on $n$ vertices and $\binom{n}{2}-t$ edges. An embedding is face-simple if its dual is simple, meaning that…

Bipartite quadrangulation conjecture. For $g>0$, the topological recursion for this spectral curve gives

Let $Q_n$ be a uniform quadrangulation with $n$ vertices, let $F(Q_n)$ be its set of faces, and for $S e othing$ with $S e F(Qn)$ let $partial S$ be the set of edges incident on on…

For any map $m$, let $\operatorname{SNC}(m)$ be the size of the shortest separating non-contractible cycle of $m$. Let $q^{(n)}$ be the random uniform high-genus quadrangulation, a…

Let $q^{(n)}$ be the random uniform high-genus quadrangulation, and let $\operatorname{syst}(q^{(n)})$ denote the size of its shortest non-contractible cycle. Systole tightness con…

Fix $\theta\in(0,1/2)$, let $g_n$ be the genus sequence used for the high-genus quadrangulations, and let $Q(n,g)$ denote the number of quadrangulations with $n$ faces and genus…

Let $q^{(n)}$ be the random uniform quadrangulation in the high-genus regime, with genus parameter determined by $\theta$, and let $\operatorname{diam}(q^{(n)})$ denote its graph d…

3-connected quadrangulation conjecture. The maximum Wiener index of a 3-connected simple quadrangulation of order $n$ is

Let $Q_{6,\arccos\frac{-1}{2\sqrt7},\frac{4\pi}{3},\arccos\frac{1}{3}}$ be the spherical $4$-gon specified by these parameters, and consider tilings of the sphere by congruent copi…

Fully simple quadrangulation formula conjecture. The number of planar fully simple quadrangulations is

Krikun's conjecture. The quantity $L_r$ grows linearly in $r$ as $r\to\infty$.

For integers $k\geq 1$ and $n\geq 2k+1$, let $SG(n,k)$ be the Schrijver graph, whose vertices are the $k$-subsets of $[n]$ containing no two cyclically consecutive elements, with e…

Tutte's bijection relates quadrangulations with $n$ faces to general planar maps with $n$ edges. The relevant metric spaces are equipped with their graph distances, and a map is as…

Let $Q$ and $Q'$ be quadrangulations related by twisting along one inner edge. Write $\gamma_Q$ and $\gamma_{Q'}$ for their undirected flip graphs, and $G_Q$ and $G_{Q'}$ for their…

Let $Q$ be a quadrangulation with $n$ inner edges. Let $F_Q(x,y)$ be its $F$-triangle and let $H_Q(x,y)$ be its $H$-triangle, defined by … where $\mathrm{SN}_Q$ is the set of serpe…

Brinkmann's three-vertex-type conjecture. There are at most three distinct vertex types in a chart of any spherical monohedral quadrangular tiling.

Let $M_n$ be a uniform planar map with $n$ edges and let $Q_n$ be a uniform planar quadrangulation with $n$ faces. Equip both maps with their graph distances, and let the Ambjørn–B…