10 problems

Brownian-sphere conjecture. The rescaled maps

Gromov--Hausdorff coupling conjecture. The variables $\mathbf U_{n_g,g}$ and $\mathbf S_g$ can be coupled so that

Let $g_n$ satisfy … and let $\mathbf{T_{n,g_n}}$ be a uniform triangulation of genus $g_n$ with $3n$ edges. An induced subgraph is a $\kappa$-expander if every set of vertices has…

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…

Let $(\mathbf{f}^{n})_{n \geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\mathbf{f}^{n},g_n)\geq 2$ for all $n\geq 1$. Assume that…

Let $S_n^O$ be the random surface in the BM model, and let $\widetilde{S}_n^O$ denote this surface conditioned on having a single vertex. Let $\widetilde{S}_n^C$ be the compactifie…

Let $(\mathcal{P}_{n})_{n \geqslant 1}$ be a good sequence of configurations. For each $n$, let $\mathbf{G}_{\mathcal{P}_{n}}$ be the associated random graph, let $\mathbb{G}_{n}$…