8 problems

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…

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 $Q_n^\dagger$ be its dual, and let $\tau_{Q_n}(\varepsilon)$, $\tau_{Q_n^\dagger}(\varepsilon)$ denote uniform mixing…

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 $Q$ 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…