6 problems

Let $g$ denote the genus of a hyperbolic surface, let $\sys(M)$ denote its systole, and let $\Nsmall$ and $\lambda_1(M)$ be the quantities used in the paper. The first quantity bel…

Let $\Sigma$ be a compact hyperbolic surface, and let $I_k(\Sigma)$ denote the maximal self-intersection number of a $k$-systole on $\Sigma$. Erlandsson–Parlier's conjecture. One h…

Let $\mathcal{M}_2$ denote the moduli space of conformal classes of closed genus-$2$ surfaces. For a conformal class $X\in\mathcal{M}_2$, let…

Dimension conjecture. For every $\varepsilon>0$, there exists an integer $g\geq 2$ such that $\mathcal{X}_g$ has dimension at least $(6-\varepsilon)g$.

Let $S_g$ be a closed orientable surface of genus $g$, let $\mathcal{M}_g$ denote its moduli space of hyperbolic metrics, and let $m_g$ be the minimum perimeter of a regular, right…

Let $\mathcal M_g$ denote the moduli space of closed hyperbolic surfaces of genus $g$, and let $\operatorname{sys}(S)$ be the systole length of $S$. Schmutz Schaller's systole-leng…