97 problems

Let $M_g$ be a closed orientable surface of genus $g$. Let $\mathrm{chr}(M_g)$ be the supremum of the natural numbers $n$ for which the complete graph on $n$ vertices can be embedd…

Let $X$ be a typical hyperbolic surface, and consider the spectral statistics of its Laplacian. Bohigas–Giannoni–Schmidt conjecture. Because the geodesic flow is time-reversal symm…

Jakobson–Naud's conjecture. They conjecture that

Zograf's conjecture. Based on numerical data, Zograf conjectured that

Buser's eigenvalue-sequence conjecture. There exist sequences of closed hyperbolic surfaces whose first eigenvalue tends to $\frac14$.

Petridis–Risager's exponential sum conjecture. For every $\epsilon>0$,

Let $(\Sigma,g)$ be a closed hyperbolic surface, and consider high-frequency limits of Laplace eigenfunctions on it. A semiclassical defect measure is a measure describing such a h…

Let $(M,g)$ be a compact hyperbolic surface, let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, and let $\gamma$ be a periodic geodesic or a geode…

A closed hyperbolic surface of genus $2$ is a compact surface equipped with a metric of constant negative curvature and genus $2$. The Bolza surface is the genus-$2$ hyperbolic sur…

Buser–Schmutz conjecture. The eigenvalue with index $2g+n-2$ satisfies

Let $S$ be a non-compact hyperbolic surface of finite area with $n$ punctures, and let $\bar S$ be the closed surface obtained by filling in its punctures. Write $\chi(S)$ and…

Let $\epsilon>0$ and let $S_1,S_2$ be two closed hyperbolic surfaces. A finite-sheeted locally isometric cover of a surface is a finite-sheeted covering surface equipped with the p…

Let $X$ be a compact hyperbolic surface, and let $(f_i)$ be an orthonormal sequence of Laplace eigenfunctions on $X$, with eigenvalues satisfying $(\Delta+\lambda_i^2)f_i=0$. Call…

Let $\mathcal{M}_{g,n}$ be the moduli space of hyperbolic surfaces of genus $g$ with $n$ punctures, and consider the discrete eigenvalues of the Laplacian below the continuous spec…

Let $X$ be a fixed compact hyperbolic surface, and let $T,L>0$ satisfy $T\to\infty$, $L\to\infty$, and $L=o(T)$. Denote by $\sigma_{\chi,\psi}^{2}$ the limiting variance appearing…

Let $X$ be a closed hyperbolic surface of genus $g$, let $\mathbb{P}_{g}$ be the Weil–Petersson probability measure on the moduli space of such surfaces, and let $\lambda_{1}(X)$ d…

Commensurability conjecture. Any two hyperbolic structures with the same length spectrum are commensurable.

Let $M=\Gamma\backslash\mathbb H^2$ be a convex co-compact hyperbolic surface, and let $\delta$ denote the dimension of the limit set of $\Gamma$. Jakobson--Naud finiteness conject…

Nielsen-type conjecture. The natural monomorphism

Let $\phi_i$ be an eigenfunction on the hyperbolic surface and let $Op(a)$ be the operator associated with a fixed symbol $a$. Write the spectral decomposition of $Op(a)\phi_i$ in…

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction with eigenvalue $\mu_i$, and let $a_n^{\sigma/\ell}(\phi_i)$ denote the Fourier coefficient associa…

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction on $Y$ with eigenvalue $\mu_i$, and let $\ell$ be a geodesic on $Y$. Write…

Let $X_0$ be the union of six planes in general position in ${\mathbb P}^3$, and let $X_t$ be a general sufficiently small deformation of $X_0$ to a sextic surface. General-deforma…

Lipnowski–Wright conjecture. A transition occurs at order $\sqrt{g}$.

Let $N_{\chi,n,b}$ denote the number of extremal disc configurations with Euler characteristic $\chi$, $n$ cusps and $b$ boundary components, and let $\Sigma_{\chi,n,b}$ denote the…