35 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…

Nielsen-type conjecture. The natural monomorphism

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…

Selberg's eigenvalue conjecture. For every $N$,

Bolza–Klein bass-note conjecture. (a) The Bolza surface uniquely maximizes the bass note among closed and orientable hyperbolic surfaces. (b) The Klein quartic uniquely maximizes t…

Uniform observability conjecture. For every $T>0$, there exists a constant $C=C(X,S,T)>0$ such that the observability inequality holds. Moreover, the constant can be chosen to depe…

Let $(X_k)_k$ be a sequence of random finite-area hyperbolic surfaces such that … Write $\lambda_1(X_k)$ for the first positive Laplace eigenvalue. Random hyperbolic-surface spectr…

Essential spectral gap conjecture. For every $\omega\in\mathcal{H}^1(X)$,

Let $X(\{\ell_n\},\{t_n\})$ be a flute surface whose cuff lengths form a non-decreasing sequence $\{\ell_n\}$, and let each twist parameter satisfy $t_n\in\{0,1/2\}$. Strong Kahn–M…

Linear small-eigenvalue conjecture. There are constants $a,b>0$ such that, whenever

Let $V_g^\epsilon$ and $\overline{V_g^\epsilon}$ be the sets defined in Theorem, and let $\mathcal{M}_g$ be the corresponding moduli space of genus-$g$ surfaces, with Weil--Peterss…

Lipnowski–Wright's conjecture. As $g\to\infty$, on most closed hyperbolic surfaces in $\mathcal{M}_g$, most geodesics of length significantly less than $\sqrt g$ are simple and non…

Jakobson–Naud's conjecture. There are only finitely many zeros in the half-plane

Quarter-gap conjecture. For every $\epsilon>0$,

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

Let $S$ and $S'$ be hyperbolic surfaces, and let the multi-set of lengths of their simple closed geodesics be called their simple length spectrum. Simple length spectrum rigidity c…

Let $\mathbb{P}_{g,k(g)}$ be the probability measure on hyperbolic surfaces of genus $g$ with $k(g)$ cusps, and let $X \in \mathcal{M}_{g,k(g)}$ be a random surface. Assume that…

Let $X$ be a random compact hyperbolic surface of genus $g$, sampled according to the Weil–Petersson probability measure. Write $N_X^\Delta(0,a)$ for the number of eigenvalues of t…

Let $X>2$, let $\Gamma$ be an arithmetic Fuchsian group defining a hyperbolic surface, and let $\mathcal{S}(T,X)$ denote the spectral exponential sum associated with that surface.…

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 closed geodesic or a geodesi…

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…

Alon and Bilu–Linial conjectures. (1) For every $\epsilon>0$, with probability $1-o(1)$ as $n\to\infty$, every new eigenvalue $\lambda$ of an $n$-cover $X'$ satisfies

Reznikov's Lindelöf-type conjecture. The geodesic period integrals on compact hyperbolic surfaces satisfy the estimate above. This would improve the known uniform bound and is pres…

Simple-closed-geodesic growth conjecture. There exists a constant $c>0$ such that

Let $S_g$ be a closed hyperbolic surface of genus $g$, and let $D_{\operatorname{inj}(S_g)}$ denote an embedded metric disk of radius equal to the injectivity radius of $S_g$. Schm…