8 problems

Square-root error conjecture. For every $\epsilon>0$,

Let $\Delta$ be the Laplacian of the hyperbolic surface $\Gamma \backslash \mathbb{H}$, and let $E(X,z,w)$ denote the error term in the spectral asymptotic formula for the counting…

Let $g\in \operatorname{SL}_{2}(\mathbb{R})$. For parameters $\ell$, $\delta$, $L$, and $\heartsuit$, consider pairs $\gamma_1,\gamma_2\in R(\ell;g)$ lying in…

Homogeneous-space counting conjecture. There exists an exhausting compact family $\mathcal{B}^Z=(B_R^Z)_{R>0}$ of $Z$ such that the quadruple…

Let $G$) be a semisimple Lie group, and let $\rho_x(G)$ denote the number of lattices in $G$ of covolume at most $x$, up to automorphism. For a lattice $\Gamma$ in $G$, let…

Polynomial-bound conjecture. There exists a constant $c=c(H)$ such that, for all sufficiently large $x$,

Belolipetsky–Lubotzky conjecture. As $x$ tends to infinity,

Let $(a,b)$ be a strongly regular semisimple collection of invariants with $a_i,b_i\in\mathcal{O}_E^{\sigma=(-1)^i}$, allowing arbitrary $b_0\in\mathcal{O}_F$. Let $\Delta_{a,b}$ b…