8 problems

McMullen's Arithmetic Chaos conjecture. There is a compact subset $\mathcal{Y}\subset\mathcal{X}$ such that, for every real quadratic field $K$, the set of closed geodesics defined…

Let $\mathfrak{X}:=\mathcal{H}/SL(2,\mathbb{Z})$ be the modular surface, let $\Delta$ be its Laplacian, and let $\phi_i$ be a sequence of $L^2$-normalized eigenfunctions of…

Let $D$ be a squarefree fundamental discriminant. For $0\leq r<R$, let $A_{r,R}(w)$ be the hyperbolic annulus centered at $w$ with inner radius $r$ and outer radius $R$, and let…

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Let $\alpha_1$ and $\alpha_2$ be oriented closed geodesics on modular surfaces, with discriminants $D_1$ and $D_2$. Their $p$-arithmetic intersection is the arithmetic intersection…

Shapes of unit lattices conjectures. The closure $\overline{\Omega}$ in the modular surface is non-compact, the closure $\overline{\Omega}$ in the modular surface has non-empty int…

Let $\phi$ be a Hecke–Maass cusp form on $\mathbb{X}$, let $\mathcal{F}_Y=\{z\in\mathbb{X}:\operatorname{Im}(z)\geq Y\}$, and let $\nu>0$. Quantitative cusp-region QUE conjecture.…

Sarnak's equidistribution conjecture. The geodesics of both the ambiguous and reciprocal families equidistribute in $\Gamma\backslash\mathbb{H}$ when ordered by discriminant.