5 problems

Let $\Gamma=SL_2(\mathbb{Z})$ be the full modular group, let $\mathbb{X}=\Gamma\backslash\mathbb{H}$ be the modular surface, and let $\phi$ be an even Hecke–Maass cusp form with ei…

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…

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…

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…