16 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…

Conjecture on ternary strong nodal domains. For any eigenfunction $f$ of $H(n,3)$, $n\geq 1$, with eigenvalue $3n$ we have $\mathrm{SND}(f)\geq n+1$.

Bírógyiköğlü–Hordijk–Leydold–Pisanski–Stadler conjecture. For every $1\leq i\leq n-2$, there is an eigenfunction $f$ of $H(n,2)$ with eigenvalue $2i$ such that $\mathrm{SND}(f)=2$.

Nodal-domain count conjecture. There are constants $c_1(\epsilon,p,q)$ and $c_2(\epsilon,p,q)$ such that, with probability $1-o_N(1)$, the $i$th eigenvector $\bm{\varphi}$ satisfie…

Let the random plane wave (RPW) be the stationary Gaussian field in the plane discussed in the paper, and consider its nodal lines and nodal domains at level zero. Let critical per…

Full-support eigenfunction conjecture. There exists a choice of eigenfunctions $\psi_n$ forming an orthonormal basis of $L^2(\mathcal{G})$ and a subsequence…

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with piecewise smooth boundary, or a compact Riemannian surface, with or without boundary. Let $\Delta$ be the Laplace–Beltram…

Small-parameter stability conjecture. There exists $h_0>0$ such that for $0<h\leq h_0$, the Courant-sharp cases for the Robin problem are the same, except the fifth one, as those f…

Let $\Gamma$ be a compact metric graph, let $\partial\Gamma$ denote its boundary, let $\beta$ be its first Betti number, and let $\omega_n$ denote the Neumann surplus associated wi…

Let $f=f_{\alpha;T}$ be a band-limited Gaussian function on an $n$-dimensional compact manifold, with $n\geq 2$ and $\alpha\leq 1$, and let $\Psi_{n;\alpha}$ be the limiting distri…

Let $(\phi_n)$ be a sequence of Laplace eigenfunctions, and let $\mu(\phi_n)$ denote the number of nodal domains of $\phi_n$. Let $Sq_1$ be the square of area $1$, with first eigen…

Hexagonal conjecture for Pleijel.

Consider the Neumann problem in the square and a sequence of eigenfunctions associated with an infinite sequence of eigenvalues. Nodal-domain divergence conjecture. The number of n…

Let $(M^n,g)$ be a manifold with boundary, and let $u_\lambda$ be a Dirichlet eigenfunction associated to the eigenvalue parameter $\lambda$. For each nodal domain $A_i$ of…

Let $\mathcal{P}$ be the collection of probability measures on the unit ball $B(1)\subseteq\mathbb{R}^{2}$, and let $c_{\max}$ denote the maximal value attained by the Nazarov–Sodi…

Let $\mathcal{P}_{Symm}$ be the family of symmetric probability measures on $\mathcal{S}^{1}$, and let $d_{\max}$ denote the maximal value attained by the Nazarov–Sodin constant…