33 problems

Let $\mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\}$, let $\mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\}$, let…

Let $\cal H_\ell$ be the eigenspace of spherical harmonics of degree $\ell$ on the sphere, and let $\mu(u)$ denote the number of nodal domains of a spherical harmonic $u$. Leydold'…

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, let $C_n^\lambda$ denote the Gegenbauer polynomial, and let $\mathbb{N}_0=\{0,1,2,\ldots\}$. For $\delta\geq\lambda+1$, any…

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, and let $({\theta}-t)_+=\max\{{\theta}-t,0\}$. For $\delta\geq\lambda+1$ and any ${\theta}\in(0,\pi)$, define … Beatson–Castell–Xu conje…

Böcherer–Sarnak–Schulze-Pillot conjecture. The basis $\{\psi_n\}$ of Hecke eigenfunctions of $L^2(\mathcal{O}^{\times}\backslash S^2)$ is quantum unique ergodic. The Jacquet–Langla…

Weak convergence conjecture. As $M\to\infty$, $\omega^M\to\omega$ in a suitably weak sense; in particular,

Consider the perturbative analysis of the wave equation with nonlinearity $r^{-2}|\mathring{\slashed{\nabla}}\phi|^2$, and let the initial data be supported on a fixed spherical ha…

Let $W$ be an interaction potential satisfying Assumption, and let $k_{\min}=\operatorname*{arg\,min}_{k\in\mathbb N}\hat W_k$. The single-component resonance condition is … where…

Let $T_{\ell;d}$ be random hyperspherical harmonics on $\mathbb S^d$, and consider the chaotic expansion of their nodal volume into its components. For $d=2$, the fourth chaotic co…

Let $q\geq 3$ be odd, and consider the variances of odd polyspectra associated with random wave models on spherical geometries. Marinucci's conjecture. The variances of odd polyspe…

Non-negativity conjecture. The coefficients $c_{m,d}(\beta)$ are non-negative.

The paper considers a more general analytic family of operators, denoted by $(3.1f)$, whose multiplier is oscillatory and denoted by $(4.29)$; here $j$ is the harmonic index, $k$ i…

Symmetry conjecture. The limit

For $n\geq 1$, let $\lambda_{n,3}$ denote the eigenvalue of the classical multidimensional-scaling kernel on $\mathbb{S}^2$ associated with spherical harmonics of degree $n$. Sign…

Let $P_{j}:=\{u\in\mathbb{R}:u\rho_{j}'(u)=0\}$, and let $f_{\ell;d}$ be a centered, unit-variance Gaussian random eigenfunction on the unit sphere $\mathbb{S}^{d}$ satisfying … Wr…

Let $Y_\ell^0$ be the degree-$\ell$, order-zero spherical harmonic, let $\eta_a$ denote the wavelet at scale $a$, and let $\kappa_0$ and $\widetilde\eta$ be the functions appearing…

Let $S_n^{\delta}(h_{\kappa}^2;f)$ be the $(C,\delta)$ means of the spherical expansion with weight $h_{\kappa}$, and let $\|\cdot\|_{p,\kappa}$ denote the corresponding weighted…

Let $d\geq 2$, let $h_{\kappa}$ be the weight function associated with the symmetric group action on the unit sphere $\mathbb{S}^{d-1}$, and let $S_n^{\delta}(h_{\kappa}^2;f)$ deno…

Let $N$ be a fixed degree, and let a random spherical harmonic of degree $N$ be defined on the three-sphere ${\mathbb S}^3$. A giant component is a nodal component distinguished fr…

Let $f,g\in\mathcal L^2(\mathcal S^n)$ have zonal expansions … For $d\in\mathbb N$, let $f^{(d)}$ and $g^{(d)}$ denote the $d$th derivatives with respect to a fixed rotation parame…

Let $\mathbb S^2$ be the unit two-sphere, let $u$ be a spherical harmonic with Laplace eigenvalue $\lambda$, and let $\mathcal N(u)$ denote its nodal set. Write $H_1(\mathbb S^2)$…

Let $f_l$ be a random spherical harmonic of degree $l$, and let $\Psi$ be the limiting distribution function prescribed by the theorem for the normalized nodal-domain areas. Its th…

Let $f_\ell$ be a centred isotropic Gaussian spherical harmonic of degree $\ell$ on ${\mathcal S}^2$, and let … be its total number of critical points. The variance conjecture. The…

Let $S^d$ be the $d$-dimensional sphere, and let $K$ be a continuous, positive definite, zonal kernel on $S^d$. Write its radial part as $K_r^d:[-1,1]\to\mathbb{R}$, so that…

Simple angular dependence conjecture. The fundamental modes of $\mathbb{B}(R)$ can be written as linear combinations of $u_1(r,\hat{\theta})$.