14 problems

Randomness conjecture for Ripley's statistic. For squarefree $nneq 7\bmod 8$,

Let $d\geq 2$, let $S^{d-1}\subseteq\mathbb{R}^d$ be the unit sphere, and let $\sigma$ be its surface measure. For a continuous function $f:S^{d-1}\to\mathbb{C}$ with…

Round-sphere optimizer conjecture. The standard round metric $g_{\rm st}$ on $\mathbb{S}^n$ is an optimizer for $\mu([g])$.

Let $\mathcal E$ be a cost function on the sphere and let the initial datum be an arbitrary probability distribution, without assuming the well-preparation condition. Well-preparat…

First Schiffer conjecture D in $\mathbb{S}^2$. If such a solution exists, then $\Omega$ is a geodesic ball. The paper states that this first-eigenvalue version is true on…

Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz, and let $\Omega\subset\mathbb{S}^2$ be a topological disk with $C^2$ boundary. Suppose the overdetermined elliptic problem … admits a…

Let $mathcal E(n)$ be the projected lattice points on $S^2$, and let $N_n$ denote their number. The covering radius $M(P_1,\dots,P_N)$ of a configuration on $S^2$ is the least…

Let $mathcal E(n)$ be the projected lattice points on $S^2$, let $N_n$ be their number, and let $sigma$ denote normalized spherical area. For a spherical cap or annulus…

Let $n\geq 2$, let ${\mathbb S}^{n-1}\subset\mathbb R^n$ be the unit sphere, and let $R(p\to q)$ denote the restriction estimate … Here $f\in\mathcal{S}(\mathbb R^n)$ and $C$ is in…

Let $S^{2n+1}$ denote the $(2n+1)$-dimensional sphere, with $n$ a positive integer. Sasaki–Einstein metric abundance conjecture. There are infinitely many families of Sasaki–Einste…

Let $(S^n,g,\operatorname{vol}_g)$ satisfy $\operatorname{Ric}_g\geq\rho g$ with $\rho>0$, and let $g^\rho_{\mathrm{can}}$ be the canonical metric on $S^n$ rescaled so that…

Let $(S^n,g,\operatorname{vol}_g)$ be a Riemannian sphere satisfying $\operatorname{Ric}_g\geq\rho g$ with $\rho>0$. Let $g^\rho_{\mathrm{can}}$ be the canonical spherical metric r…

Minimal-spacing conjecture. As $n\to\infty$,

Nearest-neighbour spacing conjecture. As $n\to\infty$ along square-free integers satisfying $n\neq7\pmod 8$,