48 problems

Let $\omega_{N,d} \subseteq \mathbb{S}^d$ be the first $N$ elements of the greedy sequence with respect to $K_{-1}$ as defined in the greedy construction. For $d$-dimensional spher…

For an integer $n>4$, let $S^n$ denote the $n$-dimensional sphere, and say that it has a unique smooth structure when every smooth manifold homeomorphic to $S^n$ is diffeomorphic t…

Proposed conjecture. For $n\equiv 6\pmod 8$ with $n\geq 14$,

An odd-dimensional sphere is the standard sphere $S^{2n+1}$, and a parallelizable manifold is a manifold with trivial tangent bundle. A Sasakian-Einstein metric is a Sasakian metri…

Let $\mathbb S^{l}(\frac{1}{2})\times\mathbb S^{l}(\frac{1}{2})$ be the generalised Clifford torus, viewed as a submanifold of $\mathbb S^{2l+2}$, and consider the associated bihar…

Let $S^{n-1}$ be the unit sphere, and let a positive definite function on $S^{n-1}$ be understood in the usual sense for functions on the sphere. Damase's continuity approximation…

Let $T_1$ be a triangulation of the $n$-sphere and let $S_{n+2}^n$ be the boundary triangulation of the $(n+1)$-simplex. For a coloring $L:V(T_1)\to V(S_{n+2}^n)$, let…

Let $S^m$ be the standard sphere, let $1\le p<\frac{2m}{m-1}$, and let $\mathcal{A}=\{A_\lambda\}_{\lambda\ge1}$ be a family of subsets of $S^m$. An $L^p$-Logvinenko-Sereda family…

Let $m$ be the mass of a material point moving on the sphere $S^{n-1}\subset\mathbb{R}^n$, let the point be placed in a constant homogeneous magnetic field described by the paramet…

The restriction conjecture for the sphere. This estimate should hold when

For each dimension $n\ge 1$ and scale $r>0$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r$. Finite CW-complex conjecture…

For $n\ge 1$, let $S^n$ be the unit $n$-sphere and let $\mathrm{\check{C}}(S^n;r)$ denote its Čech complex at scale $r\in(0,\pi)$. Write $\operatorname{conn}(K)$ for the connectivi…

Let $\mathcal{S}^{(d)}_n$ be the set of $n$-vertex triangulations of the $d$-sphere, let $M(\mathcal{S}^{(d)}_n)$ denote their minimum volume, and let $\mu^{(d)}_n$ be the median o…

Let $S^n$ be the $n$-sphere, let $r\ge 0$, and let $\mathrm{VR}(S^n;r)$ denote the Vietoris–Rips complex formed using the strict-distance convention of the paper. Finite-CW conject…

Let $S^n$ be the $n$-sphere, let $r_n$ be the scale defined in the paper, let $\mathrm{SO}(n+1)$ be the special orthogonal group, let $A_{n+2}$ be the alternating group, and let…

Let $S^n$ be the $n$-sphere, and write $\mathrm{VR}(S^n;r)$ for its Vietoris–Rips complex at scale $r\ge 0$. Critical-scale conjecture. There are countably many critical scales … s…

For each positive integer $k$, let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, and let $\mathrm{dis}(\gamma_{2k+1})$ denote it…

Let $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the odd trigonometric moment curve embedding, let $\Pi_{2k+1}:\mathbb{S}^{2k+1}\to\gamma_{2k+1}(\mathbb{S}^1)$ be the neares…

Let $\alpha:\mathbb{S}^1\to\mathbb{S}^2$ be the embedding … and let $R_\alpha$ be its induced correspondence. Variant correspondence conjecture. … This correspondence arose from co…

Let $\psi_{2k+1}:\mathbb{S}^{2k+1}\to\mathbb{S}^1$ and $\psi_{2k}:\mathbb{S}^{2k}\to\mathbb{S}^1$ be the projections associated with the trigonometric moment curve embeddings, and…

Let $k\in\mathbb{N}$, and let $\gamma_{2k}:\mathbb{S}^1\to\mathbb{S}^{2k}$ and $\gamma_{2k+1}:\mathbb{S}^1\to\mathbb{S}^{2k+1}$ be the trigonometric moment curve embeddings defined…

Finiteness conjecture. Only finitely many even-dimensional spheres admit non-round Einstein metrics of cohomogeneity one.

Sphere extension conjecture. The $L^2\to L^{(2n+4)/n}$ extension estimate should hold uniformly in $q$:

Let $T^3=S^1\times S^1\times S^1$ be the three-dimensional torus, embedded minimally in the round four-dimensional sphere $\mathbb{S}^4$. Two embeddings are considered equivalent i…

Let $\mathcal{S}_{3,1}=\{\bm{x}\in\mathbb{R}^3:x_1^2+x_2^2+x_3^2=1\}$, and let $E_{2,2}(A)$ denote the number of quadruples $(\bm{x}_1,\bm{x}_2,\bm{x}_3,\bm{x}_4)\in A^4$ satisfyin…