126 problems

A submanifold of Euclidean space is biharmonic when its inclusion map is a biharmonic map, equivalently a critical point of the bienergy functional. It is minimal when its mean cur…

Graham's conjecture. For every triangle $T$,

Borsuk's conjecture.

Let Euclidean space of any dimension be tiled by hypercubes whose positions lie in a lattice. Minkowski's conjecture. Some pair of hypercubes must meet face-to-face. This is the ge…

Maeta's generalized Chen conjecture. Any $k$-harmonic submanifold in the Euclidean space $\mathbb R^{n+1}$ is minimal.

Corsten–Frankl's conjecture. A simplex is diameter-Ramsey if and only if its circumcenter belongs to its convex hull.

A geodesic net in the Euclidean plane is a finite multigraph embedded in the plane whose edges are geodesic segments; its vertices are partitioned into balanced and unbalanced vert…

A $d$-dimensional simplex is the convex hull of $d+1$ affinely independent points. A simplex is a $k$-reptile if it can be dissected into $k$ smaller simplices, each similar to the…

Let $P_{1},\dots,P_{n}$ be distinct points in ${\mathbb R}^{3}$, and let $\operatorname{At}(P_{1},\dots,P_{n})$ denote the associated complex-valued Atiyah determinant. Atiyah's no…

A circle-center set is a subset of $\mathbb{R}^2$ that is not contained in any line and that contains the center of the circle through any three non-collinear points of the set. Gu…

Let $M$ be a connected compact embedded $H$-surface in $\mathbb{R}^3$ whose boundary is a round circle. Sphericality conjecture. The surface $M$ is spherical. In Euclidean space th…

Let $S$ be a set of $s$ points in three-dimensional Euclidean space, and let $n^2$ lines be given, each incident to $n$ points of $S$, with no $n$ lines coplanar. Bourgain's incide…

Let $\Lambda_2$, $\Lambda_8$, and $\Lambda_{24}$ be the hexagonal lattice in $\mathbb{R}^2$, the $E_8$ root lattice in $\mathbb{R}^8$, and the Leech lattice in $\mathbb{R}^{24}$, r…

Let $\gamma$ be a closed curve in Euclidean space $\mathbb{E}^d$ of length $L$. An orthogonal projection of $\gamma$ to $\mathbb{E}^{d-1}$ means the restriction to $\gamma$ of an o…

Let $n$ points be given in the Euclidean plane, and consider all angles formed by triples of these points. Minimum-angle conjecture. Among all angles formed by triples of $n$ point…

Let $S$ be an orthocentric simplex in any dimension, with facet center ${\cal F}$ and incenter ${\cal I}$. Regularity conjecture. If ${\cal F}={\cal I}$, then $S$ must be regular.…

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

Kähler-structure conjecture. Every globally defined harmonic morphism $\phi:\mathbb{R}^4\to\mathbb{C}$, whether submersive or not, is holomorphic with respect to a Kähler structure…

Invariant side-square area conjecture. The sum of the areas of the constructed squares remains invariant throughout $\mathcal P$ if and only if the circumcenter of the triangle coi…

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

Canonical Ramsey conjecture. Every Ramsey set should also be canonically Ramsey.

For a finite point set $V\subset\mathbb{R}^d$, a source $s\in V$, and $\epsilon>0$, a Steiner shallow-light tree is a tree spanning $V$ that may use Steiner points, with root stret…

Let $R$ be a finite subset of Euclidean space. It is called Ramsey if, for every number of colors $r\in\mathbb{N}$, there is a dimension $n=n(R,r)$ such that every $r$-coloring of…

Let $\mathcal{E}$ and $\mathcal{E}_c$ be nested ellipses admitting a family $\mathcal{T}$ of Poncelet triangles, and let $T=ABC$ be a triangle in $\mathcal{T}$. Let $T'=A'B'C'$ be…

Let $d$ be the metric induced on a finite subset of $\mathbb{R}^m$, let $\mathcal{I}$ be a subset of $\operatorname{peel}(d)$, and let $d|_{\mathcal{I}}$ denote the restriction of…