23 problems

Maximizer structure conjecture. The following properties hold:

Let $\mathcal{S}_{ij}$ be a Schubert variety in ${\rm Gr}(2,n)$. GD-degree conjecture. If $1<j-i<n-1$, then the Schubert variety $\mathcal{S}_{ij}$ has GD degree $2$. The paper rep…

Let $\mathcal{C}\subset\mathbb{P}^3$ be a curve, and let the Chow threefold and secant surface associated with $\mathcal{C}$ have Grassmannian-distance degree ${\rm GDdegree}$ and…

Let $\mathcal{C}\subset\mathbb{P}^3$ be a curve, and consider the associated Chow threefold, secant surface, and tangent curve of lines, together with a data line and an optimal li…

Let a surface in ${\rm Gr}(2,4)$ be defined by generic polynomials of degrees $d_1,d_2$, and let a threefold in ${\rm Gr}(2,4)$ be defined by a generic polynomial $f$ of degree $d$…

Perturbation conjecture. For every $\epsilon>0$, there is a set $S'=\{p'_1,p'_2,\ldots,p'_n\}$ in bijective correspondence with $S$ such that each corresponding pair satisfies…

Let $d\ge 2$ be an even integer, let $A,B\subseteq\mathbb F_q^d$, and define … where $\|v\|=\sum_{i=1}^dv_i^2$. Two-set Erdős–Falconer conjecture. If … for a sufficiently large con…

Even-dimensional Erdős–Falconer conjecture. If $|A|\ge Cq^{d/2}$ for a sufficiently large constant $C$ independent of $q$, then

Dot-product realisation conjecture. One has

Let $S$ be a set of $n$ points in the plane. Let $m_1(S)$ and $m_2(S)$ be the numbers of occurrences of the smallest and second smallest distances in $S$, respectively, and let…

Linear weighted-distance conjecture. The number of equilibria of the weighted Euclidean distance function defined by $n$ points with positive real weights in $\mathbb{R}^2$ is at m…

Let $d \geq 2$. A graph is a forbidden minor for $d$-flattenability when it is a minimal obstruction to $d$-flattenability. For an edge $e$ of a graph $G$, $(G \setminus e,e)$ deno…

Let $G$ be a graph, let $f$ be a nonedge of $G$, and let $G \cup f$ denote the graph obtained by adding $f$. An atom of $G \cup f$ is a graph-theoretic atom containing $f$, and an…

Let $q$ be a prime power, let $d$ be a positive integer, and let $\Delta\geq 2$. A distance tree is a tree whose edges are assigned nonzero distances in $\mathbb{F}_q^d$, and…

Bipartite rigidity conjecture. The framework $K(A,B)$ is rigid in $\mathbb{R}^d$, unless either the points in $A$ or the points in $B$ are contained in a hyperplane in…

Sharp threshold conjecture. The threshold for the existence of a reconstructible set of size linear in $n$ is sharp and occurs at

One-point deletion conjecture. There exist $p_i\in P$ and $q_i\in Q$, together with some $A\in O(2)$ and $\vec{t}\in\mathbb{R}^2$, such that

Koh–Shen conjecture. If $|E||F|\gg_C q^d$, then

Polar-duality conjecture. If the first sum attains its maximum at a point $N$, then the second sum attains its minimum at $N$, and conversely. If the first sum is independent of th…

Cross-polytope conjecture. Then $A_1,\ldots,A_{2n}$ are the vertices of an $n$-dimensional cross-polytope.

Regular simplex conjecture. Then $A_1\ldots A_{n+1}$ are the vertices of a regular simplex.

Regular polygon conjecture. Then $A_1,A_2,\ldots,A_n$ are the vertices of a regular polygon inscribed in a circle concentric to $\Gamma$.

Let $P$ and $Q$ be two disjoint nonempty closed sets in Euclidean space, and let a $k$-sector be the geometric object associated with $P$ and $Q$ that separates their distance-base…