49 problems

Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does … ex…

Let $r,k\geq 2$ be fixed. Let $A\subset\mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that, if there are at least…

Is there a set of $n$ points in $\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is …

Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\epsilon n$ with no th…

Let $N$ points be given in the Euclidean plane, and let a unit distance mean a pair of points whose Euclidean distance is $1$. The unit distance conjecture. The number of unit dist…

Is a finite locally free group scheme killed by its order?

Let $\mathbb{K}$ be a field of characteristic zero, let $nge 1$ be an integer, and let $f1,\dots,fninmathbb{K}[X1,\dots,Xn]$. Let $F:\mathbb{K}^n\to\mathbb{K}^n$ be the polynomial ma…

Let $f:X\to\Delta$ be a semistable one-parameter family of complex projective varieties over a holomorphic disk, with $X$ smooth and central fiber $X_0=f^{-1}(0)$ a reduced simple…

For every amphicheiral knot $K$, does there exist $f(z)\in\mathbb{Z}[z]$ such that …

If $C,D\subseteq\mathbb{R}^{2}$, define the distance between $C$ and $D$ by … Let $h(n)$ be the maximal number of unit distances between disjoint convex translates. That is, $h(n)$…

Let $A\subset \{x\in\mathbb{R}^2:|x|<r\}$ be a measurable set with no integer distances, that is, such that $|a-b|\notin\mathbb{Z}$ for any distinct $a,b\in A$. How large can the m…

Machado–Seade conjecture. The following statements are equivalent: $(D,0)$ is weighted homogeneous; there exists a holomorphic vector field tangent to $D$ with an isolated singular…

Are the total Cartier indices bounded for a bounded family of varieties with rational singularities?

Is the index of a numerically trivial log-canonical foliated log Calabi--Yau triple uniformly bounded in dimension three?

Let $X$ be a normal projective variety of dimension $n$, and let $\mathcal{F}$ be a foliation on $X$ of rank $r$. Let $H_{1},\ldots,H_{n-1},H$ be ample divisors on $X$, and let $C$…

Is every infinite-dimensional Lie group modelled on a complete locally convex space regular?

Does there exist an absolute constant $C>0$ such that, for every positive integer $n$ and every $n$-dimensional convex body $K$, the lattice covering density satisfies …

Nonnegativity conjecture. All stringy Hodge numbers $h^{p,q}_{\rm st}(X)$ are nonnegative.

Let $A\subset\mathbb{R}^{2}$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertex set $A$, where two vertices…

Draw $n$ squares inside the unit square with no common interior point. Let $f(n)$ be the maximum possible sum of the side-lengths of the squares. Is $f(k^2+1)=k$?

Let $G=\operatorname{conv}\{(-c,0),(c,0),(0,s)\}$ be the isosceles triangle whose equal sides have length $1$, whose base angles are $36^{\circ}$, and whose apex angle is…

How many pairwise non-overlapping infinite unit cylinders can simultaneously touch a unit ball?

Can the braid-group action on full exceptional collections fail to be transitive for $D^b(X)$ when $X$ is a smooth projective variety?

In Theorem 1.1 of the cited paper, is $\Lambda\setminus\mathcal{G}$ a pluripolar subset of $\Lambda$, where $\Lambda\simeq\mathbb{P}^{N}$?

Does there exist a constant $c>0$ such that, for all sufficiently large $n\in\mathbb{N}$, … Here $a_1(X)\geq a_2(X)\geq\cdots$ are the multiplicities of the distinct distances dete…