27 problems

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 …

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 …

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

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

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

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 $\Lambda\simeq\mathbb{P}^{N}$?

Can the torsion-free hypothesis be removed from the positivity theorem for coherent quotients of tensor powers of $\Omega^1_{\mathcal X}(\log\Delta)$?

What is the exponential rate of the Cohn--Elkies linear-programming bound for sphere packing as $d\to\infty$?

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 $C$…

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…

Let $k\geq 3$ be an odd integer. For any integer $n$, let $N_k(n)$ be the number of pairs of positive integers $(b_1,b_2)$ such that $b_1,b_2\leq (k-1)/2$, $b_1+b_2\geq (k+1)/2$, a…

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

Let $\mathbb{K}$ be a field of characteristic zero, let $n\ge 1$ be an integer, and let $f_1,\dots,f_n\in\mathbb{K}[X_1,\dots,X_n]$. Let $F:\mathbb{K}^n\to\mathbb{K}^n$ be the poly…

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…

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…

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

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

For bounded $S\subset\mathbb Z^d$, is $\beta_{\mathbb Z}(S)=2^d$ if and only if $\operatorname{conv}(S)$ is unimodularly equivalent to $[0,m]^d$?

Conjecture 6.1 (NuevaMirada). If $P$ is a simple $3$-polytope with face vector $(p_3,p_4,\ldots,p_m)$, such that $\sum_{k\geq 7}p_k\geq 3$, then

For a reducible quasi-reduced free-group word $w$, let $F_w$ be the CW complex whose cells encode crossing matchings and their resolutions. Must $F_w$ be contractible?

Let $T$ be a positive-definite weighted tree with exactly one vertex $v$ satisfying $w(v)<d(v)$. Must $T$ contain a vertex that is irreducible in its lattice $L(T)$?

For the benchmark's $G_\beta$-invariant clean triangulations and squeeze maps of the strip, prove that $\sqrt 3$ is the infimum of the realized values of $\beta$.

For semifree noncommutative differential graded algebras, are stable tame isomorphism, quasi-isomorphism, or derived Morita equivalence algorithmically decidable?