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Schacht’s and Lamaison’s uniform Turán-density questions
14 hours agoHao Lin, Jie Ma +1 more
Lin, Ma, and Zhou characterize 3-uniform hypergraphs with vanishing cherry density and compare jump behavior for the principal uniform Turán densities.
extremal combinatoricshypergraph theoryPreprint · review pending
Gersho’s crystallization conjecture
14 hours agoAta Deniz Aydin, Mikaela Iacobelli
Aydin and Iacobelli claim asymptotic local minimality of the body-centered cubic lattice under small periodic perturbations in three-dimensional quantization.
quantizationdiscrete geometryPreprint · review pending
Brandt conjecture on the binding number of K_k-free graphs
14 hours agoJifu Lin, Heng Yang
Lin and Yang claim that every finite nonempty K_k-free graph has binding number at most k−2 for k≥6, with equality examples.
extremal graph theoryPreprint · review pending
Fuglede’s conjecture
15 hours agoGal Binyamini, Avner Kiro +2 more
Binyamini, Kiro, Kozma, and Pila develop Bessel-zero definability and point-counting tools that improve density estimates motivated by Fuglede’s conjecture.
o minimalitytranscendence theoryPreprint · review pending
Eisenbud–Schreyer Ulrich-bundle existence question
15 hours agoJagadish Pine
Pine claims existence of Ulrich bundles for every polarized smooth surface over an algebraically closed field of positive characteristic and weakly Ulrich bundles in all dimensions.
algebraic geometryvector bundlesPreprint · review pending

Díaz–Park sharpness conjecture for fusion systems
Let $S$ be a finite $p$ -group, let $\mathcal{F}$ be a fusion system over $S$ , and let $M=(M_{*},M^{*})$ be a Mackey functor over $\mathcal{F}$ on $\mathbb{F}_{p}$ . Write…
open · progress reportedResearch corpusfusion systemshigher limits
Torsion conjecture
For a number field $K$ and an abelian variety $A$ over $K$ , let $A(K)$ denote the group of $K$ -rational points of $A$ and let … be its torsion subgroup, which is finite since…
openNamed problemalgebraunsolved problem
Grothendieck's homotopy hypothesis for infinity-groupoids
Grothendieck's conjecture. The functor $\overline{\Pi_\infty}$ is an equivalence of categories.
openResearch corpushomotopy theoryinfinity groupoids
P versus NP problem
Problem. Determine whether …
openMillennium Prizemillennium prizeunsolved problem
Extended states in the Anderson model in dimension ν ≥ 3
Problem (Simon's Problem 1). Prove that for every $\nu\ge3$ there are $a<b$ , with $b-a$ small, and an open interval $I\subset\mathbb R$ such that, almost surely, $H_\omega$ has pur…
openNamed problemphysics