MathDB | Open problems in research mathematics

Featured Problems

Last updated 1 hour ago

Navier–Stokes existence and smoothness
2 weeks ago
OpenAI reports that an unreleased internal model produced a finite-time blowup proof for forced Navier-Stokes flow, accompanied by a Lean formalization.
Millennium Prizepartial differential equationsfluid dynamicsAnnounced
Hodge conjecture
1 day agoSeonggil Lee
A lone-author Zenodo deposit claims a proof of the Hodge conjecture using topological confinement, chiral symmetry breaking, and a universal rough operator algebra.
Millennium Prizealgebraic geometrytopologyPreprint · review pending
Iwaniec’s Beurling–Ahlfors norm conjecture
3 days agoAndrea Agazzi, Kari Astala +13 more
A preprint claims the exact Lp norm of the Beurling–Ahlfors transform is p−1 for p≥2 and its dual value for 1<p≤2.
harmonic analysisquasiconformal analysisPreprint · review pending
Yau–Tian–Donaldson conjecture for constant scalar curvature metrics
1 month agoJihao Liu
A preprint claims to construct a polarized smooth projective fivefold that is K-polystable but has no constant-scalar-curvature Kähler metric.
complex geometryKähler geometryPreprint · review pending
Pierce–Birkhoff conjecture
2 weeks ago
A preprint claims a continuous piecewise-quadratic semialgebraic function that lacks a finite lattice-polynomial representation.
real algebraic geometryreal algebraPreprint · review pending

87,484 entries

Syracuse conjecture (Collatz conjecture)
For every positive integer $n\in\mathbb{Z}_{>0}$ , define the Collatz map $T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ by $T(n)=n/2$ if $n$ is even and $T(n)=3n+1$ if $n$ is odd. The conje…
claimed solvedNamed problemdynamical systemsnumber theory
Unique choice is Weihrauch-equivalent to choice on Baire space
Let ${\rm UC}_X$ denote the unique closed choice operation on a represented Hausdorff space $X$ , and let ${\rm C}_X$ denote closed choice on $X$ . For Baire space…
claimed progressResearch corpuschoice principlescomputable analysis
Matroid Secretary Conjecture
Conjecture. There exist an absolute constant $c\ge 1$ and an online algorithm such that, for every finite matroid $M$ and every choice of nonnegative weights, the algorithm selects…
claimed solvedNamed problemcombinatorial optimizationmatroids
Rank-three case of the Kajitani–Ueno–Miyano cyclic orderability conjecture
Problem. Is every finite uniformly dense matroid of rank $3$ cyclically orderable?
claimed solvedNamed problemcombinatoricscyclic orderings
Proportion of critical-line zeros of the Riemann zeta function
Problem. Determine $\kappa$ . In particular, is
claimed progressNamed problemanalytic number theoryasymptotic density