MathDB | Open problems in research mathematics

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Navier–Stokes existence and smoothness
4 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
Iwaniec’s Beurling–Ahlfors norm conjecture
1 week 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
3 weeks ago
A preprint claims a continuous piecewise-quadratic semialgebraic function that lacks a finite lattice-polynomial representation.
real algebraic geometryreal algebraPreprint · review pending
Schiffer and Pompeiu conjectures
1 week agoJizhou Guo
Jizhou Guo constructs convex non-ball domains disproving the Schiffer and Pompeiu conjectures in dimensions 3, 4, 6, 8, 10, and 14.
partial differential equationsspectral geometryPreprint · review pending

Knowing When to Stop: IGP-Bench and the Multi-Agent System GALOIS for the Inverse Galois Problem
Shaowu Zhang, Jan Pavel Safrata, Brent Kong, Eric Lu, Sathvik Redrouthu, Caiman Moreno-Earle, Tony Yue YU2 days ago

Explicit Realization of PSU₃(8) as a Galois Group over ℚ
This problem asks for an explicit realization of the projective special unitary group $\mathrm{PSU}_3(8)$ as a Galois group over $\mathbb{Q}$ . The goal is to construct a separable…
openNamed problem
Explicit Realization of PSU₃(4) as a Galois Group over ℚ
This problem asks for an explicit realization of the projective special unitary group $\mathrm{PSU}_3(4)$ as a Galois group over $\mathbb{Q}$ . The goal is to construct a separable…
openNamed problem
The conjecture that the Schur-degree threshold equals the Schur number
For $n\ge 2$ , let $S(n)$ be the $n$ th Schur number. Let $L(n)$ be the least length such that every sequence $A$ in $\mathbb N_+$ with $|A|=L(n)$ and average $\mu(A)\le n$ satisfies…
claimed progressResearch corpusadditive combinatoricsschur numbers
Maximum number of intersections between two simple heptagons
Problem. Determine the exact value of $f(7,7)$ . In particular, is …
openNamed problem
The conjectured first-difference sequence for the Fibonacci word
Let $f$ be the infinite Fibonacci word, let $\Sigma$ be its alphabet, and define … Here $\operatorname{Cl}(w)$ is the number of distinct closed factors of $w$ , and $F_n$ denotes th…
claimed progressResearch corpuscombinatorics on wordsfibonacci word