MathDB | Open problems in research mathematics

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Gersten conjecture
14 hours agoMorten Lüders
Morten Lüders proves the Gersten conjecture for p-adic étale Tate twists of smooth mixed-characteristic schemes in the Nisnevich topology and extends the associated cycle-class-map framework.
algebraic geometrymotivic cohomologyPeer reviewed
Fasi–Higham positivity conjecture for Padé approximants
2 days agoFrancesco Scardino
Francesco Scardino gives exact counterexamples to the unrestricted positivity conjecture and proves eventual positivity under a spectral-minimality condition.
numerical analysismatrix analysisPreprint · review pending
Betke–Henk–Wills lattice-point conjecture
2 days agoFei Xue
Fei Xue proves Malikiosis’s discrete monotonicity property for every compact convex set when the relevant lattice subgroup has rank at most two.
convex geometrydiscrete geometryPreprint · review pending
Haiman–Stembridge monomial-immanant positivity conjecture
2 days agoKeith McNulty
Keith McNulty proves Haiman’s q=1 positivity statement for monomial immanants of every hook shape.
algebraic combinatoricsrepresentation theoryPreprint · review pending
Ni–Zheng curvature conjecture for Kähler C-spaces
2 days agoRun-Qiang Jian, Zhuhong Zhang
Run-Qiang Jian and Zhuhong Zhang prove the conjectured curvature inequalities for classical Kähler C-spaces.
complex differential geometryKähler geometryPreprint · review pending

Goldbach's conjecture
Conjecture (strong, or binary, Goldbach conjecture). Every even integer $n>2$ is the sum of two primes: …
open · progress reportedCelebratedcombinatoricsnumber theory
Poincaré conjecture
Poincaré conjecture. Every such $M$ is homeomorphic to the three-sphere …
solvedNamed problemtopologyunsolved problem
Navier–Stokes existence and smoothness
Whole space. The data decay rapidly: for all multi-indices $\alpha$ , all $m\ge0$ and all $K\ge0$ , … A solution is admissible if $u,p\in C^\infty(\mathbb R^3\times[0,\infty))$ and t…
open · progress reportedMillennium Prizemillennium prizeunsolved problem
Erdős Problem #690 — Let $d_k(p)$ be the density of those integers whose $k$ th smallest prime factor is $p$ (i.
Let $d_k(p)$ be the density of those integers whose $k$ th smallest prime factor is $p$ (i.e. if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$ ). For fixed $k\geq 1$ is…
open · solution reportedErdős problemnumber theory
General-coefficient C*-algebraic Bieberbach bounds
Let $A$ be a unital C-algebra. Suppose the normalized power series … converges in the norm of $A$ and is injective on the norm-open unit ball $\{u:\|u\|<1\}$ . Must … Does the asser…
open · solution reportedNamed problem