Mathematics

Is there an efficient algorithm for the minimum-rank problem?5 questions

Completing the near-extreme min-rank classification of side-information digraphs: the value $n-2$ and min-rank two over nonbinary fields
Open for about 14 years
Lossless clique compression and provable guarantees for minrank computation in index coding
Open for about 7 years
Determining the minimum rank of the signed adjacency matrices of a graph over all signings
Open for about 9 years
Unconditional complexity of Gröbner-basis solving of the generalized MinRank problem: the Fröberg-type maximal-rank conjecture in the over-determined regime and polynomiality of F5 in the degree of the ideal
Open for about 14 years
Near-Optimal Inapproximability of Orthogonality Dimension and Minrank, and NP-Hardness of General Index Coding
Open for about 3 years

The character problem for Lie superalgebras of type

\mathfrak{osp}

6 questions

The

11/8

conjecture8 questions

2D

instantaneous frequency4 questions

The

3n+1

conjecture and complex-analytic methods5 questions

The

k

-server conjecture3 questions

P=NP

？6 questions

r=r_{\omega}

？ and

s=s_{\omega}

?6 questions

Classification of tight contact structures on three-dimensional manifolds5 questions

4D Smooth Poincaré Conjecture10 questions

Existence of Hadamard matrices7 questions

Atiyah conjecture9 questions

Arnold conjecture9 questions

Gröbner-Shirshov bases of Artin groups9 questions

Existence problem of strange attractors in the Hénon map family9 questions

Mathematical study of Einstein field equations6 questions

Bass-Quillen conjecture9 questions

Bass conjecture and Soule conjecture9 questions

The union-closed sets conjecture6 questions

The Borel conjecture: every homotopy equivalence between closed aspherical manifolds is homotopic to a homeomorphism6 questions

Pólya conjecture9 questions

Hydrodynamic limit of the Boltzmann equation7 questions

Boltzmann-Grad limit of the Boltzmann equation10 questions

Bochner-Riesz multiplier problem8 questions

Incompressible Navier-Stokes equations6 questions

Brown problem8 questions

Brennan conjecture9 questions

Bloch constant7 questions

Broué's abelian defect group conjecture10 questions

Irreducible decomposition of difference algebraic varieties9 questions

Related problems in ordinary differential equations and stochastic analysis6 questions

The canonical inner model problem for supercompact cardinals7 questions

Is there a polynomial algorithm for large number factorization?9 questions

Flux compactifications of string vacua and generalized complex geometry5 questions

Decidability problem of the theory of real numbers with exponentials7 questions

Existence of one-way functions9 questions

Completing the near-extreme min-rank classification of side-information digraphs: the value n−2n-2n−2 and min-rank two over nonbinary fields

Lossless clique compression and provable guarantees for minrank computation in index coding

Determining the minimum rank of the signed adjacency matrices of a graph over all signings

Unconditional complexity of Gröbner-basis solving of the generalized MinRank problem: the Fröberg-type maximal-rank conjecture in the over-determined regime and polynomiality of F5 in the degree of the ideal

Near-Optimal Inapproximability of Orthogonality Dimension and Minrank, and NP-Hardness of General Index Coding

Closed formula for Borel highest-weight conversion of finite-dimensional osp(2m|2n)-modules in the signed epsilon-delta sequence cases left open by Cheng–Kwon

The fork-representation character conjecture for osp(2m∣2n)\mathfrak{osp}(2m|2n)osp(2m∣2n): a positive supersymmetric Schur expansion for [0,…,0,r,p−r][0,\ldots,0,r,p-r][0,…,0,r,p−r]

Generalised Verma modules, closed character formulae, and cohomology for higher-rank orthosymplectic Lie superalgebras ospk∣2n\mathfrak{osp}_{k|2n}ospk∣2n​

Uniform closed character theory for finite-dimensional irreducible modules over the general ortho-symplectic Lie superalgebras (II-type, types B(m|n) and D(m|n))

Maximal atypical block equivalence for osp(2k+2∣2k)\mathfrak{osp}(2k+2|2k)osp(2k+2∣2k) versus osp(2k+1∣2k)\mathfrak{osp}(2k+1|2k)osp(2k+1∣2k)

Geometric underpinning of super duality for ortho-symplectic and related Lie superalgebras

The SWF–Pin(2)-surjectivity conjecture for odd-type involutions on spin rational homology spheres

Definite spin and E8E_8E8​-boundings of homology 3-spheres: Tange's invariants ds\mathfrak{ds}ds, g8g_8g8​, and the (11/8)-linked questions

Manolescu's κ\kappaκ-invariant of Dehn surgeries on knots: the odd-odd case and the ±2\pm 2±2 ambiguity

The 11/8-conjecture (Matsumoto): b2≥118∣σ∣b_2\ge\frac{11}{8}|\sigma|b2​≥811​∣σ∣ for closed smooth spin 4-manifolds

Topological criteria for Floer KGK_GKG​-split homology spheres and the K-theoretic route to the 11/8 conjecture

Intersection-form constraints for spin 4-manifolds bounded by Seifert rational homology 3-spheres: extending the μˉ\bar\muμˉ​ and κ\kappaκ inequalities to non-orientable bases and degenerate cobordisms

Topologically but not smoothly H-slice knots in punctured connected sums of K3 surfaces for n ≥ 4 (Problem 4.26 of Konno–Miyazawa–Taniguchi)

The 11/8-Conjecture for Closed Smooth Spin 4-Manifolds (Matsumoto's Conjecture)

Statistical performance theory for the wavelet-domain monogenic AM/FM estimators of the Riesz–Laplace transform: bias and variance under noise and finite sampling

Mathematical foundations of idempotent-generated multidimensional analytic signals: existence of extended Hilbert operators, necessity of multiplier idempotency conditions, and boundary-value structure

Robust multicomponent AM-FM image demodulation: stability at energy-operator singularities, quantified accuracy beyond narrowband assumptions, and principled multiscale filterbank fusion

Robust multiscale AM-FM demodulation and image reconstruction for real digital images: out-of-passband and low instantaneous frequencies, nonideal sampling, and principled filterbank selection

The Collatz (3n+1) conjecture, equivalently the nonvanishing of the arrival coefficients in Masetti's iterative functional equation

Fixed-point classification and orbit asymptotics for the Collatz operator T\mathcal{T}T: the open problems of Neklyudov's functional-analytic approach

Holomorphic solution spaces of the functional equations satisfied by generating functions of generalized Collatz dynamics

Logical strength of the linear-operator necessary condition for the Collatz conjecture (Soileau's Fourier-average condition)

The Collatz 3x−13x-13x−1 problem (the 3n+13n+13n+1 problem for negative start values): exactly three linearly independent holomorphic solutions of the Berg–Opfer functional equations

The Randomized Competitive Ratio of Weighted kkk-Server on Uniform Metrics: Closing the Gap between Ω(2k)\Omega(2^k)Ω(2k) and exp⁡(O(k2))\exp(O(k^2))exp(O(k2))

The Harmonic k-server conjecture: is the Harmonic algorithm k(k+1)2\frac{k(k+1)}{2}2k(k+1)​-competitive on every metric space?

The infinite-server problem across metric spaces: which spaces are competitive, and what is the competitive ratio on depth-D trees and layered graphs

P versus NP and the finite descriptive-complexity separations: NP versus co-NP (Σ11\Sigma^1_1Σ11​ versus Π11\Pi^1_1Π11​) and P versus Δ11\Delta^1_1Δ11​ over finite structures

Fixed-Polynomial Circuit Lower Bounds for ⊕P\oplus P⊕P, Uniform MAMAMA, PNPP^{NP}PNP, and NPNPNP

Does the polynomial hierarchy collapse? (Is the polynomial-time hierarchy infinite?)

Provability of P vs. NP in algorithmically verifiable mathematics (AV-mathematics)

Open questions of the Verifier Trade-off Theorem: certificate-length versus verification-time trade-offs beyond deterministic worst-case verification

NP versus BQP: is NP contained in bounded-error quantum polynomial time?

Weakly tight MAD families in the b<s\mathfrak{b}<\mathfrak{s}b<s regime and Sacks indestructibility

Splitting versus closed almost disjointness: does s=ℵ1\mathfrak{s} = \aleph_1s=ℵ1​, or at least sω=ℵ1\mathfrak{s}_\omega = \aleph_1sω​=ℵ1​, imply aclosed=ℵ1\mathfrak{a}_{closed} = \aleph_1aclosed​=ℵ1​?

Existence of a completely separable MAD family in ZFC (the Erdős–Shelah problem)

The dual-reaping number versus the pair-splitting number, and star-uniformity of Borel ideals versus the cofinality of the null ideal (Question 4.8 of Hrušák–Meza-Alcántara–Minami, 2010)

Open boundary cases of the negative strong polarized partition relation: the sigma-reaping number under rσ=c\mathfrak{r}_\sigma=\mathfrak{c}rσ​=c and the real-valued measurable endpoint

Is the dual reaping number rd\mathfrak{r}_drd​ at most the pair-splitting number spair\mathfrak{s}_{\mathrm{pair}}spair​?

Existence of tight contact structures on closed hyperbolic 3-manifolds (the hyperbolic case of Eliashberg's tight-existence question)

Classification of tight contact structures on the four-fiber Brieskorn spheres Σ(2,3,7,41)\Sigma(2,3,7,41)Σ(2,3,7,41) and Σ(2,3,11,13)\Sigma(2,3,11,13)Σ(2,3,11,13)

Tight contact structures on the Whitehead link surgery family M(n,r)M(n,r)M(n,r): classification beyond the resolved parameter range and universal tightness

Complete classification of tight contact structures on the remaining surgeries on the figure-eight knot and Stein fillability of the Phi-counted structures

General slice-Bennequin characterization of tight contact 3-manifolds and the tight stratum with vanishing Ozsváth–Szabó contact invariant (Hedden–Raoux)

Pochette surgery on the 4-sphere: homology-4-sphere outputs and exotic smooth structures (Suzuki–Tange Questions 5.1–5.5)

Is the Gluck twist of S^4 about an arbitrary 2-knot diffeomorphic to S^4? (Kirby problem 4.23)

Standardness of the extended Cappell–Shaneson homotopy 4-spheres and inequivalent smooth structures on the associated nonorientable 4-manifolds (Torres, 2026, Questions D–F)

Standardness of Cappell-Shaneson homotopy 4-spheres: Gompf's conjecture on Cappell-Shaneson matrices

Must a 3-manifold that embeds smoothly in a homotopy 4-sphere embed smoothly in the standard 4-sphere? (Budney-Burton, Question 8.3)

Topologically but not smoothly integrally split 2-component links in S4S^4S4 (Golla–Marengon Questions 1.4 and 1.5)

Generalized Property R Conjecture: handleslide equivalence of R-links to the unlink and the status of the generalized-square-knot families Lc/dp,qL^{p,q}_{c/d}Lc/dp,q​

Completing the PL classification of the census of closed orientable 4-manifold triangulations with up to six pentachora: the residual candidate PL classes for S4S^4S4, CP2\mathbb{C}P^2CP2, and QS4(2)QS^4(2)QS4(2)

Smooth unknottedness of turned 1-twist-spun tori and the Cochran–Davis torus in the 4-sphere

Completing the group-trisection dictionary for smooth 4-manifolds: the homeomorphism equivalence relation, group-theoretic reductions of the smooth 4-dimensional Poincaré conjecture, and trisection genus versus exotic smooth structures

Existence of periodic compatible binary sequences and circulant-core Hadamard matrices: the six conjectures of Orozco López (2018)

Cryer's growth conjecture for Hadamard matrices: is the complete-pivoting growth factor ρ(H) equal to n?

The Hadamard conjecture: existence of Hadamard matrices of every order 4m

The circulant Hadamard conjecture and the possible orders of Hadamard matrices with one circulant core

Hadamard matrices of the unresolved orders below 1000: the smallest open cases of the Hadamard and skew Hadamard conjectures

The strong skew Hadamard conjecture: existence of skew Hadamard matrices of every order divisible by 4

The Hadamard conjecture: existence of a Hadamard matrix of every order divisible by 4

The Weighted Atiyah Conjecture for Hecke–von Neumann algebras of Coxeter groups beyond the infinite dihedral group

The ppp-adic Atiyah question: integrality of ppp-adic Betti numbers

The center-valued Atiyah conjecture for congruence subgroups of SLn(Z)\mathrm{SL}_n(\mathbb{Z})SLn​(Z) and their finite extensions

Integrality of the rate of relations of a torsion-free finitely presented group: the first ℓ2\ell^2ℓ2-Betti number case of the Atiyah conjecture

Integrality of the von Neumann rank of ZΓ-modules for torsion-free amenable groups (Question 2.1 of Chung–Thom: the amenable case of Kaplansky's zero divisor conjecture and the strong Atiyah conjecture)

Closure of the Strong Atiyah Conjecture under finite group extensions

The strong Atiyah conjecture for groups with bounded torsion

The range of L2L^2L2-Betti numbers of universal covers of finite CW-complexes

The Atiyah problem for finitely presented groups and the lamplighter group: exact sets of ℓ2\ell^2ℓ2-Betti values

The degenerate Arnold conjecture on T2m×CPn\mathbb T^{2m}\times\mathbb{CP}^nT2m×CPn: the cup-length bound 2m+n+12m+n+12m+n+1 for arbitrary Hamiltonian diffeomorphisms

Clifford symplectic pencils and the multi-time Arnold conjecture beyond flat targets

Gong's two-point Gromov-Witten cup-length conjecture for degenerate symplectic fixed points (Conjecture 1.7 of arXiv:2507.04191)

A C0C^0C0 Arnold-type criterion via spectral invariants (Question 3 of Buhovsky–Humilière–Seyfaddini)

Viterbo's conjecture on the spectral norm of Lagrangians in cotangent bundles and the Arnold-type principle for Hausdorff limits of the zero section

A pair-topology Arnold conjecture for bmb^mbm-symplectic manifolds in dimensions at least four

Completing the near-extreme min-rank classification of side-information digraphs: the value $n-2$ and min-rank two over nonbinary fields

The fork-representation character conjecture for $\mathfrak{osp}(2m|2n)$ : a positive supersymmetric Schur expansion for $[0,\ldots,0,r,p-r]$

Generalised Verma modules, closed character formulae, and cohomology for higher-rank orthosymplectic Lie superalgebras $\mathfrak{osp}_{k|2n}$

Maximal atypical block equivalence for $\mathfrak{osp}(2k+2|2k)$ versus $\mathfrak{osp}(2k+1|2k)$

Definite spin and $E_8$ -boundings of homology 3-spheres: Tange's invariants $\mathfrak{ds}$ , $g_8$ , and the (11/8)-linked questions

Manolescu's $\kappa$ -invariant of Dehn surgeries on knots: the odd-odd case and the $\pm 2$ ambiguity

The 11/8-conjecture (Matsumoto): $b_2\ge\frac{11}{8}|\sigma|$ for closed smooth spin 4-manifolds

Topological criteria for Floer $K_G$ -split homology spheres and the K-theoretic route to the 11/8 conjecture

Intersection-form constraints for spin 4-manifolds bounded by Seifert rational homology 3-spheres: extending the $\bar\mu$ and $\kappa$ inequalities to non-orientable bases and degenerate cobordisms

Fixed-point classification and orbit asymptotics for the Collatz operator $\mathcal{T}$ : the open problems of Neklyudov's functional-analytic approach

The Collatz $3x-1$ problem (the $3n+1$ problem for negative start values): exactly three linearly independent holomorphic solutions of the Berg–Opfer functional equations

The Randomized Competitive Ratio of Weighted $k$ -Server on Uniform Metrics: Closing the Gap between $\Omega(2^k)$ and $\exp(O(k^2))$

The Harmonic k-server conjecture: is the Harmonic algorithm $\frac{k(k+1)}{2}$ -competitive on every metric space?

P versus NP and the finite descriptive-complexity separations: NP versus co-NP ( $\Sigma^1_1$ versus $\Pi^1_1$ ) and P versus $\Delta^1_1$ over finite structures

Fixed-Polynomial Circuit Lower Bounds for $\oplus P$ , Uniform $MA$ , $P^{NP}$ , and $NP$

Weakly tight MAD families in the $\mathfrak{b}<\mathfrak{s}$ regime and Sacks indestructibility

Splitting versus closed almost disjointness: does $\mathfrak{s} = \aleph_1$ , or at least $\mathfrak{s}_\omega = \aleph_1$ , imply $\mathfrak{a}_{closed} = \aleph_1$ ?

Open boundary cases of the negative strong polarized partition relation: the sigma-reaping number under $\mathfrak{r}_\sigma=\mathfrak{c}$ and the real-valued measurable endpoint

Is the dual reaping number $\mathfrak{r}_d$ at most the pair-splitting number $\mathfrak{s}_{\mathrm{pair}}$ ?

Classification of tight contact structures on the four-fiber Brieskorn spheres $\Sigma(2,3,7,41)$ and $\Sigma(2,3,11,13)$

Tight contact structures on the Whitehead link surgery family $M(n,r)$ : classification beyond the resolved parameter range and universal tightness

Topologically but not smoothly integrally split 2-component links in $S^4$ (Golla–Marengon Questions 1.4 and 1.5)

Generalized Property R Conjecture: handleslide equivalence of R-links to the unlink and the status of the generalized-square-knot families $L^{p,q}_{c/d}$

Completing the PL classification of the census of closed orientable 4-manifold triangulations with up to six pentachora: the residual candidate PL classes for $S^4$ , $\mathbb{C}P^2$ , and $QS^4(2)$

The $p$ -adic Atiyah question: integrality of $p$ -adic Betti numbers

The center-valued Atiyah conjecture for congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$ and their finite extensions

Integrality of the rate of relations of a torsion-free finitely presented group: the first $\ell^2$ -Betti number case of the Atiyah conjecture

The range of $L^2$ -Betti numbers of universal covers of finite CW-complexes

The Atiyah problem for finitely presented groups and the lamplighter group: exact sets of $\ell^2$ -Betti values

The degenerate Arnold conjecture on $\mathbb T^{2m}\times\mathbb{CP}^n$ : the cup-length bound $2m+n+1$ for arbitrary Hamiltonian diffeomorphisms

A $C^0$ Arnold-type criterion via spectral invariants (Question 3 of Buhovsky–Humilière–Seyfaddini)

A pair-topology Arnold conjecture for $b^m$ -symplectic manifolds in dimensions at least four

Length-preserving geodesic rewriting for Artin groups with $A_3$ or $B_3$ subdiagrams: the rank-three types $(2,3,3)$ and $(2,3,4)$

The Bokut–Fong–Ke–Shiao hypothesis on Gröbner–Shirshov bases for braid semigroups beyond type $A_n$

The lattice conjecture for the left-division order on Krammer's asymmetric Artin monoid $A_n$ (Conjecture 11.12 of ‘An asymmetric generalisation of Artin monoids’)

The nature of the classical Hénon attractor at $a=1.4$ , $b=0.3$ : strange attractor or stable periodic orbit?

The remaining parameter gap in the Devaney–Nitecki conjecture: transversal homoclinic points of the area-preserving Hénon map for $-1<a\le-1+10^{-7}$

Generic past (and hence full) $C^{0,1}_{\mathrm{loc}}$ -inextendibility of the Dafermos–Luk–Oh spacetimes (two-ended spherically symmetric Einstein–Maxwell–scalar-field perturbations of Reissner–Nordström)

Generic $W^{1,2}$ -inextendibility of the dynamical Kerr Cauchy horizon: Christodoulou's formulation of strong cosmic censorship for the Einstein vacuum equations without symmetry

Strong cosmic censorship at the $W^{1,2}_{loc}$ level for nonlinearly perturbed Reissner–Nordström black holes without symmetry (Christodoulou's formulation)

$W^{1,2}_{\mathrm{loc}}$ -formulation of strong cosmic censorship in the spherically symmetric Einstein-Maxwell-real-scalar-field model: inextendibility of all nontrivial $C^0$ extensions for generic two-ended data

Quillen's conjecture $Q_n$ : are all projective modules over the localization of a regular local ring at a regular parameter free?

Bass–Quillen for ramified mixed-characteristic test rings $R/(p-b)$ (Popescu's Question 6)

The rank- $r\ge 2$ case of the non-archimedean analytic Bass–Quillen question of Kerz–Saito–Tamme (Question 3)

The Burghelea conjecture: vanishing of $T_*(x,\mathbb{Q})$ for groups whose classifying space is a finite CW-complex

The Weil-étale cohomological formula for $\operatorname{ord}_{s=0}\zeta(\mathcal{X},s)$ and $\zeta^*(\mathcal{X},0)$ for regular arithmetic schemes (Morin's Conjecture 4.2 and its conjectural inputs)

The $L^2$ -Brown formula $E(G)(s)=\chi^{(2)}(C_G(s))$ for groups of type FP over $\mathbb{C}$ (Chatterji–Mislin Conjecture 1) and the scope of the class $\mathcal{B}_\infty$

Bass's finiteness conjecture for $K_0$ of regular rings finitely generated over $\mathbb{Z}$

The Bost conjecture: is the $\ell^1$ -assembly map an isomorphism for every countable discrete group?

Lexicographic initial segments as extremal non-FC families: Pulaj–Wood's Conjecture 1 on the threshold $FC(k,n)$

Pólya's conjecture for products $(0,1)\times\Omega$ with an interval factor and arbitrary bounded cross-section $\Omega$