Mathematics

The

g

-conjecture on the sphere6 questions

Regularity, shellability, polytopality, and ballness of the canonical stacked triangulations arising from the generalized lower bound conjecture
Open for about 22 years
Nevo's conjecture for 4-spheres without large missing faces: is $g_2 \geq g_1$ ?
Open for about 17 years
Finding an explicit non-polytopal Bier sphere: the polytopality problem for Bier's deleted-join spheres
Open for about 2 years
The Nevo–Petersen conjecture: are the γ-vectors of flag homology spheres f-vectors of balanced simplicial complexes?
Open for about 17 years
Smoothability of moment-angle manifolds over sphere triangulations
Open for about 13 years
Nonnegativity of $\gamma_2$ for flag simplicial spheres: the first nontrivial case of Gal's conjecture
Open for about 21 years

Existence of isometric immersions of surfaces into four-dimensional Euclidean space8 questions

Efficient and reliable computation of surface intersection curves5 questions

Behavior of chemotaxis model solutions and pattern formation problems4 questions

Rigidity problems of holomorphic mappings under group actions7 questions

Numerical methods for discontinuous solutions of combustion equations7 questions

Overcoming the curse of dimensionality under coarse measurement of variables3 questions

Weak Pinsker conjecture5 questions

Sacks’ problem on degree-invariant solutions to Post’s problem and Martin’s conjecture9 questions

Classification of three-dimensional manifolds9 questions

hp

-adaptive finite element methods for three-dimensional elliptic and electromagnetic field problems8 questions

Three types of special relativity and gravity and their interrelations7 questions

Thurston's virtual fibering conjecture3 questions

Schanuel conjecture9 questions

Designing practical and secure public-key cryptosystems5 questions

Uniqueness in the inverse scattering problem for acoustic and electromagnetic waves7 questions

Does there exist a strongly polynomial-time algorithm for linear programming?8 questions

Does the bin-packing problem admit an absolute approximation algorithm?2 questions

Does there exist a stable first-order complete theory with finitely many countable models greater than one?8 questions

Numerical study of binary black hole systems6 questions

Hyperbolic conjecture1 questions

The Arithmetic Persistence Conjecture: does arithmetic hyperbolicity persist under extension of the base field?
Open for about 7 years

Convergence of difference methods for hyperbolic conservation law systems with discontinuous solutions6 questions

The cycle double cover conjecture9 questions

L^{p}

boundedness of bilinear Hilbert transforms5 questions

Multiscale modeling and computation of water molecules6 questions

Smale's mean value conjecture8 questions

Ergodicity of stochastic queueing networks3 questions

Finiteness conjecture of central configurations in celestial mechanics8 questions

Convex polygons and the Erdős-Szekeres problem6 questions

The

d

-step conjecture for convex polytopes6 questions

Reconstruction conjecture for graphs8 questions

Pfaffian orientation of graphs5 questions

Turing equivalence problem8 questions

Automorphism problem of Turing degrees6 questions

Generalized Lax conjecture: a hyperbolicity cone can be represented as a section of a semidefinite cone5 questions

Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves8 questions

Regularity, shellability, polytopality, and ballness of the canonical stacked triangulations arising from the generalized lower bound conjecture

Nevo's conjecture for 4-spheres without large missing faces: is g2≥g1g_2 \geq g_1g2​≥g1​?

Finding an explicit non-polytopal Bier sphere: the polytopality problem for Bier's deleted-join spheres

The Nevo–Petersen conjecture: are the γ-vectors of flag homology spheres f-vectors of balanced simplicial complexes?

Smoothability of moment-angle manifolds over sphere triangulations

Nonnegativity of γ2\gamma_2γ2​ for flag simplicial spheres: the first nontrivial case of Gal's conjecture

Existence of a smooth isometric immersion of the entire hyperbolic plane into R4\mathbb{R}^4R4

Existence of complete isometric immersions with flat normal bundle of hyperbolic space into space forms of larger curvature

Global smooth isometric immersions of two-dimensional Riemannian manifolds into R4\mathbb{R}^4R4

Flexibility of isometric immersions of two-dimensional metrics into R4\mathbb{R}^4R4: direct convex integration on the Poznyak curvature equation

Lowering Günther's bound: smooth isometric immersions of arbitrary Riemannian manifolds into semi-Euclidean spaces of dimension below c(n)

Local well-posedness of the flag geodesic equation: geodesics on the manifold of isometric immersions of a square into R3\mathbb{R}^3R3

Global C2C^2C2 isometric immersion of the hyperbolic plane H2\mathbb{H}^2H2 into R4\mathbb{R}^4R4

Isometric regular immersion of the entire Lobachevsky (hyperbolic) plane into Euclidean 4-space E4E^4E4

Validated ODE-based approximation of surface-to-surface intersection curves: extension to tangential intersections, higher-order segment continuity, and certified step control

Certified approximate parameterization of intersection curves of general implicit surfaces: topology guarantees, computable error constants, and a priori distance bounds

Correctness and validity scope of the rotation-index stopping rule for tracing closed surface-intersection loops

Certified tangential-area bounding and error estimation for BVH-based surface–surface intersection with osculating toroidal patches

Certified topology by values for surface–surface intersection curves: validated eigenvalue criteria and conditioning analysis for Lagrange-basis Bézoutian matrix pencils

Mechanism arresting runaway aggregation and selecting stable, uniformly spaced finite aggregates in the Keller–Segel model of starved C. elegans L1 larvae

Phenotypic heterogeneity and the chemotactic instability in Keller-Segel-type models: pattern-onset thresholds and the global-existence/blow-up dichotomy for phenotype-structured autoattraction populations

Does logistic growth prevent blow-up in the fully parabolic Keller–Segel chemotaxis system in dimensions N≥3N\ge 3N≥3?

Blow-up of supercritical-mass solutions for the two-dimensional fully parabolic Keller–Segel system on the plane (Calvez–Corrias open problem)

Dynamics of subgroups of Aut(X_1) x Aut(X_2) on spaces of holomorphic maps between Hermitian symmetric spaces

Characterizing holomorphic fiber spaces whose every holomorphic self-map preserves a fiber

Infinitesimal rigidity criteria for CR mappings beyond finite nondegeneracy, and the local structure of the mapping moduli space

Holomorphic rigidity of Teichmüller space without the codimension hypothesis

Rigidity of lattice-symmetric proper holomorphic maps between balls at arbitrary Hölder boundary regularity

Structure of proper holomorphic maps between reducible bounded symmetric domains: rank-1 factors, general reducible targets, and semi-product maps beyond rank conditions

Realizable Hermitian-invariant symmetry groups Γf\Gamma_fΓf​ of rational proper maps between balls: characterization and minimal realization cost

Rigorous asymptotic solution of the one-dimensional discrete-source detonation problem in the point-source, zero-delay limit

Low-Mach-number and carbuncle-free robustness of the all-shock reactive Riemann solver for thermally perfect gases

Subcell-resolution methods for advection–reaction equations with stiff sources: non-splitting high-order-in-time integration and implicit-solver compatibility on coarse meshes

Existence and admissibility of the discontinuous steady deflagration families in the kinetic-theory recombination model (regimes 1(c) and 2(c) of Conforto–Monaco–Ricciardello)

A priori prediction of the spurious weak-detonation regime in fractional-step shock-capturing computation of stiff detonations

Global-in-time existence and wave-interaction structure of the generalized Riemann problem for the scalar Chapman–Jouguet combustion model

Predictive stability theory and convection-compatible source-term evaluation for high-order fractional-step shock-capturing methods with stiff reaction terms

Adaptive data-driven choice of the regularization parameter for deconvolution estimation of P(X<Y) with unknown error distributions

Smooth backfitting for errors-in-variables additive models with estimated error distributions and super-smooth errors

Convergence rates and optimal bandwidth matrices for kernel deconvolution density estimation with multivariate panel data

Failure of the Weak Pinsker Property for actions of non-amenable sofic groups

The weak Pinsker property beyond amenability: the sofic group boundary and a homological characterization

Kalikow's uniform martingale questions: finite-alphabet and minimal-alphabet representations of measure-preserving transformations

Does there exist a non-Bernoulli K-automorphism all of whose nontrivial splittings consist of one Bernoulli and one non-Bernoulli factor?

Finitary factors of i.i.d. processes with no finitary i.i.d. factor (Gabor's Open Problem (i))

Steel's uniformization conjecture for Turing-invariant functions: the global form of Martin's conjecture

Natural many-one degrees and the optimal set-theoretic hypotheses for the uniform Martin conjecture on m-degrees

Martin's Conjecture: the classification of Turing-invariant functions under the axiom of determinacy (parts 1 and 2)

Bard's Question 5.1: does the locally generated family Jx\mathcal{J}_{\mathbf{x}}Jx​ exhaust all uniformly Turing-invariant functions on a single Turing degree?

Steel's uniformity conjecture: is every Turing-invariant function equal on a cone to a uniformly Turing-invariant function?

The Slaman–Steel question: is Turing equivalence hyper-recursively-finite?

Martin's Conjecture in the enumeration degrees: the Wadge order on the Scott domain, jump-skip iterates, non-uniform regressive functions, and cone theories under Turing Determinacy

The regressive case of Martin's conjecture on the arithmetic degrees

Open questions from Part 1 of Martin's Conjecture for order-preserving and measure-preserving functions: the Rudin–Keisler position of the Martin measure, the hyperarithmetic degrees, and ZF provability

Is the 3-manifold homeomorphism problem in NP?

Wall's conjecture: is every finitely presented 3-dimensional Poincaré duality group the fundamental group of a closed aspherical 3-manifold?

Is the homeomorphism problem for closed 3-manifolds in NP? NP recognition of the remaining Thurston geometries

Bounded-runtime homeomorphism problem for finite-volume hyperbolic manifolds

Recognizing surface presentations of 3-manifold groups: decidability beyond the two-generator case and Whitehead degree bounds (Null's Questions 4.18 and 4.19)

Decidability of equality of hyperbolic volumes of finite-volume hyperbolic 3-manifolds

Wall's conjecture for Poincaré duality groups of dimension 3

Elementary-recursive effectivity of 3-manifold classification: Kuperberg's Conjectures 9.1-9.3 and the complexity placement of the homeomorphism problem

Cannon's conjecture: word-hyperbolic groups with 2-sphere Gromov boundary are virtually cocompact Kleinian groups

Remaining obstacles to a fully optimal two-dimensional hp-AFEM: near-best conforming hp-partitions, degree-robust broken-to-conforming equivalence, p-robust a posteriori error control, and p-uniform solvers

Uniform elliptic regularity pickup for the Neumann–Poisson problem on vertex patches of shape-regular simplicial meshes (the Dong–Ern conjecture, Assumption 3.5)

Sharp scale-resolution conditions for wavenumber-explicit hp finite element analysis of the time-harmonic Maxwell impedance problem

Wavenumber-explicit hp-FEM quasi-optimality for high-frequency Maxwell equations: general geometries and practical transparent-boundary truncations

Convergence and rate-optimality of automatic hp-adaptive finite element methods for indefinite (frequency-domain) Maxwell equations

Polynomial-degree-robust a posteriori error estimation for hp-finite element discretizations of Maxwell's equations on general nonaffine element geometries

Open problems in hp-adaptive edge-element methods for time-harmonic Maxwell's equations: automatic anisotropic refinement for edge singularities, scalable variable-p solvers, and resonant regimes

Quasi-optimal hp estimates for projection-based interpolation on hexahedral elements

Open problems in the covariant 1/c21/c^21/c2 expansion of general relativity: radiative modes, matching to full GR, and quantization of non-relativistic gravity

The Type II non-relativistic geometric trinity of gravity: action-level teleparallel and symmetric-teleparallel Newton–Cartan theories from the 1/c expansion

Open problems in the torsional Newton–Cartan large-ccc expansion of general relativity: non-twistless torsion, torsion-coupled motion, inequivalent dynamics, and physical regimes

Simultaneously Galilean- and Carrollian-compatible affine connections: the general torsionful classification and the gravitational content they support

Conditions for recovering a potential-based Carrollian gravitational theory from general Carrollian metric-affine connections

Action principle for the equation-level Carrollian electric and magnetic limits of general relativity

Quantization and non-perturbative duals of non-relativistic and Carrollian Jackiw–Teitelboim gravity

An automorphic criterion for fibrations of arithmetic hyperbolic 3-manifolds (Dunfield–Ramakrishnan, Question 1.7)

Lackenby's strong Heegaard gradient conjecture: positive strong Heegaard gradient for closed hyperbolic 3-manifolds

Nevo's conjecture for 4-spheres without large missing faces: is $g_2 \geq g_1$ ?

Nonnegativity of $\gamma_2$ for flag simplicial spheres: the first nontrivial case of Gal's conjecture

Existence of a smooth isometric immersion of the entire hyperbolic plane into $\mathbb{R}^4$

Global smooth isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^4$

Flexibility of isometric immersions of two-dimensional metrics into $\mathbb{R}^4$ : direct convex integration on the Poznyak curvature equation

Local well-posedness of the flag geodesic equation: geodesics on the manifold of isometric immersions of a square into $\mathbb{R}^3$

Global $C^2$ isometric immersion of the hyperbolic plane $\mathbb{H}^2$ into $\mathbb{R}^4$

Isometric regular immersion of the entire Lobachevsky (hyperbolic) plane into Euclidean 4-space $E^4$

Does logistic growth prevent blow-up in the fully parabolic Keller–Segel chemotaxis system in dimensions $N\ge 3$ ?

Realizable Hermitian-invariant symmetry groups $\Gamma_f$ of rational proper maps between balls: characterization and minimal realization cost

Bard's Question 5.1: does the locally generated family $\mathcal{J}_{\mathbf{x}}$ exhaust all uniformly Turing-invariant functions on a single Turing degree?

Open problems in the covariant $1/c^2$ expansion of general relativity: radiative modes, matching to full GR, and quantization of non-relativistic gravity

Open problems in the torsional Newton–Cartan large- $c$ expansion of general relativity: non-twistless torsion, torsion-coupled motion, inequivalent dynamics, and physical regimes

Hyperbolic manifolds that fiber over $S^1$ in higher odd dimensions: existence, abundance, virtual fibration, and structure of the fibrations

Pellarin's operator-theoretic Schanuel conjecture for the Carlitz exponential over Tate algebras: regular $\mu$ -independence of $\exp_C$ -values

Unconditional generic solutions of equations involving the modular $j$ -function and its iterates

Is $G$ -disjointness of subfields of $\mathbb{C}$ a symmetric relation?

Unique determination of a perfectly conducting ball by finitely many electric far-field data: the size restriction $kR<\sqrt{2}$ and the minimality of four measurements

Combinatorics of the interior-point counterexample polytope $\mathrm{CEX}_n(t)$ : the $n$ -cube conjecture and the vertex–tropical-segment correspondence

Does Lascar rank at least $\omega^\omega$ force continuum many countable models? (Tanović's conjecture for superstable theories)

Vaught's conjecture for superstable theories of arbitrary $U$ -rank

Eventual non-isolation of generic types of simple superstable groups and the $\omega^\omega$ rank threshold for countable models

Convergence of Lax–Friedrichs and Godunov schemes for $2\times2$ nonstrictly hyperbolic systems with an isolated umbilic point beyond the symmetric $(a,b)=(3,0)$ prototype

Convergence of fully discrete Godunov and Lax–Friedrichs schemes for general $n\times n$ systems of hyperbolic conservation laws

Intrinsic $L^\infty$ bounds and convergence of entropy stable shock-capturing space–time discontinuous Galerkin schemes for multi-dimensional systems of conservation laws

Does a $\Sigma_5$ -flow imply an $H_5$ -flow? Unit-sphere flows versus oriented 5-cycle double covers

Small Oriented Cycle Double Cover Conjecture: does every simple 2-connected graph other than $K_4$ and $K_6$ admit an oriented cycle double cover with at most $n-1$ directed cycles?

$L^{p}$ boundedness of the bilinear Hilbert transform in the lower range $1/2<p\leq 2/3$