Mathematics

Isospectral problem6 questions

Inverse spectral problem for the Euler–Bernoulli matrix beam with general measures: recovering the two mass measures from the Weyl function
Open for about 5 years
Global classification of isospectral Euler–Bernoulli beams with constant stiffness–mass product
Open for about 16 years
Do the first three Dirichlet eigenvalues determine a triangle?
Open for about 15 years
Non-isometric Steklov-isospectral metrics on balls in dimensions 3, 4, and 5
Open for about 8 years
Spectral determination of semi-regular polygons beyond the convex class
Open for about 8 years
The two-dimensional case of Sunada's method for the covering spectrum: does equality of covering spectra for all invariant metrics imply jump equivalence?
Open for about 17 years

Lattice embeddings in the recursively enumerable degrees and decidability of the two-quantifier theory7 questions

Cohomology of harmonic bundles7 questions

Chvátal's conjecture for independence systems4 questions

Mathematical analysis of supersonic flow around blunt bodies5 questions

Are there effective algorithms for the MQ problem in multivariate public key cryptography?7 questions

Effective solutions to systems of polynomial equations5 questions

Multiple ergodic theorems8 questions

Iitaka conjecture:

C_{n,m}

10 questions

Chern's conjecture for affine flat manifolds6 questions

Representation theory of non-semisimple Brauer algebras8 questions

Non-negative matrix factorization7 questions

Classification of non-commutative surfaces9 questions

Defining relations problem of non-symmetrizable Kac-Moody algebras1 questions

The Gabber–Kac defining-relations problem for non-symmetrizable generalized Cartan matrices
Open for about 45 years

Existence of projective planes of non-prime-power order6 questions

Existence problem for solutions of nonlinear Dirac equations6 questions

Model order reduction for nonlinear dynamical systems7 questions

High-accuracy schemes for discontinuous solutions of nonlinear partial differential equations7 questions

High-dimensional Riemann problem for nonlinear hyperbolic conservation law systems7 questions

De Giorgi and Gibbons conjectures in nonlinear elliptic equations6 questions

Fürstenberg conjecture8 questions

Computation of piecewise polynomial equation systems and related geometric problems4 questions

Von Neumann paradox6 questions

Symbolic-numeric hybrid computation4 questions

Foulkes conjecture8 questions

Variable selection problem for complex data6 questions

Restriction problems of the Fourier transform5 questions

Two problems concerning hyperfinite Borel equivalence relations5 questions

Unresolved specific issues in high-accuracy finite element methods6 questions

Analysis and computation of high-dimensional large-scale inverse scattering problems7 questions

The higher-dimensional uniformization conjecture6 questions

Gorenstein symmetry conjecture10 questions

Goldbach conjecture7 questions

Grothendieck standard conjectures9 questions

Griffiths problems6 questions

Kakeya problem5 questions

Inverse spectral problem for the Euler–Bernoulli matrix beam with general measures: recovering the two mass measures from the Weyl function

Global classification of isospectral Euler–Bernoulli beams with constant stiffness–mass product

Do the first three Dirichlet eigenvalues determine a triangle?

Non-isometric Steklov-isospectral metrics on balls in dimensions 3, 4, and 5

Spectral determination of semi-regular polygons beyond the convex class

The two-dimensional case of Sunada's method for the covering spectrum: does equality of covering spectra for all invariant metrics imply jump equivalence?

Finite final segments of the d.c.e. Turing degrees beyond the distributive case: join-semidistributive and interval-dismantlable lattices, and the general characterization

Decidability of the two-quantifier theory of the enumeration degrees

Bounded-query polynomial-time degree structures and definability of the recursive degrees: open questions in the Shore and Slaman program on the P-T degrees

Decidability of the ∀∃-theory of the effectively closed Medvedev degrees in the lattice language

Decidability of the two-quantifier (Π2\Pi_2Π2​) theory of the Σ20\Sigma^0_2Σ20​-enumeration degrees

Natural first-order definability of the totally ωn\omega^nωn-c.a. degrees in the computably enumerable degrees for finite n>1n>1n>1

Finite-lattice characterizability of the >ω2>\omega^2>ω2-fickle (not totally ω2\omega^2ω2-c.a.) computably enumerable Turing degrees

Admissibility of the limiting Hermitian–Einstein metric for reflexive Higgs sheaves on compact Kähler klt spaces: uniform Higgs-field bounds along the orbifold Hermitian–Yang–Mills flow (Question 3.6 of Zhang–Zhang–Zhang)

L2L^2L2-cohomology of wild harmonic bundles on quasi-compact Kähler manifolds: the surviving open core of the Jost–Yang–Zuo program

Compatibility of the L2L^2L2-pure Hodge structure with Saito's pure Hodge structure on intersection cohomology (Shentu–Zhao, Conjecture 8.5)

Algebraic formula for L2L^2L2 higher direct images in non-abelian Hodge theory beyond relative dimension one

Extending the Nahm transformation for parabolic Higgs bundles on the projective line beyond the generic regime: critical weights, flat factors, and non-semisimple leading term at infinity

Extending the twistor-complex–L2L^2L2-complex comparison for harmonic bundles to general singular supports and to the wild case in higher dimensions

An L2L^2L2-cohomology theory for harmonic Higgs bundles on arbitrary Galois coverings of Zariski open sets

The ddd-degree Erdős–Ko–Rado conjecture in the optimal range n≥2k+1n \ge 2k+1n≥2k+1

Star-optimality for cross-intersecting subfamilies of hereditary families: Borg's cross-intersecting analogue of Chvatal's conjecture

Sharp base-size thresholds for cross-intersecting subfamilies of levels of hereditary families

Erdős–Ko–Rado property for independent sets in disjoint unions of claws

The regular-versus-Mach reflection transition criterion for self-similar shock reflection: rigorous theory beyond the potential-flow counterexamples (full Euler constructions, validity of the weak-type/angle criterion, and dynamic stability)

General closure for the boundary-layer-edge crossflow velocity gradient on the symmetry plane of blunt bodies beyond the Newtonian hypersonic regime

Global shocks for steady supersonic flow past a curved wedge: the full Euler system, large nondecaying wedge perturbations, and the detached regime

Physical selection, uniqueness, and dynamical stability of detached shock solutions past a blunt body in supersonic potential flow

First-principles determination of bow-shock curvature in front of a blunt body: the power-law exponent and the non-axisymmetric stagnation-streamline extension

Active security of parallel-repeated MQ identification schemes without the compression hypothesis

Images of the MLD–MQ reductions and the complexity of key recovery in code-based and multivariate post-quantum cryptography

Sharpness of the zero-coefficient bounds in the Thomae–Wolf reduction for underdetermined multivariate quadratic systems

Polynomial-time completion of the Coggia-Couvreur key-recovery attack on Loidreau's rank-metric encryption scheme for masking-space dimension λ≥4\lambda \ge 4λ≥4

Classical complexity of random degree-ddd (d≥3d \ge 3d≥3) polynomial systems over F2\mathbb{F}_2F2​ with dual Reed-Solomon codeword constraints (the Briaud-Dinur-Ghosal-Jain-Lou-Sahai quantum-advantage instances)

Closing the gap between the square case and the polynomial-time regime for underdefined multivariate quadratic systems

Semi-regularity of the RMQ modelings: proving Assumption 1 and Heuristic 1 behind the complexity analysis of the Regular Multivariate Quadratic problem

Average-case and excess-dimension-parametrized aggregate separation bounds for polynomial systems (open problems of Emiris–Mourrain–Tsigaridas)

Sub-cubic polynomial system solving for arbitrary zero-dimensional ideals: removing the shape-position, radicality, regularity, and large-characteristic hypotheses

The Eisenbud–Green–Harris conjecture on Hilbert functions of ideals containing a regular sequence

Gröbner-basis complexity of critical point computation beyond maximal minors: Jacobian corank at least two and the degree of regularity via Grothendieck polynomials

Fröberg's conjecture: the Hilbert–Poincaré series of ideals generated by generic homogeneous forms

Open conjectures from Bergelson–Moreira–Richter's Hardy-field multiple ergodic averages: a joint ergodicity criterion for commuting transformations, a nilsequence structure theorem for weighted multicorrelations, and weakening Property (P) to Property (WP)

Removing the finite-index hypotheses in the Khintchine-type multiple recurrence theorem for countable abelian groups

Classification of weights for which uniform weighted convergence implies Cesàro convergence, beyond the tempered regime

Does goodness for equidistribution alone imply joint ergodicity of integer sequences (Frantzikinakis's Problem 1)?

General Structural Conjecture for joinings of sated ZD\mathbb{Z}^DZD-systems with partial invariance (Austin's Conjecture 6.0.4)

Class-uniform syndeticity of multiple recurrence beyond uniformly amenable groups: extension to all amenable groups and central-set recurrence for three or more commuting actions

Everywhere failure of simultaneous recurrence for minimal systems with commutativity not assumed (Conjectures 1–3 of Huang–Shao–Ye, 2024)

Pointwise almost-everywhere convergence of multiple ergodic averages for commuting transformations

The anticanonical Iitaka inequality Cn,m−C^{-}_{n,m}Cn,m−​ in positive characteristic without FFF-splitting hypotheses

Positive-characteristic boundary of Iitaka's subadditivity conjecture C_{n,m}

Meng's Conjecture 1.8: the equality of Kodaira dimensions κ(X)=κ(F)+κ(Y,det⁡^f∗ωX⊗m⊗ωY)\kappa(X)=\kappa(F)+\kappa(Y,\widehat{\det} f_*\omega_X^{\otimes m}\otimes\omega_Y)κ(X)=κ(F)+κ(Y,detf∗​ωX⊗m​⊗ωY​) for fibrations over bases of non-negative Kodaira dimension

The exceptional case of the Generalized Nonvanishing Conjecture for threefold pairs with κ=0\kappa=0κ=0, q=0q=0q=0, and maximal nef dimension, and the residual case of the Iitaka conjecture C7,3C_{7,3}C7,3​

Iitaka's subadditivity conjecture for the logarithmic Kodaira dimension

Anticanonical Iitaka inequality Cn,m−C^-_{n,m}Cn,m−​ in positive characteristic without Frobenius-positivity hypotheses

Iitaka's Cn,mC_{n,m}Cn,m​ conjecture: subadditivity of the Kodaira dimension for algebraic fiber spaces

The log Iitaka conjecture Cn,mC_{n,m}Cn,m​ for algebraic fiber spaces over bases of dimension at least three

Iitaka subadditivity for fibrations of threefolds over curves in arbitrary positive characteristic

The Iitaka conjecture Cn,mC_{n,m}Cn,m​: subadditivity of the Kodaira dimension for algebraic fiber spaces

Chern's conjecture for closed affine manifolds and its Euler–Satake analogue for compact affine orbifolds

Affine structures on nontrivial connected sums of closed manifolds

Chern's Conjecture on the Vanishing of the Euler Characteristic of Closed Affine Manifolds

Finite-cover product splitting for compact Hessian manifolds in dimensions seven and above

Chern's conjecture: vanishing of the Euler characteristic of closed affinely flat manifolds

Chern's conjecture for closed nilpotent affine manifolds: vanishing of the Euler characteristic

Decomposition numbers of the periplectic Brauer algebra in positive characteristic

Blocks of the Brauer algebra in positive characteristic and decomposition numbers beyond the non-dividing regime

The saturated condition for decomposition numbers of the cyclotomic Brauer algebra over the complex field: Conjecture 1 of Rui–Song

Calibrated irreducible representations and Jacobson radicals of the periplectic Brauer algebra AnA_nAn​ and its degenerate affine extension sWn\mathrm{s}\mathbb{W}_nsWn​

Arm property and closed forms for standard-module Gram determinants of the Kadar–Yu algebras (Conjectures 3.27.1, 3.31.1, 3.32.1 and 5.2.1 of Morris–Martin)

Open problems on the traceless projection of mixed tensor products: spectral-set equality, the kernel of the walled Brauer centralizer map, and trace-level idempotents

Diagrammatic identification of the radical of the Brauer algebra at orthogonal and symplectic parameters (Gavarini's Conjectures 5.10 and 5.14)

Explicit combinatorics of the non-semisimple walled Brauer algebra regime N=m+n−lN=m+n-lN=m+n−l: modified multiplicities, kernel, matrix units, and the Sm×SnS_m\times S_nSm​×Sn​-commuting subalgebra

Critical points and Euclidean distance degrees of low-rank approximation with zeros: the Kubjas–Sodomaco–Tsigaridas conjectures

Why spectral orthogonalization helps nonnegative matrix factorization: characterizing the problem properties behind Muon's advantage

Sufficient conditions for global rigidity of nonnegative matrix factorizations: a stress-matrix analogue

Polynomial-time computation of identifiable nonnegative matrix factorizations beyond separability: the sufficiently-scattered condition

Nontrivial upper bounds on the nonnegative rank of matrices of ordinary rank four and higher

Simultaneous partial identifiability of Exact NMF under conditions weaker than the selective-window (separability) assumption

The rational–real nonnegative rank gap after Cohen–Rothblum: classification, minimal examples, and certification of rational factorizations

Decidability of the two-quantifier ( $\Pi_2$ ) theory of the $\Sigma^0_2$ -enumeration degrees

Natural first-order definability of the totally $\omega^n$ -c.a. degrees in the computably enumerable degrees for finite $n>1$

Finite-lattice characterizability of the $>\omega^2$ -fickle (not totally $\omega^2$ -c.a.) computably enumerable Turing degrees

$L^2$ -cohomology of wild harmonic bundles on quasi-compact Kähler manifolds: the surviving open core of the Jost–Yang–Zuo program

Compatibility of the $L^2$ -pure Hodge structure with Saito's pure Hodge structure on intersection cohomology (Shentu–Zhao, Conjecture 8.5)

Algebraic formula for $L^2$ higher direct images in non-abelian Hodge theory beyond relative dimension one

Extending the twistor-complex– $L^2$ -complex comparison for harmonic bundles to general singular supports and to the wild case in higher dimensions

An $L^2$ -cohomology theory for harmonic Higgs bundles on arbitrary Galois coverings of Zariski open sets

The $d$ -degree Erdős–Ko–Rado conjecture in the optimal range $n \ge 2k+1$

Polynomial-time completion of the Coggia-Couvreur key-recovery attack on Loidreau's rank-metric encryption scheme for masking-space dimension $\lambda \ge 4$

Classical complexity of random degree- $d$ ( $d \ge 3$ ) polynomial systems over $\mathbb{F}_2$ with dual Reed-Solomon codeword constraints (the Briaud-Dinur-Ghosal-Jain-Lou-Sahai quantum-advantage instances)

General Structural Conjecture for joinings of sated $\mathbb{Z}^D$ -systems with partial invariance (Austin's Conjecture 6.0.4)

The anticanonical Iitaka inequality $C^{-}_{n,m}$ in positive characteristic without $F$ -splitting hypotheses

Meng's Conjecture 1.8: the equality of Kodaira dimensions $\kappa(X)=\kappa(F)+\kappa(Y,\widehat{\det} f_*\omega_X^{\otimes m}\otimes\omega_Y)$ for fibrations over bases of non-negative Kodaira dimension

The exceptional case of the Generalized Nonvanishing Conjecture for threefold pairs with $\kappa=0$ , $q=0$ , and maximal nef dimension, and the residual case of the Iitaka conjecture $C_{7,3}$

Anticanonical Iitaka inequality $C^-_{n,m}$ in positive characteristic without Frobenius-positivity hypotheses

Iitaka's $C_{n,m}$ conjecture: subadditivity of the Kodaira dimension for algebraic fiber spaces

The log Iitaka conjecture $C_{n,m}$ for algebraic fiber spaces over bases of dimension at least three

The Iitaka conjecture $C_{n,m}$ : subadditivity of the Kodaira dimension for algebraic fiber spaces

Calibrated irreducible representations and Jacobson radicals of the periplectic Brauer algebra $A_n$ and its degenerate affine extension $\mathrm{s}\mathbb{W}_n$

Explicit combinatorics of the non-semisimple walled Brauer algebra regime $N=m+n-l$ : modified multiplicities, kernel, matrix units, and the $S_m\times S_n$ -commuting subalgebra

Completing the classification of orders in the Sklyanin algebra: the exceptional case $\overline{U}=k$ , Veronese maximality of maximal orders (Question 9.4), and uniqueness of virtual blowups

Existence of a symmetric transversal design $\mathrm{STD}_2[12;6]$

Qualitative theory of $L^{2}$ -normalized solitary waves of the Maxwell-Dirac system: uniqueness, multiplicity, regularity, and stability

Normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities: the fully supercritical regime $p\ge 6$ , sharp existence thresholds, and classification of the threshold eigenvalues $\pm mc^2$

Existence and multiplicity of $L^2$ -normalized solutions for nonlinear Dirac equations with non-autonomous nonlinearities beyond the small-mass regime

Two-dimensional Riemann problem with three constant states for the system $u_t+(u^2)_x+(uv)_y=0$ , $v_t+(uv)_x+(v^2)_y=0$ : initial-data classes beyond three contact discontinuities

One-dimensionality of stable Allen–Cahn solutions with bounded energy density in $\mathbb{R}^n$ for $5\le n\le 7$ (hypothesis $(\star)$ )

De Giorgi's conjecture for the Allen–Cahn equation in intermediate dimensions $5\le N\le 8$

One-dimensional symmetry of bounded H-monotone entire solutions of the elliptic system $\Delta u=\nabla H(u)$ in dimensions $4\le N\le 8$ (Fazly–Ghoussoub system analogue of the De Giorgi conjecture)

Furstenberg's $\times p,\times q$ conjecture: uniqueness of non-atomic $\times p,\times q$ -invariant measures on the circle, equivalently uniqueness of $T_p,T_q$ -fixed homeomorphisms of $[0,1]$

Weak $C^$ -simplicity without the weak unique trace property, just-infinite-type structure theory, and an ASH model for $C^(\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2)$ (Questions 4.1-4.3 of Bruce-Scarparo)

Lifting Kloeckner's infinitesimal counterexamples to the Furstenberg $\times 2,\times 3$ rigidity problem: differentiability of push-forward maps on Wasserstein space and genuine deformations of invariant measures

Furstenberg's $\times(2,3)$ conjecture, its equivalent complex-analytic and operator-algebraic formulations, and the related periodic-approximation and equidistribution problems

Furstenberg's $\times p,\times q$ conjecture: classification of ergodic invariant measures on the circle, equivalently uniqueness of an irreducible representation of $\mathbb{Z}[1/pq]\rtimes\mathbb{Z}^2$

Furstenberg's $\times 2,\times 3$ orbit-dimension conjecture: must $\dim_H\overline{\{2^nx \bmod 1\}}+\dim_H\overline{\{3^nx \bmod 1\}}\ge 1$ hold for every irrational $x$ ?

Strongly expansive minimal models for $\times p_1,\dots,\times p_d$ -ergodic measures and the higher-dimensional complexity gap (Cyr–Kra Conjecture 3.2)